Handbook of Nuclear Engineering Dan Gabriel Cacuci

Handbook of Nuclear Engineering Dan Gabriel Cacuci (Ed.) Handbook of Nuclear Engineering With  Figures and  Tables Volume I Nuclear Engineering Fundamentals 123 Professor Dan Gabriel Cacuci Institute for Nuclear Technology and Reactor Safety Karlsruher Institut für Technologie (KIT) Gotthard-Franz-Str.   Karlsruhe Germany Library of Congress Control Number:  ISBN: ---- This publication is available also as: Electronic publication under ISBN ---- and Print and electronic bundle under ISBN ---- © Springer Science+Business Media LLC  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC.,  Spring Street, New York, NY , USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Springer is part of Springer Science+Business Media springer.com Printed on acid-free paper Preface A new Handbook of Nuclear Engineering is indeed a rare event, although the engineering sciences – and especially nuclear engineering – have progressed immensely during the fiftytwo years that have passed since the publication of the first handbook of this kind. Even if the basic principles of nuclear engineering have remained unchanged for decades, it is compelling to note that the professional practice in this sector has enjoyed great progress during this period. Professor Dan Cacuci has embarked on an ambitious task: to edit a new Handbook of Nuclear Engineering, aiming at making it as all encompassing as possible. He successfully carried out this huge task in a very short time span, bringing together specialists of the highest international reputation. Primordially, nuclear engineering draws its roots from the nuclear sciences, a field founded on the most fundamental advances in the physics of the th century. This is an exacting field conceptually, demanding knowledge not only of the intimate structure of matter, but also of the mathematical formalisms that represent this structure. A field that is, regrettably, increasingly more neglected in the general education. At the same time, nuclear engineering is also about engineering at the highest level of exigency for harnessing the extremely high power density in nuclear systems, intertwined with the absolute necessity to control and operate these systems in conditions of maximum safety and economic feasibility. Last but not least – and this is probably the source of its unique characteristics – nuclear engineering is anchored in time, in its own history, first and foremost, and especially in the extent of time over which the responsibility of nuclear engineers must last – from designing, constructing and operating reactors to the management of the ultimate destination of decommissioned reactor plants. These exceptional stakes are becoming critical now – at a time when the “third-generation” of nuclear engineers needs to be educated, a generation which is to succeed the “masterbuilders”, who themselves had, in turn, succeeded the “pioneers”. This third-generation of nuclear engineers is already being called upon to fulfill a new demand – that of integrating all of the knowledge produced by its predecessors in the quest for meeting an increasing demand for energy production under enhanced safety requirements and optimal economic conditions. Energy is a major stake of the th century. Humanity will imperatively need to manage the sustainable production, transportation, and applications of energy. Nuclear power will inevitably play a decisive role in this quest. Needs for energy, needs for competences, needs for education and training, needs for reference books in nuclear engineering: I do make the wish that the present work would contribute its share towards satisfying these needs, enlightening the young generations to which we are passing on great responsibilities. In view of the contents of this Handbook, covering widely vi Preface and deeply the vast field of nuclear engineering, and the outstanding quality of this coverage – guaranteed by the worldwide reputation of the authors – I have no doubt that this wish will come true. Laurent Turpin Director National Institute for Nuclear Science and Technology (INSTN) Commissariat a l’Energie Atomique (CEA) Centre CEA de Saclay  Gif-sur-Yvette Cedex France August  About the Editor Dan Gabriel Cacuci Institute for Nuclear Technology and Reactor Safety Karlsruher Institut für Technologie (KIT) Gotthard-Franz-Str.   Karlsruhe Germany Dan Gabriel Cacuci has received his Ph.D. degree in applied physics and nuclear engineering from Columbia University, New York, in . His career has bridged activities in the academia and multidisciplinary national research centers. His teaching and research experience as a full professor at leading academic institutions includes appointments (tenured, part-time, visiting, or adjunct) at the University of Tennessee (–), University of California at Santa Barbara (–), University of Illinois at Urbana-Champaign (–), University of Virginia (–), University of Michigan (–), University of California at Berkeley (–), Royal Institute of Technology Stockholm (), the French National Institute for Nuclear Sciences and Technologies in Paris (since ), and the University of Karlsruhe (since ). Professor Cacuci has also served as a group leader and section head at Oak Ridge National Laboratory (–), Institute Director at the Nuclear Research Center Karlsruhe in Germany (–), and acted as Scientific Director of the Nuclear Energy Directorate, Commissariat a l’Energie Atomique, France (–). Professor Cacuci’s scientific expertise encompasses several areas in nuclear engineering (reactor multi-physics, dynamics, safety and reliability), predictive best-estimate analysis of large-scale physical and engineering systems, and large-scale scientific computations. He has authored  books,  book chapters, and over  peer-reviewed articles. Since , Professor Cacuci has been the Editor of “Nuclear Science and Engineering –The Research Journal of the American Nuclear Society” (ANS). He is a member of the European Academy of Arts and Sciences, Honorary Member of the Romanian Academy, ANS Fellow, and has received many awards, including four titles of Doctor Honoris Causa, the Alexander von Humboldt Prize for Senior Scholars (Germany), the E. O. Lawrence Award and Gold Medal (US DOE), the Eugene P. Wigner Reactor Physics Award (ANS), and the Glenn Seaborg Medal (ANS). Professor Cacuci has served on numerous international committees, was founding coordinator of the EURATOM-Integrated Project NURESIM (European Platform for Nuclear Reactor Simulation, –), and founding coordinator (–) of the Coordinated Action for establishing a Sustainable Nuclear Fission Technology Platform (SNF-TP) in Europe. Table of Contents Preface................................................................................................... About the Editor ...................................................................................... Contributors ........................................................................................... Biographies of Contributors....................................................................... Introduction............................................................................................ v vii xiii xxi li Volume  Nuclear Engineering Fundamentals  Neutron Cross Section Measurements ...................................................  Robert C. Block ⋅ Yaron Danon ⋅ Frank Gunsing ⋅ Robert C. Haight  Evaluated Nuclear Data .......................................................................  Pavel Obložinský ⋅ Michal Herman ⋅ Said F. Mughabghab  Neutron Slowing Down and Thermalization ...........................................  Robert E. MacFarlane  Nuclear Data Preparation ....................................................................  Dermott E. Cullen  General Principles of Neutron Transport ................................................  Anil K. Prinja ⋅ Edward W. Larsen  Nuclear Materials and Irradiation Effects................................................  Clément Lemaignan  Mathematics for Nuclear Engineering....................................................  Dan Gabriel Cacuci ⋅ Mihaela Ionescu-Bujor Volume   Reactor Design Multigroup Neutron Transport and Diffusion Computations .....................  Alain Hébert  Lattice Physics Computations...............................................................  Dave Knott ⋅ Akio Yamamoto  Core Isotopic Depletion and Fuel Management .......................................  Paul J. Turinsky x Table of Contents  Radiation Shielding and Radiological Protection.....................................  J. Kenneth Shultis ⋅ Richard E. Faw  High Performance Computing in Nuclear Engineering..............................  Christophe Calvin ⋅ David Nowak Volume  Reactor Analysis  Analysis of Reactor Fuel Rod Behavior ...................................................  Paul Van Uffelen ⋅ Rudy J.M. Konings ⋅ Carlo Vitanza ⋅ James Tulenko  Noise Techniques in Nuclear Systems ....................................................  Imre Pázsit ⋅ Christophe Demazière  Deterministic and Probabilistic Safety Analysis.......................................  Mohammad Modarres ⋅ Inn Seock Kim  Multiphase Flows: Compressible Multi-Hydrodynamics............................  Daniel Lhuillier ⋅ Meng-Sing Liou ⋅ Theo G. Theofanous  Sensitivity and Uncertainty Analysis, Data Assimilation, and Predictive Best-Estimate Model Calibration...........................................................  Dan Gabriel Cacuci ⋅ Mihaela Ionescu-Bujor  Reactor Physics Experiments on Zero Power Reactors ..............................  Gilles Bignan ⋅ Philippe Fougeras ⋅ Patrick Blaise ⋅ Jean-Pascal Hudelot ⋅ Frédéric Mellier Volume  Reactors of Generations III and IV  Pressurized LWRs and HWRs in the Republic of Korea...............................  Nam Zin Cho ⋅ Han Gon Kim  VVER-Type Reactors of Russian Design ..................................................  Sergei B. Ryzhov ⋅ Victor A. Mokhov ⋅ Mikhail P. Nikitenko ⋅ George G. Bessalov ⋅ Alexander K. Podshibyakin ⋅ Dmitry A. Anufriev ⋅ J́anos Gadó ⋅ Ulrich Rohde  Sodium Fast Reactor Design: Fuels, Neutronics, Thermal-Hydraulics, Structural Mechanics and Safety...........................................................  Jacques Rouault ⋅ P. Chellapandi ⋅ Baldev Raj ⋅ Philippe Dufour ⋅ Christian Latge ⋅ Laurent Paret ⋅ Pierre Lo Pinto ⋅ Gilles H. Rodriguez ⋅ Guy-Marie Gautier ⋅ Gian-Luigi Fiorini ⋅ Michel Pelletier ⋅ Dominique Gosset ⋅ St éphane Bourganel ⋅ Gerard Mignot ⋅ Fr éd éric Varaine ⋅ Bernard Valentin ⋅ Patrick Masoni ⋅ Philippe Martin ⋅ Jean-Claude Queval ⋅ Daniel Broc ⋅ Nicolas Devictor  Gas-Cooled Reactors...........................................................................  Bertrand Barré Table of Contents  Lead-Cooled Fast Reactor (LFR) Design: Safety, Neutronics, Thermal Hydraulics, Structural Mechanics, Fuel, Core, and Plant Design..................  Luciano Cinotti ⋅ Craig F. Smith ⋅ Carlo Artioli ⋅ Giacomo Grasso ⋅ Giovanni Corsini  GEM∗ STAR: The Alternative Reactor Technology Comprising Graphite, Molten Salt, and Accelerators...............................................................  Charles D. Bowman ⋅ R. Bruce Vogelaar ⋅ Edward G. Bilpuch ⋅ Calvin R. Howell ⋅ Anton P. Tonchev ⋅ Werner Tornow ⋅ R.L. Walter Volume  Fuel Cycles, Decommissioning, Waste Disposal and Safeguards  Front End of the Fuel Cycle...................................................................  Bertrand Barré  Transuranium Elements in the Nuclear Fuel Cycle ....................................  Thomas Fanghänel ⋅ Jean-Paul Glatz ⋅ Rudy J.M. Konings ⋅ Vincenzo V. Rondinella ⋅ Joe Somers  Decommissioning of Nuclear Plants ......................................................  Maurizio Cumo  The Scientific Basis of Nuclear Waste Management ..................................  Bernard Bonin  Proliferation Resistance and Safeguards ................................................  Scott F. DeMuth Index......................................................................................................  xi Contributors Dmitry A. Anufriev OKB “GIDROPRESS” Joint Stock Company State Atomic Energy Corporation “Rosatom” Podolsk Moscow Region Russia Carlo Artioli Italian National Agency for New Technologies, Energy and Sustainable Economic Development v. Martiri di Monte Sole  Bologna ITALY carlo.artioli@enea.it Robert Bari Brookhaven National Laboratory Upton NY - USA bari@bnl.gov Bertrand Barré  rue des Blanchisseurs  INSTN Chaville France bcbarre@wanadoo.fr Charles G. Bathke Los Alamos National Laboratory Los Alamos NM  USA bathke@lanl.gov George G. Bessalov OKB “GIDROPRESS” Joint Stock Company State Atomic Energy Corporation “Rosatom” Podolsk Moscow Region Russia Gilles Bignan French Atomic Energy Commission Reactor Studies Department Cadarache Research Center Saint Paul Lez Durance France gilles.bignan@cea.fr Edward G. Bilpuch Triangle Universities Nuclear Laboratory Duke University Durham NC  USA Patrick Blaise French Atomic Energy Commission Reactor Studies Department Cadarache Research Center Saint Paul Lez Durance France patrick.blaisse@cea.fr Robert C. Block Gaerttner LINAC Laboratory Mechanical, Aerospace & Nuclear Engineering Department Rensselaer Polytechnic Institute Troy, NY - USA blockr@rpi.edu Bernard Bonin Commissariat à l’Energie Atomique France Nuclear Energy Directorate CEA Saclay Gif-sur-Yvette Cedex France Bernard.Bonin@cea.fr xiv Contributors Stéphane Bourganel CEA Saclay Centre d’Etudes Nucléaires de Saclay  Gif-sur-Yvette France stephane.bourganel@cea.fr Charles D. Bowman ADNA Corporation  Los Pueblos Los Alamos NM  USA cbowman@cybermesa.com Brian Boyer Los Alamos National Laboratory Los Alamos NM  USA bboyer@lanl.gov Daniel Broc French Atomic Energy Commission (CEA) Saint Paul Lez Durance France daniel.broc@cea.fr Thomas Burr Los Alamos National Laboratory Los Alamos NM  USA tburr@lanl.gov Dan Gabriel Cacuci Institute for Nuclear Technology and Reactor Safety Karlsruher Institut für Technologie (KIT) Karlsruhe Germany dan.cacuci@kit.edu Christophe Calvin CEA/DEN/DANS/DMS CEA Saclay  Gif-sur-Yvette France christophe.calvin@cea.fr P. Chellapandi Indira Gandhi Center for Atomic Research Tamilnadu India pcp@igcar.gov.in Nam Zin Cho Department of Nuclear and Quantum Engineering Korea Advanced Institute of Science and Technology Daejeon Korea nzcho@kaist.ac.kr Luciano Cinotti MERIVUS srl V. Orazio  Roma ITALY luciano.cinotti@gmail.com Giovanni Corsini Giovanni.Corsini@tele.it Dermott E. Cullen Lawrence Livermore National Laboratory Livermore CA  USA redcullen@comcast.net Maurizio L. Cumo University of Rome Corso Vittorio Emanuele II  Rome  Rome Italy maurizio.cumo@uniroma.it Yaron Danon Rensselaer Polytechnic Institute Troy NY USA Contributors Christophe Demazière Department of Nuclear Engineering Chalmers University of Technology Göteborg Sweden Scott F. DeMuth Los Alamos National Laboratory Los Alamos NM  USA scott.demuth@comcast.net Nicolas Devictor French Atomic Energy Commission (CEA) Saint Paul Lez Durance France nicolas.devictor@cea.fr Philippe Dufour Chargé demission RNA-Na CEA/DEN/CAD/DER/SESI French Atomic Energy Commission (CEA) Saint Paul Lez Durance France philippe.dufour@cea.fr Michael Ehinger Oak Ridge National Laboratory Oak Ridge TN - USA ehingermh@ornl.gov Thomas Fanghänel European Commission Joint Research Centre Institute for Transuranium Elements Karlsruhe Germany thomas.fanghaenel@ec.europa.eu Richard E. Faw Department of Mechanical and Nuclear Engineering Kansas State University Manhattan KS  USA fawre@triad.rr.com Gian-Luigi Fiorini French Atomic Energy Commission (CEA) Saint Paul Lez Durance France gian-luigi.fiorini@cea.fr Philippe Fougeras French Atomic Energy Commission (CEA) Reactor Studies Department Cadarache Research Center Saint Paul Lez Durance France philippe.fougeras@cea.fr János Gadó KfKI, Atomic Energy Research Institute of the Hungarian Academy of Sciences Budapest Hungary Joint Stock Company State Atomic Energy Corporation “Rosatom” Podolsk Moscow Region Russia goolo@sunseru.kfki.hu Guy-Marie Gautier French Atomic Energy Commission (CEA) Saint Paul Lez Durance France guy-marie.gautier@cea.fr Jean-Paul Glatz European Commission Joint Research Centre Institute for Transuranium Elements Karlsruhe Germany jean-paul.glatz@ec.europa.eu Dominique Gosset French Atomic Energy Commission (CEA) Saint Paul Lez Durance France dominique.gosset@cea.fr xv xvi Contributors Giacomo Grasso Italian National Agency for New Technologies Energy and Sustainable Economic Development v. Martiri di Monte Sole  Bologna ITALY giacomo.grasso@enea.it Frank Gunsing CEA Saclay - Irfu France frank.gunsing@cea.fr Robert C. Haight Los Alamos National Laboratory Los Alamos NM  USA Alain Hébert Institut de Génie Nucléaire École Polytechnique de Montréal Montréal Québec Canada alain.hebert@polymtl.ca John Howell Department of Mechanical Engineering, Systems and Control University of Glasgow Glasgow QQ UK j.howell@mech.gla.ac.uk Jean-Pascal Hudelot French Atomic Energy Commission (CEA) Reactor Studies Department Cadarache Research Center Saint Paul Lez Durance France jean-pascal.hudelot@cea.fr Mihaela Ionescu-Bujor Fusion Program Karlsruher Institut für Technologie (KIT) Karlsruhe Germany mihaela.ionescu-bujor@kit.edu Frédérie Jasserand CEA Saclay Gif-Sur-Yvelte France Magnus Hedberg Institute for Transuranium Elements Karlsruhe Germany Magnus.HEDBERG@ec.europa.eu Han Gon Kim Advanced Plant Development Office Korea Hydro and Nuclear Power Co., Ltd Daejeon Korea kimhangon@khnp.co.kr Michal Herman National Nuclear Data Center Brookhaven National Laboratory Upton NY - USA mwherman@bnl.gov Inn Seock Kim ISSA Technology, Inc.  Seneca Crossing Drive Germantown MD  USA innseockkim@gmail.com Calvin R. Howell Triangle Universities Nuclear Laboratory Duke University Durham NC  USA Dave Knott Studsvik Scandpower Inc. Wilmington NC USA Dave.Knott@studsvik.com Contributors Rudy J.M. Konings European Commission Joint Research Centre Institute for Transuranium Elements Karlsruhe Germany rudy.konings@ec.europa.eu Klaus Lützenkirchen Institute for Transuranium Elements Karlsruhe Germany Klaus-Richard.Luetzenkirchen@ ec.europa.eu Edward W. Larsen Department of Nuclear Engineering and Radiological Sciences University of Michigan Ann Arbor MI USA edlarsen@umich.edu Robert E. MacFarlane Nuclear and Particle Physics, Astrophysics and Cosmology Theoretical Division Los Alamos National Laboratory Los Alamos NM  USA ryxm@lanl.gov Christian Latge French Atomic Energy Commission (CEA) Saint Paul Lez Durance France christian.latge@cea.fr Philippe Martin French Atomic Energy Commission (CEA) Saint Paul Lez Durance France philippe.martin@drncad.cea.fr Clément Lemaignan CEA-INSTN and CEA-DEN Grenoble France clement.lemaignan@cea.fr Patrick Masoni French Atomic Energy Commission (CEA) Saint Paul Lez Durance France patrick.masoni@cea.fr Daniel Lhuillier Institut Jean le Rond d’Alembert CNRS and University Paris  Paris France daniel.lhuillier@upmc.fr Klaus Mayer Institute for Transuranium Elements Karlsruhe Germany Klaus.Mayer@ec.europa.eu Meng-Sing Liou NASA Glenn Research Center Cleveland, OH USA meng-sing.liou@nasa.gov Frédéric Mellier French Atomic Energy Commission (CEA) Reactor Studies Department Cadarache Research Center Saint Paul Lez Durance France Pierre Lo Pinto French Atomic Energy Commission (CEA) Saint Paul Lez Durance France pierre.lopinto@cea.fr Gerard Mignot French Atomic Energy Commission (CEA) Saint Paul Lez Durance France gerard.mignot@cea.fr xvii xviii Contributors Mohammad Modarres Department of Mechanical Engineering University of Maryland Martin Hall College Park MD  USA modarres@umd.edu Victor A. Mokhov OKB “GIDROPRESS” Joint Stock Company State Atomic Energy Corporation “Rosatom” Podolsk Moscow Region Russia Said F. Mughabghab National Nuclear Data Center Brookhaven National Laboratory Upton NY - USA mugabgab@bnl.gov Mikhail P. Nikitenko OKB “GIDROPRESS” Joint Stock Company State Atomic Energy Corporation “Rosatom” Podolsk Moscow Region Russia David Nowak Mathematics and Computer Science Division Argonne National Laboratory  South Cass Avenue Argonne IL - USA nowak@mcs.anl.gov Pavel Obložinský National Nuclear Data Center Brookhaven National Laboratory Upton NY - USA oblozinsky@bnl.gov Laurent Paret French Atomic Energy Commission (CEA) Saint Paul Lez Durance France laurent.paret@cea.fr Imre Pázsit Department of Nuclear Engineering Chalmers University of Technology Göteborg Sweden Department of Nuclear Engineering and Radiological Sciences University of Michigan Ann Arbor Michigan USA imre@chalmers.se Michel Pelletier French Atomic Energy Commission (CEA) Saint Paul Lez Durance France michel.pelletier@cea.fr Alexander K. Podshibyakin OKB “GIDROPRESS” Joint Stock Company State Atomic Energy Corporation “Rosatom” Podolsk Moscow Region Russia Anil K. Prinja Chemical and Nuclear Engineering Department University of New Mexico Albuquerque NM USA prinja@unm.edu Jean-Claude Queval French Atomic Energy Commission (CEA) Saint Paul Lez Durance France jean-claude.queval@cea.fr Contributors Baldev Raj Indira Gandhi Center for Atomic Research Tamilnadu India dir@igcar.gov.in Gilles H. Rodriguez French Atomic Energy Commission (CEA) Saint Paul Lez Durance France gilles.rodriguez@cea.fr Ulrich Rohde Institute of Safety Research Research Center Dresden-Rossendorf Germany u.rohde@fzd.de Vincenzo V. Rondinella European Commission Joint Research Centre Institute for Transuranium Elements Karlsruhe Germany Jacques Rouault French Atomic Energy Commission (CEA) Saint Paul Lez Durance France jacques.rouault@cea.fr Sergei B. Ryzhov OKB “GIDROPRESS” Joint Stock Company State Atomic Energy Corporation “Rosatom” Podolsk Moscow Region Russia Mark Schanfein Idaho National Laboratory Idaho Falls ID  USA Mark.Schanfein@inl.gov J. Kenneth Shultis Department of Mechanical and Nuclear Engineering Kansas State University Manhattan KS  USA jks@ksu.edu Craig F. Smith Naval Postgraduate School  University Circle Monterey CA USA cfsmith@nps.edu Joe Somers European Commission Joint Research Centre Institute for Transuranium Elements Karlsruhe Germany Ugo Spezia SOGIN SpA Roma Italy Rebecca Stevens Los Alamos National Laboratory Los Alamos NM  USA rstevens@lanl.gov Theo G. Theofanous Department of Chemical Engineering Department of Mechanical Engineering Center for Risk Studies and Safety University of California Santa Barbara CA USA theo@theofanous.net; theo@engineering.ucsb.edu Anton P. Tonchev Triangle Universities Nuclear Laboratory Duke University Durham NC  USA xix xx Contributors Werner Tornow Triangle Universities Nuclear Laboratory Duke University Durham NC  USA Ivo Tripputi NEA Working Party on Decommissioning and Dismantling (WPDD) SOGIN SpA Roma Italy tripputi@sogin.it James Tulenko  Nuclear Science Center University of Florida Gainesville FL USA tulenko@ufl.edu Paul J. Turinsky Department of Nuclear Engineering North Carolina State University Raleigh NC USA turinsky@ncsu.edu Paul Van Uffelen European Commission Joint Research Centre Institute for Transuranium Elements Karlsruhe Germany paul.van-uffelen@ec.europa.eu Bernard Valentin French Atomic Energy Commission (CEA) Saint Paul Lez Durance France bernard.valentin@cea.fr Frédéric Varaine French Atomic Energy Commission (CEA) Saint Paul Lez Durance France frederic.varaine@cea.fr Carlo Vitanza OECD Halden Reactor Project Halden Norway Carlo.Vitanza@hrp.no R. Bruce Vogelaar Virginia Tech Blacksburg VA  USA Richard Wallace Los Alamos National Laboratory Los Alamos NM  USA rwallace@lanl.gov Maria Wallenius Institute for Transuranium Elements Karlsruhe Germany Maria-S.Wallenius@ec.europa.eu R.L. Walter Triangle Universities Nuclear Laboratory Duke University Durham NC  USA Akio Yamamoto Graduate School of Engineering Nagoya University Nagoya Japan a-yamamoto@nucl.nagoya-u.ac.jp Biographies of Contributors Dmitry A. Anufriev Dmitry A. Anufriev was born on June , , in Krivandino, Moscow region. He graduated in  as an engineer in thermal physics from the Moscow Power Engineering University. Since , he has worked at OKB “GIDROPRESS” in positions of increasing responsibilities, as head of group, and deputy head of the department. His professional interests are focused on VVER reactor plant design development, elaboration of the schemes and processes of reactor plant operation, protection and control system operation algorithms, and safety analyses. Carlo Artioli Dr. Ing. Carlo Artioli, received his degree in nuclear engineering (summa cum laude) at the University of Bologna, Italy, in . He joined the former CNEN (Nuclear Energy National Committee) where he worked as core designer for fast reactors. He took part in the European activities for the harmonization of nuclear codes and in the programs for Plutonium management and Minor Actinides reduction. Currently serving as researcher for the Italian National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA, former CNEN) in Bologna, he has been core design leader for the ADS EFIT within the European IP EUROTRANS and core neutronic design for the ELSY reactor, as well as in charge of definition and designing of the DEMO LFR in the Italian program. Also, he is professor of Neutronic Design of the Reactor in the Postgraduate Master on Advanced Nuclear Systems in the Bologna University. He is author of several papers; mainly about new concepts related to LFR and lead coolant ADS, in international conferences. Robert Bari Dr. Robert Bari has over  years of experience in the field of nuclear energy. He is a senior physicist and a senior advisor at Brookhaven National Laboratory and has directed numerous studies of advanced nuclear energy concepts involving subjects such as nuclear energy technology performance, safety, nonproliferation, economics, and waste management. He has lectured widely on nuclear technology and has published more than  papers. For over  years, Dr. Bari has served at various levels of management at Brookhaven National Laboratory and has directed numerous programs for the US Department of Energy and the US Nuclear Regulatory Commission. Dr. Bari is currently international cochairman of the working group that has developed a comprehensive methodology for evaluation of proliferation resistance and physical protection of new nuclear energy concepts proposed within the multinational Generation IV International Forum. He has served on the Board of Directors of the American Nuclear Society, and is the past president of the International Association for Probabilistic Safety Assessment and Management. He has served as an adjunct faculty member and advisor to several major universities in the field of nuclear technology and received his doctorate in physics from Brandeis University () and his bachelor’s degree in physics from Rutgers University (). xxii Biographies of Contributors Bertrand Barré Bertrand Barré is the scientific advisor to the Chairperson of the AREVA group. He is also professor emeritus of nuclear engineering at the Institut National des Sciences et Techniques Nucléaires, INSTN. From  to , he was the head of the Nuclear Reactor Directorate of the French Atomic Energy commission, CEA, and from  to , he was Vice-president in charge of R&D in COGEMA (now AREVA NC). Past President of the European Nuclear Society (ENS) and of the International Nuclear Societies Council (INSC), Past chairman of the Standing Advisory Group on Nuclear Energy of the IAEA, Bertrand Barré has just left the Chair of the International Nuclear Energy Academy (INEA). His recent publications include Nuclear Power, understanding the future (with P.R. Bauquis – Hirlé, . Atlas des energies – Autrement . Les  mots du nucléaire – PUF . His personal Web site (French/English) is www.bertrandbarre.com. Charles G. Bathke Dr. Charles G. Bathke is a staff member in the D- “International and Nuclear Systems Engineering” group at Los Alamos National Laboratory (LANL). His knowledge of the nuclear fuel cycle derives from his past work on Accelerator Transmutation of Waste (ATW) and its successor the US Advanced Fuel Cycle Initiative (AFCI), where he developed the Nuclear Fuel Cycle Simulation (NFCSim) code that is used to simulate the civilian nuclear fuel cycle from cradle (mining) to grave (waste repository). In the course of his career, he has performed systems analyses of reactors based upon various magnetic fusion confinement schemes, proton accelerators used to generate tritium, electron accelerators used for X-ray radiography, and terrorist-induced biological events. For the past three years, his research interests have gravitated to the nonproliferation arena, where he has been analyzing the material attractiveness of nuclear materials associated with reprocessing. He received his Ph.D. in Nuclear Engineering from the University of Illinois in , followed by a postdoc at the Princeton Plasma Physics Laboratory. He has been at LANL since . Georgy G. Bessalov Georgy G. Bessalov was born on September , , in Skopin, Ryazan region. He was graduated in  as a mechanical engineer from Moscow Power Engineer University. Since  he held the following positions at OKB “GIDROPRESS”: engineer, head of group, and head of department. Reactor plant design development, commissioning tests at NPPs with VVER-, and safety analyses are his professional interests. At present, he is a leading engineer on reactor plants with VVER- reactors. Gilles Bignan Gilles Bignan earned his Ph.D. in nuclear physics and instrumentation from the Caen University of Sciences () and the degree of Engineer in nuclear energy from the Institute of nuclear sciences in Caen. He started his career at CEA in the development of nuclear measurement devices for various applications (online power estimation of a power reactor, fuel monitoring for safety-criticality risk assessment, and nuclear material control for Safeguards). Since , he is working in the Research Reactor field for neutron physic and Safety test and he is currently involved in the new International Material Testing Reactor Project called Jules Biographies of Contributors Horowitz Reactor as the User Facility Manager. He is an expert for the IAEA in the field of research reactors. Edward G. Bilpuch Edward G. Bilpuch is a Henry W. Newson professor of Physics at Duke University. He did his undergraduate studies at the University North Carolina at Chapel Hill and earned a Ph.D. degree in physics from University of North Carolina at Chapel Hill in . He began his career at Duke in  as a research associate, subsequently was appointed to the faculty, and was promoted to professor in . He was director of the Triangle Universities Nuclear Laboratory from  through , and named Henry W. Nelson professor of Physics in . Professor Bilpuch perfected high-resolution techniques in nuclear physics using linear accelerators. These endeavors culminated in an experiment that set new standards for measurements of properties of atomic nuclei. Professor Bilpuch has authored or coauthored  articles published in professional journals and has given many seminars in the USA and throughout the world. He has directed or codirected over  Ph.D. graduate students associated with the nuclear laboratory. He was a guest professor at the University of Frankfurt, Germany,  and , and at Fudan University of Shanghai, China,  and . He was the recipient of a Senior US Scientist Humboldt Award in Germany in . In , he was named an honorary professor at Fudan University and in  he was the recipient of the Distinguished Alumnus Award from the University of North Carolina in Chapel Hill. In , he was awarded an honorary doctorate by Frankfurt University in Germany. Patrick Blaise Patrick Blaise, Ph.D., is an industrial engineer in Nuclear Energy from Brussels, and got his Ph.D. in reactor physics from Marseille University in . He was involved in designing and conducting experimental programs in the EOLE critical Facility in Cadarache from  to , mainly in the frame of % MOX LWRs, and responsible for international collaborations in experimental reactor physics. Since , he is a senior expert in Reactor Physics Experimental programs. He is currently in charge of code qualification for Advanced LWRs at the section for reactor physics and cycle at Cadarache. Robert C. Block Dr. Robert C. Block is professor emeritus of Nuclear Engineering at Rensselaer Polytechnic Institute (RPI). During his -year career at RPI, he served as Director of the Gaerttner LINAC Laboratory, Chairman of the Department of Nuclear Engineering and Engineering Physics, and Associate Dean of Engineering. Before joining RPI, he was a staff member at the Oak Ridge National Laboratory. He has had sabbaticals at the AERE Laboratory at Harwell, UK; the Kyoto University Research Reactor Laboratory in Kumatori, Japan; and the Sandia National Laboratory. He is currently employed by the Knolls Atomic Power Laboratory and is a consultant to national laboratories and industry. Dr. Block has served on numerous advisory committees and has almost  publications in journals and conference proceedings to his credit. He was awarded the ANS Seaborg Medal in xxiii xxiv Biographies of Contributors  from the American Nuclear Society, the Glenn Murphy Award in  from the American Society for Engineering Education, and the William H. Wiley Distinguished Faculty Award in  from RPI. Bernard Bonin Dr. Bernard Bonin has a background in fundamental research in high energy physics and materials physics. Between  and , he was head of a Service of Research and Studies on Nuclear Waste, within CEA’s Institute for Nuclear Protection and Safety. His studies then aimed at obtaining an overall view of the scientific basis for nuclear waste management, with special interest in the migration of radioactive contaminants in the underground environment. In , he was appointed assistant to the Director of Research and Development in COGEMA, in charge of the organization of the R&D on the front end of the nuclear fuel cycle, and on future nuclear energy systems. He received the Areva Innovation Award in  for the development of a radon detection system. Since , he is Deputy Scientific Director in the Nuclear Energy Division of the Commissariat a l’Energie Atomique and professor at the French Institute for Nuclear Sciences and Techniques. Stéphane Bourganel An engineer and researcher at CEA/Saclay, Stéphane Bourganel earned a Ph.D. from the INPG School in Grenoble, France and contributes to R&D in the field of the PWR lifetime. He is involved in design and analysis of experiments dedicated to decay heat issues. He worked on the first fluence calculations for the fourth reactor generation lifetime. Furthermore, he takes part, as a teacher, in training course on the Monte Carlo computer code TRIPOLI-. Charles Daniel Bowman Dr. Charles Daniel Bowman was awarded a Ph.D. from Duke University in  in neutron physics. He led the development of world-class accelerator-based neutron sources and their application to basic and applied science at Lawrence Livermore National Laboratory, the National Institute of Science and Technology, and the Los Alamos National Laboratory. In , he became a Fellow of the American Physical Society; in , he was awarded the U S Department of Commerce Silver Medal; and in , he was honored as a Fellow of the Los Alamos National Laboratory. In , Dr. Bowman became Project leader for the Los Alamos Accelerator-Driven Transmutation Technology (ADTT) program. When the US-funded International Science and Technology Center (ISTC) was formed in Moscow in , Dr. Bowman also led Project  of the ISTC, which employed about  Russian scientists on the ADTT concept. In , he organized ADNA Corporation (Accelerator-Driven Neutron Applications) to pursue the application of accelerator sources of neutrons in nuclear energy culminating in GEM*STAR. Christophe Calvin Christophe Calvin is leading a laboratory in the Nuclear Energy Division of the French Atomic Commission (CEA) in charge of the development of reactor physics simulation codes. He Biographies of Contributors received a Ph.D. in applied mathematics and parallel computing and worked, at CEA Grenoble, in a CFD code development team in charge of the code parallelization. He has also led the development of a new generation of reactor physics simulation code at CEA Saclay. He currently has an expert position in HPC at CEA Nuclear Energy Division, participating to international collaboration with European research institutes and with Japan academic laboratories and institutes in reactor physics simulation and high-performance computing. He has published more than  papers in international conferences or journals on scientific code architecture, parallel algorithms, and scientific code parallelization. P. Chellapandi An outstanding scientist of repute, P. Chellapandi is the director of Safety Group and associate director of Nuclear Engineering Group. He joined the Indira Gandhi Centre for Atomic Research, Kalpakkam, (IGCAR) in . Since then, he has been working on the design and development of  MWe prototype fast breeder reactor (PFBR), over a wide spectrum of design, namely, conceptualization, development of sophisticated computer codes, detailed analysis, design validation, preparation of preliminary safety analysis reports, execution of R&D activities involving national academic institutions, and R&D establishments in the country. He is also the convener of a task force, which is coordinating for the manufacture and erection of reactor assembly components. He earned his B.E. (Hons.) in Mechanical Engineering from Madras University in  securing the first rank, and M.Tech. in Engineering Mechanics (Gold Medalist with CGPA ), and Ph.D. in Applied Mechanics from IIT Madras. He is a professor at the Homi Bhabha National Institute, and adjunct professor of PSG college, Coimbatore, and Sathyabama University, Chennai. He has guided more than  postgraduate students and published about  papers in journals, national and international conferences. He is a Fellow of Indian National Academy of Engineering. Nam Zin Cho Prof. Nam Zin Cho received his B.S. in nuclear engineering from Seoul National University and Ph.D. in nuclear engineering from University of California at Berkeley. He worked at Science Applications, Inc. in Palo Alto from  to , and at Brookhaven National Laboratory in Long Island from  to . In , he joined the faculty at Korea Advanced Institute of Science and Technology (KAIST) in Korea, where he is mostly involved in teaching and research in reactor physics and neutron transport computation. He is Fellow of American Nuclear Society and Associate Editor of Nuclear Science and Engineering. He was the Technical Program Chair for the PHYSOR , a series of the ANS topical meetings on reactor physics, held in Seoul, Korea. From  to , he served as Commissioner of the Atomic Energy Commission of the Republic of Korea. From  to , he was President of Korean Nuclear Society. Luciano Cinotti Luciano Cinotti has been student at the Scuola Superiore degli Studi Universitari e Perfezionamento S. Anna, the renowned School linked to the University of Pisa, Italy, where he received his Dr. Ing. degree in . He has served  years in France in the French-Italian Ansaldo-Novatome xxv xxvi Biographies of Contributors team in charge of the development of the large sodium-cooled fast reactors. After returning to Italy, he has led the Ansaldo Nucleare activities for the improved European fast reactor (EFR) and the innovative reactors PIUS and PRISM, while conceiving ISIS, a full-passive reactor for the combined generation of electricity and heat. He has been responsible of the department of Nuclear Technology of Ansaldo Nuclear Division till  and has been the coordinator of the ELSY STREP in the proposal preparation phase. He is currently involved in the development and promotion of the LFR and Euratom Representative and is the chairman in the GIF LFR Provisional Steering Committee. He is author of several papers on new reactors published in learned journals. Giovanni Corsini Giovanni Corsini received his degree of Dott. Ing. in chemical engineering in  at the University of Pisa. After  years as a chemical project engineer in Foster Wheeler Italy, he joined Ansaldo in , where he has gained experience in the field of pressurized heavy water reactors and in the field of fast reactors (SPX and particularly EFR, where he has been coleader of the decay heat removal subsystem). Subsequently, he has participated in the ECC-funded TACIS project for the safety improvement of the Russian VVER -type Novovoronezh and Kola power plants. He worked in the Ansaldo team for the design of the experimental accelerator-driven system (XADS), in particular for issues relevant to corrosion protection in LBE. While continuing the participation in the ADS field, and in the European ELSY project, since  he also worked as project engineer of the Intermediate Cooling Loop of Heat Removal System and the Cover Gas System of Megapie. At the end , he left Ansaldo Nucleare but continues to work as an independent professional in the field of lead technology. Dermott E. Cullen Dermott E. Cullen earned his Ph.D. in nuclear engineering from Columbia University, New York City, in . From  through , he worked at Brookhaven National Laboratory, Long Island, New York as a nuclear physicist. From  through , he worked at Lawrence Livermore National Laboratory, Livermore, California. From  through , he worked at the Nuclear Data Section, International Atomic Energy Agency, Vienna, Austria. From  through , he worked at Lawrence Livermore National Laboratory, Livermore, California. He is now retired and lives in Livermore, California. He has more than  years of experience with nuclear and atomic data, involving its preparation and use in particle transport calculations. He is the author of over  papers on this subject, as can be seen in his online resume located at http://home.comcast.net/∼redcullen/RESUME.pdf Maurizio Luigi Cumo Full professor of nuclear plants in the University of Rome Sapienza since , Maurizio Luigi Cumo was engaged in experimental and theoretical research in the field of thermofluidynamics of nuclear plants systems and components in the ENEA Research Centre, Casaccia. He has published  books and over  scientific publications. Furthermore, he was engaged in nuclear safety researches as president of the Italian SOGIN Company for Nuclear Installations Decommissioning and chairman of the International Scientific Council of the Nuclear Energy Directorate of the French Commissariat à l’Energie Atomique. He is a member of the Italian Biographies of Contributors Academy of Sciences, said of the forty, and of the European Academy of Sciences and Arts, leading as president the Italian Society for the Advancement of Sciences. Yaron Danon Dr. Yaron Danon is a professor in the department of Mechanical Aerospace and Nuclear Engineering at Rensselaer Polytechnic Institute (RPI) in Troy, New York. He is also the director of the Gaerttner Electron Linear Accelerator (LINAC) Laboratory at RPI, Dr. Danon is a member of the US Cross Section Evaluation Working Group and chairman of the Measurements Committee. He is also a member of the Nuclear Data Advisory Group (NDAG), which advises the US Nuclear Criticality Safety Program (NCSP) on issues relating to nuclear data. Dr. Danon has over  publications in archived journals and conference proceedings. Christophe Demazière Christophe Demazière (Ph.D. in reactor physics, Chalmers University of Technology, Sweden, ; Engineering degree, Hautes Etudes d’Ingénieur, France, ) is an associate professor in the department of Nuclear Engineering, Chalmers University of Technology, Gothenburg, Sweden, lecturing in nuclear reactor physics (reactor physics and nuclear thermal-hydraulics) and nuclear reactor modeling (deterministic methods). He is a member of the American Nuclear Society (Reactor Physics Division, Mathematics and Computation Division, Thermal Hydraulics Division, Education and Training Division) and of the Swedish Nuclear Society. His research interests include pressurized water reactor and boiling water reactor physics, reactor dynamics applied to both critical and subcritical systems, power reactor noise and noise diagnostics, signal processing and data analysis (linear, nonlinear, wavelet, and fractal analysis), nuclear reactor modeling and calculations, (including coupled neutronics/thermal-hydraulics), and computational methods applied to nuclear reactor modeling. Scott DeMuth Dr. Scott DeMuth has a Ph.D. in chemical engineering and has been a staff member at Los Alamos National Laboratory for the past  years. Prior to Los Alamos, he was a research engineer at Oak Ridge National Laboratory for  years. His specialty is nuclear materials processing and safeguards. He has worked for research and development in all areas of the nuclear fuel cycle, including uranium enrichment, fuel fabrication, thermal and fast reactors, fuel reprocessing, and waste management. Currently he is on assignment in Washington, DC working for the US Department of Energy’s Office of International Regimes and Agreements (NA-). Nicolas Devictor After receiving a Ph.D. in applied mathematics, Nicolas Devictor worked from  to  on structural reliability, probabilistic safety assessment, and risk management. He is one of the editors of Uncertainty in Industrial Practice – A Guide to Quantitative Uncertainty Management, published by Wiley in . He was in charge of the coordination of the project “Safety and Reliability of the facility,” a transversal project in support of the sodium fast reactor and gas-cooled fast reactor, and served as the CEA-representative in the Working Group RISK of OECD/NEA. From  to , Nicolas Devictor was head of the laboratory in charge of the preconceptual xxvii xxviii Biographies of Contributors design of sodium fast-cooled reactors, and related safety analysis. Since , Nicolas Devictor has been project manager for the R&D leaded by CEA team in support to SFR and Astrid project. Philippe Dufour After graduating from “Ecole Centrale de Paris,” Dufour joined CEA (Commissariat à l’Energie Atomique) in . The beginning of his career was devoted to thermal-hydraulical studies for the Superphenix fast reactor. Subsequently, he was involved in other fields such as, neutronics, safety, development of computer codes, economy, and optimization, for various reactor types. From  to , Philippe Dufour was head of the laboratory in charge of the preconceptual design of sodium fast-cooled reactors, and related safety analysis. He is currently project advisor at the Department for Reactor Studies at Cadarache (DEN/DER/SESI). Mike Ehinger Mike Ehinger has worked in the areas of safeguards and nuclear nonproliferation for  years. He is currently at Oak Ridge National Laboratory as a senior program development manager working principally in the area of International Safeguards. From  to , he worked at the International Atomic Energy Agency (IAEA) in Vienna on the development, installation, and implementation of equipment and procedures for IAEA inspections at the Rokkasho Reprocessing Plant in Japan. Prior to going to Vienna, he was manager of safeguards programs at Oak Ridge. While at Oak Ridge, the Russian Material, Protection, Control and Accounting (MPC&A) program was among his responsibilities and he actively participated in the tasks related to the Russian fuel reprocessing sites Mayak and Tomsk. He was also involved with the US and Russian HEU down-blend program serving as a monitor at enrichment facilities. He has broad international experience having participated in activities at many reprocessing facilities throughout the world including England, France, and India. Prior to joining Oak Ridge in , he was a senior nuclear material control engineer at the Barnwell Reprocessing Plant where he installed the Nuclear Material Accountancy system and demonstrated its capabilities to US and international regulatory, development, and operating organizations. From  to , he was the accountability supervisor during the operating years of the West Valley Reprocessing Plant. During this period, he hosted the very first IAEA inspection at a reprocessing facility. Thomas Fanghänel Prof. Dr. Thomas Fanghänel currently serves as the director of the Institute for Transuranium Elements (ITU), which is one of the seven institutes of the European Commission’s Joint Research Centre located in Karlsruhe, Germany. It is ITU’s mission to provide the scientific foundation for the protection of the European citizen against risks associated with the handling and storage of highly radioactive elements. Research is focused on () basic actinide science and applications, () safety of conventional and advanced nuclear fuel cycle including spent fuel disposal, and () safeguards and nuclear forensics. Since , he is professor of Radiochemistry at the Ruprecht-Karls-University Heidelberg. Prof. Fanghänel has degrees in Chemistry (Diplomchemiker), a Ph.D. and Habilitation in Inorganic and Physical Chemistry from the Technical University Bergakademie Freiberg. Before he was appointed as the director of ITU, he had been director of the Institute for Nuclear Waste Biographies of Contributors Disposal (INE) of the former Forschungszentrum Karlsruhe (now Karlsruhe Institute of Technology, KIT), director of the Institute of Radiochemistry, Forschungszentrum Rossendorf, and professor of Radiochemistry at the University of Dresden. He has more than  years of research experience with special expertise in Actinide chemistry and long-term safety of nuclear waste disposal. Since , he is a member of the Reactor Safety Commission of the German Federal Ministry for the Environment, Nature Conservation, and Nuclear Safety. Richard E. Faw Richard E. Faw, born in rural Ohio, was educated at the University of Cincinnati (B.S., Chemical Engineering, ) and at the University of Minnesota (Ph.D., Chemical Engineering, ). In , he joined the nuclear engineering faculty at Kansas State University, where he served until . Appointments at the university include department head, –, director of the Radiation Shielding Laboratory, –, and director of the Nuclear Reactor Laboratory, –. He is licensed by the Nuclear Regulatory Commission as a senior reactor operator and by Kansas and Ohio as a professional engineer. He is a Fellow of the American Nuclear Society and recipient of university awards for teaching and research. Dr. Faw is author or coauthor of four textbooks on radiation shielding and radiological assessment. Two of these books are in second editions. Temporary assignments include service in the US Army Combat Developments Command from  to , and research appointments at Argonne National Laboratory (), UKAEA Culham Laboratory (–), and Oak Ridge National Laboratory (). Research and teaching interests include radiation shielding and dosimetry, boiling heat transfer, and nuclear reactor accident analysis. Dr. Faw was employed part time with Black & Veatch in ABWR plant design from  to  and with GE-Hitachi in ESBWR safety analysis from  to . Dr. Faw now makes his home in Winston-Salem, North Carolina. Gian Luigi Fiorini Gian Luigi Fiorini graduated in nuclear engineering (Pisa, Italy ) and joined the CEA in . His professional experience spans  years of activities related to the definition, the realization, and the management of experimental and theoretical activities concerning the operation and the safety of fission reactors. Since the beginning, until the s he worked essentially on the sodium LMFBR. Since  years, he has been engaged within the innovation program at CEA addressing all the nuclear technologies, including ADS and ITER. He is a former member of the Gen IV Roadmap Integration Team, and currently serves as a “Chargé de Mission Generation IV” at the CEA Department for Reactor Studies (DEN/DER/SESI). Strongly involved within the Gen IV Initiative implementation, he was member of the “Gen IV Sodium Fast Reactor (SFR) Steering committee,” cochairman of the SFR Design & Safety Management board and of the Gen IV Expert Group. He is currently cochairman of the “Gen IV Risk & Safety Working Group,” and member of the French Advisory Group on Safety (GCFS – AREVA, CEA, EDF). Philippe Fougeras Philippe Fougeras received his Ph.D. in reactor physics from the Orsay Faculty of Science in . He started his career at CEA as physicist in the code validation team for Pu recycling in light water reactors (LWRs). He was in charge of the EOLE Critical Facility from  to , xxix xxx Biographies of Contributors covering the MISTRAL program, and was head of the Experimental Programs Laboratory of the CEA Cadarache from  to , supervising programs on EOLE, MINERVE, and MASURCA facilities, also involved in international collaborations and development of new experimental techniques. He is currently deputy head of the section for reactor physics and fuel cycles at Cadarache. János Gadó Dr. János Gadó is director of the KFKI Atomic Energy Research Institute, Budapest, since . The institute belongs to the Hungarian Academy of Sciences. Dr. Gadó graduated at the Roland Eötvös University, Budapest, as a physicist (). He started to work in the field of reactor physics, and later he moved to the area of nuclear safety. He was project manager of various projects related to the safety of the Paks NPP, and represents Hungary on several international committees and projects. Dr. Gadó became Doctor of the Academy in . He was awarded various medals and prizes. Guy-Marie Gautier Guy-Marie Gautier graduated in  from “Ecole Nationale Supérieure d’Electricité et de Mécanique de Nancy,” and has over  years experience on the management of experimental facilities and the definition of operating points for sodium fast reactor (SFR) and pressurized water reactor (PWR). Since , Mr. Gautier has been involved in the definition of the French Atomic Commission (CEA) Innovative Program. He was the coordinator of the innovative activities of the safeguard systems of PWRs, and he was in charge of the definition of the design options of innovative concepts, especially of PWR and now for SFR, including economic assessment studies. Jean-Paul Glatz Jean-Paul Glatz received his Ph.D. in analytical and radiochemistry, and has been working since  at the European Commission’s Joint Research Center (JRC) Institute for Transuranium Elements (ITU), in Karlsruhe, in various research fields. From  to , he was head of the “Hot Cells” department at ITU responsible for all postirradiation examination work on irradiated fuel and other highly active materials, related to the safety of nuclear fuel, spent fuel characterization in view of storage and partitioning and transmutation (P&T). Since , he has been head of the “Nuclear Chemistry” department, responsible for projects on chemical characterization of irradiated fuel and other highly radioactive materials. The corresponding projects include reprocessing studies, the behavior of spent nuclear fuel under repository conditions, and the behavior of radionuclides in the environment and the use of short-lived alpha-nuclides for cancer therapy. Dominique Gosset After receiving a Ph.D. in  in the Yves Quéré’s Irradiated Solids Laboratory, Dominique Gosset joined the Absorber Materials Laboratory in CEA. This laboratory was in charge of fundamental and project studies on the absorber materials to be used in PWR (Ag–In–Cd alloy, hafnium compounds) and mainly in FBRs (boron carbide, moderators) and was a support unit for development laboratories in CEA, including international collaborations (Russia, Japan, Biographies of Contributors and the USA). D. Gosset was especially involved in structural and microstructural analysis of the materials (microscopy, X-ray diffraction), development of new materials and concepts and postirradiation examinations. In the s, his main interests shifted toward fundamental analysis of the behavior of nuclear ceramics under irradiation, focused on microstructure evolutions (e.g., phase transformation in Zirconia and in spinels), through a close collaboration with a CNRS-Ecole Centrale Paris laboratory. More recently, part of his fields of interest come back to the carbides (ZrC, SiC), as potential materials for Gen-IV reactors. He authored or coauthored over  referred papers and patents. Giacomo Grasso Giacomo Grasso received his MS degree in nuclear engineering (summa cum laude) from the University of Bologna, Italy, in . Subsequently, he received his Ph.D. in nuclear reactor physics at the Nuclear Engineering Laboratory (LIN) of Montecuccolino, Department of Energy and Nuclear Engineering and of Environmental Control (DIENCA), University of Bologna, in . He currently serves as researcher in the Italian National Agency for New Technologies, Energy, and Sustainable Economic Development (ENEA) in Bologna, Italy. He worked on the modeling of neutron transport in nuclear reactors, Generation IV nuclear reactor design, computational methods for particle transport, plasma physics and complex system dynamics, and nuclear fuel cycle scenarios. He participated in the ELSY project as a neutronics core designer, and also participated in several European projects for fuel cycle scenario studies. He has authored  technical papers and articles in international conferences and journals. Frank Gunsing Dr. Frank Gunsing is a research team leader in the Nuclear Data Measurements Group of the Nuclear Physics Division (Irfu/SPhN) at the “Commissariat à l’Energie Atomique et aux Energies Alternatives” (CEA) in Saclay, France. Following his Ph.D. degree in nuclear physics from Delft University in the Netherlands in , he worked at the IRMM in Belgium and joined the French CEA in . He obtained the Habilitation to Direct Research from the University of Paris VII in . He is interested in nuclear data and measurement techniques for applications in nuclear technology, astrophysics, and nuclear structure. He is involved in experimental work among others at the IRMM in Geel and at the n_TOF facility at CERN. He is active in European Framework research programs and serves on several scientific advisory committees. Robert C. Haight Dr. Robert C. Haight is a research team leader in the Neutron and Nuclear Science Group of the Los Alamos Neutron Science Center (LANSCE). Following his Ph.D. degree in nuclear physics from Princeton University, he entered experimental neutron physics  years ago and has held various positions at Los Alamos and at the Lawrence Livermore National Laboratory. He has used neutron sources based on cyclotrons, Van de Graaff accelerators, intense -MeV neutron generators with rotating tritium targets, and the spallation neutron sources at LANSCE. He has concentrated on charged-particle production in neutron-induced reactions, fission neutron spectra, and the physics of polarized neutrons. He is interested in applications of nuclear techniques and nuclear data to defense, medicine, space, and fission and fusion energy. He is a Fellow of the American Physical Society. xxxi xxxii Biographies of Contributors Alain Hébert Alain Hébert has been a professor of the Institut de Génie Nucléaire at École Polytechnique de Montréal since . From  to , he worked at the Commissariat à l’Énergie Atomique, located in Saclay, France. During this period, he led the development team of the APOLLO lattice code, an important component of the Science™ and Arcadia™ packages at Areva. Back in Montréal, he participated in the development of the DRAGON lattice and TRIVAC reactor codes, both available as Open Source software. Alain Hébert is the author of Applied Reactor Physics (), a reference textbook in the domain. He is also a contributing author in more than  full papers and  conference papers, published between  and now. Magnus Hedberg Magnus Hedberg is a project manager for one of the European Commissions Safeguards actions; “Nuclear and Trace analysis for Safeguards Purposes (NTAS).” His special interest is in particle analysis for nuclear Safeguards purposes. Hedberg was previously, for a period of  years, the unit head for a mass spectrometry group at the International Atomic Energy Agency laboratories in Austria. Hedberg is originally from Sweden where he earned a M.Sc. degree in electrical engineering at the technical faculty of Lund University in . Michal Herman Michal Herman earned his M.S. in nuclear physics from the University of Warsaw, Warsaw in , Ph.D. in nuclear physics from the Institute of Nuclear Research in Warsaw, Warsaw in . His scientific carrier is dedicated to low-energy nuclear reactions, and is best known for contributions to multistep compound reaction theory. He is the main developer of the nuclear reaction model code EMPIRE widely used as nuclear reaction data evaluation tool. He worked as a research scientist in the Institute of Nuclear Physics Research, Warsaw in –, research scientist in ENEA, Bologna in –; nuclear data physicist in the Nuclear Data Section, IAEA Vienna in –; in , he joined the National Nuclear Data Center at BNL. In January , he took over US nuclear data leadership as the head of the National Nuclear Data Center and chair of the Cross Section Evaluation Working Group. Calvin R. Howell Calvin R. Howell received his B.S. degree from Davidson College in  and obtained his Ph.D. degree from Duke University in . He is a professor of physics at Duke University and the director of the Triangle Universities Nuclear Laboratory. He holds adjunct professorships in the Medical Physics Program at Duke University and in the physics department at North Carolina Central University. He is a Fellow of the American Physical Society (APS). His research includes the study of fundamental properties of nuclear systems, plant physiology using radioisotopes, and applications of nuclear physics in the areas of national security, nuclear energy, and medicine. He has coauthored more than  articles in scientific journals and has held visiting scientist positions at Los Alamos National Laboratory, the Stanford Linear Accelerator Center, and Jefferson Laboratory. Since joining the faculty at Duke University in , he has served as faculty coordinator for the Carolina Ohio Science Education Network, as the faculty coordinator for the Mellon Minority Undergraduate Fellowship Program, and as the academic coordinator for the Summer Medical and Dental Education Program at the Duke Biographies of Contributors University Medical Center. Professor Howell has served the physics community extensively as a Nuclear Physics Program Director at the National Science Foundation (NSF), as a member of the Department of Energy (DOE)/NSF Nuclear Science Advisory Committee, as a member of the Executive Committee of the Division of Nuclear Physics of the APS, as chair of the Executive Committee of the Southeastern Section of the APS, as chair of the APS Committee on Minorities, and as a member of numerous NSF and DOE review and planning panels. John Howell Dr. John Howell has been with the Faculty of Engineering, University of Glasgow in the UK since . Prior to this position, he was with the Control and Instrumentation Division of the United Kingdom Atomic Energy Authority. He started his career as a control and dynamics engineer with responsibilities for the assessment, design, and development of control and measurement systems for the UK’s prototype fast reactor and associated reprocessing plant, which were located at Dounreay. Dynamic modeling was a core component of these activities. His research focus over the past  years has been on the evaluation of monitoring data collected from process plants, in general. The emphasis has been on isolation and diagnosis. Although his roots are very much in model-based reasoning, he has also researched into approaches including ICA, rule-based reasoning, and signed-graph analysis. He has contributed to the development of a number of solution monitoring systems for the IAEA. Jean-Pascal Hudelot Jean-Pascal Hudelot received his Ph.D. in reactor physics from the Grenoble Faculty of Science (). He started his career at CEA as a physicist in the development of innovative neutron and gamma measurement techniques on experimental facilities. He was in charge of the MINERVE Facility from  to , covering the OSMOSE, HTC, VALMONT, and OCEAN programs, and was also involved in international collaborations. He is now head of the Nuclear Project Laboratory at CEA Cadarache, supervising the development, validation and qualification of neutronics, and photonics calculation tools for experimental and irradiation reactors, including JHR, OSIRIS, CABRI. Mihaela Ionescu-Bujor Mihaela Ionescu-Bujor received her Dr.-Ing. degree from the Institute for Nuclear Technology and Reactor Safety, Faculty of Mechanical Engineering, University of Karlsruhe, Germany, in , with the grade of “Excellent with Distinction.” For  years prior to her doctoral degree, she was a staff researcher at Siemens AG, KWU, Erlangen, working in the reactor physics and thermal-hydraulics section. During –, she served as group leader for sensitivity and uncertainty analysis in the Institute for Reactor Safety at Forschungszentrum Karlsruhe (FZK), performing original research on and supervising the implementation of the adjoint sensitivity analysis procedure (ASAP) into various large-scale multi-physics computational code systems. Typical systems included the development of the coupled two-fluid and heat structure adjoint models for the reactor safety code system RELAP/MOD., and sensitivity and uncertainty analysis of dynamic reliability of large-scale systems modeled by Markov chains. During –, she served as a task force leader in the FUSION Program at FZK, providing management and scientific leadership for the design and construction of the Helium Loop Karlsruhe (HELOKA) facility at FZK. Since , Dr. Ionescu-Bujor has been program manager at xxxiii xxxiv Biographies of Contributors the Karlsruhe Institute of Technology/Forschungszentrum Karlsruhe (KIT/FZK), responsible for planning KIT/FZK fusion projects, including scheduling, resource allocation, cost control, budget management, and quality management. Dr. Ionescu-Bujor has extensive expertise in sensitivity and uncertainty analysis of largescale systems; consistent assimilation of experimental and computational data for obtaining best estimate results with reduced uncertainties, reliability analysis, numerical methods for multiphysics (thermal-hydraulics, structural, neutronics) code systems, and computer software. Dr. Ionescu-Bujor has coauthored  books,  book chapters, and over  peer-reviewed articles. Frédéric Jasserand Frédéric Jasserand is an engineer at CEA in the radioprotection field, performing shielding studies, including dosimetry studies for fabrication and recycling facilities, and the conception of transport casks. Han Gon Kim Dr. Han Gon Kim received his B.S. in nuclear engineering from Seoul National University and Ph.D. in nuclear engineering from Korea Advanced Institute of Science and Technology (KAIST). In , he became a member of Korea Hydro & Nuclear Power Co., Ltd. (KHNP) as a senior researcher, where he is actively involved in research and development in nuclear reactor thermal hydraulics. From  until now, he is a team leader being responsible for design of nuclear steam supply system (NSSS) of advanced nuclear power plants. He is a member of Korea Nuclear Society. Inn Seock Kim Inn Seock Kim received his Ph.D. from the University of Maryland in the area of process diagnostics for large technological systems such as nuclear power plants, and his work in this area has been acknowledged by many organizations, including the Halden Reactor Project of the OECD and the Korea Atomic Energy Research Institute. Dr. Kim served on the scientific staff of Brookhaven National Laboratory from  through , and also used to teach many courses related to reliability engineering and probabilistic analysis of systems safety while serving on the Nuclear Engineering faculty of Hanyang University in Seoul, Republic of Korea. Inn Seock Kim is particularly interested in helping secure safety in both operating and future nuclear power plants. He has been involved in research activities for enhancing decisionmaking infrastructure, licensing framework, and regulatory effectiveness by use of both deterministic and risk-informed methodologies. He has consulted to numerous organizations throughout the world, such as the US Nuclear Regulatory Commission, Swiss nuclear regulatory body, Spanish nuclear regulator, European Space Agency, and Korean nuclear institutions including Korea Institute of Nuclear Safety. He recently founded International System Safety Analysis (ISSA) Technology, Inc., in Maryland, which specializes in probabilistic safety assessment and applications, accident management, instrumentation and controls, and human factors analysis. Dave Knott Dave Knott received his Ph.D. in nuclear engineering, from the Pennsylvania State University in , for development of the lattice physics code KRAM to model the detailed depletion Biographies of Contributors of Gadolinium isotopes from boiling water reactor (BWR) fuel designs. He has also received a Master of Computer Science from Cleveland State University in . His career includes employment at the Perry Nuclear Power Plant (–, in-core analysis); Studsvik of America, Inc. (–, development of the lattice physics code CASMO-); First Energy Nuclear Operating Company (–, creation of the core design and physics support group at the Perry Nuclear Power Plant); Global Nuclear Fuels (General Electric, –, development of the lattice physics code LANCER); Studsvik Scandpower, Inc., (–Present, development of the safety-related software MARLA for optimizing fuel movement during refueling outages and during dry storage cask loading campaigns). In his current position as a senior nuclear engineer at Studsvik Scandpower, Inc., in Wilmington, North Carolina, he has focused on developing software for improving capacity factors at boiling water reactors (BWRs) and pressurized water reactors (PWRs). He has served as a research adjunct associate professor of nuclear engineering, University of Cincinnati, –. He has also served as a visiting scholar in nuclear engineering, Ohio State University, , and an external advisor, Ph.D. thesis committee, Pennsylvania State University, . Rudy J. M. Konings Rudy J. M. Konings received his Ph.D. in chemistry from the University of Amsterdam in , working since  for ECN, the Energy Research Centre of the Netherlands, on various topics related to nuclear fuel, such as fission product chemistry and transmutation. In , he became the manager of the product group Fuels, Actinides and Isotopes at the Nuclear Research and consultancy Group (NRG), a partnership firm created by the merger nuclear activities of ECN and KEMA. A year later, he left NRG for a position at the European Commission’s Joint Research Centre (JRC), Institute for Transuranium Elements (ITU). There he continued working on nuclear fuels, with a strong interest in thermodynamic studies. Since , he is head of the Materials Research department of ITU, focusing on scientific research in support of the development of safe advanced fuel. Rudy Konings was appointed editor of Journal of Nuclear Materials in . Edward W. Larsen Edward W. Larsen is a professor in the department of Nuclear Engineering and Radiological Sciences at the University of Michigan. In , he obtained a Ph.D. in Mathematics from Rensselaer Polytechnic Institute and joined the faculty of the Department of Mathematics at New York University. In , he became an associate professor in the Department of Mathematics at the University of Delaware. In , he joined the Transport Theory Group at Los Alamos National Laboratory, and in , he became a professor in the Department of Nuclear Engineering and Radiological Sciences at the University of Michigan. Professor Larsen’s research involves the development of advanced mathematical algorithms for solving particle transport problems. His work has included neutron transport methods for nuclear reactor applications, thermal radiation transport methods, and charged particle transport methods for medical physics applications. His research has helped extend the theory and simulation of neutral and charged particle transport processes through, for example, the development of improved discretization methods for deterministic transport calculations, improved acceleration methods for speeding up the iterative convergence of deterministic calculations, and new hybrid Monte Carlo-deterministic algorithms. xxxv xxxvi Biographies of Contributors Professor Larsen is a Fellow of the American Nuclear Society (ANS). For his contributions to nuclear engineering, he received the US Department of Energy E.O. Lawrence Award (), the ANS Arthur Holly Compton Award (), and the ANS Eugene P. Wigner Reactor Physicist Award (). Christian Latge Christian Latge received his doctorate in chemical engineering, and engaged in R&D related to sodium fast reactor technology and SuperPhenix start-up at CEA, in . During the period –, he was involved in the thermonuclear project ITER for the design of tritium systems. From  to , he was head of Laboratory “Process Studies and consultancy” for nuclear reactor technologies. During –, he served as the director of the Sodium School, while during –, he was head of service for Process studies for Decontamination and Nuclear Waste Conditioning. Since , he has been a research director at CEA, serving also as project director (–) of the International Project Megapie (Spallation target for waste transmutation). In addition, he currently serves as professor and International Expert in liquid metal technologies, involved in various European projects, including education and training. Clément Lemaignan Clément Lemaignan is a research director at CEA and professor at INSTN on nuclear metallurgy, physics of fracture and material science. Prof. Lemaignan holds  patents, and has authored  books,  review book chapters, and over  peer-reviewed articles. He has been editor for the Journal of Nuclear Materials for more than  years. He is an officer of the Palmes Académiques and has received many other distinctions. Daniel Lhuillier Daniel Lhuillier is a senior researcher with the French National Center for Scientific Research (CNRS). After graduating from the Ecole Normale Supérieure, he obtained a Ph.D. in physics from the University Pierre and Marie Curie (Paris, France). He began his research work at the Laboratory of Aerodynamics (Orsay, France), then moved to the Laboratory for Modeling in Mechanics and is presently a member of the Institute Jean le Rond d’Alembert (IJLRA) of the University Pierre and Marie Curie. Dr Lhuillier is a specialist of thermo-mechanics of continuous media, with applications to superfluid helium, polymer solutions, suspensions of particles, two-phase mixtures, and granular materials. He received the bronze medal of CNRS and the E.A. Brun prize of the French Academy of Sciences. Meng-Sing Liou Meng-Sing Liou is a senior technologist of NASA Glenn Research Center. He received B.S. in mechanical engineering from National Cheng Kung University, Taiwan and Ph.D. in aerospace engineering from the University of Michigan. His research interest has been centered about computational fluid dynamics, including development of numerical algorithms and applications to topical areas of high-speed aerodynamics, air-breathing propulsion, and multiphase flows. Multidisciplinary design optimization for aeronautical systems is another interest currently. Dr. Liou has received several major NASA awards, including Abe Silverstein Medal (), Exceptional Achievement Medal (), and Exceptional Scientific Achievement Medal (). Biographies of Contributors Pierre Lo Pinto Pierre Lo Pinto is a senior engineer with over  years experience in the area of nuclear safety and fast neutron reactor system. He worked on analyzing fast sodium reactors (Phenix, Superphenix, etc.), and coordinated international collaborations (Japan, Russia, etc.) on several innovative design or safety studies in particular in the fields of SFR (e.g., ECRA, CAPRA). More recently, he has been in charge of designing innovative SFR concepts at CEA, and safety-related topics, and is involved as a CEA representative in the SMFR (small modular fast reactor) project involving ANL, CEA, and JAEA. Klaus Lützenkirchen Since , Dr. Klaus Lützenkirchen has been head of the Nuclear Safeguards and Security Department of the European Commission’s Joint Research Centre Institute for Transuranium Elements in Karlsruhe, Germany. His activities are focused on nuclear forensics, nuclear safeguards, and nonproliferation. He has previously worked at GSI (Gesellschaft für Schwerionenforschung) in Darmstadt, Germany, at the Weizmann Institute in Rehovot, Israel, as an assistant professor at the University of Mainz, and as professor of nuclear chemistry at the University of Strasbourg, France. He holds a doctorate in nuclear chemistry from the University of Mainz, Germany. Robert E. MacFarlane Robert E. MacFarlane has received his Ph.D. from the Carnegie Institute of Technology in  and then joined the Physics Division at Los Alamos National Laboratory, where he did experimental studies of the lattice dynamics of bismuth and antimony with inelastic neutron scattering. When that program shut down, he moved to the Theoretical Division to work on nuclear data. There he led the development of the NJOY nuclear data processing system and worked on the production of data libraries for a wide range of applications, such as fast reactors, thermal reactors, fusion systems, accelerator systems, weapons, and cold-neutron facilities. During this period, he also had several administrative positions, including a number of years as group leader, he managed the group’s network of workstations on the side, and he provided his group with an early presence on the World Wide Web. MacFarlane was also active in the Cross Section Evaluation Working Group (CSEWG), which manages the US-evaluated nuclear data files (ENDF/B). He was involved in the ENDF/B-V, VI, and VII libraries. He worked on the development of a number of formats, including the ENDF- structure, energy-angle distributions, charged-particle representations, atomic data, and thermal scattering data. He was heavily involved in data testing for the various ENDF/B versions, and he also participated in the evaluation effort. Some significant components of this were his work on () improving the performance of ENDF/B for fast critical assemblies, including Godiva, Jezebel, and Bigten; () energy balance issues for heating and damage; () thermal neutron scattering data for important moderators. MacFarlane officially retired in , but remains active by continuing to work part time. Philippe Martin Philippe Martin has received his degree of Doctor Engineer in , and has spent his professional career (–) at the Atomic Energy Commission (CEA), France. He served as head of the Components and Structures Design Laboratory and a member of the French Committee xxxvii xxxviii Biographies of Contributors for rules to be applied to Fast Reactors Design (–); head of the life extension Project of the Phénix reactor (–); senior expert (–); head of the Innovative reactors & systems Section (–); head of the Simulation of fuel behavior Section (–); deputy head of the Fuel Department in charge of development of fuels for fourth-generation reactors; and CEA’s representative to the Gen-IV SFR Project arrangement negotiation (–). He is a member of the GEN IV Sodium Fast Reactors Steering Committee of GENIV International Organization (–) and contributor to CEA’s Nuclear Energy Division SFR program. Since , he has served as an engineer-consultant for the Nuclear Energy Division of CEA. Patrick Masoni Patrick Masoni is an engineer at CEA specializing in thermo-mechanical analysis of structures. During –, he analyzed the thermomechanical static behavior of the cores of the French fast breeder reactors Phenix and Superphenix. During –, he performed theoretical and experimental dynamic mechanical studies of light water reactor (LWR) fuel pins and fuel assemblies, and the mechanical behavior of the Phenix reactor core (vibrations, shocks, and seismic calculations). Since , he has performed theoretical and experimental thermomechanical studies regarding the behavior of fuel pins for future (Gen-IV) reactors (GFR and SFR), and also assessed the static thermomechanical behavior of existing French fast breeder reactors. Klaus Mayer Dr. Klaus Mayer obtained his Ph.D. in  in the field of radiochemistry and analytical chemistry from the University of Karlsruhe, Germany. He then worked for  years at the Institute for Transuranium Elements (ITU) as postdoctoral researcher. In , he started working with the European Commission at IRMM Geel (Belgium) on actinide isotopic reference materials, high accuracy mass spectrometric measurements of U, Pu, and Th, the organization of an external quality control program for nuclear material measurements, and the coordination of support activities to the Euratom safeguards office and to the IAEA. In , he moved to ITU Karlsruhe (Germany) for working on the development and application of analytical methods for nuclear safeguards purposes. He is chair of the ESARDA (European Safeguards Research and Development Association) Working Group on Destructive Analysis. Currently, he is in charge of ITU’s activities on combating illicit trafficking and nuclear forensics, and is cochairman of the Nuclear Smuggling International Technical Working Group (ITWG). Frederic Mellier Frederic Mellier is a research engineer at CEA. He started his career participating to neutron physic studies for SUPERPHENIX before moving toward activities in support of PHENIX operating and irradiation program. He was in charge of the development and maintenance of the PHENIX core management tools and was involved in experiments for absorber rod reactivity worth measurements. He joined the MASURCA facility in  as the person responsible for experimental programs. He coordinated the MUSE- program (funded by the European Commission within the framework of the fifth EURATOM/FP) and is now involved in the GUINEVERE project (sixth FP), both programs aiming at studying the reactivity measurement issue in subcritical systems driven by an external source. Biographies of Contributors Gerard Mignot Gerard Mignot obtained his Ph.D. in  at the Aix-Marseille University, option energy science. He joined CEA as an engineer, working on thermal-hydraulics with applications on different nuclear reactor types: sodium fast reactor, pressurized water reactor, and high temperature reactor. From  to , he was in charge of studies on light water reactor innovative fuels and systems in the service of innovative reactor studies in Cadarache. In , he changed fields and coordinated core design studies for the future sodium-cooled fast reactor. Since , he serves as a project manager in the Reactor Studies Department, in charge of the core and reactor design for sodium fast reactor within the CEA SFR program. Mohammad Modarres Mohammad Modarres is a University of Maryland distinguished scholar–teacher, a professor and director of Nuclear Engineering, and director of Reliability Engineering at the University of Maryland; he is also a Fellow of the American Nuclear Society. He received his Ph.D. in nuclear engineering from MIT in  and M.S. in mechanical engineering also from MIT. Professor Modarres’ research areas are probabilistic risk assessment, uncertainty analysis, and physics of failure degradation modeling. He has served as a consultant to several governmental agencies, private organizations, and national laboratories in areas related to probabilistic risk assessment. Professor Modarres has authored or coauthored over  papers in archival journals and proceedings of conferences,  books,  handbook,  edited books, and  book chapters in various areas of risk and reliability engineering. Dr. Modarres’ interests in energy technologies are in the next-generation nuclear reactors and nuclear fuel cycle safety, risk assessment, and management. These areas include research to advance probabilistic risk and safety assessment techniques and applications to complex energy technologies and addressing possible safety issues of the next generation of nuclear power designs in the context of the current combined risk-informed and traditional defensein-depth regulatory paradigm. His research focuses on understanding safety and regulatory implications of the advanced nuclear reactors. Prof. Modarres is also performing research on the following areas: () technology-neutral nuclear power plant regulation; () hazard assessment of fire in advanced nuclear power plants; () fatigue and corrosion-based degradation assessment of reactor vessels and piping of advanced nuclear power plants using probabilistic modeling based on physics of failure with characterization of uncertainties; () best estimate thermalhydraulic analyses of reactor transients using probabilistic methods to account for variability and uncertainties; and () risk and performance-based maintenance techniques for monitoring and assessing nuclear plant system health. Victor A. Mokhov Dr. Victor A. Mokhov was born on February , , in Sverdlovsk. In , he graduated as a mechanical engineer from the Bauman Moscow State Technical University and joined OKB “GIDROPRESS” as an engineer. His career progressed through the positions of design engineer, deputy head of department, and head of department. He is currently the chief designer of OKB “GIDROPRESS,” and is an honored designer of the Russian Federation. His professional interests include the following: reactor plant design, accident analysis, probabilistic safety analysis, and the development of thermal-hydraulic codes for reactor plant transient analyses. xxxix xl Biographies of Contributors Said Mughabghab Dr. Said Mughabghab received his M.S. in nuclear physics from the American University of Beirut, Beirut in , followed by Ph.D. in nuclear physics from the University of Pennsylvania, Philadelphia in . He dedicated his scientific carrier to physics of nuclear reactions in the low-energy region, and is best known for numerous contributions to physics of neutron resonances and reaction mechanisms. He has gained worldwide reputation as key author of compendia of neutron resonance parameters and thermal cross sections commonly known as BNL-, its fifth edition published as Atlas of Neutron Resonances in . He has been a research scientist in the National Nuclear Data Center at Brookhaven National Laboratory since , involved in neutron physics research, high-flux beam reactor experiments, neutron resonances, neutron radiative capture and nuclear level densities, compilation of neutron resonance parameters and thermal cross sections. From  to , he worked in the BNL Reactor System Division with J. Powell on the Space Nuclear Thermal Propulsion program to determine the feasibility of using a small reactor for nuclear rocket propulsion for possible future mission to Mars. Mikhail P. Nikitenko Dr. Mikhail P. Nikitenko was born on July , , in Mariinsk, Khabarovsk region, and graduated as a physical engineer from the Tomsk Polytechnical University, in . He joined OKB “GIDROPRESS” as an engineer in , and became head of group, and head of department. Currently he is the deputy chief designer of OKB “GIDROPRESS,” and is an honored designer of the Russian Federation. His professional interests include reactor plant design development, safety analysis, and reactor process analysis. David Nowak David Nowak graduated from Rensselaer Polytechnic Institute with a B.S. degree in physics and from the University of Chicago with a Ph.D. in theoretical physics. His current position at Argonne National Laboratory involves working with the Department of Energy to develop a modeling and simulation effort in support of a revitalized nuclear enterprise in the USA. Prior to Argonne, Dr. Nowak held a series of senior management positions at Lawrence Livermore National Laboratory (LLNL) which included a broad spectrum of highly visible national security responsibilities. Until June , he was deputy associate director for Defense and Nuclear Technologies. He was a founding member of and the first LLNL Program Leader for the Department of Energy’s Accelerated Strategic Computing Initiative (ASCI) Program. He led the team that developed the acquisition strategy that culminated in the -Teraflops Purple system and the -Teraflops Blue Gene system. He was one of the principals developing and executing the ASCI University Alliance Program. Dr. Nowak served as advisor to the Assistant Secretary for Defense Programs in the Department of Energy. Currently, Dr. Nowak splits his time between Chicago and Paris. Pavel Obložinský Pavel Obložinský received his M.S. in nuclear physics from the Czech Technical University, Prague, in , and his Ph.D. in nuclear physics from the Slovak Academy of Sciences, Bratislava, in . His scientific carrier has been dedicated to low-energy nuclear reactions, Biographies of Contributors focusing on studies of neutron-induced reactions. He is best known for numerous contributions to pre-equilibrium nuclear reaction theory and nuclear reaction data evaluation projects. Dr. Obložinský held positions as head of Nuclear Physics Department, Slovak Academy of Sciences, Bratislava (–); deputy head of Nuclear Data Section of the International Atomic Energy Agency, Vienna (–); head of the National Nuclear Data Center, Brookhaven National Laboratory (–); chair of the US Nuclear Data Program (–); chair of the US Cross Section Evaluation Working Group (CSEWG) (–). He was responsible for the release of the US-evaluated nuclear data library ENDF/B-VII. in . Laurent Paret Laurent Paret graduated from the Département Génie mécanique, Université de Technologie de Compiègne, in . After working at AREVA NC as the engineer in charge of deployment of a software to manage MOX fuel fabrication, he joined the CEA, in , to work on MOX-fuels fabrication for experimental irradiations. Since , he has been performing fuels studies for Gen-IV reactors (GFR and SFR). Currently, he is project manager in charge of the sodium fast reactor (SFR) fuels development. Imre Pázsit Imre Pázsit (M.Sc., ; Ph.D., , Budapest, Hungary) is professor and chair of the Department of Nuclear Engineering, Chalmers University of Technology, Göteborg, Sweden. Former employments include the Central Research Institute for Physics, Budapest (–) and the nuclear research center Studsvik, Nyköping, Sweden (–). Imre Pázsit is a Fellow of the American Nuclear Society (), and a member of the Executive Committee on the Mathematics and Computation Division of ANS. He is a member of the Royal Swedish Academy of Engineering Sciences () and a working member of the Royal Academy of Arts and Sciences in Göteborg (). Since , he has also been an adjunct professor in the Department of Nuclear Engineering and Radiological Sciences of the University of Michigan, Ann Arbor, USA. Between and  he was the head of the Section for Mathematical Physics of the Swedish Physical Society. His research interests include the following: fluctuations in neutron transport; atomic collision cascades and fusion plasmas; reactor dynamics; neutron noise analysis applied to reactor diagnostics and nondestructive analysis in nuclear safeguards; transport theory of neutral and charged particles; intelligent computing methods such as artificial neural networks and wavelet analysis; diagnostics of two-phase flow and fusion plasma; positron annihilation spectroscopy and positron transport. He coauthored with L. Pál a book entitled Neutron Fluctuations – a Treatise on the Physics of Branching Processes, Elsevier Science Ltd., . He has published over  articles in international journals; several book chapters, numerous reports, conference proceedings, popular science booklets, and articles. He is a member of the Editorial Board of the Annals of Nuclear Energy, the International Journal of Nuclear Energy Science and Technology, and the Nuclear Technology and Radiation Protection Journal. Michel Pelletier Michel Pelletier received his Ph.D. from the University of Paris IX (Orsay), in materials sciences, in , and joined CEA, Cadarache working in experimental fuel studies (definition of irradiation, PIE, and synthesis). During –, he was a member of the SFR fuel-safety xli xlii Biographies of Contributors working group (CABRI program). A senior expert since , he is also lecturer at the “Institut National des Sciences et Techniques Nucléaires” and at the School of plutonium (Cadarache), specializing in the modeling and design of reactor fuel behavior. Alexander K. Podshibyakin Dr. Alexander K. Podshibyakin was born on June , , in Podolsk, Moscow region, and graduated as an engineer in thermal physics from the Moscow Power Engineering University, in . He joined OKB “GIDROPRESS,” and progressed through a sequence of positions of increased responsibility: leading engineer, head of department, first deputy chief designer of OKB “GIDROPRESS.” Currently, he is the chief specialist on VVER NSSS, and is an honored designer of the Russian Federation. His professional interests include the following: reactor plant design development, analytical and theoretical studies, elaboration of the schemes and processes of reactor plant operation, commissioning tests at NPPs, safety analyses. Anil K. Prinja Anil K. Prinja is a professor and associate chair in the Department of Chemical and Nuclear Engineering at the University of New Mexico (UNM), USA. He obtained his B.Sc. (“st Class Honors,” ) and Ph.D. () in nuclear engineering from Queen Mary College, University of London, UK, and held a research staff appointment at the University of California, Los Angeles (UCLA), before joining UNM in . He has since held visiting professor’s appointments at Chalmers University, Sweden, and at UCLA, and has an ongoing affiliate appointment at Los Alamos National Laboratory, USA. Professor Prinja’s research interests include development of efficient solution techniques for stochastic and deterministic computation of high-energy charged particle transport, formulating models for radiation transport in random media, application of stochastic methods to uncertainty quantification in radiation transport, and investigating neutron branching processes in multiplying media. Professor Prinja is a Fellow of the American Nuclear Society. From  to , he was the associate editor of Annals of Nuclear Energy and presently serves on the Editorial Boards of Transport Theory and Statistical Physics, and Annals of Nuclear Energy. Jean Claude Queval Jean Claude Queval, born in , graduated as an aerospace engineer in , has a very extensive experience in dynamic and seismic behavior of equipment and structures, for  years. He is in charge of the experimental part of the EMSI Laboratory at CEA Saclay. He is the author of many papers in R&D and seismic qualification tests, fluid-structure interaction computation, and fuel assembly modeling. Baldev Raj Dr. Baldev Raj (born in ; B.E., Ph.D., D.Sc.) holds memberships in the International Nuclear Energy Academy, German National Academy of Sciences; he is a Fellow of the Third World Academy of Sciences, and Fellow of all Engineering and Science Academies in India. He is a distinguished scientist and director, Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu. His specializations include materials characterization, testing and evaluation using nondestructive evaluation methodologies, materials development and performance assessment Biographies of Contributors and technology management. He has more than  publications in leading refereed journals and books. He has coauthored  books and coedited  books and special journal volumes. He has  Indian Standards and  patents to his credit. He is editor-in-chief of two series of books: one related to NDE Science and Technology and another related to Metallurgy and Material Science. He is on the editorial boards of national and international journals. He is the member of many national and international committees and commissions. He has been invited to deliver plenary and panel speeches in the most eminent international forums and more than  occasions in  countries. He has won many national and international awards and honors. He has a passion for teaching, communications, and mentoring. His other interests include science and technology of cultural heritage and theosophy. Ulrich Rohde Ulrich Rohde studied physics at the University of Minsk (Byelorussia), obtaining a master’s degree in . During –, he worked as a research scientist in the Central Institute for Nuclear Research in Rossendorf near Dresden, the later Forschungszentrum DresdenRossendorf (FZD). In June , he obtained a doctorate in physics in the field of two-phase flow modeling and boiling water reactor (BWR) stability analysis. Since , he has been the head of the Safety Analysis Department of the Institute of Safety Research of the FZD, performing research in reactor dynamics, thermal hydraulics, and fluid dynamics. Dr. Rohde gained particular experience in reactor dynamics and safety analysis of VVERtype reactors. He contributed to the development and validation of the Rossendorf reactor dynamics code DYND, which was initially designed for VVER type reactors, and is currently used as a transient-analysis tool in most of the countries with operating VVER reactors; the code is a part of the European software platform NURSIM. He has participated on scientific and management level in several international projects on safety analyses and validation of coupled neutronics/thermal hydraulics codes for VVER. From  until , he was a member of the Scientific Council of Atomic Energy Research (AER), an international association on reactor safety and reactor safety of VVER. Gilles H. Rodriguez Gilles H. Rodriguez (Engineer, chemistry, University of Lyon, France, ; Master of Science in chemical and process engineering, Polytechnic University of Toulouse, France, ) is a senior expert engineer at CEA/CADARACHE (French Atomic Energy Commission/Cadarache center). He is working on Generation-IV Fast Reactor research program. His areas of expertise are fast reactor technology, liquid metal processes, and process engineering. Since , he has been a project leader of sodium technology and components, within the CEA SFR project organization, and is representing CEA/France on the GEN IV SFR Project Management Board (component design and balance of plant). Vincenzo V. Rondinella Vincenzo V. Rondinella has received his Ph.D. in materials sciences from Rutgers University, New Jersey. Currently, he is the head of the Hot Cells (HC) department at the Institute of Transuranium Elements (ITU), the European Commission’s Joint Research Centre (JRC) located in Karlsruhe, Germany. The HC department performs postirradiation examinations on light water reactor (LWR) and advanced reactor fuels and cladding materials, focusing xliii xliv Biographies of Contributors on properties and behavior related to various stages of the nuclear fuel cycle, including fuel behavior in-pile, effects associated with different irradiation history (burn up, temperature, power, etc.) and behavior/evolution during storage and disposal. Dr. Rondinella’s main areas of interest include the follwing: radiation damage, properties of high burn up nuclear fuels, mechanical properties of the fuel rod system (fuel + cladding/coatings), and the development and characterization of advanced fuels for new reactors. Jacques Rouault Jacques Rouault graduated in  from Ecole Centrale de Paris, and joined CEA (Commissariat à l’Energie Atomique) in . He spent most of his career in the field of fast reactors. He was successively involved in the field of fast reactor fuel behavior (postirradiation examinations, modeling). From  to , he was the CAPRA project manager (Feasibility studies of a fast reactor optimized to burn Pu) in the context of a large international collaboration. From  to , he managed the Section of fuel development (FBR, PWR), covering postirradiation examinations, code development, experimental irradiation in Phénix and other reactors, and the material program for Minor Actinides transmutation. From  to , he directed the Section of Innovative Systems Study mainly focusing on GEN IV system design. Since , he has responsibilities in the CEA Gen-IV program management with emphasis on the sodium fast reactor. Sergei B. Ryzhov Dr. Sergei B. Ryzhov was born on November , , in Chekhov, Moscow region. He was graduated in  as a mechanical engineer, from the Bauman Moscow State Technical University, and joined OKB “GIDROPRESS” as an engineer. Since then, he progressed through appointments as design engineer, head of group, head of department, and chief designer. Currently, he is the director-generaldesigner of OKB “GIDROPRESS,” and has been designated an honored designer of the Russian Federation. His professional interests include the following: reactor plant design development, and participation in research and experimental activities. Mark Schanfein Mark Schanfein joined Idaho National Laboratory (INL) in September  as their senior nonproliferation advisor, after a -year career at Los Alamos National Laboratory (LANL). His current focus is on leveraging INL technology, facilities, and nuclear material to build an international safeguards program. He has over  years of experience in international and domestic safeguards. This includes his  years at Los Alamos National Laboratory where he most recently served as the program manager for Nonproliferation and Security Technology, a portfolio that included international and domestic safeguards and security. He also spent  years at the LANL Plutonium Facility as the team leader for all NDA measurements at this facility, and the Chemistry and Material Research facility, totalling over  instruments. He has over  years of experience working at the International Atomic Energy Agency, where he served  years as a safeguards inspector and inspection group leader covering inspections on a diverse set of facilities, and another  years as the unit head for Unattended Monitoring Systems. Biographies of Contributors J. Kenneth Shultis J. Kenneth Shultis, born in Toronto, Canada, graduated from the University of Toronto with a BA.Sc. degree in engineering physics (). He gained his M.S. () and Ph.D. () degrees in nuclear science and engineering from the University of Michigan. After a postdoctoral year at the Mathematics Institute of the University of Groningen, the Netherlands, he joined the Nuclear Engineering faculty at Kansas State University in  where he presently holds the Black and Veatch Distinguished Professorship. He teaches and conducts research in radiation transport, radiation shielding, reactor physics, numerical analysis, particle combustion, remote sensing, and utility energy and economic analyses. He is a Fellow of the American Nuclear Society, and has received many awards for his teaching and research. Dr. Shultis is the author or coauthor of six textbooks on radiation shielding, radiological assessment, nuclear science and technology, and Monte Carlo methods. He has written over  research papers and reports, and served as a consultant to many private and governmental organizations. Craig F. Smith Dr. Craig F. Smith earned his B.S. degree in engineering (summa cum laude) at the University of California, Los Angeles (UCLA) in  and his Ph.D. in nuclear science and engineering, also at UCLA, in . He is a Fellow of the American Nuclear Society (ANS) and of the American Association for the Advancement of Science (AAAS). He is currently serving on assignment as the Lawrence Livermore National Laboratory (LLNL) chair professor of Physics at the Naval Postgraduate School (NPS) in Monterey. He also serves as adjunct faculty at the Monterey Institute of International Studies. In addition to his faculty role, Dr. Smith leads research efforts in nuclear energy technology at LLNL. He is the US member of the Generation IV International Forum (GIF) Provisional System Steering Committee on lead-cooled fast reactors, and has worked internationally in research related to nuclear energy technology, radiation detection, and automated systems. He has authored more than  technical papers and articles and  books, including the recently published Connections: Patterns of Discovery, (Wiley, ). Joe Somers Joe Somers hails from Dublin, Ireland, where he completed a chemistry degree at Trinity College Dublin in . Thereafter, he concluded a Ph.D. on surface chemistry (), before joining the Fritz Haber Institut der Max Planck Gesellschaft in Berlin. There, his investigations centered on the use of UV and soft X-ray photoelectron and absorption spectroscopy at the synchrotron radiation source BESSY for the elucidation of geometric and electronic structure of adsorbates on metal and semiconductor surfaces. In , he joined the scientific staff of the European Commission’s Joint Research Centre at the Institute for Transuranium Elements (JRC-ITU) in Karlsruhe, initially investigating the agglomeration of airborne particles using very high intensity ultra sonic waves. For the last  years, his research interests have concentrated on the development and testing of ceramic materials for nuclear fuel applications. This research has covered a wide variety of nuclear fuels (UO , MOX, Minor actinide fuels) for a variety of reactor systems (LWR, HTR, SFR, GFR, ADS), and has been performed through institutional and international programs. In addition, the local structure of actinide-bearing compounds and its evolution due to irradiation damage xlv xlvi Biographies of Contributors has been a subject of particular interest. Currently, he is head of the Nuclear Fuels department at the JRC-ITU. Ugo Spezia Ugo Spezia has a master’s degree in nuclear engineering, and has been active in the Italian nuclear industry since , working on the Italian Unified Nuclear Project. Since , he has been the secretary-general of the Italian Nuclear Association (AIN), advisor of the Italian Nuclear Regulatory Agency (ANPA), and, since , technical manager of the Italian Nuclear Plant Management Company (SOGIN), the state-owned company charged of the Italian nuclear plant decommissioning. Since , he has been technical manager of the SOGIN Safety Project Control. He is author of several publications on nuclear energy and nuclear technology and coauthor of the book Nuclear Plant Decommissioning (). Theofanis G. Theofanous Theofanis (Theo) G. Theofanous is professor in the Chemical Engineering and Mechanical Engineering Departments at UCSB, and founding director of the Center for Risk Studies and Safety (CRSS). He is a graduate of the National Technical University of Athens, Greece, and holds a Ph.D. from the University of Minnesota, both in chemical engineering. Prior to coming to UCSB in , he taught at Purdue’s Chemical Engineering and Nuclear Engineering Departments, where he was also the founding director of the Nuclear Reactor Safety Laboratory. Professor Theofanous has worked on the physics and computation of multiphase flows and on methodologies for addressing uncertainties in risk management; he championed a key-physics-based approach to addressing complexity as a basis for decision making; and he brought applications to completion on important safety issues in both the chemical and nuclear industries, and as of recently in the domain of National Defense. He is consulted extensively, and internationally, for industrial and governmental organizations, and served on a number of National Research Council Panels, including the one that assessed the safety of the Nation’s Research and Production (Defense, Nuclear) Reactors in the aftermath of the Chernobyl accident. His practical credits include the following: the Risk Oriented Accident Analysis Methodology (ROAAM), the In-Vessel Retention (IVR) design concept for PWRs, and the Basemat-Internal Melt Arrest and Coolability (BiMAC) device for BWRs and large PWRs. He is a member of the National Academy of Engineering, a fellow of the American Nuclear Society, and was awarded an honorary doctorate from the University of Lappeenranta in Finland. In , he received the E. O. Lawrence Medal from the US Department of Energy for his work on managing risks of severe accidents in nuclear power reactors. Anton Tonchev Anton Tonchev is an assistant research professor at Duke University. He received his Ph.D. degree in  from the Joint Institute for Nuclear Research, Dubna, Russia. His main area of research is nuclear structure and nuclear astrophysics, in particular the study of the nuclear dipole response and the different low-energy modes of excitation. These low-energy modes of excitation address important issues such as how protons and neutrons arrange themselves Biographies of Contributors in neutron-rich nuclei under electric dipole absorption. His other research interests include nuclear forensics and neutron science related to national security. Werner Tornow Werner Tornow is a nuclear physics experimentalist who received his doctorate in  from the University of Tuebingen, Germany. He is a full professor at Duke University and served as director of the Triangle Universities Nuclear Laboratory from  to . His expertise is in neutron-induced reactions ranging from few-body systems to heavy nuclei. More recently, he has also been involved in photon-induced reaction studies, and in neutrino physics and doublebeta decay searches. Ivo Tripputi Ivo Tripputi is a nuclear engineer who is in the nuclear industry for more than  years. He played several roles in the engineering department of ENEL for the development of new reactor projects for the Italian utility. Later he was involved in the development of utility requirements for advanced plants in USA and in Europe with leading roles. Recently he was decommissioning manager for all Italian nuclear fuel cycle plants and manager for R&D, innovation and special projects. He has been IAEA expert for development of safety guides and on special assistance projects in various parts of the world. He is currently chair of the Working Party for Decommissioning and Dismantling of OECD/NEA. James S. Tulenko James S. Tulenko is emeritus professor in the Department of Nuclear and Radiological Engineering at the University of Florida (UF) in Gainesville, Florida and served as chairman of the Department for  years (–). He currently is the director of the Laboratory for Development of Advanced Nuclear Fuels and Materials at UF. Prior to his academic career, Professor Tulenko spent  years in the nuclear industry as manager, Nuclear Fuel Engineering at Babcock and Wilcox; manager of Physics at Nuclear Materials and Equipment Corp; and manager, Nuclear Development at United Nuclear Corporation. Professor Tulenko has numerous fields of interest in the nuclear area, most of which involve nuclear fuel and the nuclear fuel cycle. He was presented with the Silver Anniversary Award of the American Nuclear Society (ANS) for his contributions to the nuclear fuel cycle in the Society’s first  years. He received the Mishma Award of the ANS in recognition of his many contributions to nuclear material research. He also received the Arthur Holly Compton Award for his contributions to nuclear science and technology, and the Glenn Murphy Award of the American Society for Engineering Education for his outstanding contributions to engineering education. Elected a Fellow of the ANS for his contributions to the nuclear fuel cycle, Professor Tulenko is a past president of the ANS, having served from  to . Paul J. Turinsky Paul J. Turinsky is a professor of Nuclear Engineering at North Carolina State University, where he also services as coordinator of the Interdisciplinary Graduate Program on Computational Engineering and Sciences, and university representative to and chair of the Battelle Energy Alliance Nuclear University Collaborators, which is associated with the operation of Idaho xlvii xlviii Biographies of Contributors National Laboratory. His area of expertise is computational reactor physics, with a focus on nuclear fuel management optimization, space-time kinetics, sensitivity/uncertainty analysis, and adaptive core simulation. He is a Fellow of the American Nuclear Society and recipient of numerous awards including the Glenn Murphy Award (ASEE), E. O. Lawrence Award in Nuclear Energy (US Department of Energy), Eugene P. Wigner Reactor Physics Award (ANS), and Arthur Holly Compton Award (ANS). He serves on several advisory committees and retains an active consulting practice supporting industry and government. Bernard Valentin Bernard Valentin graduated with a master’s degree in heat transfer from National Institute of Nuclear Science and Technic (INSTN), and joined CEA in . He has worked on modeling and computations related to safety of SFR, particularly on PHENIX and SPX reactors. He is now a specialist of SFR core and subassembly cooling, developing the ELOGE platform for design and thermo-hydraulics computations of such cores and subassemblies. Paul Van Uffelen Paul Van Uffelen studied nuclear engineering in Belgium and in France. He obtained his Ph.D. in the field of nuclear engineering at the University of Liège. Since he started his research career at the Belgian nuclear research centre (SCKCEN) in , Paul has been involved in research for light water reactor (LWR) fuel, more precisely in the field of fission gas behavior and heat transfer. During –, he was a visiting scientist at the OECD Halden Reactor Project in Norway, where he worked under the supervision of Carlo Vitanza and Wolfgang Wiesenack on in-pile experiments. Since , Paul is leading the modeling team at the Materials Research department of the European Commission’s Joint Research Centre (JRC), Institute for Transuranium Elements. His present research activities cover modeling of LWR fuel under normal operating and design basis accident conditions, as well as advanced fuel behavior modeling by means of the TRANSURANUS fuel performance code. Frédéric Varaine Frédéric Varaine has over  years experience in the area of nuclear physics and core design for fast reactor. He was involved in the restart of SUPER-PHENIX in , having performed core physics calculation, safety, and neutronics tests. Subsequently he became responsible for irradiations studies performed in the PHENIX reactor for the transmutation demonstration. From  to , he was a project manager for transmutation studies under the  French law on the management of radioactive waste. Since , he has been a group leader of the core design laboratory, which is conducting all neutronics studies for reactor design of Generation-IV systems and especially gas and sodium fast reactor. He participates in several collaborative projects on SFR with Japan, India, the USA, Russia, and China. Carlo Vitanza Carlo Vitanza graduated in nuclear physics at the University of Milan, Italy, in . Until , he worked in a nuclear fuel company in Italy. He then moved to the OECD Halden Reactor Project, Norway, where he worked for  years, dealing primarily with the large nuclear fuel program carried out through the Halden reactor experiments. During –, he was the director of the Halden research establishment. Biographies of Contributors In , he moved to the OECD Nuclear Energy Agency (NEA) in Paris, France, where among others he was in charge of the international research projects conducted under the OECD sponsorship in the nuclear safety area. During his stay at the OECD-NEA – he initiated a number of such international projects, mainly in the area of thermal-hydraulics, severe accidents, and fuel safety. In , he returned at Halden, where he is currently responsible for new development and marketing. R. Bruce Vogelaar Dr. R. Bruce Vogelaar received his Ph.D. in accelerator-based nuclear astrophysics from the California Institute of Technology in , and then became head of cyclotron operations at Princeton University, where he later joined their faculty as an assistant professor. He subsequently moved to Virginia Tech in  becoming a professor with research in fundamental weak-interaction physics, using ultracold neutrons at the Los Alamos National Laboratory and very large neutrino detectors at the INFN underground laboratory in Gran Sasso, Italy. He led a team to advance Kimballton as a potential national underground facility, and is currently the founding director of the Kimballton Underground Research Facility in addition to leading an active National Science Foundation (NSF)-funded research group. Dr. Vogelaar encouraged and participated in some of the earliest neutron transport experiments underlying the foundations of GEM*STAR and conducted GEM*STAR’s most recent neutron transport and fuel burn-up studies using the code MNCPX. Richard Wallace Dr. Richard Wallace has a Ph.D. in nuclear astrophysics from the University of California. He has  years experience in nuclear weapons analysis, nuclear materials use and protection, nuclear safeguards systems, and technical program management. Currently, he is a group leader for the N- Safeguards Systems Group at Los Alamos National Laboratory (LANL), overseeing a staff of experts in advanced safeguards systems development, nonproliferation policy analysis, international engagement activities related to the nuclear fuel cycle and safeguards, and IAEA activities related to developing potential proliferation indicators. From  to , Dr. Wallace was a senior analyst with the International Atomic Energy Agency (IAEA) in Vienna, Austria, working to collect, evaluate and analyze open source and proprietary information to identify and assess indicators of potential clandestine nuclear weapons activities. He shared in the  Nobel Peace Prize that was awarded to the IAEA. From  to , he was a project leader for the US–Russian Nuclear Materials Protection, Control, and Accounting program at LANL and acting program manager for Russian Nonproliferation Programs. In , he provided technical advice to Department of Energy (DOE) during negotiations for the Comprehensive Test Ban Treaty. From  to , he was involved in nuclear weapons physics simulation modeling. R. L. Walter Prof. R. L. Walter is a full professor in the department of Nuclear Physics at Duke University, and conducted most of his research at the Triangle University Nuclear Laboratory (TUNL) in Durham, North Carolina. He is an international leader in polarization in nuclear physics, and has supervised over  Ph.D. students, performing theses (mostly) on this subject. His studies xlix l Biographies of Contributors include the production of polarized charged particles and neutrons, and the use of such particle beams in studies of nuclear structure. Akio Yamamoto Dr. Akio Yamamoto received his Ph.D. in energy science at Kyoto University, in , for his work on loading pattern optimization methods for light water reactors (LWRs). During –, he was in charge of in-core fuel management and related methodology development for commercial LWRs at Nuclear Fuel Industries, Ltd., Japan. Currently, he is an associate professor in the Graduate School of Engineering, Nagoya University, Japan. His research is focusing on the development of advanced nuclear design methods for current and Gen-IV reactors, largescale simulations using parallel/distributed computing, in-core fuel optimizations, education of reactor physics and energy policy/strategies. He is a member of both the Atomic Energy Society of Japan (AESJ) and American Nuclear Society (ANS). Introduction In , when nuclear engineering was at the dawn of becoming an international profession, McGraw-Hill Book Company published the first Nuclear Engineering Handbook, edited by Harold Etherington. It contained  chapters, contributed by  specialists, all from the USA, spanning , pages, and served, for decades, as the reference book for nuclear engineers worldwide. Following the role model provided by that pioneering work, but surpassing it considerably in depth and extent, the present Handbook of Nuclear Engineering comprises  chapters in  volumes, spanning over , pages. The Editor is very grateful to the  international experts who contributed to turning this handbook, in a very short time, from a paper project into a comprehensive and inspiring reference work, at a time when the field of nuclear engineering and technology appears to be at the dawn of a worldwide renaissance, after some  decades of stagnation and even decline, in some countries. Each of the five volumes of this handbook is devoted to a representative segment of the field of nuclear engineering and technology, as indicated by their respective titles. They commence with a presentation of the fundamental sciences underlying nuclear engineering and move successively on to reactor design and analysis, fuel cycles, decommissioning, waste disposal, and safeguards of nuclear materials. As a whole, the handbook strives, of course, to cover all of the representative aspects of this field. The five volumes of this handbook are: Volume , Volume , Volume , Volume , Volume , entitled Nuclear Engineering Fundamentals, comprising Chaps.  through  entitled Reactor Design, comprising Chaps.  through  entitled Reactor Analysis, comprising Chaps.  through  entitled Reactors of Generations III and IV, comprising Chaps.  through  entitled Fuel Cycles, Decommissioning, Waste Disposal and Safeguards, comprising Chaps.  through  Each chapter strives to be self-contained, covering the current state-of-the-art and open issues in the respective area of nuclear science and engineering. An expert reader could go directly to the chapter of interest. Graduate students, on the other hand, may wish to consult the chapters in the first volume, since knowledge of the contents of those chapters will greatly facilitate the understanding of the material presented in the subsequent chapters/volumes. To assist the reader, the remainder of this Introduction briefly summarizes the contents of the handbook’s chapters. Chapter , entitled Neutron Cross Section Measurements, gives an overview of neutroninduced cross-section measurements, both past and present. Neutron cross sections are the key quantities required to calculate neutron reactions (e.g., in reactors, shields, nuclear explosions, detectors, stars). This chapter presents the principal characteristics of time-of-flight and mono-energetic fast neutron facilities and explains the physics of typical neutron cross sections and their measurements. In particular, the chapter provides an overview of the R-matrix formalism, which is the fundamental theory for describing resonance reactions. The chapter ends with a brief overview of the current libraries of evaluated cross sections, mentioning not only the major general purpose libraries, e.g., the ENDF (USA evaluated nuclear data library), the JEFF (European joint evaluated fission and fusion library), the JENDL (Japanese evaluated nuclear data library), the CENDL (Chinese evaluated nuclear data library), and the BROND lii Introduction (Russian evaluated neutron reaction data library) but also the most important special purpose and derived libraries. Chapter , entitled Evaluated Nuclear Data, describes the status of evaluated nuclear data for nuclear technology applications. The chapter commences with a presentation of the evaluation procedures for neutron-induced reactions, focusing on incident energies from thermal energy up to  MeV, although higher energies are also mentioned. The status of evaluated neutron data for actinides is discussed next, followed by paradigm examples of neutron-evaluated cross-section data for coolants/moderators, structural materials, and fission products. Neutron covariance data characterizing uncertainties and correlations are presented next, highlighting the procedures for validating evaluated data libraries against integral benchmark experiments. Additional information of importance for nuclear technology, including fission yields, thermal neutron scattering, and decay data is also presented. The chapter concludes with a brief introduction to current web retrieval systems, which allow easy access to a vast amount of up-to-date evaluated data for nuclear engineering and technology, including the latest versions of the major libraries ENDF/B-VII., JEFF-., and JENDL-.. Chapter  is entitled Neutron Slowing Down and Thermalization and presents the theory underlying the generation of thermal cross sections, concentrating on the phonon expansion method. Paradigm examples are given for graphite, water, heavy water, and zirconium hydride. The graphite example demonstrates incoherent inelastic scattering and coherent elastic scattering for crystalline solids. The water example demonstrates incoherent inelastic scattering for liquids with diffusive translations. Heavy water adds a treatment for intermolecular coherence. Zirconium hydride shows the effects of the “Einstein oscillations” of the hydrogen atoms in a cage of zirconium atoms, and it also demonstrates incoherent elastic scattering. Steadystate slowing down is illustrated for typical cross-section data, highlighting slowing down by elastic scattering, inelastic scattering, and resonance cross sections in the narrow resonance approximation. Intermediate resonance self-shielding effects are introduced using the NJOY flux calculator and the WIMS implementation. The effects of time and space on slowing down are demonstrated using Monte Carlo simulations. Neutron thermalization is modeled first by using Monte Carlo simulations of several systems, followed by modeling using multigroup discrete-ordinates and collision-probability methods. The chapter also demonstrates size effects in thermalization and concludes by reviewing open issues that warrant further theoretical and experimental investigations. Chapter , entitled Nuclear Data Preparation, focuses on data-processing codes that translate and manipulate the evaluated cross-section and other nuclear data (e.g., photon interactions, thermal-scattering laws) from the universally used ENDF/B format to a variety of formats used by individual particle transport and diffusion code systems for various applications. Data processing codes prepare nuclear data for use in continuous energy Monte Carlo codes as well as in multigroup Monte Carlo and deterministic codes. The main tasks performed by data processing codes are the reconstruction of energy-dependent cross sections from resonance parameters and Doppler broadening to a variety of temperatures encountered in practical systems. The chapter underscores the importance of the multiband method for enhancing the accuracy of reactor criticality and shielding computations. Chapter  is entitled General Principles of Neutron Transport and highlights the theory underlying the forward and the adjoint transport equations as well as the scaled transport and neutron precursor equations. In particular, the chapter includes a discussion of the lack of smoothness of the angular flux in multidimensional geometries, which impacts negative numerical simulations. The chapter also presents the derivation of the time-dependent Introduction integral transport equation as well as derivations of the transport equation in specialized one-dimensional (D), two-dimensional, and three-dimensional geometries. Since the exact transport equation cannot be solved for large-scale reactor physics applications, various approximations have been developed in time, in parallel with the development of computational resources. The chapter also features, in a more rigorous manner than hitherto presented in the literature, derivations of the transport equation’s approximate forms used for practical applications, highlighting: () an asymptotic derivation of the point kinetics equation, () an asymptotic derivation of the multigroup P and diffusion equations, () derivation of the spherical harmonics (PN ) and simplified spherical harmonic (SPN ) approximations, and () an asymptotic derivation of the point kinetics equations. Computational neutron transport methods are also discussed briefly, underscoring the salient features of the most popular deterministic methods, Monte Carlo methods, and hybrid Monte Carlo/deterministic methods. Chapter  is entitled Nuclear Materials and Irradiation Effects and highlights the physical transformations induced in materials by neutron irradiation. Irradiation damage is caused by (n,α) reactions, as well as by elastic interactions of neutrons with atoms, leading to displacement cascades and generation of point defects. In turn, the migration and clustering of point defects induce major changes in microstructures. The physical mechanisms leading to irradiation hardening, reduction of ductility, swelling, and irradiation creep and growth are described in detail for the alloys (e.g., structural and stainless steels, aluminum, zirconium, and vanadium alloys) and ceramics used not only in operating reactors but also envisaged for use in future fission and fusion reactors. The chapter also describes water radiolysis and changes in electrical properties of insulating ceramics. Chapter , entitled Mathematics for Nuclear Engineering, summarizes the main mathematical concepts and tools customarily used in nuclear science and engineering, thus facilitating the understanding of the mathematical derivations in the other chapters of the handbook. The material presented in this chapter highlights the following topics: vectors and vector spaces, matrices and matrix methods, linear operators and their adjoints in finite and infinite dimensional vector spaces, differential calculus in vector spaces, optimization, least squares estimation, special functions of mathematical physics, integral transforms, and probability theory. Chapter , entitled Multigroup Neutron Transport and Diffusion Computations, reviews the traditional methods for solving the steady-state transport equation, including the spherical harmonics expansion method, the collision probability method, the discrete ordinates method, and the method of characteristics. The chapter also highlights the most popular discretization methods for solving the neutron diffusion equation numerically. Chapter  is entitled Lattice Physics Computations and presents a detailed description of the elements that comprise a so-called “lattice physics code.” Lattice physics codes analyze axial segments of fuel assemblies, referred to as “lattices,” to determine the detailed spatial and spectral distribution of neutrons and photons within and across the segment. The major components of a lattice physics code include a corresponding cross-section library, and various modules for performing computations in the cross sections’ resonance region, fine-mesh transport calculations within the heterogeneous lattice geometry, and burnup computations. The flux distribution obtained from lattice computations is used to condense and homogenize the cross-section data into the structure needed for coarser-level “nodal” codes, in which each lattice is treated as a “node.” The nodal codes are used to model the coupled neutronics and thermal-hydraulics behavior of the entire reactor core during steady-state and transient operation. liii liv Introduction Chapter  is entitled Core Isotopic Depletion and Fuel Management. It commences by discussing numerical methods for solving the Bateman equation, which models the transient behavior of the isotopes (fissile, fertile, burnable poison, and fission products) produced during reactor operation. After introducing the concepts of breeding, conversion, and transmutation, the chapter discusses out-of-core and in-core nuclear fuel management, emphasizing fuel management for light water reactors (LWRs). Out-of-core fuel management aims at optimizing decisions regarding: () fuel cycle length; () stretch-out operations; and () feed fuel number, fissile enrichment, burnable poison loading, and partially burnt fuel to be reinserted, for each cycle in the planning horizon. On the other hand, in-core fuel management requires decisions for determining the loading pattern, control rod program, lattice design, and assembly design. The chapter continues with a brief review of in-core fuel management decisions for heavy water reactors, very high temperature gas-cooled reactors, and advanced recycle reactors. The mathematical optimization techniques and the tools for accomplishing the computations which are required to support decision making in nuclear fuel management are discussed with a view toward further enhancing the capabilities in these areas. The chapter concludes by summarizing the current state of fuel depletion and management capabilities, while discussing the avenues for further progress in these areas. Chapter , entitled Radiation Shielding and Radiological Protection, deals with shielding against gamma rays and neutrons with energies up to  MeV, while assessing the health effects from exposure to such radiation. The chapter commences by describing the features of radiation sources and fields, including the mathematical modeling of the energy and directional dependence of the radiation intensity. The interaction of neutrons and gamma rays with matter is presented next, highlighting the computation of various types of radiation doses stemming from radiation intensity and target characteristics. This discussion leads to a detailed description of photon radiation attenuation and neutron shielding calculations, as well as corresponding dose evaluations. The chapter presents the basic concepts of buildup factors and point-kernel methodology for photon attenuation computations, as well as the established concepts and computational methods for designing shielding against fast, intermediate, and thermal energy neutrons. The chapter also highlights the special cases of albedo, skyshine, and streaming dose calculations and concludes with a discussion of shielding materials, radiological assessments, and risk calculations. Chapter  is entitled High Performance Computing in Nuclear Engineering and comprises an introduction to high-performance computer and processor architectures, highlighting the current parallelism models. The chapter then presents the key concepts and requirements for designing parallel programs. This presentation is followed by presentations of paradigm applications of high-performance computing in reactor physics, nuclear material sciences, and thermal-hydraulic nuclear engineering. The chapter concludes with a discussion of open issues and possible paths forward in this rapidly developing field. Chapter , entitled Analysis of Reactor Fuel Rod Behavior, focuses on the behavior of light water reactor (LWR) fuel rods. Following a presentation of the main properties of fuel and cladding materials, this chapter systematically describes the thermal and mechanical behavior under irradiation as well as the behavior of the fission gas produced in the fuel. The chapter also highlights the typical phenomena and issues of interest for the design and licensing of LWR fuels, namely: the high burnup structure, pellet-cladding interaction, pellet-coolant interaction, loss-of-coolant accidents (LOCA), and reactivity-initiated accidents (RIA). Chapter  is entitled Noise Techniques in Nuclear Systems and deals with neutron fluctuations in nuclear systems. Such neutron fluctuations, called “neutron noise,” provide valuable Introduction information regarding the behavior of the reactor system. This chapter focuses on the concepts and methodologies for extracting nonintrusive information from neutron noise, aiming at the detection, identification, and quantification of potential operational anomalies, at the earliest possible stage. Chapter  is entitled Deterministic and Probabilistic Safety Analysis and highlights the evolution of deterministic and probabilistic safety analyses, which play a crucial role for assuring public health and safety in the peaceful uses of nuclear power. The chapter commences with a review of the origins of nuclear power safety analysis, comprising both deterministic and probabilistic methods. Deterministic approaches, including the defense-in-depth and safety margin concepts, provide methodologies for quantifying deterministically uncertainties associated with the adequacy of safety features. The chapter also discusses in detail probabilistic safety assessment methods and their uses in nuclear power safety analysis and safety-related decision making, reflecting the maturing and widening applications of such methods. Chapter , entitled Multiphase Flows: Compressible Multi-Hydrodynamics, presents a conceptual framework for modeling three-dimensional multiphase flows in terms of a local disperse system description (bubbles/drops in a continuous liquid/vapor phase). This framework is based on a new, “hybrid,” effective-field method (EFM) that incorporates features of a statistical approach and reveals more clearly the nature of phase interactions at the individual particle scale. The resulting multiphase flow formulation is amenable to numerical simulations based on the direct solution of the Navier–Stokes equations resolved at the particle scale. The basic constitutive treatment concerns pseudo-turbulent fluctuations of the continuous phase, and the resulting systems of equations are fully closed and hyperbolic (and hence directly usable for computations) even in their non-dissipative form, except for a non-hyperbolic corridor around the transonic region. The results thus obtained are discussed in relation to formulations that form the basis of current numerical tools (codes) employed in nuclear reactor design and safety analyses (mostly addressing bubbly flows) as well as the formulations found in other contexts. The numerical implementation of this new mathematical formulation emphasizes flow compressibility, focusing on capturing shocks and contact discontinuities robustly, for all flow speeds and at arbitrarily high spatial resolutions. A key role is played by “upwinding,” applied within the context of a scheme that emphasizes conservative discretization, extending thereby the ideas underlying the Advection Upstream Splitting Method to compressible multi-hydrodynamics (including the EFM). Chapter  is entitled Sensitivity and Uncertainty Analysis, Data Assimilation, and Predictive Best-Estimate Model Calibration. This chapter provides the theoretical and practical means of dealing with the discrepancies between experimental and computational results and is therefore of paramount importance for validating the design tools used in all aspects of nuclear engineering. Such discrepancies motivate the activities of model verification, validation, and predictive estimation. The chapter presents the modern statistical and deterministic methods for computing response sensitivities to model parameters, highlighting, in particular, the adjoint sensitivity analysis procedure (ASAP) for nonlinear large-scale systems with feedback. Response sensitivities to parameters and the corresponding uncertainties are the fundamental ingredients for predictive estimation (PE), which aims at providing a probabilistic description of possible future outcomes based on all recognized errors and uncertainties. The key PE activity is model calibration, which uses data assimilation procedures for integrating computational and experimental data in order to update (calibrate or adjust) the values of selected parameters (such as cross sections and correlations) and responses (such as temperatures, pressures, reaction rates, or effective multiplication factors) in the simulation model. This chapter also presents a lv lvi Introduction state-of-the-art mathematical framework for time-dependent data assimilation and model calibration, using sensitivities and covariance matrices together with the maximum entropy principle and information theory to construct a prior distribution that encompasses all the available information (including correlated parameters and responses), while minimizing (in the sense of quadratic loss) the introduction of spurious information. When the experimental information is consistent with computational results, the posterior probability density function yields reduced best-estimate uncertainties for the best-estimate model parameters and responses. Open issues (e.g., explicit treatment of modeling errors, reducing the computational burden, treating non-Gaussian distributions) are addressed in the concluding section of this chapter. Chapter  presents Reactor Physics Experiments on Zero Power Reactors, performed in the EOLE, MINERVE, and MASURCA Zero Power Reactors (ZPRs), operated by the Commissariat a l’Energie Atomique (CEA), France. The experimental programs in these ZPRs have long played a crucial role in the validation of neutron physics codes and nuclear data by reducing the uncertainties of the experimental databases. These ZPRs also provide accurate data regarding: () innovative materials for reactor (fuels, absorbers, coolants, and moderators); () new reactor concepts (advanced BWRs, sodium and gas-cooled Generation-IV reactors, as well as hybrid systems, involving a subcritical reactor coupled to an external accelerator); () plutonium and waste management (involving heavy nuclides and long-lived fission products); () transmutation of long-lived nuclides; and () data for the new materials testing reactor “Jules Horowitz,” currently under construction in Cadarache. Chapter  is entitled Pressurized LWRs and HWRs in the Republic of Korea. It describes Korean experiences and accomplishments in the design, operation, and construction of two Generation-III pressurized light water reactor (LWR) plants (OPR and APR, respectively), which are currently in service and/or under construction in the Republic of Korea. The chapter also presents the Korean experience regarding the pressurized heavy water reactors (CANDUs) built and operated in the Republic of Korea. Chapter , entitled VVER-Type Reactors of Russian Design, describes the design and technical layout of the Russian VVER- and VVER- reactor plants. VVER reactors are a special design of pressurized water reactors, featuring a hexagonal geometry for the fuel assemblies, with fuel rods arranged in a triangular grid. The cladding for the fuel rods is manufactured from a zirconium–niobium alloy. The large-sized equipment can be advantageously transported by railway to enable a complete manufacturing process under factory conditions, but this design feature imposes a limit on the outer diameter of the reactor pressure vessel. The design of the horizontal steam generators with a tube sheet in the form of two cylindrical heads is also original. Reactors with the designations RP V-, V-, and V– are Generation-III reactors. Comprehensive sections are devoted to the primary circuit systems and equipment, reactor coolant system, reactor core, main circulation pumps, pressurizer, steam generators, chemical and volume control systems, and secondary circuit components (main steam line system, main feedwater system, turbine, generator, and moisture separator reheater), for both VVER- and VVER- reactor plants. Chapter  is comprehensively entitled Sodium Fast Reactor Design: Fuels, Neutronics, Thermal-Hydraulics, Structural Mechanics and Safety. The chapter highlights the fundamental motivations for building a fast reactor system: effective utilization of uranium resources through the judicious exploitation (transmuting, converting, or breeding) of fertile material and a sustainable closed fuel cycle, permitting a flexible management of actinides to minimize high-level Introduction waste and reduce the burden on deep geological storages. These advantages had been envisaged as early as in  by Enrico Fermi, who demonstrated the breeding principle and stated: “The people who will develop SFR technology will lead the world in the future.” The chapter provides a comprehensive review of the development of sodium fast reactors (SFR) in the USA, Russia, France, the UK, Germany, Italy, Belgium, the Netherlands, Japan, India, China, and Korea, highlighting the motivation and challenge underlying the two basic design choices, namely “pool design” and “loop design.” The objectives, scope, and levels of the “defense in depth” safety principles are emphasized, aiming at the “practical elimination” of initiating events and sequences leading to hypothetical severe plant conditions. The considerations underlying the choice of materials (for fuel, structures, absorbers, shielding) are presented in depth, since these are paramount for optimizing the reactor core design (including geometrical parameters, neutronics, thermal-hydraulics, structural mechanics, reactivity effects) and performance objectives. The chapter also underscores several important specific issues regarding the design and analysis of the mechanical integrity of the reactor (comprising thermal striping, stratification, seismic-induced forces, fluid–structure interactions, buckling of thin shells). Design basis and design extension conditions, including residual risks, are analyzed for typical anticipated transients without scram. The vast French licensing experience with the Phenix, SPX-, SPX- sodium-cooled fast reactors is discussed with a view toward innovations, leading to enhanced safety of future innovative SFR designs. The chapter concludes with presentations of the ongoing SFR activities in France and India. Chapter  is entitled Gas-Cooled Reactors and presents the main features of reactors that use a gas (carbon dioxide or helium) as the primary fluid for cooling the core. The oldest reactors of this type were the British Magnox and the French natural uranium graphite gas (NUGG) reactors, which used carbon dioxide as coolant in graphite-moderated cores fueled with natural uranium. The availability of low-enriched uranium fuel allowed the British to develop the advanced gas-cooled reactor as a successor to the Magnox reactor. In a world progressively dominated by the water-cooled reactors, mostly PWR and BWR, helium remained under consideration as coolant for the prismatic and pebble-bed high-temperature reactors (HTR) moderated with graphite. Both of these HTRs use a very innovative fuel element – the coated particle. As recalled in this chapter, the same type of fuel was also used in the NERVA US-program, which aimed at developing nuclear propulsion for rockets. Helium cooling is also used for two envisaged Generation-IV concepts, namely the very high temperature reactor (VHTR) system aimed at both electricity generation and hydrogen production and the gas-cooled fast-neutrons reactor (GFR). Chapter  is entitled Lead-Cooled Fast Reactor (LFR) Design: Safety, Neutronics, Thermal Hydraulics, Structural Mechanics, Fuel, Core, and Plant Design. The lead-cooled fast reactor (LFR) has both a long history and a currency of innovation. Early work on such reactors, dating back to the s, was devoted to submarine propulsion, as Russian scientists pioneered the development of reactors cooled by heavy liquid metals (HLM). More recently, there has been substantial interest in both critical and subcritical reactors cooled by lead (Pb) or lead– bismuth eutectic (LBE), not only in Russia (BREST-, SVBR-) but also in Europe, Asia, and the USA. This chapter reviews the historical development of the LFR and provides detailed descriptions of the current initiatives to design LFRs for various missions. Currently, the leading designs are for: () accelerator-driven subcritical (ADS) systems for nuclear materials management, () small modular systems for deployment in remote locations, and () central station plants for integration into developed power grids. The chapter describes: design criteria and lvii lviii Introduction system specifications; specific LFR features regarding neutronics, coolant properties, and material compatibility issues; core and reactor plant design; and considerations related to the balance of plant and plant layout for Generation-IV LFR and HLM-cooled ADS concepts (SSTAR, ELSY, MYRRHA, EFIT). Chapter  is entitled GEM∗STAR: The Alternative Reactor Technology Comprising Graphite, Molten Salt, and Accelerators, and it illustrates conceptually the possible benefits of implementing supplementary neutrons from accelerators in a futuristic reactor concept called GEM∗STAR (Green Energy Multiplier∗Subcritical Technology for Alternative Reactors) by its authors. This chapter does not aim at providing a complete history of molten salt, graphite, and accelerator technologies but rather a description of how these elements of nuclear power development might be integrated to address the main barriers that constrain the full deployment of today’s nuclear power technology. The basis for the GEM∗STAR concept is a subcritical (initial multiplication factor = .) thermal-spectrum reactor, operating with a continuous flow of molten salt fuel in a graphite matrix. If sufficient external neutrons are available, GEM∗STAR may operate with natural uranium and un-reprocessed LWR spent fuel, recycling its own fuel several times without needing external reprocessing. Chapter  is entitled Front End of the Fuel Cycle and describes the complete set of industrial operations needed to produce a functional fuel element ready to be loaded in a nuclear reactor. This chapter commences by providing data concerning the element uranium (its abundance and relevant properties) and continues by describing the uranium exploration, mining, concentration, and site rehabilitation processes. Light water reactors, which constitute the vast majority of currently operating and under-construction nuclear reactors, must use uranium enriched in the isotope U-. The enrichment process requires that uranium be converted into a gaseous compound. The enriched uranium is subsequently fabricated into solid ceramic pellets and assembled into leak tight metallic pins, which are, in turn, assembled to form the fresh fuel element. The chapter also provides valuable information on mixed uranium-plutonium oxide (MOX) fuel assemblies for recycling plutonium in LWR as well as basic data on plutonium and thorium. The chapter also provides an explanation of the fascinating Oklo Phenomenon, which occurred almost two billion years ago in a particular uranium deposit in Gabon. Chapter  is entitled Transuranium Elements in the Nuclear Fuel Cycle. The transuranium elements neptunium, plutonium, americium, and curium are produced from actinides through neutron capture processes and therefore arise mostly as by-products of fuel irradiation during the operation of a nuclear reactor. The nuclear properties of transuranium (TRU) elements significantly affect the nuclear fuel cycle, largely determining the requirements and procedures for handling, storing, reprocessing, and disposing the spent fuel and high-level waste. Socioeconomical and political considerations have so far prevented the establishment of a standard, universally accepted route for the treatment of TRU-elements. In particular, it is still a matter of political debate if plutonium is waste or a resource for the production of energy. This chapter provides an overview of past and current experiences and perspectives regarding the recovery and incorporation of TRU-elements in fuels and targets for advanced nuclear fuel cycles, as well as the disposal of TRU-elements as the main components of high-level nuclear waste. In particular, this chapter highlights the main properties of TRU fuels, the specific requirements for their fabrication, their irradiation behavior, and their impact on the back end of the fuel cycle. For the latter, a major issue is the development of options for reprocessing and separation of TRU-elements from spent fuel, in order to make these elements available for further treatment. The chapter also discusses the effects on long-term storage and final disposal caused by the presence of TRU-elements in irradiated fuel and high-level nuclear waste. At this time, Introduction the final destination for TRU-elements continues to remain an open issue, as various countries pursue a diversified set of options. The comprehensive understanding of the worldwide knowledge and experience presented in this chapter provides an essential basis for developing viable, safe, and technologically effective options for the treatment of TRU-elements, which, in turn, are crucial for ensuring that nuclear energy remains a key component in a sustainable mix of energy production. Chapter  is entitled Decommissioning of Nuclear Plants and covers all aspects related to the closure of the operating life of nuclear plants, providing a description of all activities and tools involved in the decision-making and operative processes of decommissioning. The main stages involved in the decommissioning process include the termination of operations, the withdrawal of the nuclear reactor plant from service, and the transformation of the plant into an out-of service state without radiological risks. In some cases, decommissioning leads to the complete removal of the plant from its original site. Nuclear plant decommissioning comprises a complex, long, and highly specialized spectrum of activities including technological tools, industrial safety, environmental impact minimization, licensing, safety analysis, structural analysis, short- and long-term planning, calculation of cash flow and financing, waste disposal, and spent fuel strategy. All of these activities must be performed in a cost-effective manner, assigning top priority to the health and safety of the workers on site as well as the public and the environment. In some countries, decommissioning is actually called “deconstruction,” because it resembles, in many respects, the construction activity. Unlike the construction activities, though, decommissioning also deals with partially activated and contaminated structures. The chapter draws technical information from the direct experience of nuclear operators as well as from the results produced by working groups, special studies, comparisons of technologies, and recommendations from the OECD-NEA, IAEA, US-NRC, and the European Commission. Chapter  is entitled The Scientific Basis of Nuclear Waste Management and provides the scientific concepts and data underlying the management of the highly radioactive waste produced at various stages of the nuclear fuel cycle. In a context in which science and technology interact strongly with social and economical issues, the scientific basis for high-level radioactive waste management continues to evolve. Within a closed fuel cycle, the management of high-level waste, from its production to its final destination, appears as a chain, whose links are treatment, recycling, conditioning, storage, and disposal. When the fuel cycle remains open, the first two links (treatment and recycling) are absent. The chapter commences by describing the origin, nature, volume, and flux of nuclear waste, while briefly presenting various management options. Waste conditioning is addressed next, emphasizing the elaboration and long-term behavior of two important conditioning matrices: cement-like materials and glass. Since some countries consider spent fuel to be “waste” that must therefore be conditioned as such, the chapter devotes a special section to this controversial issue. The chapter also describes in detail the design and properties of installations for interim storage of long-lived waste, since such installations are already operational in several countries that exploit nuclear power plants. Since the definition of “ultimate waste” varies from country to country due to socioeconomical issues, the means and ways for the disposal of such waste in deep geological repositories is not universally resolved, even though the basic scientific concepts and technological issues involved are largely known. Therefore, this chapter also emphasizes the mechanisms, models, and orders of magnitude of the main physical and chemical phenomena that govern the long-term evolution of various types of possible geological repositories for high-level radioactive ultimate waste. The chapter concludes with a short description of the methodology used for evaluating the safety of radioactive waste disposal installations. lix lx Introduction Chapter  is entitled Proliferation Resistance and Safeguards and presents the status of the international activities regarding these very complex issues. As is well known, the Nuclear Nonproliferation Treaty (NNPT or NPT) is the primary cornerstone of international efforts to prevent the proliferation of nuclear weapons. Currently,  countries are party to the treaty, with only four sovereign states abstaining: India, Israel, Pakistan, and North Korea. The NPT is broadly interpreted to comprise three pillars: nonproliferation, disarmament, and the right to the peaceful use of nuclear technology. This chapter addresses each of these NPT-pillars and commences by analyzing the significance of the Strategic Arms Reduction Treaty (START), which was signed between the USA and the former Soviet Union on  July . This treaty, considered by many to be the largest and most complex arms control treaty in history, has led to a significant reduction in the number of deployed warheads for both the USA and Soviet Union. Furthermore, the US and Russian presidents signed a preliminary agreement on  July  to reduce further the number of active nuclear weapons to between , and , by . Although these treaties have been the backbone of joint US and Russian efforts toward nuclear disarmament, they have not addressed the discontinuation of weapons-grade fissile material production and disposition of excess weapons-grade materials. The current situation is that the USA, France, and the UK have ceased the production of weapons-grade materials. In , the USA and Russia signed the Plutonium Production Reactor Agreement to cease the production of plutonium for weapons production. Although Russia still operates nuclear reactors used previously for production of weapons material, to generate heat and electricity, it does not reprocess the spent fuel correspondingly, and plans to decommission these reactors. While India and Pakistan apparently are still producing weapons-grade material, unsubstantiated reports indicate that China has also instituted a moratorium on production, while Israel’s position is unclear. In September , the USA and Russia, each formally agreed to transform  metric tons of excess military plutonium into a more proliferation-resistant form over the course of  years, by irradiating it in nuclear power reactors. Currently, Russia favors irradiation in a new generation of fast reactors yet to be developed, and the USA favors irradiation in their existing commercial LWR fleet. Additionally, a joint program has been developed by the USA and Russia to disposition excess highly enriched uranium (HEU) by blending it with natural uranium to produce low enriched uranium (LEU) for commercial power reactor fuel. This chapter presents a broad overview of experimental and statistical methods and tools for assessing nonproliferation compliance at declared facilities. In the broader nonproliferation context, monitoring for undeclared activities involves many difficult additional statistical issues. The chapter concludes by presenting a paradigm example of validating a safeguard design for the detection of abrupt diversion. Although this handbook represents the most comprehensive snapshot of the current state of nuclear engineering and technology worldwide, several topics were not treated comprehensively as they could have been, due to scheduling constraints. Thus, the Editor believes that full stand-alone chapters would be warranted for presenting in adequate detail topics such as: Monte Carlo and variational methods for reactor physics and shielding computations, reactor dynamics and control, nuclear instrumentation, corrosion under irradiation, life extension of Generation-II LWRs, advanced supercritical water reactors, and research and materials testing reactors. The next edition of this handbook envisages a comprehensive exposition of these and related topics, along with updates of the topics covered in the present  chapters. The Editor commenced work on this handbook in the fall of , while serving as the Scientific Director of the Nuclear Energy Directorate of the Commissariat a l’Energie Atomique (CEA), France. The extensive yet focused work required during  to finalize this handbook Introduction has been greatly facilitated by the support of Mr. Laurent Turpin, Director of the National Institute for Nuclear Science and Technology (INSTN) at CEA, and the Editor wishes to express his heartfelt appreciation for this support. The Editor also wishes to acknowledge with distinctive pleasure the very cordial and efficient collaboration with the publication team of SpringerVerlag, led by Mr. Alex Green (Editorial Director for Engineering) in the USA. The Editor also wishes to express special thanks to Dr. Sylvia Blago, Lydia Mueller, and Simone Giesler, for their particularly dedicated professional and very friendly collaboration in weekly interactions over the almost  years dedicated to this handbook project. Dan G. Cacuci Institute for Nuclear Technology and Reactor Safety Karlsruhe Institute for Technology Germany lxi  Neutron Cross Section Measurements Robert C. Block ⋅ Yaron Danon ⋅ Frank Gunsing ⋅ Robert C. Haight  Rensselaer Polytechnic Institute, Troy (NY), USA  CEA Saclay - Irfu, France  Los Alamos National Laboratory, Los Alamos (NM), USA  Introduction . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . .   History . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . .   . .. .. .. .. .. .. .. .. . .. .. .. .. Currently Active Laboratories .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Time-of-Flight Laboratories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Gaerttner LINAC Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Los Alamos Neutron Science Center . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The ORELA Laboratory at Oak Ridge National Laboratory . . .. . . . . . . . . . . . . . . GELINA at the JRC-IRMM in Geel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The n_TOF Facility at CERN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The IREN Facility at Dubna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The PNF Laboratory at Pohang. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Electron Linac at Kyoto University Research ReactorInstitute, KURRI . . . . . Monoenergetic Fast Neutron Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Neutron Energies Below  MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Neutron Energies in the MeV Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Neutron Energies Near  MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Neutron Energies Above  MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .                 . . . . . Neutron Cross Sections .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Total Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Partial Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Resonance Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . High Energy Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .        . .. . . . .. . Cross Section Measurements . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Thermal Energy Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Thermal Flux Averaged Cross Section. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Resonance Energy Region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Unresolved Resonance and Continuum Energy Region . . . . . . . .. . . . . . . . . . . . . . . The Neutron Time of Flight Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Neutron Density and Flux Distributions at Thermal Energies .. . . . . . . . . . . . . . . Surrogate Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .         Dan Gabriel Cacuci (ed.), Handbook of Nuclear Engineering, DOI ./----_, © Springer Science+Business Media LLC    Neutron Cross Section Measurements . Cross Section Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . .. .. .. .. .. .. . .. .. .. . Nuclear Resonances and the R-Matrix Formalism . . .. . . .. . . .. . . .. . . .. . . .. . Introduction. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Channel Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Wave Function in the External Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Collision Matrix U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Relation Between the Cross Sections andthe Collision Matrix U . . . . . . . The Wave Function in the Internal Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Relation Between the R-Matrix and the Collision Matrix U . . . . . . . . . . . . . Approximations of the R-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Breit–Wigner Single Level Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Breit–Wigner Multi Level Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Reich–Moore Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .               Concluding Remarks . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  References . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  Neutron Cross Section Measurements  Abstract: This chapter gives an overview of neutron-induced cross section measurements, both past and present. A selection of the principal characteristics of time-of-flight and monoenergetic fast neutron facilities is given together with several examples of measurements. The physics of typical neutron cross sections and their measurements are explained in detail. Finally an overview of the R-matrix formalism, which is at the basis of resonance reactions, is given. The many references provide a starting point for the interested reader.  Introduction When a beam of particles impinges upon a nucleus, the ratio of the reaction rate (in units of s− ) to the incident flux (in units of cm− s− ) is defined as the “cross section” (in units of cm or barn with  barn = − cm ). The cross section is thus a measure of the strength of a reaction. Neutron cross sections are the key quantities required to calculate neutron reactions taking place in reactors, shields, transmuters, nuclear explosions, detectors, stars, etc. Neutron cross sections have been measured over the past  years and techniques are being continually developed to improve the accuracy and completeness of neutron cross section data for both stable and radioactive nuclei. As new reactors, neutron sources, waste transmutation devices, and medical applications emerge, there is need for improved neutron cross section measurements. There is an ongoing major international effort to provide a complete-as-possible compilation of evalulated cross section data for nuclear calculations. Major evaluation compilations are ENDF (USA evaluated nuclear data library) (Chadwick et al. ), JEFF (European joint evaluated fission and fusion library) (Koning et al. ), JENDL (Japanese evaluated nuclear data library) (Shibata et al. ), CENDL (Chinese evaluated nuclear data library) (Ge Zhigang et al. ), and BROND (Russian evaluated neutron reaction data library) (Ignatyuk and Fursov ). Since evaluated data can be no better than the measurements upon which they are based, thus there is an ongoing need for better measurements. In addition experimental compilations like references Mughabghab (), Sukhoruchkin et al. () or the international database EXFOR (previously also known as CSISRS) Zerkin et al. and available on the websites http://www.nds.iaea.org/exfor, http://www.nea.fr/html/dbdata/x, http://www.nndc.bnl.lgov/. It is a difficult task to review such a large effort as neutron cross section measurements without committing errors of commission or omission. If we erroneously reported on your laboratory and/or its publications, we apologize in advance as it was never our intent to be in error. Also, if we either left you out of our summaries or did not give as much attention to your contributions as you deem we should have, again we apologize with the excuse that it was impossible to present all the work that has been and is now going on. We admit that we presented many details from the laboratories that the authors are working with – this was only because the authors are more familiar with these works and does not imply that they are superior to other laboratory efforts. Finally, we did not include any “figure-of-merit” listings that were so popular in the past when comparing neutron sources. In retrospect, these figures-of-merit frequently wound up making one’s own neutron source look superior to another’s, whereas the total merit of a neutron source can never be summarized by a single number. It is the total impact of the cross sections measured, the number of published articles and conference proceedings, the impact on the field, etc. that need to be taken into account in evaluating a neutron source. To paraphrase British Prime Minister Benjamin Disraeli’s famous statement “There are lies, damned lies and figures-of-merit.” (“There are lies, damned lies and statistics” is attributed to Prime    Neutron Cross Section Measurements Minister Benjamin Disraeli in the nineteen century. Mark Twain again made it famous in the twentieth century. Wikipedia: http://en.wikipedia.org/wiki/Lies,_damned_lies,_and_statistics) ().  History The neutron was discovered by Chadwick in  from the interaction of alpha particles incident on beryllium or boron (Chadwick ). Neutrons were soon recognized as excellent particles to probe nuclei since their lack of charge enabled them to readily penetrate deeply into matter and into the charged nucleus. Fermi and coworkers used neutrons generated from natural alpha sources to study nuclear reactions over a large range of nuclei and found that the strength of interaction varied greatly from nucleus to nucleus and that in many cases the interaction can be enhanced by moderating the neutron energy with hydrogen (Amaldi and Fermi ). Ultimately the neutron was discovered to produce fission in uranium and this led to the major effort in the s to produce nuclear reactors and weapons. The design of these neutron devices critically depended on neutron cross section data from thermal to tens of MeV energy and major laboratories in the US and Europe were set up to make these measurements. In the s and s cross section measurements utilized the neutrons from accelerators and early reactors. Activation with and without Cd covering and reactivity measurements using devices such as the pile oscillator provided thermal-spectrum-averaged cross sections. Crystal spectrometers, neutron velocity selectors, and neutron choppers using the time-of-flight (TOF) method provided sufficient energy resolution to measure resonance energy cross sections typically up to ∼ eV. In the s positive ion accelerators utilized nuclear reactions to produce monoenergetic neutron beams and these provided cross section data from the keV to MeV energy range. Pulsed accelerators for electrons and protons were used to produce intense bursts of neutrons which enhanced the TOF measurements well beyond what fast choppers could provide. Even nuclear explosions were used to provide a very intense pulse of neutrons for time-of-flight experiments. Since the s cross sections were mainly obtained either by pulsed accelerator TOF methods (Ray and Good ) or neutrons derived from positive-ion-induced nuclear reactions. In the s and s neutron cross section measurement laboratories were set up in USA, Europe, and Japan. TOF measurements were carried out at the General Atomics Corporation (Orphan et al. ), Oak Ridge National Laboratory linac (Dabbs ) and Van de Graaff (Good et al. ), Rensselaer Polytechnic Institute (Hockenbury et al. ), the Lawrence Livermore National Laboratory (LLNL) linac (Behrens et al. ) and D-T source (Hansen et al. ), the LLNL cyclograaff (Davis ), National Bureau of Standards (Carlson et al. ), the linacs at Harwell (Firk et al. ), Saclay (Blons ), and Geel (Kohler et al. ) the Karlsruhe cyclotron (Cierjacks et al. ) and Van de Graaff (Wisshak and Kappeler , ), the Frascati Neutron Generator (Pillon et al. ), the pulsed reactor at Dubna (Frank and Pacher ), Kyoto University Reseach Reactor Institute (Fujita ) and the linac of the Japan Atomic Energy Research Institute at Tokai Mura (Asami ). Light-ion-based accelerator measurements were made at Los Alamos National Laboratory (Woods et al. ), Duke University (Farrell and Pineo ), Ohio University (Finlay et al. ), Lowell University (Kegel ), Argonne National Laboratory (Dudey et al. ), Bruyères-le-Châtel (Joly et al. Neutron Cross Section Measurements  ), the Forschungszentrum Jülich (Qaim et al. ; Grallert et al. ), Research Institute National Defence, Stockholm (Nystrom et al. ), the Institute of Physics and Power Engineering at Obninsk  MeV source (Lychagin et al. ), and pulsed Van de Graaff (Kononov et al. ; Kononov et al. ; Kornilov and Kagalenko ; Zhuravlev et al. ) and the DT source of Kossuth University at Debrecen, Hungary (Biro et al. ). Recently the pulsed spallation sources at Los Alamos (Lisowski and Schoenberg ) and CERN (Borcea et al. ) have been utilized for TOF cross section measurements where their intense neutron pulses are well suited for both high resolution and high neutron intensity measurements. The high energy protons that produce these neutrons enables measurements up to s of MeV. Other initiatives concern the time-of-flight facility nELBE at the Forschungzentrum in Dresden-Rossendorf (Klug et al. ), the neutron source FRANZ at the Stern Gerlach Zentrum in Frankfurt (Petrich et al. ) and the neutron source at JPARC at the Japan Atomic Energy Research Institute in Tokai (Futakawa et al. ) and in the future at SPIRAL at GANIL in Caen (Fadil and Rannou ). The above list of laboratories is not exhaustive but the references will provide a good starting point for the interested reader. Neutron cross section measurements increased rapidly into the s and s as many laboratories were set up throughout the world for these measurements. This activity peaked in the s but decreased from  to today, as shown in > Fig. . The upper bar graph 250 Total number of references Measurement references 200 150 100 50 0 08 20 06 20 4 0 20 02 20 0 0 20 98 19 6 9 19 4 9 19 2 9 19 90 19 8 8 19 86 19 4 8 19 2 8 19 80 19 8 7 19 76 19 4 7 19 72 19 0 7 19 Year ⊡ Figure  The plot labeled “Total number of references” shows the total number of references published per year pertaining to neutron cross sections; these publications refer to both evaluation and experiments. The plot labeled “Measurement references” are for publications which contain information on neutron cross section measurements    Neutron Cross Section Measurements is a plot of the number of journal references per year pertaining to neutron cross sections while the lower bar graph is a plot of the number which pertain to measurements of neutron cross sections. This plot was obtained by using the Engineering Village search program http://www.engineeringvillage.org, Elsevier, Inc. (). Here we see that from  to  the “activity” in this field dropped by about a factor of two. This drop reflects the decrease in the funding in this field with the corresponding decrease in the number of laboratories working in this area. From  to  the activity increased ≈% and, with the renewed interest in new power reactors, it is predicted that this rate will continue to rise in the next decade or so.  Currently Active Laboratories . Time-of-Flight Laboratories Major laboratories for time-of-flight cross section measurements are located in the USA, Europe, and Asia. All of these facilities utilize accelerators for electrons or protons to the produce intense pulses of neutrons and typically have long flight paths to obtain good energy resolution. A non-exhausitive summary of these facilities is listed in > Table . The three major laboratories in USA are the Gaerttner LINAC Laboratory at Rensselaer Polytechnic Institute (RPI), the LANSCE Laboratory at Los Alamos National Laboratory and the ORELA Laboratory at Oak Ridge National Laboratory The ORELA and Gaerttner facilities utilize an electron linear accelerator while the LANSCE facility uses a high-energy proton linear accelerator. In Europe the GELINA facility of the Joint Research Center IRMM in Belgium has a long record of measurements. The new facility IREN in Russia is under construction but operational and has replaced the pulsed reactor IBR-. The facility nELBE in Dresden-Rossendorf is a relatively new facility. The neutron source n_TOF at CERN, Geneva, is operational since . For Asia we include PNF at PAL in Pohang, Korea, and KURRI in Kumatori, Japan. Also the neutron source MLF-NNRI at J-PARC has been put in operation. Many of these laboratories have several flight paths such that several experiments can be set up and acquire data without interfering with each other. The table lists the particle, its energy and the target used to generate the neutrons. Beam parameters like pulse width, beam power, and repetition rate are also given as well as the range of flight paths available. Finally the number of neutrons produced per pulse is given. Other parameters like the average current (beam power divided by the particle acceleration voltage, equivalent to its energy in eV), the pulse charge (average current divided by the frequency) or the number of neutrons per second (neutrons per pulse times frequency) are sometimes given in other facility descriptions, for example, in Koehler (); Lisowski and Schoenberg (); Klug et al. (). They can be easily derived from the listed parameters. > Table  does not contain information about resolution components due to the targetmoderator system. Neutron time-of-flight resolution reflects the distribution of the measured neutron time-of-flight for a given neutron energy. Its origin has several components, like the pulse width, the moderator geometry, and the detector used. This distribution known as the resolution function is usually highly non-Gaussian and changes with neutron energy. In the following some of the facilities are described in more detail. Location RPI, Troy, USA ORNL, Oak Ridge, USA EC-JRC-IRMM, Geel, Belgium FZD, Rossendorf, Germany JINR, Dubna, Russia PAL, Pohang, Korea Kumatori Japan LANL, Los Alamos, USA LANL, Los Alamos, USA CERN, Geneva, Switzerland J-PARC, Tokai, Japan Facility RPI ORELA GELINA nELBE IREN PNF KURRI LANSCE-MLNSC LANSCE-WNR n_TOF MLF-NNRI p p p , ,    ep      e- e- e- e- e-   ee-  e- Particle Beam energy (MeV) Hg Pb W W Ta Ta Ta W L-Pb U Ta Ta Ta Neutron target ,  .  ,  ,  .  – ,  Pulse width (ns) ,  .   . . .    . ,      , - –,  >   Pulse frequency (Hz) . Beam power (kW) Parameters of several neutron time-of-flight facilities. For the new facilities parameters still may improve ⊡ Table    – – , ,  , ,   –  – – ,  – Flight path lengths (m) . ×   ×   ×   ×   ×   ×  . ×  . ×  . ×  . ×   ×  . ×  . ×  Neutron production (n/pulse) Neutron Cross Section Measurements    Neutron Cross Section Measurements .. The Gaerttner LINAC Laboratory The Gaerttner LINAC Laboratory started its operation in December of  and is since used for research and teaching at Rensselaer Polytechnic Institute (RPI). The laboratory and LINAC were designed and built in order to perform time-of-flight (TOF) measurements in the neutron energy range from thermal to  MeV. As such, it is a pulsed L band electron LINAC capable of delivering up to  MeV electrons at a repetition rate of up to  Hz and a pulse width that can vary from  ns to  μs with up to  kW of average electron beam power. The facility is equipped with several flight paths ranging from  to  m which enables measurements in a broad energy range and high energy resolution. In addition, it is equipped with a lead slowing-down spectrometer (LSDS) which provides a high neutron flux utilized for fission and (n,α) cross section measurements of small samples (<μg) or samples with small cross sections (<μb). A layout of the facility is shown in > Fig. . Neutrons are produced by interaction of the electrons with a water- (or air-) cooled tantalum target which produces a high Bremsstrahlung flux that interacts with the tantalum to produce neutrons with an evaporation energy spectrum. Different neutron targets were constructed to produce tailored flux shapes for the different experiments (Danon et al. ; Overberg et al. ). Accelerator room 30 deg Port Capture detector Modular EJ-301 transmission detector Modular Li_glass transmission detector LINAC  Energy analyzing magnet 15 m transmission detector Neutron producing target 250 m station 100 m station Neutron scattering detector Transmission detector Lead slowing down spectromrter ⊡ Figure  Layout of the RPI LINAC and beam ports and detectors Neutron Cross Section Measurements  ⊡ Figure  A picture of a modular neutron detector for neutron transmission measurement at a TOF path length of  m For transmission measurements in the neutron energy range from thermal to a few hundreds of keV,  Li-glass detectors are utilized (Barry ) at flight paths of , , , and  m. At the  m flight station, a modular  Li-glass detector shown in > Fig.  was installed to enable high energy resolution transmission measurements. In order to improve the detector resolution the design employs photomultipliers outside the neutron beam which reduces neutron scattering back to the  Li-glass. For measurements of neutrons with energies above . MeV a modular liquid scintillator is utilized at  m or  m flight path (Rapp et al. ). An example of Mo total cross section data measured with the liquid scintillator detector positioned at  m is shown in > Fig. . A -section NaI multiplicity detector (Block et al. ) is used on one of the flight paths at about  m. Capture, scattering, and fission data are obtained as a function of the total energy deposited in the detector and the number of NaI segments which detect each event. Scattering events are detected from neutron interactions with a ceramic  B C liner surrounding the sample. The  keV gammas from  B (n,αγ) interactions are used to provide a distinct signature of neutron scattering into the liner. > Figure  shows an example of natural Hf capture and transmission data obtained with this detector and a  Li-glass detector at  m (Trbovich et al. ). A fast-neutron scattering detector system was installed at a  m flight path. The system includes eight liquid scintillator detectors surrounding a sample with a digital data acquisition system capable of pulse shape analysis to separate neutron from gamma events and has  ns TOF resolution (Saglime et al. ). Measurements with this system serve as benchmark data to qualify cross section libraries by simulation of the whole experimental setup (Saglime et al. ).   Neutron Cross Section Measurements 10 7 Natural Mo Natural Mo RPI-8 cm sample ENDF/B-VII.0 ENDF/B-VI.8 RPI-8 cm sample ENDF/B-VII.0 ENDF/B-VI.8 6 Cross section [bam] 9 Cross section [bam]  8 5 4 7 3 6 0.4 0.5 0.6 0.7 0.8 Energy [Mev] 0.9 1.0 1.0 5.0 10.0 15.0 Energy [Mev] 20.0 ⊡ Figure  Transmission of  cm natural Mo sample and evaluated data. Left – showing cross section structure at the low energy range. Right – showing the high energy range ⊡ Figure  An example of nat Hf transmission (top) and capture (lower) data measured, respectively, with the Li-glass and multiplicity detectors (Trbovich et al. ). Note that the results of four transmission and four capture measurements have been simultaneously fitted with one set of resonance parameters  Neutron Cross Section Measurements  A lead slowing-down spectrometer driven by the LINAC provides a neutron flux intensity which is about  orders of magnitude higher than a TOF experiment with a flight path producing an equivalent energy-time relation (. m). The resolution of the spectrometer is about % (FHWM) in the energy range from  eV to  keV. Measurements with the LSDS include fission cross sections of small samples (< µg) for actinides such as  Es,  Cf and  Cm (Danon et al. ). Recently the LSDS was used for neutron energy dependent fission fragment spectroscopy and demonstrated the method for  U and  Pu samples (Romano et al. ). In addition methods were developed for (n,α) and (n,p) measurement of samples with small cross sections and the (n,α) cross section of , Sm isotopes were successfully measured. The large target room and flexibility of the facility were used to develop methods for high accuracy transmission measurements using an iron filtered beam (Danon et al. a), and neutron resonance scattering to provide benchmark data for scattering kernel development (Danon et al. b). .. The Los Alamos Neutron Science Center The Los Alamos Neutron Science Center (LANSCE) at the Los Alamos National Laboratory features three separate neutron sources that are used for nuclear data measurements. All are driven by  MeV protons from a linear accelerator, which bombard tungsten targets of different designs. The resulting spallation spectrum of neutrons is similar to a fission neutron spectrum with a high energy tail that extends in principle to  MeV and, in practice, has been used up to  MeV. This spectrum can be moderated to lower neutron energies by water or liquid hydrogen moderators (Lujan Neutron Scattering Center (MLNSC) facility), taken directly (WNR facility), or slowly moderated by an assembly of pure lead (lead slowing-down spectrometer). The beam lines and detector stations are shown in > Fig. . The Lujan facility at LANSCE (LANSCE-MLNSC) provides white sources of neutrons based on a moderated spallation source (Mocko et al. ). The  MeV proton beam from the linac, which is several hundred microseconds long, is compressed by a proton storage ring into a  ns (FWHM) bunch and then is directed to a two-stage tungsten target. Spallation neutrons are moderated by water or liquid hydrogen moderators. The target-moderator-reflector system (TMRS) includes neutron reflectors to increase the neutron flux in the flight paths. The repetition rate of the source is  Hz with . ×  protons per burst, making approximately .×  neutrons per second. Of the  flight paths,  are used for nuclear reaction experiments including neutron capture, fission, and fundamental neutron properties. The neutron energies span the range of cold neutrons (a few milli-electron volts) to several hundred keV. The detector for advanced neutron capture experiments (DANCE) is a nearly π highly-segmented calorimeter of BaF scintillators with very high efficiency that permits neutron capture cross section measurements on samples of  mg or less (Jandel et al. ). Fission experiments are carried out on another flight path with parallel-plate ionization chambers. The Weapons Neutron Research (LANSCE-WNR) facility is based on micropulses from the linear accelerator providing bunched  MeV proton bursts on a bare tungsten target. The macrostructure of the proton beam consists of trains of micropulses, approximately  μs long, with a macropulse frequency of  Hz. Internal to the macropulse are micropulses of up to  ×  protons with spacings adjustable in units of  ns. Common micropulse spacings are . and . μs. The proton micropulse width is approximately . ns and therefore allows    Neutron Cross Section Measurements ⊡ Figure  The beam lines and detector stations at the Los Alamos LANSCE facility high-energy experiments with excellent timing. The unmoderated spallation neutrons are collimated in six flight paths with production angles from ○ to ○ . Experimental stations at flight paths of – m are set up for high-resolution gamma-ray detection (GEANIE array with  HPGe detectors), neutron emission (FIGARO array with  liquid scintillation neutron detectors), charged-particle emission (NZ with four charged-particle telescopes), medium-energy few-nucleon research, long-flight path experiments, industrial irradiations of electronic components (ICEHOUSE), and fission cross section measurements. A representative measurement is shown in > Fig.  for the ratio of the  Np to  U fission cross sections. Detailed information on the beams and research is given on the LANSCE web site and references (Tovesson et al. ; Bernstein et al. ; Rochman et al. ). A lead slowing-down spectrometer is used for high neutron flux experiments. It is driven by the proton beam, either the micropulse beam usually used at WNR or the beam from the proton storage ring. The beam hits a tungsten target at the center of a -ton cube of pure lead, . m on a side. Materials and detectors are placed in channels in the cube and are subjected to very large fluxes of neutrons for cross section measurements as functions of neutron energy. The reaction rate is measured as a function of time, which can be converted into average neutron energy in the range . eV– keV. The conversion is equivalent to a time-of-flight experiment with a flight path of . m. The trade-off for this approach is that the energy resolution is only about % (FWHM) in ΔE/E (Rochman et al. ). With the very large neutron flux, cross sections have been measured with samples less than  ng. .. The ORELA Laboratory at Oak Ridge National Laboratory The Oak Ridge electron linear accelerator (ORELA) is a high power, high resolution, and highly versatile white neutron source (Dabbs ). Over the course of the last  years, ORELA Neutron Cross Section Measurements Fission cross section ratio 237Np/235U  Fission cross section ratio versus energy 1 10–1 10–2 10–3 JENDAL 3.3 ENDF/B-VI 10–4 10–5 10–6 10–7 10–2 LANSCE data 2007 10–1 1 10 102 103 104 En (eV) 105 106 107 108 ⊡ Figure  Ratio of fission cross section for  Np to that of  U over  orders of magnitude in neutron energy from data taken at LANSCE (Tovesson and Hill , ). The experimental values are compared with data in the JENDL. and ENDF/B-VI libraries and were used to improve the data for  Np in the new ENDF/B-VII library experiments have contributed the majority of neutron nuclear data that are used for applied and basic nuclear physics programs around the world such as nuclear criticality safety, nuclear reactor physics, neutron shielding, nuclear medicine, and nuclear astrophysics. ORELA consists of a  MeV electron linear accelerator, neutron producing targets, underground and evacuated flight tubes, sophisticated detectors, and data acquisition systems. Simultaneous measurements are possible at  detector stations on  separate flight paths at distances between  and  m from the neutron source. An artist’s view of the laboratory is shown in > Fig. . Neutron capture, (n,γ), measurements routinely are performed using a pair of deuterated benzene scintillation detectors on flight path  at the -m station. Improvement in this apparatus over the last few years has resulted in substantially reduced background from scattered neutrons and ORELA measurements with this new system have demonstrated that previous (n,γ) cross-sections are in error, sometimes substantially, due to this effect (see > Fig. ). Total cross section measurements can be performed with an NE- plastic (above  keV) or  Li glass (below  keV) scintillator. The sample can be cooled to  K to significantly improve the resolution of closely spaced resonances by reducing the Doppler broadening as shown in > Fig. . Fission measurements have been performed using a variety of ionization chambers, such as a small hemispherical ion chamber for high alpha-activity rejection. A new type of detector, called a compensated ion chamber, was pioneered at ORELA and made possible the first measurements of (n,α) cross sections on intermediate- to heavy-mass nuclides at astrophysical relevant energies. These data demonstrated that the latest nuclear models used to calculate astrophysical (α,γ) and (α,p) reaction rates for explosive nucleosynthesis studies are in need of serious revision. .. GELINA at the JRC-IRMM in Geel The neutron time-of-flight facility GELINA (Geel Linear Electron Accelerator) of the European Commission’s Joint Research Center IRMM in Geel, Belgium, has been operational for more  40 ER T ME I FL T GH ST IO R TE F T H LIG ON I AT ST ESCAPE HATCH ME 20 N NEUTRON-PRODUCING TARGET AT ET RG M TA OO R GROUND LEVEL EL N RO T EC L CE AC R TO LA DU MO 35-METER FLIGHT STATION OM RO OM RO R TO A ER Cutaway view of ORELA showing linac, cylindrical target room, and flight paths ⊡ Figure  TO 85-METER FLIGHT STATION 10 150-METER FLIGHT STATION G IN ILD H BU ORT N CENTRAL DATA AREA TO 80 AND 200-METER FLIGHT STATIONS OM RO OFFICES AND LABORATORIES  SHIELD TEST STATION  Neutron Cross Section Measurements Capture cross section (b) Neutron Cross Section Measurements  ORELA Si data Si ENDF/B-VI 0.04 0.03 0.02 0.01 0.00 60 55 50 65 Neutron energy (KeV) 75 70 ⊡ Figure  Comparison of the ENDF/B-VI and recent ORELA  Si capture cross section from  to  keV (Guber et al. ). Due to the neutron sensitivity of the experimental set up the previous  Si(n,γ) experiments (on which ENDF/B-VI is based) seriously overestimated the capture cross section 400 ORELA 1998 Data233U (n, tot) Pattenden et al. 1963 350 233 U (n, tot) [b] 300 250 200 150 100 50 0 30 40 50 60 Neutron Energy [eV] 70 80 ⊡ Figure  Comparison of recent ORELA measurements with previous  U total cross sections. Using a cryogenically cooled sample significantly improved the resolution than  years. Neutron-induced reaction data have been measured for a variety of applications related to nuclear technology and nuclear science. The facility is based on an electron LINAC providing a pulsed electron beam of  MeV average energy and a typical power of  kW, impinging on a rotating uranium target cooled by a flow of mercury. Typical operation frequencies range from  to  Hz. The initial electron burst, consisting of a train of micropulses with energies from  to  MeV, is compressed by a    Neutron Cross Section Measurements Flight path station Neutron beam lines Uranium target Electron beam ⊡ Figure  A schematic view of the neutron beam lines of GELINA post-acceleration ○ bending magnet system to a bunch of less than  ns FWHM, preserving the average current. The Bremsstrahlung induced in the uranium target by the electron beam produces neutrons by (γ,n) and (γ,f) reactions. The neutron distribution emitted by the uranium target has a typical fission-evaporation spectrum peaked at around  MeV with a small intensity of low-energy neutrons. Two moderators consisting of slabs of  cm thick water canned in beryllium are placed under and above the uranium target. They modify this fast spectrum to a partial moderated spectrum containing a Maxwellian peak at thermal energies and an approximate /E energy dependence at higher energies. At  Hz the average neutron production rate at the source is . ×  neutrons/s. Twelve neutron flight paths leading to experimental stations at distances from  to  m depart from the neutron source under angles which are multiples of ○ , as is shown in > Fig. . Each flight path can be shielded either from the uranium neutron production target or from the water moderators by  cm long shadow bars consisting of lead and copper, in order to obtain a fast or moderated neutron spectrum. More details on the facility can be found, for example, in Tronc et al. () and Flaska et al. (). The experimental stations are equiped with measurement setups for neutron capture with Ge, BGO, or low neutron-sensitive C D detectors in combination with the weighting function technique (Borella et al. a; Schut et al. ), for transmission experiments using  Liglass detectors (Kopecky and Brusegan , Borella et al. b), for fisison measurements with dedicated fission chambers (Wagemans et al. ), and for (n,xnγ) measurements HPGe detectors (Mihailescu et al. ) and other experiments. In > Fig.  an example of a neutron capture and transmission experiment on an enriched  Pb sample is shown in the left panel. The capture yield shows the measured yield, the background contribution, and the measured scattered neutron contribution. The transmission factor and a zoom on the  keV s-wave resonance are shown in the right panel. Resonance parameters were obtained by a simultaneous analysis of the capture and transmission data (Borella et al. b). .. The n_TOF Facility at CERN The construction and commissioning of the neutron time-of-flight facility at CERN, Switzerland, after an initial proposal (Rubbia et al. ), was finished in  when the facility became operational.  f Neutron Cross Section Measurements e e ⊡ Figure  The neutron capture yield spectrum of  Pb (left panel), together with the transmission factor (right panel), both measured at GELINA (Data are from Borella et al. b) The facility uses a  ns wide,  GeV/c proton beam with up to  ×  protons per pulse hitting a lead target, yielding about  neutrons per incident proton. A water slab surrounding the target serves as coolant and as moderator. At present a single flight path is available with an experimental station located at  m. A . T sweeping magnet is placed at a distance of  m from the spallation target to remove residual charged particles. One collimator with an inner diameter of  cm is placed at  m while a second collimator with a variable diameter of either . cm or  cm is situated at  m from the production target. The main elements of the neutron beam line are shown in > Fig. . The repetition period of the proton pulses from CERN’s PS acelerator is a multiple of . s, which allows to cover the energy range down to subthermal energies without overlapping of slow neutrons in subsequent cycles. A full description of the characteristics and performances of the facility is described elsewhere (Abbondanno et al. ). The facility is mainly used for capture and fission measurements. The energy distribution of the incident neutron flux is continuously measured during the experiments with an in-beam neutron flux detector (Marrone et al. ). The spatial distribution has been obtained with a MicroMegas-based detector (Pancin et al. ). The developed data acquisition system F N S L N P C S ⊡ Figure  A schematic view of the neutron beam line of n_TOF at CERN S   Neutron Cross Section Measurements (Abbondanno et al. ) uses sampling of the detector signals in order to extract the deposited energy and the time of flight. In-house developed deuterated benzene C D gamma-ray detectors contained in a cylindrical low mass carbon fibre housing (Plag et al. ) have been used for neutron capture measurements. Samples are kept in position by a carbon fiber sample changer. The low neutron capture cross sections of both carbon and deuterium assure a low contribution to the background from sample scattered neutrons. Since this detector does not measure the full gamma-ray cascade following neutron capture, it requires the use of weighting functions to reconstruct the neutron capture yield (Abbondanno et al. ; Borella et al. a). Although the detection efficiency for a single detector is only about % for a  MeV gamma-ray, due to the gamma-ray multiplicity after neutron capture, in the order of – for medium and high mass nuclei, the efficiency to detect capture event is roughly % for the set of two detectors. A second neutron capture detection system consists of a π % efficiency total absorption capture detector, made up of  BaF crystals contained in  B loaded carbon fiber capsules, coupled to XPB photomultipliers equipped with (for this purpose) especially designed voltage dividers. Samples are surrounded by a C H O ( Li) neutron absorber which moderates and absorbs sample scattered neutrons. Fission experiments have been performed with two different detector systems. Two fission ionization chambers (FIC) use deposits of fissile isotopes on  µm thick aluminum foils. The FIC- detector was used for the low activity samples while the FIC- detector was used for the samples with higher activity (Calviani et al. , ). As an example the  U(n,f) cross section measured at n_TOF with the FIC detector is shown (Calviani et al. ) in > Fig. . In > Fig.  the neutron capture spectrum of  Th measured with the optimized C D gamma-ray detectors is shown (n_TOF Collaboration ). The second type of fission detector is based on parallel plate avalanche counters (PPACs), developed with target deposits on . μm thin mylar or  μm aluminum foils, allowing to detect the two fission fragments in coincidence (Paradela et al. ). 60 50 n_TOF ENDF/B-VII.0 Guber 2000 Weston 1968 40 σf (barn)  30 20 10 0 580 590 600 610 620 630 Neutron energy (eV) 640 650 ⊡ Figure  Part of the measured  U(n,f) cross section at n_TOF at CERN (Data from Calviani et al. ) Neutron Cross Section Measurements  104 232 Experimental Th(n,γ) spectrum 232 Experimental radioactivity Th 232 Fitted radioactivity Th Background scattered neutrons Weighted counts 103 102 10 1 10−1 1 10 102 103 104 Neutron energy (eV) 105 106 ⊡ Figure  The count rate spectrum of the  Th(n,γ) reaction and the calculated background contribution from sample scattered neutrons. For comparison also the contribution of the natural radioactivity of  Th is shown (Data are from n_TOF Collaboration ) The facility has been operating from  to  (phase-I) with neutron capture and fission measurements and was upgraded in  with a new spallation target (phase-II). Future plans include the construction of a short flight path (phase-III). .. The IREN Facility at Dubna The pulsed neutron source research complex IREN is a new facility which will be realized in several stages. Eventually, the facility will comprise a  MeV electron linac delivering  kW beam power, and impinging on a tungsten electron-neutron converter located in the center of a subcritical neutron multiplying target. This target will consist of about  kg of highly enriched (more than %  Pu) metallic plutonium, resulting in a neutron yield of  n/s and a pulse width of  ns. These are target values that will be achieved gradually. By the middle of  a  MeV electron linac together with a non-multiplying tungsten target had been put in operation, making the installation ready in the first stage for high resolution neutron spectroscopy experiments in the energy range up to several s of keV. Planned experiments are foreseen in the field of nuclear astrophysics, nuclear data, nuclear structure and fundamental symmetries, as well as neutron and gamma activation analysis and medical radioactive isotope production. The neutron source parameters of this first stage have been included in > Table . The neutron source has replaced the phased-out facility IBR-, which was also an accelerator-driven subcritical assembly. IREN uses the existing flight path infrastructure of eight neutron beams with lengths from  to , m. In > Fig.  a schematic view of the future subcritical core is shown. An example of a measurement of the first stage is the  Ta(n,γ) reaction with both the IBR and IREN neutron source, shown in > Fig. , measured with a six times  l segmented liquid scintillator detector located at the  m flight path.    Neutron Cross Section Measurements ⊡ Figure  A schematic view of the IREN facility (left panel) and its subcritical core (right panel) (Figure from Ananiev et al. ) .. The PNF Laboratory at Pohang The Pohang neutron facility (PNF) (Kim et al. ; Wang et al. ) consists of a  MeV electron linac, a water-cooled Ta target, and an -m long TOF path which has recently been set up in Pohang, Korea. The maximum electron energy for TOF measurements is  MeV, and the measured peak beam currents at the entrance of the first accelerating structure and at the end of linac are  and  mA, respectively. The duration of electron beam pulses are – μs, and the pulse repetition rate is  Hz. The measured energy spread is ±% at its minimum. The energy spread is reduced when optimizing the RF phase of the RF-gun and the magnetic field strength of the alpha magnet. The neutron target is water cooled and composed of ten .-cm diameter Ta disks with different thickness and a total Ta thickness of . cm. This target is set at the center of a cylindrical water moderator contained in an aluminum cylinder with a diameter of  cm and a height of  cm. The distributions of neutrons with and without water moderator are described elsewhere. The photoneutrons produced in the giant dipole resonance region consist of a large number of evaporated neutrons and a small fraction of directly emitted neutrons, which dominate at higher energies. The MCNP calculated neutron yield per kW of beam power for electron energies above  MeV at the Ta target is . ×  n/s (Nguyen et al. ), which is consistent with the calculated value based on Swanson’s formula, . ×  Z . , where Z is the atomic Neutron Cross Section Measurements  Counts / bin 4000 IREN 3000 2000 1000 0 Counts / bin 6000 IBR-30 5000 4000 3000 2000 1000 0 10 100 Neutron energy (eV) ⊡ Figure  Example of a measurement of the  Ta(n,γ) neutron capture reaction performed during the commissioning phase in  with a six times  l segmented liquid scintillator detector located at the  m flight path using both the IBR- (lower panel) and IREN (upper panel) neutron source (Data from Belikov et al. ) number of the target material. The total neutron yield per kW of beam power is measured to be (. ± .) ×  n/s by using the multiple-foil activation technique. Neutron flight tubes are constructed of stainless steel and placed perpendicularly to the electron beam. The collimation system is mainly composed of H BO , Pb, and Fe collimators. Transmission measurements are made with a  Li-ZnS(Ag) scintillator (BC) with a diameter of . cm and a thickness of . cm. A typical TOF spectrum is shown in > Fig.  where the upper curve is with no sample in the beam and the black curve has Co, In and Cd in the beam to determine the background at strong resonances. A large volume bismuth germanate (Bi Ge O ; BGO) detector is under construction for the measurement of neutron capture cross sections. This detector is an assembly of BGO bricks and will have a total volume of . l. .. Electron Linac at Kyoto University Research Reactor Institute, KURRI The electron linear accelerator (KURRI-LINAC) at the Kyoto University Research Institute in Kumatori, Japan, was installed in  as a  MeV machine and subsequently upgraded to  MeV. The accelerator operates at L-band (, MHz) and provides electron pulse widths from  ns to  μs. The research covers a wide range of neutron measurements using the TOF   Neutron Cross Section Measurements 400 59Co (132 eV) y = A1*exp(–I/t1) + y0 y0 = 1 300 115In Counts  115In 200 A1 = 64.20 (9.04 eV) t1 = 355.12 (3.85 eV) 115In (1.457 eV) Cd (<0.025 eV) 100 Fitting function 0 0 4000 1000 2000 3000 Channel number [0.5 µs/ch] ⊡ Figure  Background level determination with Co, In, and Cd samples in the beam at Pohang. The background was determined by fitting to the counts at the blacked-out resonances produced by these samples method. Currently two flight paths are used for TOF experiments with detector stations at ., ., and . m. In particular, neutron capture measurements are made with a π total absorption BGO spectrometer, another BGO detector, a C D liquid scintillation detector and a π Ge spectrometer (on loan from JAEA) (Kobayashi et al. ; Shcherbakov et al. ; Kobayashi et al. ; Hori et al. ). > Figure  shows the BGO total absorption spectrometer where  blocks of BGO surround a  LiF shielded capture sample and > Fig.  shows a measurement of the  I capture v o c t c ´ c n b b ´ s ´ v o c ⊡ Figure  The BGO total absorption spectrometer for neutron capture measurements at the Kyoto linac Neutron Cross Section Measurements  Neutron capture cross section (b) 104 JENDL–3.2 ENDF/B–VI JEF–2.2 Popov (1962) Gabbard (1959) Gibbons (1961) Weston (1960) Block (1961) Linenberger (1946) Macklin (1983) Yamamuro (1980) Stavisskii (1961) Shorin (1975) Present 103 102 101 100 10–1 10–2 10–2 10–1 100 101 102 103 Neutron energy (eV) 104 105 106 ⊡ Figure  Capture cross section of  I measured at the Kyoto linac (Hori  private communication) 24.2 m Linac shielding wall Electron beam Water moderator neutron BF3-counter Lead shadow bar Neutron producing target Neutron beam filters Sample Collimators Evacuated flight tubes Lead shielding Borated paraffin BGO Detector 10.5 m Lead Concrete H2BO3 Iron Borated paraffin ⊡ Figure  Experimental arrangement for a BGO detector capture cross section measurement at the . m flight station at the Kyoto linac cross section obtained with this detector. In > Fig.  the geometry is shown used for the  Np capture cross section measurement and > Fig.  shows the results of this measurement. A lead slowing-down spectrometer is also installed at KURRI to facilitate fission cross section measurements of small samples.   Neutron Cross Section Measurements 10,000 Capfure cross section (barn)  Weston & Todd (1981) Kobayashi et al. (2002) JENDL -3.3 ENDF/B - VI 1,000 ENDF/B - VI Present (t = 100, t n ch = 400 ns x 20) 1,000 Present (t = 100 ns) n Present (t = 3ms) n 100 100 10 10 0.01 1 0.1 1 Neutron energy (eV) 1 10 100 Neutron energy (eV) ⊡ Figure   Np capture cross section below  eV (left panel) and above  eV (right panel). “Present data” are from the Kyoto linac . Monoenergetic Fast Neutron Facilities Cross section measurements in the energy range  keV to many MeV are frequently carried out with light-ion-induced nuclear reactions that lead to monoenergetic or quasi-monoenergetic neutrons. Typically, a source reaction is chosen to produce neutrons of one energy (with a small energy spread), and these neutrons can either be continuous in time or pulsed. For the latter type of source, TOF techniques can be used to reject parasitic neutrons of other energies. TOF is also commonly used in neutron scattering and neutron-emission measurements where the energy of the outgoing neutron is determined by the flight time from the sample to the detector. Data on the neutron source reactions are available from the International Atomic Energy Agency (IAEA) (Drosg ). Many low energy accelerator facilities have produced monoenergetic neutron cross section data over the past  years. Although several of these facilities are no longer operational, some are still very productive. Facilities that specialize in TOF measurements typically have special capabilities including pulsed beams with time spreads approaching  ns and flexible pulse spacing. Most facilities feature relatively high beam currents in order to produce usable neutron fluxes, neutron-production targets that can withstand the high beam current, gas production targets with pressures of several atmospheres of  H or  H ,  Li,  Be targets targets, and experimental areas that have low-mass floors and large target rooms to minimize room backscattering into the sample or the detectors. The available accelerator beams and beam energies determine the range of neutron energies. Unique detector capabilities at certain facilities give them a special status. Here, we give examples of facilities that are optimized for monoenergetic measurements with neutrons in the s and s of keV range, others in the range – MeV, specialized -MeV neutron facilities, and sources that produce neutrons in the – MeV range. A selection of monoenergetic neutron source facilities is given in > Table . Examples are given of measurements performed in this wide energy range. .. Neutron Energies Below  MeV Neutron capture reactions have been studied extensively in the keV region at the Van de Graaff laboratories at the Forschungszentrum Karlsruhe (FZK) and in Japan at the Tokyo Institute of Neutron Cross Section Measurements  ⊡ Table  Table of monoenergetic or quasi-monoenergetic neutron sources Laboratory Location Accelerator Particlea Beam Energyb (max) Common Reaction Neutron Energies (typical) Tokyo Inst. of Techn. JAEA Tokyo, Japan  MV Pelletron p  MeV   keV– MeV Tokai, Japan  MV Pelletron p  MeV  Li(p,n) Li(p,n) H(d,n)  H(d,n)  keV– MeV  FZKc Karlsruhe, Germany Frankfurt, Germany Lowell (MA), USA Sendai, Japan . MV Van de Graaff RFQ/DTLd p . MeV  Li(p,n)  keV–. MeV p . MeV  Li(p,n) – keV . MV Van de Graaff . MV Dynamitron p . MeV  Li(p,n)  keV– MeV p, d . MeV  H(d,n) – MeV Ohio Univ. Athens (OH), USA . MV Tandem VdG p, d  MeV IRMM Geel Belgium  MV Van de Graaff p, d  MeV Univ. Frankfurt U. Mass Lowell Tohoku Univ. N(d,n)  MeV  H(d,n) –, – MeV  H(d,n)  H(p,n)  H(d,n) H(d,n)   TUNL PTB CIAE Durham (NC), USA Braunschweig, Germany  MV Tandem VdG Cyclotron Beijing, China  MV Van de Graaff p, d  MeV  H(d,n) – MeV p d  MeV . MeV  Be(p,n) H(d,n) – MeV d  MeV  H(p,n)  MeV  H(d,n) H(d,n)   Osaka OCTAVIAN JAEA- FNS CIAE Osaka, Japan Electrostatic d  keV  H(d,n)  MeV Tokai-Mura, Japan Beijing, China Electrostatic d  keV  H(d,n)  MeV CockroftWalton Cyclotron d  keV  H(d,n)  MeV p  MeV  Li(p,n) – MeV Cyclotron p  MeV  Li(p,n) Li(p,n) – MeV Cyclotron p  MeV  Li(p,n) – MeV S-Cyclotron p  MeV  Li(p,n) – MeV S-Cyclotron p  MeV  Tohoku Univ. CYRIC Sendai, Japan Louvain Lovain, Belgium Tokai-Mura, Japan Uppsala, Sweden Faure, South Africa  JAEA-TIARA TSL-Uppsala iThemba a particles: proton (p), deuteron (d) Max beam energy for nuclear data measurements c recently shut down, included here for recent data and technique d Radio-frequency quadrupole (RFQ), Drift tube linac (DTL) b .– MeV Be(p,xn) – MeV   Neutron Cross Section Measurements Technology and the Japan Atomic Energy Agency laboratory in Tokai. All of these facilities use the  Li(p,n) reaction with proton beams in the .– MeV range to produce monoenergetic neutrons in the range  keV– MeV. Pulsed beams of ≈  ns width can be used with TOF techniques to separate neutrons from gamma rays produced at the source. Although the FZK facility is no longer operational, an upgraded facility at the Goethe University, Frankfurt, Germany, is expected to begin operation soon with neutron fluxes increased by – orders of magnitude relative to the FZK flux. Neutron capture is detected by a single, well-shielded NaI(Tl) gamma-ray detector at Tokyo, (Igashira et al. ), ( > Fig. ) and by a π calorimeter of BaF crystal scintillators at Karlsruhe ( > Fig. ) (Wisshak et al. ). The former is used to measure discrete high energy gamma rays whereas the latter detects the full energy of the gamma-ray cascade, which is the Q-value of the capture plus the incident neutron energy in the center of mass, a specificity that can help distinguish capture in the sample from background captures in other materials. TOF techniques can be used to deduce the capture cross section or capture-to-fission ratio as a function of neutron energy, for example, (Beer and Kappeler ) and (Wisshak et al. ). For activation experiments, a neutron continuum spectrum approximating a Maxwellian spectrum with kT ≈  keV can be produced with the  Li(p,n) reaction with . MeV protons on a stopping lithium target (Beer and Kappeler ). This approach has been used in favorable cases to measure the neutron capture cross section for radioactive isotopes such as  Eu (Jaag and Kappeler ). This particular temperature for the Maxwellian is of importance in understanding s-process nucleosynthesis in stars. 6 5 150 mm 7 Sample 5 12 1 76f x152 mm Nal(TL) mm 3 90 2 4 1 10  2 Annular Nal(TL) 3 Plug Nal(TL) 4 Photo–mul. 5 Li–target 6 Insulator 7 Capacitive pick off Lead 80% Paraffin + 20% H3BO3 ⊡ Figure  Typical experimental arrangement used in keV-neutron capture gamma-ray measurements at the Tokyo Institute of Technology (Igashira et al. ) Neutron Cross Section Measurements 10B γ Collimated n-beam γ Sample 6LiCO γ  + araldite 3 P-beam γ Pb Neutron target Flight path 77 cm ⊡ Figure  Calorimetric array of BaF scintillators at Karlsruhe covering close to the full π solid angle. Neutrons produced by the  Li(p,n) reaction are incident on a sample in the middle of the array and the cascade of gamma rays following neutron capture is detected by the scintillators (Wisshak et al. ) .. Neutron Energies in the MeV Region At the University of Kentucky, a vertical electrostatic accelerator, extensively rebuilt from its original Van de Graaff accelerator, provides beams of protons, deuterons,  He ions, and  He ions from . to  MeV. Neutron fluxes are provided by the  H(p,n) He reaction from . to  MeV, by the  Li(p,n) Be reaction from . keV to . MeV, by the  H(d,n) He reaction from  to . MeV, and from  to  MeV by the  H(d,n) He reaction. The Van de Graaff accelerator terminal provides continuous beams, those pulsed to  ns, and pulsed and bunched beams of  ns burst width at a repetition rate of . MHz. Almost all neutron-induced reactions have been done with ≈  ns pulsed and bunched beams. For neutron-detection experiments, a post-acceleration buncher provides sub-nanosecond pulses. Most neutron experiments take place in a “neutron hall,” with a thin metal floor above a pit of . m depth, and an area . m × . m. Shielded detectors for pulsed-beam neutron detection are used for TOF detection with flight paths up to  m. Gamma-ray detection usually uses shorter flight paths of about . m. Gamma-ray detection is available with HPGe detectors, actively shielded with BGO annulus detectors. The whole assembly is housed in a large shield. A variety of liquid scintillation neutron detectors is available with diameters of . cm or . cm and thickness from . to . cm. Four large HPGe detectors are housed in a coincidence array (KEGS) for gamma-gamma correlation experiments. An example of inelastic scattering cross section data is shown in > Fig.  where the excitation of specific levels as a function of incident neutron energy was determined by detecting their gamma-ray decay (Lesher et al. ). Analysis of the Doppler shift of the gamma rays gives information on the lifetimes of the states. Another single-end Van de Graaff used for neutron research is at the University of Massachusetts Lowell Radiation Laboratory. A typical neutron source is produced by the  Li(p,n) reaction with a tightly bunched proton beam with a maximum energy of . MeV. Neutron emission is measured with this facility, an example being the spectrum of neutrons emitted   Neutron Cross Section Measurements 60 2736.0 keV J=2 J=3 J=4 Cross section (mb/sr) 50 40 30 20 10 0 3 4 En (MeV) ⊡ Figure  Data for the level at . keV in  Mo are compared with statistical model calculations for three different spin possibilities. These data are from the University of Kentucky. The best fit for this level is J =  (Lesher et al. ) 1.E+06 Neutrons per MeV (arb norm)  Fission neutron spectra 235 U + n (En) 1.E+05 1.E+04 1.E+03 En = 0.5 MeV En = 2.5 MeV Data * 0.1 1.E+02 1.E+01 Curves from ENDF/B-VII.0 1.E+00 0 2 4 6 8 18 12 14 16 Fission neutron energy (MeV) ⊡ Figure  Fission neutron spectral shapes for  U(n,f) at two different incident energies. The data are from University of Massachusetts at Lowell (Staples et al. ) and are compared with the shapes evaluated in ENDF/B-VII in the fission of actinides. > Figure  gives a result for the neutron emission specta from neutron-induced fission of  U (Staples et al. ). In the European Union, high intensity quasi mono-energetic neutron sources are produced at the Institute for Reference Materials and Measurements (IRMM) at Geel, Belgium, by a vertical  MV Van de Graaff accelerator with either continuous or pulsed ion beams (Reimer et al. ; Semkova and Plompen ). The installation has two pulsing systems: () a fast beam pulsing generating a minimum ion beam pulse width of  ns and pulse repetition rates of ., . or . MHz, and () a slow pulsing system giving a minimum pulsing width of  μs at an Neutron Cross Section Measurements  ⊡ Figure  Cross sections for the  Am(n,n)  Am reaction with recent data from the VdG Geel (from Sage et al. ) adjustable frequency up to  kHz. Neutron fields that have well-defined energies are produced by the nuclear reactions  Li(p,n),  H(p,n),  H(d,n) or  H(d,n) giving neutrons within the energy regions . to . MeV and . to  MeV. This facility houses six experimental set-ups in two large laboratory halls. In addition to TOF experiments, many neutron activation experiments have been carried out. An example of activation cross section results is shown in > Fig.  for the  Am(n,n) Am reaction (Sage et al. ). At Ohio University, a high current tandem Van de Graaff accelerator with a terminal voltage up to . MV accelerates hydrogen ions to . MeV and other ions to higher energies. Beams of protons, deuterons,  He,  He,  Li,  Li,  Be,  B,  B,  C, and  C are produced. A central focus of the laboratory is neutron physics, and a pulsing and bunching system is available to produce beams of about  ns width for protons and deuterons, and .– ns width for Li through O. A particularly important piece of experimental equipment is a “beam swinger,” which allows the beam incident on a target to be rotated through a range of angles (Finlay et al. ). One flight path with length of  to  m is used with the swinger to investigate neutron scattering from ○ to ○ . The laboratory is equipped with gas cell target assemblies that can be used with  H or  H gas. Solid targets containing tritium are also available. Efficiency calibrations of neutron detectors can be carried out with a  Cf source or with previously measured spectra from deuteron bombardment of stopping targets of B or Al (Massey et al. ; Di Lullo et al. ). This facility ( > Fig. ) has been used for neutron elastic and inelastic scattering studies. An example of the angular distribution of elastic scattering of neutrons is shown in > Fig. . The data from this facility have greatly improved the understanding of the neutron-nucleus optical model. Somewhat higher neutron energies are provided by the Triangle Universities’  MV FN tandem Van de Graaff accelerator at Duke University. It is equipped with a variety of ion sources,    Neutron Cross Section Measurements Large target room Edwards Accelerator Laboratory Ohio University, Athens, Ohio XY Steerer X Steerer Y Steerer & Chopper Einzel lens Slits Quadrupole lens Pumping station Faraday cup Gate valve BPM Small target room 4.5 MV Tandem van de Graaff HV Beam swinger Produces beam 0 to 160 deg relative to TOF tunnel Low-energy area © Catain matel 30 m Time-of-flight tunnel ⊡ Figure  Schematic layout of the Ohio University Tandem Van de Graaff facility showing the accelerator, the beam swinger, the -m time-of-flight tunnel and other beam lines beam lines, and target stations. The associated polarized ion source is the most intense source of dc polarized H+ and D+ ions in the world. Unpolarizedbeams of protons and deuterons are available from a direct-extraction negative-ion source. These beams are being used in a wide range of nuclear reaction studies including few-body reactions, radiative capture, and polarized neutron induced reactions. An example of neutron scattering from  Li is given in > Fig.  (Hogue et al. ). At the Physikalish-Technische Bundesanstalt (PTB) standards laboratory in Braunschweig, Germany, a cyclotron is used to produce standard neutron fields and to make basic cross section measurements of activation and differential scattering. High current beams of protons up to  MeV, deuteron beams up to . MeV and alpha particles up to  MeV are produced here. This very well characterized facility is unique in that the cyclotron is moved around the scattering sample in order to change the set of scattering angles ( > Fig. ) (Mannhart and Schmidt ; Schmidt ). A high-energy tandem Van de Graaff with a terminal voltage up to  MV is used at the Chinese Institute of Atomic Energy in Beijing, China. A multi-detector array of liquid scintillators detects the scattered neutrons. Another approach, the so-called “abnormal” TOF spectrometer is used to overcome difficulties of source breakup neutron interference in the  H(d,n) reaction in the – MeV region (Schmidt et al. ). In the latter, the neutrons are collimated to a narrow beam and the scattering sample is placed approximately  m from the source. Detectors for the scattered neutrons are located – cm from the sample. In this method, neutron emission from the sample can be studied down to  MeV in the emitted neutron energies. Neutron Cross Section Measurements 54Fe  (n,n) 104 104 Present work 26 MeV dσ / dΩ (mb/sr) 104 24 MeV 104 22 MeV 103 20 MeV 102 101 0 20 40 60 80 100 θcm (deg) 120 140 160 180 ⊡ Figure  Elastic differential cross section for neutron scattering from  Fe at three incident energies taken at Ohio University compared with phenomenological optical model fits (Mellema et al. ) .. Neutron Energies Near  MeV A special neutron energy is  MeV, where neutrons can be produced with high intensity by the  H(d,n) He reaction which has a large cross section near the wide resonance just above  keV and which is, therefore, accessible to low energy deuteron accelerators. This  MeV source reaction (also called a D-T source reaction for deuterons plus tritons) is also the most favorable for fusion energy, and therefore data at this energy are very important for those applications. Many laboratories, both small and large, have used  MeV neutrons for cross section measurements. An example is the Octavian facility at Osaka University and one type of measurement made there is neutron emission spectra and angular distributions ( > Fig. ). Other types of measurements made at laboratories of this type are neutron activation (of importance to waste disposal for fusion reactors), hydrogen and helium production (important for radiation damage of structural materials), standard cross section measurements, and integral tests of neutron transport, such as the pulsed sphere measurements carried out at several laboratories (Hansen et al. ).   Neutron Cross Section Measurements 103 8 Li(n, n0) 103 13.94 MeV Dlferential scattering cross section, 0(θ) (mb/sr)  10 3 12.94 103 12.04 10 3 10 3 10 3 10.95 9.96 8.96 103 7.47 0 30 60 90 120 θcm (deg) 150 ⊡ Figure  Angular distribution of elastic scattering of neutrons from  Li (Hogue et al. ). The data are from the Triangle Universities Nuclear Laboratory For much higher neutron-production rates, rotating tritium-loaded targets are used to handle beam currents on the order of s or even  mA. Present facilities at the Japan Atomic Energy Agency (fast neutron source – FNS) can produce -MeV neutron fluxes well in excess of  n/cm /s for samples close to the source. With a source of very high intensity, neutrons can be collimated in very narrow beams, sometimes called “pencil beams,” and still retain enough intensity for cross section measurements. The advantages of this approach are described in Kondo et al. (). One special technique, referred to as “neutron-tagging” or “associated-particle” technique, is particularly appropriate for D-T neutron sources when the number of neutrons incident on a sample needs to be known precisely. The reaction at low incident neutron energy produces an alpha particle in addition to the neutron, and the two are correlated in angle by two-body kinematics. The alpha particle is usually detected at a back angle and the direction of the associated Neutron Cross Section Measurements 0 1 2 3 4  5m ⊡ Figure  Cyclotron and time-of-flight paths for neutron scattering experiments at PTB. Note that the cyclotron can be moved to change the set of scattering angles neutron is known exactly, to within the angular resolution of the alpha particle detector. This technique is used to measure cross sections without the need to refer to another, so-called “standard,” reaction cross section. This technique is not limited to the D-T reaction and  MeV neutrons but other reactions, such as  H(d,n) He also provide an associated charged particle in addition to the neutron. This technique is not used much at present due to the improvements in “standard” cross sections, but it is still very useful when very precise neutron detector calibrations or absolute cross section measurements are desired. .. Neutron Energies Above  MeV The Svedberg Laboratory (TSL) (Klug et al. ) at Uppsala, Sweden, is a good example of a laboratory using neutrons in the s of MeV region and higher. It consists of a cyclotron that can accelerate protons up to  MeV, deuterons up to  MeV, alpha particles up to  MeV and all the way up to  Xe to  MeV. > Figure  shows the neutron energy distribution for a quasimonoenergetic peak around  MeV obtained from the  Li(p,n) Be reaction. Approximately, half of the neutrons generated are in the peak. The low energy tail can be removed in the data analysis by TOF techniques.   Neutron Cross Section Measurements 1.E+00 DDX – 30° 1.E–01 cross section (b/MeV/sr) Lead 1.E–02 1.E–03 Silicon *0.1 1.E–04 1.E–05 Carbon *0.01 1.E–06 1.E–07 0 2 4 6 8 10 12 Emitted neutron energy (MeV) 14 16 ⊡ Figure  Neutron emission spectra at ○ from . MeV neutron bombardment of carbon (data times .), silicon (data times .), and lead measured at Osaka (Takahashi et al. ). The data are expressed as double-differential cross sections. Elastically scattered neutrons are in the peaks near  MeV. Inelastic scattering to resolved levels in carbon with excitation energy of . MeV is observed with an emitted neutron energy of about  MeV. Other resolved states are seen in carbon and silicon, and to some extent in lead Without TOF Yied/MeV (arb. units)  With TOF 60 70 80 90 En (MeV) 100 110 ⊡ Figure  The quasi-monoenergetic  MeV neutron beam distribution at the The Svedberg Laboratory (TSL) at Uppsala, Sweden, obtained from the  Li(p,n) Be reaction. The upper histogram is for all the neutrons generated. For the lower histogram, time of flight was used to reduce the low energy tail. Approximately half of the neutrons are contained in the peak Neutron Cross Section Measurements  Other facilities for neutrons in this energy range include the CYCLONE cyclotron (protons to  MeV, high mass particles up to Xe) at Louvain-la-Neuve, Belgium (Jongen and Ryckewaert ), the TIARA cyclotron (K =  AVF) at the JAEA Laboratory at Tokai-Mura, Japan (Baba et al. ; Ibaraki et al. ), the AVF cyclotron ( MeV protons,  MeV deuterons and higher masses) at Tohoku University, Japan (Baba ), and the cyclotrons at the iThemba Laboratory for accelerator based sciences (LABS) at Faure, South Africa (up to  MeV protons) (http://www.tlabs.ac.za).  Neutron Cross Sections . Introduction In the interaction between neutrons and nuclei, the kinetic energy of the neutron determines the nature of the interaction. A characteristic quantity is the reduced de Broglie wave length λ = λ/π of the neutron-nucleus center-of-mass system, defined by √ λ= ħ mE () where m is the reduced mass, E the kinetic energy in the center of mass frame, and ħ = h/π is the reduced Planck constant. If the nucleus has a mass A times that of the neutron m n , then m= A × mn . A+ () At very low neutron energies, typcially in the meV range and below, this wave length has a size of the order of the spacing between the nuclei in their material, for example, a crystalline structure. The neutron then does not see individual nuclei but interacts by scattering from the crystalline structure. Since this is a wave phenomenon one does not refer to the neutron kinetic energy, but rather to the neutron wavelength. This technique is commonly used in neutron diffraction. In the energy range between roughly  meV and  MeV, the wave length varies from the distance between the atoms to the size of a single nucleon, therefore covering approximately the same order of magnitude as the size of the nucleus. Since the electrically neutral neutron has no Coulomb barrier to overcome, and has a negligible interaction with the electrons in matter, it can directly penetrate the atomic nucleus and interact with it. For these energies, reactions often go through the formation of a compound nucleus. The compound nucleus model was introduced by Bohr () to explain the observed resonances in neutron-nucleus reactions. In this theory the formation of the compound nucleus is decoupled from its decay, such that the reaction cross section σn,x can be factored into the product of the cross section for forming the compound nucleus, σc , times the probability of decay via reaction x as shown in the equation Γx σ n,x = σ c . () Γ Here Γx /Γ, the probability of decay via reaction x, is expressed as the partial width Γx divided by the total width Γ (of the resonance). In this picture, the neutron binding energy which becomes    Neutron Cross Section Measurements available to the compound nucleus is rearranged among all nucleons, and gives rise to a complex configuration corresponding to a well-defined nuclear state with an energy, spin, and parity. Within Fermi’s description of excitations of particle-hole configurations, such a state would correspond to an extremely complicated configuration of a many particle, many hole state. At the high excitation energies above the neutron binding energies, for most nuclei the nuclear system is extremely complex and no nuclear model is capable of predicting the position and other properties of these excited states. For a heavy nucleus the level density in this region near the neutron binding energy is very high. A neighboring eigenstate can be excited by only a small change in excitation energy and may have a completely different wave function. This is a manifestation of what is also called chaotic behavior. Due to extreme configuration mixing, the nucleus in this regime above the neutron binding energy has a statistical behavior. This is expressed by the assumption that the matrix elements, relating nuclear states, have a random character, governed by a Gaussian distribution with zero mean. This statistical model of the compound nucleus is referred to as the Gaussian orthogonal ensemble (GOE) (Lynn ; Mehta ; Haq et al. ; Bohigas et al. ; Mehta ). The statistical model has direct consequences on the observables of the reaction cross sections. The channel widths are proportional to the square of the matrix elements and have, therefore, a chi-squared distribution with one degree of freedom, also called the Porter-Thomas distribution (Porter and Thomas ). The observed gamma width of a resonance is the sum of many, for heavy nuclei, several tens of thousand, individual gamma widths, and each of these widths varies over the resonances according to a a Porter-Thomas distribution. Therefore, the total radiation width for many heavy nuclei is of a similar order of magnitude. Observed fission widths correspond to a relatively small number of fission channels, at maximum three or four. The resulting distribution can be approximated by an effective chi-squared distribution with a small, fractional number of degrees of freedom. With increasing excitation energy the Γ widths of the states start to overlap and the resulting cross sections become smooth. The properties of the eigenstates, like the decay widths, fluctuating from one state to another, become apparent as values averaged over many resonances. These average values now can be predicted by nuclear models, parameterized with average properties. Measured average cross sections can therefore finetune the parametrization of these models. At even higher excitation energies, many more decay channels open up. Some reaction cross sections may only be accessible by nuclear model calculations.  > Figure  shows some typical neutron-induced cross sections for the nucleus U. In addition to the total, capture, and fission cross sections showing resolved resonances up to several keV, some threshold reactions are also shown. The cross sections for inelastic scattering leaving the nucleus in the first, second, and third excited state are shown, as well as three (n,xn) reactions with x = , , . In the same > Fig.  typical neutron spectra are shown in the energy range from − to   eV. The energy region around a few tens of meV is called the thermal region and is of importance in reactor physics where the water moderated neutrons are in thermal equilibrium with the water and have Maxwell–Boltzmann distributed velocities peaked at an equivalent kinetic energy of kT = . meV. A different energy distribution is found for neutrons in certain stars where the synthesis of the isotopes heavier than about A =  takes place (Wallerstein et al. ). The neutrons are present as a hot gas and also have a Maxwellian kinetic energy distribution in the figure shown for temperatures with kT ranging from  to  keV in Asymptotic Giant Branch stars. The velocities of neutrons from  U thermal neutron-induced fission follow in good agreement a Maxwell–Boltzmann energy distribution, peaked at about  MeV as  Cross section (barn) Neutron Cross Section Measurements 104 102 100 10 –2 10 –4 10 –6 Total Fission Capture (n,n’) (n,xn) 10 –8 Neutron flux (dn/dlnE) 10–5 10–4 10–3 10–2 10–1 100 101 102 103 104 Neutron energy (eV) 1.0 Water moderated 105 Stellar 106 107 108 109 Fission 0.5 Spallation 0.0 10–5 10–4 10–3 10–2 10–1 100 101 102 103 104 Neutron energy (eV) 105 106 107 108 109 ⊡ Figure  Typical neutron-induced cross sections (upper panel) here for the nucleus  U over an energy range from − to  eV, together with on the same scale some typical neutron energy distributions (lower panel), from fully moderated neutrons, stellar spectra at several temperatures, fission neutrons, and a typical spallation spectrum (also see text) shown in > Fig. . A typical spallation neutron spectrum in an accelerator-driven subcritical system, in this case MEGAPIE at PSI in Switzerland (Panebianco et al. ), is shown as well. But at excitation energies just above the neutron binding energy, usually just a few decay channels are open. In > Fig.  the cross sections for  U + n are shown for incident neutrons up to  eV. Only the capture, elastic scattering, and fission channels are open for these energies. The eigenstates are visible as resonances at the same energy in each of the cross sections. Since these states are not bound, the compound nucleus decays eventually through the emission of a gamma ray(s), a neutron, a charged particle, or the scission into mostly two fission fragments. The way of decay and the decay probability of the compound nucleus are considered to be independent from the way the compound nucleus was formed, but respecting conservation of energy and angular momentum. The decay probability through a decay channel c with width Γc is the branching ratio Γc /Γ (). At low energy (in non-fissile nuclei) such a channel corresponds mainly to the emission of gamma rays or a neutron. Typical widths Γ of measured resonances are in the order of electron volts. According to Heisenberg’s uncertainty principle, the corresponding life time of the compound nucleus is in the order of τ = ħ/Γ ≃ − s, several orders of magnitude larger than the typical time needed by a neutron to cross a nucleus without interaction. In > Fig.  a picture of the compound nucleus reaction is sketched. After the formation of the highly excited state by an incident neutron, the compound nucleus can decay by emission of gamma radiation, which is called radiative neutron capture, or by emission of a neutron, which is elastic scattering. If the kinetic energy   Neutron Cross Section Measurements 235 Total Elastic scattering Fission Capture 103 Cross section (b)  U+n 102 101 100 0 1 2 3 4 5 6 7 Neutron energy (eV) 8 9 10 ⊡ Figure  The neutron capture, elastic, and fission cross section for  U up to  eV, showing the resonance structure s ⊡ Figure  Schematic view of the formation and decay of a compound nucleus with the orders of magnitude of the level spacing and neutron separation energy for a heavy mass nucleus. The resonances observed in the reaction cross section correspond to the excitation of nuclear levels Neutron Cross Section Measurements  of the neutron is high enough, threshold reactions are possible, like inelastic scattering, leaving the target nucleus in an excited state. Heavier nuclei are fissionable since the nuclear potential energy becomes lower than that of the ground state at large deformation. If the nucleus has an excitation energy higher than the barrier height, the nucleus can fission. This may be the case when a compound nucleus is formed, even with a neutron with nearly zero kinetic energy. But also when the compound nucleus is in a state below the fission barrier height, fission can occur through tunneling (subthrehold fission). If the nucleus in its ground state fissions, we speak of spontaneous fission. In direct reactions, the opposite reaction mechanism to compound nucleus reactions, the incident neutron interacts directly with one or a few nucleons without forming a compound nucleus. The time scale of direct reactions is of the order of − s, a much shorter time than for compound-nucleus resonance reactions. Direct reactions become important for the heavier nuclei at neutron energies higher than about  MeV where the de Broglie wavelength of the neutron becomes comparable to the size of nucleons. But also at lower neutron energies, mainly for light A or closed shell nuclei, direct reactions may contribute significantly to the total cross section. The width of an isolated resonance in a reaction cross section has in good approximation a Breit–Wigner shape (Breit and Wigner ), which is the typical shape for any quantummechanical state with a finite lifetime. This can be derived from the time dependence of the wave function Ψ(t) of a non-stationary state with an energy E and a lifetime τ. The time dependence of this wave function is () Ψ(t) = Ψ()e −i E  t/ħ e−t/τ is observed as an exponential decay in time, like for example the familiar decay of the activity of a radioactive source. The squared absolute value of the Fourier transform of Ψ(t), and defining Γ = ħ/τ gives the energy distribution P(E) having the Breit–Wigner form P(E) ∝ Γ (E − E  ) + Γ  / () This Breit–Wigner form is present in the formulas for resonance cross sections. Also the more exact R-matrix expressions result in Breit–Wigner shapes in the limiting cases. . Total Cross Section In principle, the “simplest” cross section to measure is the total cross section using the transmission method. > Figure  shows a typical transmission method setup where a neutron beam impinges on a detector and a sample of the nuclei of interest is cycled in and out of the beam. The neutron transmission T is given by T= sample in counting rate = e−N σ t sample out counting rate () where N is the areal sample thickness (in units of atoms/barn) and σ t is the neutron total cross section (in units of barns, where  barn equals − cm ). It should be noted that this measurement does not require accurate knowledge of the neutron flux or the detector efficiency. It    Neutron Cross Section Measurements n Neutron source Evacuated flight tube Sample Evacuated flight tube Neutron detector Flight length ⊡ Figure  The principle of the transmission experiment setup. The neutron detector is located far from the sample in order to detect only neutrons that have not undergone an interaction in the sample (sample in position). The neutron flux incident on the sample is measured by removing the sample from the beam (sample out position) only requires a neutron source that is constant in time or one that can be normalized for equal numbers of neutrons for the sample-in and sample-out runs. Atoms inside the target are in thermal motion so that the measured total cross section σ t in () is actually a Doppler broadened cross section. Thus, the transmission at neutron energy E is ′ ′ ′ T(E) = exp (−N∫ σ t (E )PE (E , t)dE ) () where T(E) is the transmission at incident neutron energy E; PE (E ′ , t) is the probability, the result of thermal motion of the target atoms at temperature t, that the interaction takes place with energy E ′ ; and the integral is over all energies. The integral is called the Doppler-broadened total cross section σΔ,t . For some materials like metals, the free gas model (Lamb ) with an effective temperature T describes in good approximation Doppler broadening by a Gaussian with a standard deviation √ √ Mm kT E σD = kT E ≃ ()  (M + m) A with M the mass of the target nucleus, m that of the incident neutron, and A the atomic number, close to M/m for medium and √ heavy nuclei. Note that in the literature we often see the Doppler width Δ D defined as Δ D = σ D . For other materials, like poly-atomic crystalline lattices, a more complicated description is sometimes needed (Meister ; Naberejnev et al. ; Dagan ). The Doppler broadening effect is symmetric and will, therefore, not result in an apparent shift of the resonance energy. In most measurements the incident neutrons are not at energy E but rather have a distribution of energy about E. Thus, the measured transmission Tm at energy E and temperature t is Tm (E, t) = ∫ R(E, E ′′)T(E ′′)dE ′′ ′′ ′ ′ ′ = ∫ R(E, E ) exp (−N∫ σ t (E )PE ′′ (E , t)dE ) dE ′′ () where R(E, E ′′) is the resolution function and is the probability that at nominal energy E the neutron has energy E ′′ . Both integrals are taken over all energies.  Neutron Cross Section Measurements Detector Reation product n Neutron source Evacuated flight tube Sample Flight length ⊡ Figure  The principle of the reaction experiment setup. The detector for the reaction products may be in the beam together with the sample, for example, for (n,f) or (n,α) experiments, or outside the beam, like for capture or scattering reactions. The off-beam detector can be close to π configuration or cover a smaller solid angle . Partial Cross Section The total cross section is the sum of all the partial cross sections such as scattering cross section, capture cross section, fission cross section, etc. A typical partial cross section experiment is shown in > Fig.  where a beam of neutrons is incident upon a sample and the reactions of interest are observed in a detector. The yield for reaction x is defined as Yx = number of x-reactions number of incident neutrons () Note that Yx is the number of occurred reactions, not the number of detected reactions which depend largely on experimental factors as the detector efficiency and the solid angle covered. The primary yield Yp is defined as the yield for first interactions. It is defined as the ratio of the reaction to total cross section σ x /σ t times the fraction of neutrons ( − T) inducing a reaction. Therefore, σx σx = [ − e−N σ t ] () σt σt where σx is the partial cross section. For simplicity, Doppler effects and resolution broadening are ignored in (). The term within the brackets is the fraction of the incident beam, which interacts in the sample and the ratio of cross sections is the fraction of interactions which are of type x. In addition to the primary yield there is multiple scattering in the sample which produces additional x reactions in the sample. Thus, the total yield Yx is Yx, p = [ − T] Yx = Yx, p + Yx,m () where Yx,m is the result of multiple scattering and subsequent reactions of type x in the sample. For thin samples where N σ t ≪ , multiple scattering becomes negligible and Yx, p can be expanded as Yx = [ −  + N σ t − (N σ t ) /! + (N σ t ) /! + ⋯] σx ≈ N σx σt ()    Neutron Cross Section Measurements or Yx () N Thus, for thin samples the partial cross section is equal to the partial yield divided by the areal sample thickness. For fission measurements, the sample can be very thin and the detector counts the fission reactions. For this case, the measured partial cross section of type x is merely the resolution broadened cross section. Applying both Doppler and resolution broadening, one obtains for thin samples the measured cross section at nominal energy E and temperature t σx ≈ ′′ ′ ′ ′ σ m (E, t) = ∫ R(E, E ) [∫ σ x (E )PE ′′ (E , t)dE ] dE . ′′ () Resonance Cross Section Neutron cross sections exhibit many resonances in the energy region above thermal energies. The cross section of an isolated resonance can be described by the single-level Breit–Wigner formula. At low energies where the neutron has zero orbital angular momentum (s-wave), the partial reaction cross section of an isolated s-wave resonance is given by σ x (E) = πλ g Γn Γx (E − E  ) + (Γ/) () where E is the neutron energy (in the center of mass system), λ is the reduced neutron wavelength, g is the statistical weight factor, Γn is the neutron width, Γx is the reaction width, Γ is the total width (full width at half maximum), and E  is the neutron energy at the peak of the resonance. The statistical weight factor g is given by g= J +  (I + ) () where J is the total angular momentum (resonance spin) of the compound nucleus and I is the total angular momentum (spin) of the target nucleus. The neutron width Γn is energy dependent and can be represented by √ E () Γn (E) = Γn (E  ) E Equation () can be transformed by dividing numerator and denominator by (Γ/) and by introducing the peak cross section at resonance to become Γx σx (E) = σ Γ √  E ) ( E  + y () Γn (E  ) Γ () where the peak cross section σ is given by  σ = πλ g Neutron Cross Section Measurements  Γn (E  ) is the neutron width at the resonance energy, λ is the reduced neutron wavelength at E  , and y is given by E − E y= () Γ/ The total cross section is the sum of the scattering cross section and the sum of all the partial cross sections. The total cross section for an isolated s-wave resonance is Γn Γ + Γn (E − E  )R/λ  + πR (E − E  ) + (Γ/) √ √ y R  E E = σ ) + σ ) ( ( + πR  E  + y E  + y λ  σ t (E) = πλ g () where the last term is the potential scattering cross section and R is the potential scattering radius. The second term represents the resonance-potential interference and, in some cases, leads to almost zero total cross section. Thus, the neutron cross section of an isolated resonance can be calculated from a set of parameters termed “resonance parameters.” These parameters are the resonance energy E, total width Γ, neutron width Γn , reaction width(s) Γx , total angular momentum J, target nucleus spin I, and the orbital angular momentum of the neutron. For non-fissile resonances the only reaction width is the radiative width Γγ . From these parameters and the Debye temperature of the target nucleus the Doppler-broadened cross section can be calculated. To account for nearby or overlapping resonances the Breit–Wigner formula is no longer valid and an R-matrix formalism, explained in more detail in > Sect. , is used to calculate the neutron cross sections. However, the same resonance parameters are used for these calculations. Thus, the measured transmission and/or partial cross sections which include the effects of Doppler broadening and resolution broadening can still be interpreted in terms of these resonance parameters. Originally, the Breit–Wigner formalism was used to fit the measured transmission and partial cross sections to obtain these parameters. Today, there are two major codes, REFIT (Moxon and Brisland ) and SAMMY (Larson ) based on the R-matrix formalism, which can extract resonance parameters from the measured data. These programs calculate the Doppler-broadened cross section, apply resolution broadening to simulate the observed transmission and partial yield data, and then determine the resonance parameters that provide the best fit to all of the measured data. It is these parameters that are used to calculate neutron cross sections for reactor and other applications. . High Energy Cross Section With increasing neutron energy the average spacing between levels ⟨D⟩ decreases (the result of level density ρ(E) increasing with energy) while the average resonance width ⟨Γ⟩ increases. In the energy region where ⟨Γ⟩ is comparable to ⟨D⟩ the cross section exhibits structure caused by unresolved clusters of partially overlapping resonances. This is a very difficult region to express analytically, so statistical representations based on average parameters are typically used in neutronic calculations. At higher energies where ⟨Γ⟩ is much larger than ⟨D⟩, the cross section is in a continuum region and is typically represented by optical and collective model expressions based on a Hauser–Feshbach description with width fluctuation corrections (see also > Sect. .).   Neutron Cross Section Measurements 106 Resonances, R-matrix 105 Total cross section (b)  Thermal RRR URR Optical model 104 103 102 101 100 –4 10 10–3 10–2 10–1 100 101 102 103 Neutron energy (eV) 104 105 106 107 ⊡ Figure  The ENDF/B-VII. evaluated total cross section of  Au  Cross Section Measurements Cross sections of interest to nuclear engineering cover the range from subthermal energies in the millielectron volts to tens of millions of electron volts. A typical example of a cross section is shown in > Fig.  where the ENDF/B evaluated total cross section of Au is plotted (Chadwick et al. ). For convenience this large energy span can be divided into four energy ranges: thermal, resolved resonance, unresolved resonance and continuum energy regions. In this figure the thermal region extends up to ∼. eV and the resolved resonance region falls between approximately . eV and . keV. Note that above . keV the cross section is represented as a smooth curve. However, immediately above . keV the cross section is still dominated by resonance structure but is considered “unresolved.” This structure is considered unresolved because the experimental resolutions could not resolve individual resonances and/or, even if the experimental resolution were “perfect” and truly resolved all the structure in the cross section, the resonances partially overlap and cannot be clearly distinguished from each other. Even though this unresolved resonance region is represented in the evaluation as a smooth average cross section, the cross section fluctuations about the average result in fluctuations in self-shielding and must be taken into account in neutronic calculations in this energy region. In the MeV energy region the Au resonances have overlapped to the extent that the cross section is now a smooth continuum and here, we only see very wide optical diffraction peaks in the cross section above ∼ MeV; this is the continuum energy region. The boundaries between the different regions, thermal, resolved resonances, unresolved resonances, and continuum, are different for each nucleus and depend on the level density of the Neutron Cross Section Measurements  104 6Li 100 102 27Al 100 103 55Mn 100 Total cross section (b) 103 100 104 107Ag 197Au 100 101 208Pb 100 104 235U 1004 10 241Am 100 10–2 10–1 100 101 102 103 104 Neutron energy (eV) 105 106 107 ⊡ Figure  The neutron total cross section of several nuclei showing large differences in the resonance spacings compound nucleus. In fact, resolved neutron resonances are an important source to obtain level density information. In > Fig.  the neutron-induced total cross section is shown for several nuclei of increasing mass. Since the level density increases with mass, resonances are more closely spaced for heavier nuclei. But near a double closed shell nucleus like  Pb, the level density is much lower, and we observe a much higher level spacing. Also nuclei with neighboring masses may show large differences in the level densitiy due to shell effects. As a result of the different structure in each of these energy regions, the cross section measurement techniques in each region can be quite different. These techniques are reviewed below.    . Neutron Cross Section Measurements Thermal Energy Region This region typically extends from a fraction of an eV down into the meV region. The cross section here is equal to the sum of the scattering cross section and the absorption cross section. The absorption cross section typically varies as /v where v is the velocity of the neutron. This can √ be seen from () and (), where at low energies the cross sections are proportional to / E (or /v); this is also seen in the low energy  Au total cross section shown in > Fig. . Some “famous” absorption cross sections at . eV, which corresponds to the maximum at , m/s in the Maxwell–Boltzmann distribution of the neutron velocities at a temperature of . K, are , barns for the (n,α) reaction in  B, , barns for the (n,γ) reaction in nat Cd, and  barns for the (n,f) reaction in  U. The scattering cross section can be more complex as solid state effects take place. Neutrons can undergo Bragg scattering by any crystalline material present, undergo paramagnetic scattering with the target nuclei and neutrons can be up-scattered or down-scattered in energy from the thermal motion of the nuclei they interact with. Thermal reactors have been used for many of the early absorption cross section measurements. Samples of the material of interest were placed inside the thermal spectrum where the induced radioactivity in the samples or the change in reactivity of the reactor were interpreted in terms of the sample absorption cross section. Reactors were used to provide monoenergetic neutron beams using crystal spectrometers or velocity selectors for cross section measurements. Fast choppers were used to produce pulses of neutrons for TOF measurements. More recently, bent tubes have been used to provide extremely pure neutron beams in the millielectron volt range for precise sub-thermal cross section measurements. For the past  years the energy-dependent cross sections in the thermal region have been measured predominantly by TOF techniques using pulsed accelerators. This technique provides good resolution well into the resolved resonance region so that in a single measurement, both thermal and resonance regions data can be obtained. Since the thermal absorption cross sections of most materials is the result of the tails of resonances, the accelerator time-of-flight method provides data over the whole energy region which contributes to the thermal cross section. An example of a measurement of the total cross in the thermal region is shown in > Fig.  for  Ho (Danon et al. ). The TOF transmission method was used for the RPI measurement. A paramagnetic scattering cross section of . barns at . eV was calculated for  Ho and this had to be accounted for in analyzing  Ho for both thermal absorption cross section and total cross section. .. Thermal Flux Averaged Cross Section The flux averaged cross section σ = ∫ σ(E)φ E (E)dE of a reaction is a useful quantity for irradiation experiments. If the cross section has a smooth shape of the form σ(E) = ( E α ) σr Er () where σr is the cross section at energy E r , then we can calculate the flux averaged cross section as σ(E)ϕ(E)dE kT α = ( ) σr Γ(α + ) σ=∫ () Er ∫ ϕ(E)dE Neutron Cross Section Measurements  Total cross section (barns) 103 RPI Zimmerman Schermer Knorr 102 101 10–4 10–3 10–2 Energy (eV) 10–1 100 ⊡ Figure  Total cross section of  Ho in the thermal energy region where Γ(x) is here the gamma function. This is particulary useful at low, thermal energies where −/ the cross section has a /v shape, so σ(E n ) ∝ E n , that is, α = −/. The flux averaged cross section becomes then √ √ π Er σ= () σr  kT In this case we often choose E r such that we have the cross section at v = ,  m/s, that is, at E r = . meV. . Resonance Energy Region This is the region in which resonances can be resolved by the technique being used for the measurement. This energy typically spans an energy region from ∼ eV to several MeV for light nuclei like Be and C, into the s of keV for the structural materials like Fe and Ni, into the keV region for rare earth nuclei like Ho or Gd and into the s of eV to keV region for heavy nuclei like U. It is in the resonance region where the TOF measurements at pulsed accelerators have made the major contributions. Each resonance represents an excited state in the compound nucleus formed by the incident neutron. The focus of these measurements is the extraction of resonance parameters both for understanding the underlying science of nuclear levels and for the practical need of being able to calculate cross sections at any operating temperature (such as in a reactor). Typical resonance parameters are the resonance energy E  , resonance spin J, orbital angular momentum of the incoming neutron ℓ, resonance total width Γ, the always present neutron width Γn , and, if applicable, other, partial widths like the radiative width Γγ or the fission width Γ f .    Neutron Cross Section Measurements ⊡ Figure  Capture yield and transmission of metallic Hf samples near the . eV resonance in  Hf ⊡ Figure  Cutaway view of the -section NaI multiplicity detector. A capture sample is placed in the center of the detector and the neutron beam is collimated along the detector axis An example of data from a TOF measurement optimized to measure resonance parameters is shown in > Fig. . These data were obtained from low energy capture and transmission measurements at the Rensselaer Polytechnic Institute LINAC with elemental Hf (Trbovich et al. ). A high-efficiency calorimeter-type detector was used for the capture measurements and is shown in > Fig. . A  Li glass scintillator detector was used for the transmission measurements. Here, both transmission and capture measurements were performed on many sample thicknesses, and the Bayesian-based SAMMY code (Larson ) was used to obtain a single set of parameters to fit all the data. In this region the resonances are reasonably well separated Neutron Cross Section Measurements  1.0 Transmission 0.9 0.8 0.508–mm Data 0.889–mm Data 1.27–mm Data 2.54–mm Data 5.08–mm Data 1.02–cm Data RPI ENDF 1.02–cm 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 –0.1 200 205 210 215 220 225 Energy, (eV) 0.7 0.051– mm Data 0.127– mm Data 2.54 – mm Data RPI ENDF 1.127– mm 0.6 Capture yield 0.5 0.4 0.3 0.2 0.1 0.0 200 205 210 215 220 225 Energy, (eV) ⊡ Figure  Transmission and capture yield from multiple samples of elemental Gd and it is relatively easy to obtain a good fit. In the example of a higher-energy measurement, as shown in > Fig.  for elemental Gd (Leinweber et al. ), the resonances are beginning to overlap and it is more difficult to make the fit. Eventually, the resonances overlap so much that it becomes very difficult to resolve them from each other. In this case better TOF resolution and/or the use of separated isotopes can extend the energy range for clearer separation of resonances and ultimate determination of resonance parameters.  > Figure  shows an example of a capture measurement of isotopically-enriched Zr at the n_TOF facility at CERN (Tagliente et al. ). This measurement spans the energy range from  eV to  keV and shows the transition from clearly separated  Zr resonances at lower energies to the keV region where the resonances are only partially resolved. These data were taken with a C D scintillator detector that was designed with minimum absorbing material to reduce its sensitivity to neutrons scattered by the capture sample. Resonance parameters were determined up to  keV; above this energy the fluctuations in the yield clearly show the effects of unresolved resonances. . Unresolved Resonance and Continuum Energy Region An example of the change from the unresolved to continuum energy region is shown by the recently measured (at RPI) total cross section of elemental Zr above . MeV shown in > Fig. . Here, we clearly see the effects of unresolved resonance structure up to ∼ MeV. Above ∼ MeV are seen the effect of nuclear optical structure with a very broad peak near  MeV;   Neutron Cross Section Measurements Capture yield 10 –1 10 –2 10 –3 10 –4 0 1 2 3 log 10(En(eV)) 4 5 2 2.5 3 3.5 log 10(En(eV)) 4 4.5 10 –1 Capture yield  10 –2 10 –3 10 –4 ⊡ Figure  Capture yield (black) and background (gray) of  Zr between  eV and  keV measured at n_TOF (Tagliente et al. ) this is clearly in the continuum energy region. This type of structure below  MeV can be interpreted in terms of the fluctuations in the observed cross section; an excellent example of this type of analysis has been applied to total cross section measurements of  P,  S,  K,  V,  Fe, and  Co between . and . MeV (Abfalterer et al. ) to obtain average level densities and resonance widths as a function of excitation energy. To represent the cross section in the unresolved resonance region for neutronic applications, two distinct approaches have been successful: ladders of pseudo resonances and probability lookup tables. Both of these approaches select resonances stochastically from a level spacing and a level strength distribution, which are based on the average spacings and strengths extrapolated from the resolved resonance region. In the ladder, case a set of parameters is stochastically selected over a given energy range and a pseudo cross section is calculated from these parameters. The pseudo cross section is then used in neutronic calculations over this energy range. In the probability lookup case a pseudo cross is calculated over a given energy range and the probability distribution of a given value of cross section is determined over this energy range. For neutronic calculations, the probability of a given cross section is stochastically sampled from the probability distribution. Neutron Cross Section Measurements  ⊡ Figure  The measured and evaluated total cross section of nat Zr from . to  MeV . The Neutron Time of Flight Method The principle of neutron time of flight measurements is based on the pulsed neutron source which produces at a time t  neutrons in a wide energy range. For reaction measurements a sample is put in the neutron beam at a well know distance L and the reactions are observed with a detector. The detection of the reaction determines the time of arrival t n of the neutron at the sample and therefore its velocity v = (t n − t  )/L, which gives the kinetic energy of the neutron. The idea is illustrated in > Fig. . Accelerator-based pulsed white neutron sources are in general either electron-based sources where the neutrons are produced via Bremsstrahlung on a high Z target, or proton-based sources where the neutrons are produced by spallation on a target of heavy nuclei. The neutrons created by the pulsed source travel along the flight path with length L during a time t before possibly undergoing a reaction in a capture, scattering or fission setup, or before getting detected in a transmission experiment (see > Fig. ). The neutron kinetic energy is determined relativistically from the neutron velocity v = L/t and momentum p = mv as E n = E tot − mc  = √ c  p + m  c  − mc  = mc  (γ − ) ⊡ Figure  The principle of the neutron time-of-flight method ()    Neutron Cross Section Measurements with γ = ( − v  /c  )−/ and where c is the speed of light. For low energies where v ≪ c and therefore γ → , the relativistic expression can be conveniently written as a series expansion of γ γ =+  v  v   v  v  + (  ) + (  ) + O(  )  c  c  c c () For resolved resonances the first two terms of the series expansion of the relativistic expression are usually sufficient, which results in the classical definition of kinetic energy    L E n ≈ mv = α  .  t ()  Taking the definition of the speed of light √ c = , ,  m/s and taking m = . MeV/c for the neutron mass, we get α = . µs eV/m when using units eV, m, and µs for E n , L, and t respectively. See also, for example, Foderaro (), Krane (), Knoll (), and Reuss () for a wider discussion on neutron physics and associated instrumentation. .. Neutron Density and Flux Distributions at Thermal Energies Fully thermalized neutrons behave like an ideal gas and can, therefore, be described in good approximation by Maxwell–Boltzmann statistics. The neutron density, that is, the number of neutrons per unit of volume, has a Maxwell–Boltzmann velocity distribution n v (v) of the form nv (v) = π ( m /  mv  ) v exp ( − ) πkT kT () where k is the Boltzmann constant, m the neutron √mass, and T the temperature. This distribution shows a maximum at v = kT/m corresponding to a kinetic energy of Emax = kT. For a velocity of , m/s, used as a reference for thermal neutrons, this gives Emax = . meV, λ = . nm, and T = mv  /k = . K, which is practically room temperature. From the velocity distribution nv (v) we can obtain the distribution n E (E) of the kinetic energy E, the distribution n t (t) of the time-of-flight t, and the distribution n λ (λ) of the wavelength λ using the relations nv (v)dv = n E (E)dE = n t (t)dt = n λ (λ)d λ () where dv, dE, dt, and d λ are obtained from taking the derivates of the expressions   L  E = mv = m    t and λ= h ht = mv mL () () Neutron Cross Section Measurements  One is often interested in the distribution of the neutron flux. The velocity distribution of the neutron flux φv (v) is related to nv (v) as φv (v) = Cv × v × nv (v) () where C v is the appropriate normalization constant to normalize the integral to unity √ ∫ φv (v)dv =  (Cv = kT/(πm) in this case). In a similar way as in () the kinetic energy, time-of-flight, and wavelength distributions of the neutron flux can be obtained using φv (v)dv = φ E (E)dE = φ t (t)dt = φ λ (λ)d λ () Applying this and by introducing the characteristic variables √ vT = kT m E T = kT √ m tT = L kT √ h λT = mkT () () () () we obtain the expressions for the neutron density and neutron flux distributions as given in > Table . This table is similar to the one in (Molnar ). Here, we also add the time-offlight distributions which are analogous to the wavelength distributions with the characteristic variables. Note from the equivalent maximum values that the energy distribution of the flux has a maximum at E = E T = kT and that the time-of-flight distribution, which is typically measured at a TOF facility, shows the maximum at flight time which corresponds to an energy of E = (/)kT. So a room temperature water moderated neutron source with a thermal peak of  meV has a peak in a time-of-flight spectrum at  meV. . Surrogate Reactions A different approach to study neutron-induced reactions in some cases is to use surrogate reactions. This technique, developed by Cramer and Britt () uses light charged particles, typically, but not only deuterons, tritons,  He or α particles, to form a compound nucleus by a few-nucleon transfer reaction. The target nucleus and charged particle is selected in such a way that the same compound nucleus is formed as in the neutron-induced reaction. In this way, the decay of the compound nucleus for a particular decay channel can be measured. The formation of the compound nucleus for the neutron-induced reaction has to be calculated using optical model calculations. The reactions are usually described using Hauser–Feshbach calculations, which in some energy ranges can be done in the Weisskopf–Ewing approximation where spin and orbital angular momentum are ignored (Weisskopf and Ewing ; Ait-Tahar and Hodgson ).    Neutron Cross Section Measurements ⊡ Table  Neutron density and flux distributions for Maxwell–Boltzmann distributed neutrons at temperature T, as a function of velocity v, energy E, wavelength λ, and time-of-flight t Most probable value Equivalent in ET v  v − v nv (v) = √  e T π vT vmax = vT ET  < v >= √ vT π E  −    E e ET nE (E) = √  ET π  Emax = ET   ET   < E >= ET   tmax = √ tT  ET  < t >= √ tT π  λ max = √ λ T  ET  < λ >= √ λ T π Distribution Mean value  tT  tT −  nt (t) = √  e t πt λ T  λ T −  n λ (λ) = √  e λ πλ v v − v φv (v) =   e T vT E E −E φE (E) =  e T ET tT tT −  φt (t) =   e t t λ T λ T −  φ λ (λ) =   e λ λ √ √  ET  < φv >= ET < φE >= ET tmax =  ET  < φt >= λ max  ET  < φ λ >= vmax =  vT  Emax = ET √  tT  √  λT =   vT  √ π tT  √ π λT  The method has gained renewed interest by several recent measurements, especially for fission and capture reactions. In > Table  a few examples of surrogate reactions and their equivalent neutron-induced reaction, are given. Because of the different involved spins and orbital momenta, in some cases a different spinparity distribution may influence the decay probabilities if the same compound nucleus has been formed by a neutron or by a charged particle. This effect can often be neglected (Younes and Britt ; Escher and Dietrich ). More details and references may be found in refs (Petit et al. ; Plettner et al. ; Boyer et al. ; Lyles et al. ; Hatarik et al. ). In > Fig.  the surrogate reaction  Th( He,p) Pa* is shown. From the measured gamma decay the  Pa(n,γ) cross section is derived and compared to evaluated data (Boyer et al. ). The surrogate ratio  U(α,α’f)/ U(α,α’f) is shown in > Fig. . These reactions are well above their thresholds and are in a range where the Weisskopf-Ewing approximation to a Hauser-Feshbach calculation is expected to represent the reactions rather well. The comparison of the known  U(n,γ) and (n,f) cross section ratio as evaluated in ENDF/B-VII to that measured in the surrogate reactions  U(d,pγ) and  U(d,p fission)  Neutron Cross Section Measurements ⊡ Table  Some examples of surrogate reactions for actinides and their equivalent neutron induced reaction, where the asterisk denotes a compound nucleus Neutron induced reaction Surrogate reaction  U+n→ U*  U(t,p) U*  U + n →  U*  U(d,pγ) U*  U + n →  U*  U(α,α’) U*  Pa + n →  Pa*  Th( He,p) Pa* Am + n →  Am*  Am( He,α) Am* Cm + n →  Am( He,d) Cm*     Cm* 2.5 σ (n,γ) (barns) Petit et al. endf 6.8 jendl 3.3 Our results 2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 Neutron energy (Mev) 0.8 1.0 ⊡ Figure  The  Pa(n,γ) cross section measured with the surrogate reaction  Th( He,p) Pa* (Data are from Boyer et al. ) (Allmond et al. ) is shown in > Fig. . This comparison shows the data obtained in surrogate reactions at equivalent low incident neutron energies. In > Fig.  the  U(n,f) cross section as determined by the surrogate ratio method is compared with values from the ENDF/B-VII evaluated data (Lyles et al. ). This comparison shows that the surrogate method does well for fission cross sections a few MeV above threshold.   Neutron Cross Section Measurements 2.5 233U(n,f)/ 235U(n,f) from ENDF-B7 234U(a,a¢f)/ 236U(a,a¢f) 2 Fission ratio Ratio from STARS 1.5 1 0.5 0 7 12 17 Excitation energy (MeV) 22 ⊡ Figure  Results of the surrogate ratio  U(α,α’f)/ U(α,α’f) compared to the ENDF-BVII ratio of U(n,f)/ U(n,f) (Burke et al. )  Contains portion of En < 0 0.3 0.25 P(d,pg) s(n,g) VS. P(d,pf) s(n,f)  lnner bar = sys.err. Outer bar = tot.err. 0.2 0.15 0.1 0.05 0 0 0.5 1.0 1.5 2.0 En (MeV) 2.5 3.0 3.5 ⊡ Figure  Comparison of the known  U(n,γ) and (n,f) cross section ratio as evaluated in ENDF/B-VII to that measured in the surrogate reactions,  U(d,pγ) and  U(d,p fission) (Allmond et al. ) Neutron Cross Section Measurements  2 s(n,f)(b) 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 Equivalent neutron energy (MeV) 18 20 ⊡ Figure  The  U(n,f) cross section as determined by the surrogate ratio method (Lyles et al. ) is compared with values from the ENDF/B-VII evaluated data . Cross Section Standards Many cross sections are measured relative to well understood, well measured, “standard” cross sections (Carlson et al. ). If the standard is measured along with the reaction of interest, then there is no need to measure the neutron flux. The particular reactions that are considered standards have been agreed to internationally and the standards data are maintained by the Nuclear Data Section of the International Atomic Energy (http://www.nds.org/standards/). Standards data include scattering, capture, and fission cross sections. In addition, for activation, there is the International Reactor Dosimetry File with recommended cross sections (not officially “standards”) (http://www-pub.iaea.org/MTCD/publications/TRS_web.pdf).  Nuclear Resonances and the R-Matrix Formalism . Introduction If the wave functions of the nuclear system before and after the reaction were known, one could calculate the cross section with the usual concepts of reaction theory. While the incoming waves are known, the reaction modifies the outgoing wave functions in a generally unknown way. The idea behind the R-matrix formalism is to use the wave function of the nuclear system of two particles when they are so close that they form a compound nucleus. Although the wave function of the compound nucleus is extremely complicated, one can expand it in its eigenstates. Matching, then, the incoming and outgoing waves to the internal wave function provides a way to describe the cross section of the reaction in terms of the properties of the eigenstates of the compound nucleus. These properties are basically the energy, spin, parity, and a set of partial widths related to the widths of the decay modes of the compound nucleus.    Neutron Cross Section Measurements This method of describing a reaction cross section using only the properties of nuclear excitation levels, is at the same time also the most important limitation. No information of the forces inside the nucleus is needed or can be extracted. The nucleus is treated as a black box of which the properties of the eigenstates have to be measured in order to describe the cross sections. The binary nuclear reactions proceeding from one system of two particles to another system of two particles can be described with the general R-matrix theory. Not only in neutroninduced reactions, but also in other cases, such a reaction goes often through the formation of a compound nucleus X ∗ . A + a → X∗ → B + b () The R-matrix formalism does not only apply to compound nucleus reactions. Both direct and indirect reactions can be described with it. The inclusion of the Coulomb interaction allows us to use it also for charged particle reactions. But the theory is applicable only in a general way for binary reactions, which is appropriate for neutron-induced reactions up to energies of several tens of MeV. In a very general way, the cross section of a two-body nuclear reaction could be calculated if the nuclear wave functions were known. The wave functions could be calculated by solving the Schrödinger equation for the nuclear system. This requires that the nuclear potential is known. When the two particles are far away, the interaction can be considered absent for neutral particles or to be the Coulomb interaction for charged particles. In these cases it is indeed possible to calculate the wave functions. When the two particles are so close to each other that a nuclear reaction takes place, the potential of the interaction is extremely complicated. For certain energy ranges and reactions this potential can still be approximated or calculated (Bauge et al. ) and the wave functions and cross sections can be calculated. In other cases however, and especially in the resolved resonance region, the complexity of the reacting system does not allow this. The first step is to consider that the reaction process can be split up geometrically into two regions for each channel where a channel is the precise constellation of particles and their spins. If the separation is smaller than the channel radius a c , all nucleons involved in the reaction are close to each other and form a compound nucleus. Although the wave function of the compound nucleus is extremely complicated, it can be expanded as a linear combination of its eigenstates without solving explicitly the Schrödinger equation of the system. In the external region, at distances larger than a c , the potential is zero for neutral particles or is the Coulomb interaction for charged particles and the Schrödinger equation of the system can be solved. The properties of the eigenstates of the compound nucleus are included in the R-matrix. Equating the values and derivatives of the wave functions at the boundary of the internal and external region assures a smooth wave function and the cross sections can be calculated as is illustrated in > Fig. . The exact internal wave function is not needed, only the values and derivatives at the nuclear surface. In the following we describe in more detail the R-matrix formalism which links the properties of the nuclear states to the cross sections. The cross section in the thermal energy region is also described by the R-matrix formalism. Reaction cross sections at thermal energy are the sum of the contributions of all nuclear states, that is, the resonances but also the bound states, sometimes referred to as “negative energy” resonances. Other reaction formalisms have been used in the past, like the K-matrix formalism (Payne and Schlessinger ) still in use for particle physics (Shyam and Scholten ), but for neutron-induced resonance reactions the Neutron Cross Section Measurements c  r (r) internal external region match value and derivative r 0 ac ⊡ Figure  Schematic view of the wave function of a channel as a function of the separation distance r. The wave function in the internal region r < ac is an expansion of the eigenstates of the compound nucleus. The full internal (r < ac ) wave function is not needed, only the value and derivative at r = ac where it matches the known external (r > ac ) wave function which is related to the Bessel functions R-matrix formalism, and in particular one of its approximations, is nowadays the preferred formalism. The R-matrix formalism was first introduced by Wigner and Eisenbud (). A most extensive and detailed overview has been given by Lane and Thomas () and by Lynn (). Recently Fröhner () summarized the R-matrix formalism together with other useful considerations on nuclear data evaluation. Other related references of interest can be found elsewhere (Humblet and Rosenfeld ; Vogt ; Schmittroth and Tobocman ; Foderaro ; Adler ; Feshbach ; Luk’yanov and Yaneva ; Brune ). A brief outline of the formalism will be given in order to understand its basic principles. .. Channel Representation It is customary to use the concept of channels in the description of nuclear reactions, which will be limited to two particle reactions in the following. The entrance channel c consists of a particular initial constellation of particles and all the quantum numbers necessary to describe the corresponding partial wave function. The type of the two particles α  and α  , with their spins I α  and I α  , and their states of internal excitation are denoted by α. Four quantum numbers are needed to include the spins of the particles in a channel. The most appropriate combination is the orbital angular momentum ℓ, the channel spin j, which is the combined spin of the two particles j = I α  + Iα  , () J=j+ℓ () the total angular momentum J    Neutron Cross Section Measurements and its projection on the z-axis m J . So the entrance channel c can be designated by the set c = {α, ℓ, j, J, m J} () ′ ′ ′ ′ ′ ′ c = {α , ℓ , j , J , m J } () Similarly, the exit channel is given by The reaction α → α ′ may go through the formation of a compound nucleus, like often the case with neutron-induced reactions. The reaction can then be written as α → A∗ → α ′ . The spin and parity are of course conserved in all stages of the reaction and the compound nucleus has its defined spin J and parity π. The conservation of spin and parity puts restrictions on the entrance channels that are open to form the compound nucleus or the exit channels open for the decay of the compound nucleus. For neutrons and protons the intrinsic spin is / and the intrinsic parity is positive. Conservation of angular momentum gives the vector addition: J = I α  + Iα  + ℓ = Iα ′ + Iα ′ + ℓ ′ () and conservation of parity gives, using + for positive and − for negative parity: π = π I α  × π I α  × (−)ℓ = π I α ′ × π I α ′ × (−)ℓ  ′ ()  The conservation of angular momentum has important consequences for cross section calculations based on channels. The total number of possible combinations to sum the spins and orbital momentum is (I α  + )(I α  + )(ℓ + ). Only J +  orientations of them add up to J. For this reason in expressions for cross sections of the formation of a compound nucleus level with spin J for a given ℓ the statistical factor g(J) g(J) = J +  (I α  + )(I α  + ) () is taken into account. The boundary r = a c is the limit between the internal region, where all the nucleons interact, and the external region where the incident and target particles do not have a nuclear interaction, other than possibly a Coulomb interaction. Although there is no sharp limit, in practice the channel radius a c can be taken just slightly larger than the radius R ′ = R  A/ of a spherical nuclear volume with A = A α  + A α  nucleons, and where for R  usually the value . fm is used. This scattering radius can be used as a first approximation of the low-energy potential scattering cross section σpot with the relation σpot = πR ′ () Experimental values of R ′ show larger structures around the smooth curve R ′ = R  A/ , which can be well described with optical model calculations. In evaluated nuclear libraries, the channel radius a c can be defined to have either the numerical value of a possibly energy-dependent scattering radius R ′ , or an energy-independent, mass-dependent channel radius given by a c = . + .A′/ fm () Neutron Cross Section Measurements  where A′ is the ratio of the isotope mass to the mass of the neutron. The channel is defined in the center of mass and the reduced mass of the particles is mc = mα = mα mα mα + mα () and the wave number k, related to the de Broglie wavelength λ, is  kc = kα = = λc √ m α E α ħ () and the relative velocity is v c = v α = ħk c /m c () The dimensionless distance ρ c is used to indicate the distance r c in measures of de Broglie wavelengths. () ρc = ρα = kc rc .. The Wave Function in the External Region The system of the two particles interacting through a central potential V(r) can be described by the Schrödinger equation of the motion of the reduced mass particle. Also, using spherical coordinates, the solution ψ(r, θ, ϕ) can, in case of a central potential, be separated in a radial and an angular part ψ(r, θ, ϕ) = R(r)Θ(θ)Φ(ϕ) () The radial part R(r) although still depends on the nonnegative integer solutions ℓ(ℓ + ) of Θ(θ). The integers appearing in the solution of Φ(ϕ) are m ℓ = , ±, ±, . . . , ±ℓ. The solutions of the angular part Θ(θ)Φ(ϕ) do not depend on the central potential and are the spherical harmonics Ymℓ ℓ (θ, ϕ). Only the solution R(r) of the radial part depends on the potential V(r). The radial Schrödinger equation [ ℓ(ℓ + ) m c d − −  (V(r) − E)] R(r) =  dr  r ħ () can be solved for the case of the Coulomb potential V(r) = −Z α  Z α  e  /(πє  r). The general solution is a linear combination of regular and irregular Coulomb wave functions. In the special case that V(r) = , such as for neutrons, (), after a rearrangement in dimensionless form, is called the spherical Bessel equation. The solution consists of a linear combination of spherical Bessel functions of the first type j ℓ (ρ), and of the second type n ℓ (ρ) (or Neumann functions). Two linearly independent complex combinations of j ℓ and n ℓ are known as spherical Bessel functions of the third type (or Hankel functions) h +ℓ (ρ) and h −ℓ (ρ) (Abramowitz ; Zwillinger ). These are functions of the dimensionless parameter ρ = kr. Although n ℓ (ρ) → −∞ for r → , this irregular solution should be included because we only need this   Neutron Cross Section Measurements ⊡ Table  The spherical Bessel functions and the incoming and outgoing waves from equations () and (). Derived quantities are given in > Table  O ℓ = I∗ℓ ℓ jℓ nℓ  sin ρ ρ − cos ρ ρ eiρ  sin ρ cos ρ − ρ ρ − cos ρ sin ρ − ρ ρ eiρ ( ℓ (−)ℓ ρ ℓ (  d ℓ sin ρ ) ρ dρ ρ − (−)ℓ ρ ℓ (  − i) ρ  d ℓ cos ρ ) ρ dρ ρ Neutron energy for A = 238 (eV) 2.9×10–1 2.9×101 2.9×103 2.9×105 2.9×107 1.0 L =0 L =1 L =2 L =3 0.8 0.6 jL(r)  0.4 0.2 0.0 – 0.2 10–3 10–2 10–1 100 r (dimensionless) 101 ⊡ Figure  The Bessel function jℓ (ρ) for ℓ = , , ,  is shown as a function of ρ and as a function of equivalent energy for a nucleus with mass A =  solution in the external region r > a c . The appropriate solution for a channel c is a linear combination of waves corresponding to incoming I c (r) and outgoing O c (r) waves for a free particle, R(r) = R ℓ (r) = y ℓ I ℓ (r) + x ℓ O ℓ (r), with and I c (r) = I ℓ (r) = −iρh −ℓ (ρ) = −iρ(j ℓ (ρ) − in ℓ (ρ)) () O c (r) = O ℓ (r) = −iρh +ℓ (ρ) = iρ( j ℓ (ρ) + in ℓ (ρ)) () The functions j ℓ (ρ) and n ℓ (ρ) together with O ℓ (ρ) are given in > Table . In > Fig.  the function j ℓ (ρ) is shown as a function of ρ and as a function of equivalent energy for a nucleus with mass A = . Neutron Cross Section Measurements ..  The Collision Matrix U The total wave function Ψ in the external region can be expressed as the superposition of all incoming and outgoing partial waves Ic and Oc , with amplitudes y c and x c , and summed over all possible channels c. ′ Ψ = ∑ y c Ic + ∑ x c ′ Oc () c′ c The complete wave functions in the channel, I c and Oc , contain the radial parts I c and O c , but also the angular part of relative motion Ymℓ ℓ , as well as the internal wave functions of the particles and the channel spin, combined in φ c , and are written as √ Ic = I c r − φ c i ℓ Ymℓ ℓ (θ, ϕ)/ v c and () √ Oc = O c r − φ c i ℓ Ymℓ ℓ (θ, ϕ)/ v c () √ The factor / v c normalizes the waves to unit flux. The physical process of the reaction will result in a modification of the outgoing waves. In the reaction, the coefficients x c of the outgoing waves, depending on the details of the reaction which are observable in the cross section, have to be determined with respect to the coefficients of the incoming waves y c . The collision matrix U cc ′ is now defined as the relation between the coefficients of the incoming and outgoing waves: xc′ ≡ − ∑ U c′ c y c () c All the physics of the reaction is contained in the elements of the collision matrix. The collision matrix has two important properties. From the conservation of probability flux in the reaction it follows that the collision matrix is unitary, which means that its complex conjugate equals its reciprocal, U∗ = U− or ∗ () ∑ U cc ′ U cc ′′ = δ c′ c′′ c The second property follows from time reversal conservation and implies that the collision matrix is symmetric, U cc ′ = U c ′ c . Finally, we can express the total wave function of () in terms of the collision matrix: Ψ = ∑ y c (Ic − ∑ U cc ′ Oc ′ ) c () c′ which is a linear combination of the wave functions for each channel c, consisting of an ingoing wave and the modified outgoing waves summed over all channels c ′ . .. The Relation Between the Cross Sections and the Collision Matrix U The relation between reaction cross sections and wave functions, describing a probability, is based on the conservation of probability density. The probability density of an incident plain    Neutron Cross Section Measurements wave, which is the flux of particles j φ is given by the quantum mechanical expression jφ = ħ (ψ ∗ ∇ψ − ψ∇ψ ∗) mi () The connection with the cross section is best illustrated by considering a flux of incident particles j inc , represented by a plain wave ψ inc , which can be expanded in a series of partial radial waves, scattering elastically at a point r =  because of an unknown physical process. The scattered wave, originating at r =  is a radial wave ψ sc and far from the scattering center at a distance r in a solid angle element dΩ the current of scattered particles across the surface r  dΩ is j sc . The total wave ψ = ψ inc + ψ sc is a solution of the Schrödinger equation for this system. The cross section of this reaction, which is a differential cross section, is defined as jsc  r dΩ j inc dσ = () Integrating over dΩ gives the total scattering cross section. If elastic scattering were the only process to occur, the total current of ingoing particles equals that of the outgoing particles. Any reaction, defined as any other process than elastic scattering, means that there is a difference in the absolute values of the ingoing and outgoing current. In the more general description of channels the total wave function is (). Elastic scattering means here that the entrance and exit channel are the same. A change of channel in the outgoing wave is considered as a reaction. With a similar approach, including the expansion of the incoming plane wave into an infinite sum of partial waves ℓ, and using the full description of the channel wave functions, the angular differential cross section for the reaction α → α ′ has been worked out by Blatt and Biedenharn (). For zero Coulomb interaction the expression is ∞  dσ  ′ λ ∑ B ℓ (c, c )Pℓ (cos θ) = dΩ  j +  ℓ= () The coefficients B ℓ (c, c ′) are rather complicated factors and contain the collision matrix elements U cc ′ and relations containing Clebsch–Gordan coefficients for the spin bookkeeping, eliminating most of the terms in the infinite sum over ℓ. The cross section for an interaction from channel c to channel c ′ is then  σ cc ′ = πλc ∣δ c ′ c − U c ′ c ∣  () If the interaction occurs without a change in the channel c then the process is called elastic scattering. The cross section is, putting c ′ = c  σcc = πλ c ∣ − U cc ∣  () and the cross section for a channel reaction, i.e. any interaction which is not elastic scattering, is obtained by summing () over all c ′ except c σcr = πλc ( − ∣U cc ∣ ) () and the total cross section is obtained by summing all channels c ′ σ c,T = σc = πλc ( − Re U cc ) () Neutron Cross Section Measurements  In practice, channel to channel cross sections are not useful. One would like to have the cross sections of α → α ′ for the component of total angular momentum J. The total reaction cross section is obtained by integrating () over the full solid angle to obtain to total cross section for the component of total angular momentum J σ αα ′ (J) = πλα g(J) ∑ ∣δ j j′ ℓℓ′ − U jℓ, j′ ℓ′ ∣ () j, j′ ,ℓ,ℓ′ and the total cross section by summing over all α ′ σ α,T (J) = πλα g(J) ∑( − Re U jℓ, jℓ ) () j,ℓ .. The Wave Function in the Internal Region The complete wave function Ψ can be described as the product of the function of relative motion and the channel-spin function, giving the internal states of the particles α  and α  and their combined spin. From the function of relative motion the radial part R(r) is separated and the remaining part is combined with the channel-spin function to give the channel surface function φ c Ψ = ∑ φ c R c (a c ) . () c The surface functions φ c have the property of orthonormality over the surface S c given by r = a c . This will be exploited to expand certain quantities in terms of surface functions. It follows immediately that R c (a c ) = ∫ φ∗c ΨdS c . () The integration over a surface, instead of integrating over a volume, is particularly useful in deriving the R-matrix relation using Green’s theorem, expressing a volume integral in a surface integral. At the channel surface r = a c the radial wave function for the internal and external region should match. The value Vc and derivative D c are defined with a normalization constant as √ ħ u c (a c ) Vc = m c a c () √ ħ ∗ = φ c ΨdS c m c a c ∫ and √ Dc = √ du c ħ ac ( ) m c a c dr r=a c ħ ∗ φ c ∇n (rΨ)dS c m c a c ∫ √ ħ ∗ = Vc + a c φ c dS c m c a c ∫ = ()    Neutron Cross Section Measurements In the internal region the wave function cannot be calculated readily by solving the Schrödinger equation since the nuclear potential is in general very complicated and the nucleus has many interacting nucleons. But the wave function can be expressed as an expansion in eigenfunctions X λ and eigenvalues E λ Ψ = ∑ Aλ Xλ () λ and the coefficients A λ can be expressed as A λ = ∫ X ∗λ Ψdτ () where the integration goes over the volume dτ of the internal region given by r < a c . The values and derivatives on the surface r = a c are defined, analog to () and (), as √ γ λc = and √ δ λc = γ λc + ħ ∗ φ c X λ dS c m c a c ∫ ħ a c φ ∗c ∇n (X λ )dS c . m c a c ∫ () () The boundary conditions to be satisfied on the channel surface are taken identical for all λ B c = δ λc /γ λc . () Applying Green’s theorem to () gives A λ = ∫ X ∗λ Ψdτ = (E λ − E)− ħ (X ∗λ ∇n (Ψ) − Ψ∇n (X ∗λ )) dS c m c ∫ () − = (E λ − E) ∑(D c − B c Vc )γ λc c using (), (), (), (), and (). The expression () for the wavefunction can now be written as X λ γ λc () ] (D c − B c Vc ) . Ψ = ∑ [∑ E λ −E c λ By multiplying each side of () by φ c ′ , integrating over the surface r = a c and using () one obtains Vc ′ = ∑ R cc ′ (D c − B c Vc ) () c with R cc ′ = ∑ λ γ λc γ λc ′ . Eλ − E () The quantity R cc ′ is the R-matrix and contains the properties E λ and γ λc of the eigenstates λ. The boundary constant B c can be chosen freely. Neutron Cross Section Measurements ..  The Relation Between the R-Matrix and the Collision Matrix U The values and derivatives of the internal wave function are given by the R-matrix relation (). The external wave function is given by () and is known except for the boundary conditions. The boundary condition is that both the internal and external wave functions have the same value and radial derivative at r = a c in order to have a smooth transition. By matching these conditions and after considerable rearrangements, the collision matrix U cc ′ can be given explicitly as a function of the R-matrix in matrix notation by U = ΩP/ [ − R(L − B)]− [ − R(L∗ − B)]P−/ Ω . () The introduced complex matrix L is given by L c = S c + iPc = ( ρ dO c ) O c d ρ r=a c () where real matrices S c is called the shift factor and Pc the penetrability factor. The matrix Ω c is Ωc = ( Ic ) O c r=a c () which can be reduced for neutral particles, using () and (), to Ω c = exp(−iϕ c ) () from which ϕ c follows ϕ c = arg O c (a c ) = arctan ( j ℓ (ρ) Im O c ) = arctan (− ) Re O c n ℓ (ρ) () All matrices in () are diagonal matrices except U and R. A table of Pℓ , S ℓ , and ϕ ℓ is given in > Table . They are directly related to the solution of the Schrödinger equation in the external region, which are the spherical Bessel and Neumann functions j ℓ (ρ) and n ℓ (ρ) for neutral particles, and can be derived from the quantities listed in > Table . ⊡ Table  The penetrability P ℓ , the level shift S ℓ and the hard-sphere phase shift ϕ ℓ for reaction channels without Coulomb interaction, as a function of ρ = kac . These parameters are derived from the quantities in > Table  ℓ Pℓ Sℓ  ρ  ρ /( + ρ ) ρ     ℓ ϕℓ ρ P ℓ− (ℓ − S ℓ− ) + Pℓ− −/( + ρ ) ρ − arctanρ ρ (ℓ − S ℓ− ) −ℓ (ℓ − S ℓ− ) + Pℓ− ϕ ℓ− − arctan   P ℓ− ℓ − S ℓ−   Neutron Cross Section Measurements Neutron energy for A = 238 (eV) –1 2.9×10 101 2.9×10 1 2.9×103 2.9×105 2.9×107 PL(r) 100 L=0 L=1 L=2 L=3 10–1 10–2 10–3 10–4 Sl ( r) 0.0 –1.0 –2.0 –3.0 101 100 Fl (ρ)  10–1 10–2 10–3 10–4 –3 10 10–2 10–1 ρ (Dimensionless) 100 101 ⊡ Figure  The functions P ℓ (ρ), S ℓ (ρ), and ϕ ℓ (ρ) for ℓ = , , ,  shown as a function of ρ and as a function of equivalent energy for a nucleus with mass A =  If the boundary conditions B c , defined by (), are real, then the δ λc and the γ λc are real and hence R is real. In addition R is symmetrical. A common choice is to take Bc = Sc () which eliminates the shift factor for s-waves, but introduces an energy dependence. The choice B c = −ℓ has also been proposed (Fröhner ). At low energy this is equivalent as can be seen in > Fig. , where Pℓ , S ℓ and ϕ ℓ are plotted as a function of ρ and as a function of equivalent energy for a nucleus with mass A = . So () defines the collision matrix in terms of the parameters of the R-matrix, γ λc and E λ , representing the physical process of the reaction, and the quantities Pc , S c , ϕ c , describing the known incoming and outgoing waves I c and O c , outside a sphere with radius a c . The values B c determine the boundary conditions at the matching point of the internal and external region, and are free to be chosen. The unknowns of the R-matrix, γ λc and E λ , need to be determined in order to know the U-matrix and subsequently the cross sections. Neutron Cross Section Measurements .  Approximations of the R-Matrix Several approximations of the R-matrix have been developed in the past in order to overcome the complications of inverting the matrix [ − R(L − B)]− appearing in (). Except in the case where only one or two channels are involved, the inversion is in general impossible without additional assumptions. The problem can be put in terms of the inversion of a level matrix A of which the elements refer to the properties of the levels λ of the system. The problem of inverting a matrix concerning all channels is now put in a problem of inverting a matrix concerning levels. The level matrix A λ μ is introduced by putting the following form ([ − R(L − B)]− ) cc ′ = δ cc ′ + ∑ γ λc γ μc ′ (L c ′ − B c ′ )A λ μ () λμ from which the elements of the inverse of A are (A− ) λ μ = (E λ − E)δ λ μ − ∑γ λc γ μc (L c − B c ) c  = (E λ − E)δ λ μ − Δ λ μ − iΓλ μ  () with the quantities Δ λ μ and Γλ μ defined by Δ λ μ = ∑(S c − B c )γ λc γ μc () c and Γλ μ =  ∑ Pc γ λc γ μc () c Now the collision matrix from () can be expressed in terms of A √ ⎛ ⎞ U cc ′ = Ω c Ω c ′ δ cc ′ + i Pc Pc ′ ∑ A λ μ γ λc γ μc ′ ⎝ ⎠ λμ () Additional approximations have been formulated in order to simplify this expression. The most illustrative is the Breit and Wigner single level (SLBW) approximation where only one level is considered. It can be extended to several, independent levels, which is the Breit and Wigner multi level (MLBW) approximation. The formalism of Reich and Moore () neglects only the off-diagonal contributions of the photon channels, which is an accurate approximation for medium and heavy nuclei. It takes into account the interference between levels and reduces to the SLBW approximation in the limit of a single level. These three formalisms will be described in some more detail. Other formalisms exist of which we mention here the formalisms of Kapur and Peierls (), Wigner and Eisenbud (), Adler (), Hwang (), and more recently Luk’yanov and Yaneva ().    .. Neutron Cross Section Measurements The Breit–Wigner Single Level Approximation The expression () can be simplified if only a single level is present. In that case the matrix contains only a single element. Therefore, (A− ) λ μ = A− = E λ − E + Δ λ − iΓλ / () with  Δ λ = Δ λλ = − ∑(S c − B c )γ λc () c and Γλ = Γλλ = ∑ Γλc = ∑ Pc γ λc c () c Substituting these expressions in () gives the collision matrix √ i Γλc Γμc ′ ⎞ −i(ϕ c +ϕ c ′ ) ⎛ U cc ′ = e δ cc ′ + E λ + Δ λ − E − iΓλ / ⎠ ⎝ () From the collision matrix the cross sections can be calculated. For the total cross section this results in   σ c = πλ c g c ( sin ϕ c + Γλ Γλc cosϕ c + (E − E λ − Δ λ )Γλc sinϕ c ) (E − E λ − Δ λ ) + Γλ / () The first part of the total cross section is the potential scattering or hard sphere scattering cross section σ p = πλc g c sin ϕ c . It is associated with the elastic scattering of the incoming neutron from the potential of the nucleus without forming a compound state. The term with the factor sinϕ c is the interference of the potential scattering and the resonant elastic scattering through formation of a compound nucleus. Finally, the term with cosϕ c describes the resonance cross sections of the channels. In a more practical case, we can see what the cross sections becomes for a neutron entrance channel c = n. We assume that the only open channels are elastic scattering and neutron capture, Γλ = Γ = Γn + Γγ . A series expansion of the trigonometric factors gives for ℓ =  at low energy in good approximation sinϕ c = ρ = ka c and sinϕ c =  for ℓ > . The cosine term can be approximated by cosϕ c =  for all ℓ. In the same way, the reaction cross section is  σcc ′ = πλ c g c Γλc Γλc ′ (E − E λ − Δ λ ) + Γλ / () and the shift Δ λ results from the boundary condition. .. The Breit–Wigner Multi Level Approximation Several resonances can be taken into account as a sum of Breit and Wigner single level cross sections. This is the most simple treatment of cross sections of many resonances. It neglects any possible interference between channels and levels (resonances). Neutron Cross Section Measurements  The Breit and Wigner multi level (BWML) approach uses a sum over the levels in the collision matrix. In the inverse of the level matrix A all off-diagonal elements A− λ μ are neglected, which means neglecting all interference terms between channels, but not between levels. .. (A− ) λ μ = (E λ − E + Δ λ − iΓλ /)δ λ μ () √ i Γλc Γμc ′ ⎛ ⎞ U cc ′ = e−i(ϕ c +ϕ c ′ ) δ cc ′ + ∑ E + Δ − E − iΓ / ⎝ ⎠ λ λ λ λ () The Reich–Moore Approximation In the approximation of Reich and Moore () it is assumed that the amplitudes γ λc are uncorrelated and have a Gaussian distribution with zero mean. This is a consequence of the chaotic behavior of the compound nucleus, except for the very light nuclei. This is known as the Gaussian orthogonal ensemble (Lynn ; Mehta , ). In medium and heavy nuclei, the number of photon channels is very large. And since the amplitudes are supposed to have a random distribution with zero mean, the expectation value of the product of two amplitudes is zero for λ ≠ μ, that is, < γ λc γ μc >= γ λc δ λ μ . Summing over the photon channels gives γ λc γ μc = ∑ c∈photon γ λc δ λ μ = Γλγ δ λ μ ∑ () c∈photon Therefore, the general expression for A− , (), can be simplified for the photon channels and becomes (A− ) λ μ = (E λ − E)δ λ μ − ∑ γ λc γ μc (L c − B c ) − c∈photon ∑ γ λc γ μc (L c − B c ) c∉photon = (E λ − E)δ λ μ − Γλγ (L c − B c )δ λ μ − γ λc γ μc (L c − B c ) ∑ c∉photon = (E λ − E + Δ λ − iΓλγ /)δ λ μ − ∑ () γ λc γ μc (L c − B c ) c∉photon Comparing this to (), the approximation may be written as a reduced R-matrix in the sense that the photon channels are excluded and the eigenvalue E λ is replaced by E λ − iΓλγ /. This Reich–Moore R-matrix is R cc ′ = ∑ λ γ λc γ λc ′ E λ − E − iΓλγ / c ∉ photon () The number of energy levels, which may be over hundreds of thousands in heavy nuclei, determines the number of possible photon decay channels. Excluding them reduces largely the number of channels and, therefore, the matrix inversion needed in the relation between the R-matrix and the cross sections. In the often occurring case at low energy that only the elastic scattering and neutron capture channels are open, the number of channels in the R-matrix is one, namely, that of the neutron channel, the photon channels being excluded explicitly. The    Neutron Cross Section Measurements total radiation width is present however in the denominator of (). The R-matrix becomes in this case an R-function of which the inversion is trivial. Including other channels, like one or two fission channels, keeps the number of channels low and makes the inversion still feasible. This approximation of the general R-matrix is the most accurate one used. . Average Cross Sections At higher energies the widths of the resonances overlap and the cross sections appear smooth and with a slow variation with energy. The total and scattering cross sections without sharply separated or observed resonances can be adequately described by representing the particlenucleus interaction by a complex potential. This optical potential, so called because mathematically analogous to the scattering and absorption of light in a medium (cloudy crystal ball), results in the partial scattering or absorption of the beam. The solution of the Schrödinger equation, usually numerically, with a given potential gives the wave functions from which the cross sections can be obtained (Fernbach et al. ). Much progress has been made since in the theoretical development and parametrization of a suitable optical model potentials, see for example Camarda et al. (), Leeb and Wilmsen (), Bauge et al. (), Dietrich et al. (), Capote et al. () and Quesada et al. (). By making averages over resonances, the energy averaged collision matrix U cc can be related to the energy-averaged cross sections σ. The development of a given shape of the optical model potential results in a value for U cc . From the usual R-matrix expressions we can formulate a number of cross sections as follows. By analogy to () the average scattering cross section σ cc can be written as σ cc = πλc g c ∣ − U cc ∣ () which can be split up into an average shape elastic scattering cross section se = πλ g ∣ − U ∣ σ cc cc c c () associated with potential scattering, and an average compound elastic scattering cross section due to resonance scattering ce σ cc = πλc g c (∣U cc ∣ − ∣U cc ∣ ) () and after () the average reaction cross section σ cr , corresponding to all nonelastic partial cross sections, as σ cr = πλc g c ( − ∣U cc ∣ ) () and following () the average total cross section σ c , T can be written as  σ c,T = πλ c g c ( − Re U cc ) . () ce and the average reacThe sum of the average compound elastic scattering cross section σ cc tion cross section σ cr can be considered as the cross section for the formation of the compound nucleus σ c , and can be written as ce σ c = σ cc + σ cr = πλc g c (∣U cc ∣ − ∣U cc ∣ +  − ∣U cc ∣ ) = πλc g c ( − ∣U cc ∣ ) () Neutron Cross Section Measurements  Then the sum of this compound nucleus formation cross section σ c and the average shape elastic se equals the total cross section σ scattering cross section σ cc c,T , which can be checked by se σ c + σ cc = πλc g c ( − ∣U cc ∣ + ∣ − U cc ∣ ) = πλc g c ( − ∣U cc ∣ +  −  Re U cc + ∣U cc ∣ ) = σ c,T () From the above expressions, only the total, shape elastic, and compound nucleus formation se cross sections σ c,T σ cc , and σ c contain the elements U cc , calculated by optical model, without other terms like ∣U cc ∣ which cannot be extracted from optical model calculations. For a direct comparison with experimental data, only the calculated average total cross section () can be used in a general way. The shape elastic scattering cross section cannot be distinguished from the compound elastic scattering. The calculated compound nucleus formation cross section () is also not directly observable, but can be used in combination with measured decay channels, like in the surrogate measurements. Finally, the average cross section for a single reaction σ cc ′ is  σ cc ′ = πλ c g c ∣δ cc ′ − U cc ′ ∣ () which contains the nearly impossible averaging over ∣U cc ′ ∣ . When we introduce the transmission coefficient Tc =  − ∣U cc ∣  () the compound nucleus formation cross section (unaveraged) can be written as σ c = πλc g c Tc () Using the usual concepts in nuclear reaction theory (reciprocity, time-reversal invariance), the probability of decay through channel c ′ as Tc′ /ΣTi the cross section for the reaction c → c ′ is then Tc ′  () σ cc ′ = πλ c g c Tc ΣTi where the sum runs over all possible channels. Averaging over a small energy interval with many resonances, taking into account shape elastic scattering in addition to compound reactions and redefining Tc as  () Tc =  − ∣U cc ∣ results in the Hauser–Feshbach formula (see also Hauser and Feshbach ; Feshbach et al. ; Hodgson ; Moldauer b,a , ; Fröhner ; Sirakov et al.  for more details) ′ se δ ′ + πλ g Tc Tc W ′ σ cc ′ = σ cc () cc cc c c ΣTi where the factor Wcc ′ is factor that includes elastic enhancement and a correction for width fluctuations, which can be written as (see for example Moldauer ) Wcc ′ = ( Γ Γc Γc ′ ) Γ Γc Γc′ ()    Neutron Cross Section Measurements The width fluctuations can be calculated most accurately using the GOE triple integral (Verbaarschot et al. ; Verbaarschot ), but also with simpler approximations. The transmission coefficients for particle channels are given by (). Two other channels exist which are the photon and fission channels. Their transmission coefficients, related to the average widths and level spacing, are defined as Tγ = π Γγ D () and Γf () D Dedicated modelizations on photon strength functions, level densities, and fission models, are used for the photon and fission transmission coefficients, but are beyond the scope of this overview. Good starting points for further reading are the user guides of specialized computer codes like EMPIRE (Herman et al. ), TALYS (Koning et al. ), and others. T f = π  Concluding Remarks The importance of neutron-induced reaction data is evident in a wide variety of research fields, ranging from stellar nucleosynthesis and nuclear structure to applications of nuclear technology. The present chapter has sketched out an impression of neutron cross section measurements at time-of-flight facilities and at monoenergetic fast neutron sources. The principal measurement techniques have been summarized. Also details on the R-matrix formalism linking measurable nuclear properties to useable nuclear cross sections have been presented. Many details were intentionally omitted but the provided references should form a good starting point for the interested reader. References Abbondanno U et al () CERN n_TOF facility: performance report. Tech. Rep. ERNSL- ECT Abbondanno U et al () New experimental validation of the pulse height weighting technique for capture cross-section measurements. Nucl Instrum Methods Phys Res Sect A (–):– Abbondanno U et al () The data acquisition system of the neutron time-of-fight facility n_TOF at CERN. Nucl Instrum Methods Phys Res Sect A (–): Abfalterer WP, Finlay RW, Grimes SM () Level widths and level densities of nuclei in the  ≤ A ≤  mass region inferred from fluctuation analysis of total neutron cross sections. Phys Rev C (): Abramowitz M () Handbook of mathematical functions with formulas, graphs, and mathematical table. Dover Publications, New York Adler DB, Adler FT () Uniqueness of R-matrix parameters in the analysis of low-energy neutron cross sections of fissile nuclei. Phys Rev C :– Ait-Tahar S, Hodgson PE () Weisskopf-Ewing calculations – neutron-induced reactions. J Phys G ():– Allmond JM, Bernstein LA, Beausang CW et al () Relative U-(n,gamma) and (n,f ) cross sections from U-(d,p gamma) and (d,pf ). Phys Rev C (): Amaldi E, Fermi E () On the absorption and the diffusion of slow neutrons. Phys Rev :  Neutron Cross Section Measurements Ananiev VD et al () Intense resonance neutron source (IREN) – new pulsed source for nuclear physical and applied investigations. Phys Elementary Part At Nuclei (PEPAN) : – Asami A () JAERI new linac. J At Energy Soc Jpn :– Baba M () Experimental studies on particle and radionuclide production cross sections for tens of MeV neutrons and protons. AIP : – Baba M, Ibaraki M, Miura T et al () Experiments on neutron scattering and fission neutron spectra. J Nucl Sci Technol (suppl. ): – Barry DP () Neodymium neutron transmission and capture measurements and development of a new transmission detector. PhD thesis, Rensselaer Polytechnic Institute Bauge E, Delaroche JP, Giro M () Laneconsistent, semimicroscopic nucleon-nucleus optical model. Phys Rev C : Beer H, Kappeler F () Capture-to-fission ratio of U- in the neutron energy-range from  to  keV. Phys Rev C ():– Beer H, Kappeler F () Neutron-capture crosssections on Ba-, Ce-, Ce-, Lu-, Lu-, and Ta- at  keV – prerequisite for investigation of the Lu- cosmic clock. Phys Rev C ():– Behrens JW, Browne JC, Ables E () Measurement of the neutron-induced fission cross-section of Th- relative to U- from . to  MeV. Nucl Sci Eng ():– Belikov OV et al () Physical start-up of the first stage of IREN facility. J Phys Conference Series : Bernstein LA et al () Pu-(n,n) Pu- cross section deduced using a combination of experiment and theory. Phys Rev C ():  Biro T, Sudar S, Miligy, Z, Dezso Z, and Csikai, J () Investigations of (n,t) cross sections at . MeV. J Inorg Nucl Chem (–):–. Blatt JM, Biedenharn LC () The angular distribution of scattering and reaction cross sections. Rev Mod Phys ():– Block RC, Marano PJ, Drindak NJ et al () A multiplicity detector for accurate low-energy neutron capture measurements. In: Proceedings of the international conference on nuclear data for science and technology, Mito, p  Blons J () High-resolution measurements of neutron-induced fission cross-sections for U- U- Pu- and Pu- below  keV. Nucl Sci Eng ():–  Bohigas O, Giannoni MJ, Schmit C () Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys Rev Lett : – Bohr N () Neutron capture and nuclear constitution. Nature : Borcea C et al () Results from the commissioning of the n_TOF spallation neutron source at CERN. Nucl Instrum Methods Phys Res Sect A :– Borella A, Aerts G, Gunsing F et al (a) The use of C D  detectors for neutron induced capture cross-section measurements in the resonance region. Nucl Instrum Methods Phys Res Sect A ():– Borella A, Gunsing F, Moxon M et al (b) Highresolution neutron transmission and capture measurements of the nucleus Pb-. Phys Rev C (): Boyer S, Dassie D, Wilson JN et al () Determination of the Pa(n,gamma) capture cross section up to neutron energies of  MeV using the transfer reaction Th(He,p)Pa ∗ . Nucl Phys A (–):– Breit G, Wigner EP () Capture of slow neutrons. Phys Rev ():– Burke JT, Bernstein LA, Scielzo ND et al () Surrogate Reactions in the Actinide Region, AIP Conference Proceedings :– Calviani M, Cennini P, Karadimos D et al () A fast ionization chamber for fission cross-section measurements at n_TOF. Nucl Instrum Methods Phys Res Sect A ():– Calviani M, Praena J, Abbondanno U et al () High-accuracy U(n,f ) cross-section measurement at the white-neutron source n_TOF from near-thermal to  MeV neutron energy. Phys Rev C (): Camarda HS, Dietrich FS, Phillips TW () Microscopic optical-model calculations of neutron total cross-sections and cross-section differences. Phys Rev C ():– Capote R, Soukhovitskii ES, Quesada JM et al () Is a global coupled-channel dispersive optical model potential for actinides feasible? Phys Rev C (): Carlson AD, Behrens JW () Measurement of the  U(n,f ) cross section from . to . MeV using the NBS electron linac, Proceedings of the International Conference on Nuclear Data for Science and Technology, – Sept. , Antwerp, Belgium, ed. K. H. Bockhoff, (Dordrecht, Netherlands; Reidel, ) pp. – Carlson AD et al () International evaluation of neutron cross section standards. Nucl Data Sheets ():–    Neutron Cross Section Measurements Chadwick J () Possible existence of a neutron. Nature : Chadwick MB et al (). ENDF/B-VII.: Next generation evaluated nuclear data library for nuclear science and technology. ENDF data can be accessed at the National Nuclear Data Center at Brookhaven National Laboratory [www.nndc.bnl.gov]. Nucl Data Sheets ():– Cierjacks S, Duelli B, Forti P et al () Fast neutron time-of-fight spectrometer used with the karlsruhe isochronous cyclotron. Rev Sci Instrum ():– Cramer JD, Britt HC () Neutron fission cross sections for Th-, Th-, U-, U-, U-, Pu-, and Pu- from . to . MeV using (t, pf ) reactions. Nucl Sci Eng (): Dabbs JWT () Neutron cross section measurements at ORELA. In: Nuclear cross sections for technology proceedings of the international conference on nuclear cross sections for technology, Knoxville, p  Dagan R () On the angular distribution of the ideal gas scattering kernel. Ann Nucl Energy :– Danon Y, Slovacek RE, Block RC et al () Fission cross-section measurements of Cm-, Es-, and Cf- from . eV to  keV. Nucl Sci Eng ():– Danon Y, Block RC, Slovacek RE () Design and construction of a thermal-neutron target for the RPI linac. Nucl Instrum Methods Phys Res Sect A ():– Danon Y, Werner CJ, Youk G et al () Neutron total cross-section measurements and resonance parameter analysis of holmium, thulium, and erbium from . to  eV. Nucl Sci Eng ():– Danon Y, Block RC, Rapp MJ et al (a) Beryllium and graphite high-accuracy total cross-section measurements in the energy range from  to  keV. Nucl Sci Eng ():– Danon Y, Liu E, Barry D et al (b) Benchmark experiment of neutron resonance scattering models in Monte Carlo codes. International Conference on Mathematics, Computational Methods & Reactor Physics (M&C ) Saratoga Springs, New York, May -, , on CD-ROM, American Nuclear Society, LaGrange Park, IL (). Davis JC () The LLNL multi-user tandem laboratory. Nucl Instrum Methods B –(Part ):– Di Lullo AR, Massey TN, Grimes SM et al () A fission chamber measurement of the B-nat(d,n) cross section for use in neutron detector calibration. Nucl Sci Eng ():– Dietrich FS, Anderson JD, Bauer RW et al () Importance of isovector effects in reproducing neutron total cross section differences in the W isotopes. Phys Rev C (): Drosg M () Drosg-, codes and database for  neutron source reactions, Tech. Rep. IAEA report IAEA-NDS-, Rev.  (May ). http://www-nds.iaea.org/drosg. html. Dudey ND, Heinrich RR, Madson AA () Reaction cross sections of Rb-(n,gamma)Rb-m, Rb-(n,gamma)Rb-, and Y-(n,gamma) Y-m between . MeV and . MeV. J Nucl Energy (): Elsevier, Inc. Engineering village (). http://www.engineeringvillage.org Escher JE, Dietrich FS () Determining (n, f ) cross sections for actinide nuclei indirectly: examination of the surrogate ratio method. Phys Rev C (): Fadil M, Rannou B () About the production rates and the activation of the uranium carbide target for SPIRAL. Nucl Instrum Methods B  (–):– Farrell JA and Pineo WFE () Neutron cross sections of  Li in the kilovolt region. In: Proceedings of the conference on neutron cross sections and technology. NBS Special Publication ():– Fernbach S, Serber R, Taylor TB () The scattering of high energy neutrons by nuclei. Phys Rev ():– Feshbach H, Porter CE, Weisskopf VF () Model for nuclear reactions with neutrons. Phys Rev ():– Finlay RW, Brient CE, Carter DE et al () The Ohio-University beam swinger facility. Nucl Instrum Methods Phys Res (–):–  Firk F, Whittaker J, Bowey E et al () A nanosecond neutron time-of-fight system for the Harwell  MeV electron linac. Nucl Instrum Methods :– Flaska M, Borella A, Lathouwers D et al () Modeling of the GELINA neutron target using coupled electron-photon-neutron transport with the MCNPC code. Nucl Instrum Methods phys Sect A ():– Foderaro A () The elements of neutron interaction theory. MIT Press, Cambridge Frank IM, Pacher P ()  st experience on the high-intensity pulsed reactor IBR-. Physica B (–):– Neutron Cross Section Measurements Fröhner F () Evaluation and analysis of nuclear resonance data. Tech. Rep. JEFF Report , OECD/NEA Fujita Y () Neutron TOF spectrometer at KURRI electron linac. In: Proceedings of the  seminar on nuclear data (JAERI-M-–), Kyoto, p – Futakawa M, Haga K, Wakui T et al () Development of the HP target in the J-PARC neutron source. Nucl Instrum Methods A (): – Ge Zhigang et al () The updated version of the Chinese Evaluated Nuclear Data Library (CENDL-.) and China nuclear data evaluation activities In: Proceedings of the international conference on nuclear data for science and technology ND, EDP Sciences,  Good WM, Neiler JH, Gibbons JH () Neutron total cross sections in the keV region by fast time-of-fight measurements. Phys Rev ():– Grallert A, Csikai J, Qaim SM et al () Recommended target materials for d-d neutron sources. Nucl Instrum Methods Phys Res Sect A ():– Guber KH et al () New neutron cross-section measurements at ORELA for improved nuclear data calculations. In: Haight RC, Chadwick MB, Kawano T, Talou P (eds) International conference on nuclear data for science and technology, AIP, vol , Santa Fe, p  Gunsing F n_TOF Collaboration et al () Measurement of the  Th(n,γ ) resonance reaction at the n_TOF facility at CERN, unpublished Hansen LF, Anderson JD, Brown PS et al () Measurements and calculations of neutron-spectra from iron bombarded with -MeV neutrons. Nucl Sci Eng ():– Hansen LF, Wong C, Komoto TT et al () Measurements and calculations of the neutron emission-spectra from materials used in fusion-fission reactors. Nucl Technol (): – Haq RU, Pandey A, Bohigas O () Fluctuations properties of nuclear energy levels: do theory and experiment agree? Phys Rev Lett (): – Hatarik R, Bernstein LA, Burke JT et al () Using (d,p gamma) as a surrogate reaction for (n,gamma) AIP Conference Proceedings () Vol., p. – Hauser W, Feshbach H () The inelastic scattering of neutrons. Phys Rev ():–  Herman M, Capote R, Carlson BV et al () EMPIRE: Nuclear reaction model code system for data evaluation. Nucl Data Sheets ():– Hockenbury RW, Bartolome ZM, Tatarczuk JR et al () Neutron radiative capture in Na, Al, Fe, and Ni from  to  kEV. Phys Rev (): – Hodgson PE () Nuclear reactions and nuclear structure. Clarendon Press, Oxford Hogue HH, Vonbehren PL, Glasgow DW et al () Elastic and inelastic-scattering of -MeV to -MeV neutrons from Li- and Li-. Nucl Sci Eng ():– Hori J et al () Neutron capture cross section measurement on  Am with a π Ge spectrometer. In: Proceedings of the  symposium on nuclear data, Tokai, pp. – Humblet J, Rosenfeld L () Theory of nuclear reactions: I. resonant states and collision matrix. Nucl Phys ():– Hwang RN () Efficient methods for treatment of resonance cross-sections. Nucl Sci Eng ():– Ibaraki M, Baba M, Miura T et al () Experimental method for neutron elastic scattering cross-section measurement in – MeV region at TIARA. Nucl Instrum Methods Phys Res Sect A ():– Igashira M, Kitazawa H, Yamamuro N () A heavy shield for the gamma-ray detector used in fast-neutron experiments. Nucl Instrum Methods Phys Res Sect A (–): – Ignatyuk AV, Fursov BI () The latest BROND- developments. In: Proceedings of the international conference on nuclear data for science and technology ND, EDP Sciences,  Jaag S, Kappeler F () Stellar (n,gamma) crosssection of the unstable isotope Eu-. Phys Rev C ():– Jandel M et al () Neutron capture cross section of Am-. Phys Rev C ():  Joly S, Voignier J, Grenier G et al () Measurement of fast-neutron capture cross-sections using a Nal spectrometer. Nucl Instrum Methods (–):– Jongen Y, Ryckewaert G () Heavy-ion acceleration at CYCLONE (Belgium). IEEE Trans Nucl Sci ():– Kapur PL, Peierls RE () Dispersion formula for nuclear reactions. Proc R Soc A:– Kegel GHR () Fast-neutron generation with a type Cn Van De graaff accelerator. Nucl Instrum Methods Phys Res Sect B –:–    Neutron Cross Section Measurements Kim GN et al () Measurement of photoneutron spectrum at Pohang neutron facility. Nucl Instrum Methods Phys Res Sect A (): – Klug J et al () SCANDAL – a facility for elastic neutron scattering studies in the – MeV range. Nucl Instrum Methods Phys Res Sect A (–):– Klug J, Altstadt E, Beckert C et al () Development of a neutron time-of-fight source at the ELBE accelerator. Nucl Instrum Methods Phys Res Sect A ():– Knoll G () Radiation detection & measurement. John Wiley & Sons, New York Kobayashi K, Lee S, Yamamoto S et al () Measurement of neutron capture cross section of Np- by linac time-of-fight method and with linac-driven lead slowing-down spectrometer. J Nucl Sci Technol ():– Kobayashi K, Lee S, Yamamoto S () Neutron capture cross-section measurement of Tc- by linac time-of-fight method and the resonance analysis. Nucl Sci Eng ():– Koehler PE () Comparison of white neutron sources for nuclear astrophysics experiments using very small samples. Nucl Instrum Methods A :– Kohler R, Mewissen L, Poortmans F et al () Highresolution neutron resonance spectroscopy. AIP Conf Proc :– Kondo K, Murata I, Ochiai K et al () Chargedparticle spectrometry using a pencil-beam DT neutron source for double-differential crosssection measurement. Nucl Instrum Methods Phys Res Sect A ():– Koning A et al () The JEFF- nuclear data library, JEFF report . Tech. rep., JEFF data can be accessed at the OECD Nuclear Energy Agency Koning AJ, Hilaire S, Duijvestijn M () TALYS a nuclear reaction program. Tech. rep. Kononov VN et al () Fast-neutron radiativecapture cross-sections and d-wave strength functions. Proceedings of the th international symposium on capture gamma-ray spectroscopy and related topics, London, pp – Kononov VN, Poletaev ED, Timokhov VM et al () Fast-neutron radiative-capture cross-sections and transmissions for tungsten isotopes. Sov J Nucl Phys ():– Kopecky S, Brusegan A () The total neutron cross section of Ni-. Nucl Phys A (–): – Kornilov NV, Kagalenko AB () Inelastic neutronscattering by U- and U- nuclei. Nucl Sci Eng ():– Krane K () Introductory nuclear physics. John Wiley & Sons, New York Lamb WE () Capture of neutrons by atoms in a crystal. Phys Rev ():– Lane AM, Thomas RG () R-matrix theory of nuclear reactions. Rev Mod Phys (): – Larson NM () Updated users’ guide for SAMMY: Multilevel R-matrix fits to neutron data using Bayes’ equations. SAMMY, computer code Report ORNL/TM-/R, Oak Ridge National Laboratory Leeb H, Wilmsen S () Violation of pseudospin symmetry in nucleon-nucleus scattering: Exact relations. Phys Rev C (): Leinweber G, Barry DP, Trbovich MJ et al () Neutron capture and total cross-section measurements and resonance parameters of gadolinium. Nucl Sci Eng ():– Lesher SR, McKay CJ, Mynk M et al () Low-spin structure of Mo studied with the (n; n′ γ) reaction. Phys Rev C (): Lisowski PW, Schoenberg KF () The Los Alamos neutron science center. Nucl Instrum Methods Phys Res sect A ():– Luk’yanov AA, Yaneva NB () Multilevel parametrization of resonance neutron cross sections. Phys Part Nucl :– Lychagin AA, Simakov SP, Devkin BV et al () Study of the Fe(n, n′ gamma) reaction with .-MeV neutrons. Sov J Nucl Phys (): – Lyles BF, Bernstein LA, Burke JT et al () Absolute and relative surrogate measurements of the U-(n,f ) cross section as probe of angular momentum effects. Phys Rev C (): Lynn JE () The theory of neutron resonance reactions, Clarendon Press, Oxford Mannhart W, Schmidt D () Measurement of the  Si(n,p),  Si(n,p) and  Si(n,alpha) cross sections between . and . MeV. J Nucl Sci Technol :– Mannhart W, Schmidt D () Measurement of neutron reaction sections between  and  MeV. AIP :– Marrone S et al () A low background neutron flux monitor for the n_TOF facility at CERN. Nucl Instrum Methods Phys Res Sect A  (–): Massey TN, A-Quraishi S, Brient CE et al () A measurement of the A- (d,n) spectrum for use in neutron detector calibration. Nucl Sci Eng ():– Mehta ML () On the statistical properties of the level-spacings in nuclear spectra. Nucl Phys : Neutron Cross Section Measurements Mehta ML () Random matrices. Academic, Boston Meister A () Calculations on lattice vibration effects in the doppler broadening of the . eV Cd neutron resonance cross section. Tech. Rep. CE/R/VG//, JRC-IRMM Mellema S, Finlay RW, Dietrich FS et al () Microscopic and conventional opticalmodel analysis of fast-neutron scattering from Fe-,Fe-. Phys Rev C ():–  Mihailescu LC, Borcea C, Koning AJ et al () High resolution measurement of neutron inelastic scattering and (n,n) cross-sections for Bi-. Nucl Phys A :– Mocko M, Muhrer G, Tovesson F () Advantages and limitations of nuclear physics experiments at an ISIS-class spallation neutron source. Nucl Instrum Methods Phys Res Sect A (): – Moldauer PA (b) Why Hauser-Feshbach formula works. Phys Rev C ():– Moldauer PA (a) Direct reaction effects on compound cross-sections. Phys Rev C (): – Moldauer PA () Evaluation of fluctuation enhancement factor. Phys Rev C (): – Moldauer PA () Statistics and the average cross-section. Nucl Phys A ():– Molnar G () Handbook of prompt gamma activation analysis. Springer, Berlin Moxon MC, Brisland JB () REFIT, a least squares fitting program for resonance analysis of neutron transmission and capture data computer code. Tech. rep., United Kingdom Atomic Energy Authority Mughabghab SF () Atlas of neutron resonances. Elsevier Science, Amsterdam Naberejnev DG, Mounier C, Sanchez R () The influence of crystalline binding on resonant absorption and reaction rates. Nucl Sci Eng : Nguyen VD et al () Measurements of neutron and photon distributions by using an activation technique at the Pohang neutron facility. J Korean Phys Soc ():– Nuclear data standards for nuclear measurements () NEANDC/INDC standards file. Tech. Rep. NEANDC-. http://www nds.iaea.org/standards/ Nystrom G, Bergqvis I, Lundberg B () Neutroncapture cross-sections in F, Mg, Al, Si, P and S from  to  keV. Phys Scr (): Orphan V, Hoot C, Carlson A et al () A facility for measuring cross sections of (n,xgamma)  reactions using an electron linac. Nucl Instrum Methods ():– Overberg ME, Moretti BE, Slovacek RE et al () Photoneutron target development for the RPI linear accelerator. Nucl Instrum Methods Phys Res Sect A (–):– Pancin J et al () Measurement of the n_TOF beam profile with a micromegas detector. Nucl Instrum Methods Phys Res Sect A (–): Panebianco S, Berg K, David JC et al () Neutronic characterization of the MEGAPIE target. Ann Nucl Energy ():– Paradela C et al () Neutron induced fission cross section of U and Np measured at the n_TOF facility. unpublished Payne GL, Schlessinger L () Properties of the K-matrix in nuclear-reaction theory. Phys Rev C ():– Petit M et al () Determination of the  Pa reaction cross section from . to  MeV neutron energy using the transfer reaction  Th( He, p) Pa. Nucl Phys A :– Petrich D, Neil M, Kappeler F et al () A neutron production target for FRANZ. Nucl Instrum Methods A ():– Pillon M, Angelone M, Martone M, Rado V () Characterization of the source neutrons produced by the Frascati neutron generator. Fus Eng Des :– Plag R et al () An optimized C D  detector for studies of resonance-dominated (n,γ) crosssections. Nucl Instrum Methods Phys Res Sect A : Plettner C, Ai H, Beausang CW et al () Estimation of (n,f ) cross sections by measuring reaction probability ratios. Phys Rev C (): Porter CE, Thomas RG () Fluctuations of nuclear reaction widths. Phys Rev ():– Qaim SM, Wolfle R, Rahman MM et al () Measurement of (n,p) and (n,alpha) reaction cross-sections on some isotopes of nickel in the energy region of  to  MeV using a deuterium gas-target at a compact cyclotron. Nucl Sci Eng ():– Quesada JM, Capote R, Soukhovitskii ES et al () Approximate Lane consistency of the dispersive coupled-channels potential for actinides. Phys Rev C (): Rapp M, Danon Y, Block RC et al () High energy neutron time of fight measurements of carbon and beryllium samples at the RPI linac. In: Society AN (ed) International conference on mathematics, computational methods & reactor physics , Saratoga Springs Ray ER, Good WM () Experimental neutron resonance spectroscopy, Academic, New    Neutron Cross Section Measurements York, chap. Pulsed Accelerator Time-of-Flight Spectrometers Reich CW, Moore MS () Multilevel formula for the fission process. Phys Rev ():– Reimer P, Koning AJ, Plompen AJM et al () Neutron induced reaction cross sections for the radioactive target nucleus Tc-. Nucl Phys A :– Reuss P () Neutron physics. EDP Sciences, Oakland Rochman D, Haight RC, O’Donnell JM et al () Neutron-induced reaction studies at FIGARO using a spallation source. Nucl Instrum Methods Phys Res Sect A (–):– Rochman D et al () Characteristics of a lead slowing-down spectrometer coupled to the LANSCE accelerator. Nucl Instrum Methods Phys Res Sect A (–):– Romano C, Danon Y, Block R et al () Measurements of fission fragment properties using RPI’s lead slowing-down spectrometer. In: Bersillon, O, Gunsing, F, Bauge, E, Jacqmin, R, Leray, S (eds), Proceedings of the international conference on nuclear data for science and technology, Nice, France, EDP Sciences Vol. , pp. – Rubbia C et al () A high resolution spallation driven facility at the CERN-PS to measure neutron cross sections in the interval from  eV to  MeV. Tech. Rep. CERN/LHC/–, CERN Sage C, Semkova V, Bouland O et al () High resolution measurements of the  Am(n,n) reaction cross section. unpublished Saglime FJ III, Danon Y, Block R () Digital data acquisition system for time of fight neutron beam measurements. In: The American Nuclear Society’s th Biennial Topical Meeting of the Radiation Protection and Shielding Division, Carlsbad, p  Saglime F, Danon Y, Block R et al () High energy neutron scattering benchmark of Monte Carlo computations. In: International conference on mathematics, computational methods & reactor physics (M&C ), Saratoga Springs Schmidt D () Determination of neutron scattering cross sections with high precision at PTB in the energy region  to  MeV. Nucl Sci Eng :– Schmidt D, Zhou ZY, Ruan XC et al () Application of non-monoenergetic sources in fast neutron scattering measurements. Nucl Instrum Methods Phys Res Sect A ():– Schmittroth F, Tobocman W () Comparison of the R-matrix nuclear reaction theories. Phys Rev C :– Schut PAC, Kockelmann W, Postma H et al () Neutron resonance capture and neutron diffraction analysis of roman bronze water taps. J Radioanal Nucl Chem ():– Semkova V, Plompen AJM () Neutron-induced dosimetry reaction cross-section measurements from the threshold up to  MeV. Radiat Prot Dosimetry (–):– Shcherbakov O, Furutaka K, Nakamura S et al () Measurement of neutron capture cross section of Np- from . to  eV. J Nucl Sci Technol ():– Shibata K et al () Japanese evaluated nuclear data library version  revision-: JENDL-.. J Nucl Sci Technol ():– Shyam R, Scholten O () Photoproduction of eta mesons within a coupled-channels K-matrix approach. Phys Rev C (): Sirakov I, Capote R, Gunsing F et al () An ENDF compatible evaluation for neutron induced reactions of Th- in the unresolved resonance region. Ann Nucl Energy ():– Staples P, Egan JJ, Kegel GHR et al () Prompt fission neutron energy-spectra induced by fast-neutrons. Nucl Phys A ():– Sukhoruchkin SI, Soroko ZN, Deriglazov VV () Low energy neutron physics, tables of neutron resonance parameters, vol I/B. Springer, Landolt-Börnstein, Berlin Tagliente G et al () Experimental study of the Zr-(n, gamma) reaction up to  keV. Phys Rev C (): Takahashi A, Ichimura E, Sasaki Y et al () Measurement of double differential neutron emission cross-sections for incident neutrons of -MeV. J Nucl Sci Technol ():– Tovesson F, Hill TS () Neutron induced fission cross section of Np- from  keV to  MeV. Phys Rev C (): Tovesson F, Hill TS () Subthreshold fission cross section of Np. Nucl Sci Eng : Tovesson F, Hill TS, Mocko M et al () Neutron induced fission of Pu-,Pu- from  eV to  MeV. Phys Rev C (): Trbovich MJ, Barry DP, Slovacek RE et al () Hafnium resonance parameter analysis using neutron capture and transmission experiments. Nucl Sci Eng ():– Tronc D, Salome JM, Bockhoff KH () A new pulse-compression system for intense relativistic electron-beams. Nucl Instrum Methods Phys Res Sect A (–):– Verbaarschot JJM () Investigation of the formula for the average of two S-matrix elements in compound nucleus reactions. Ann Phys (): – Neutron Cross Section Measurements Verbaarschot JJM, Weidenmuller HA, Zirnbauer MR () Grassmann integration in stochastic quantum physics – the case of compound nucleus scattering. Phys Rep Rev Sect Phys Lett ():– Vogt E () Theory of low energy nuclear reactions. Rev Mod Phys ():– Wagemans C, De Smet L, Vermote S et al () Measurement of the U-(n, f ) cross section in the neutron energy range from . eV up to  keV. Nucl Sci Eng ():– Wallerstein G et al () Synthesis of the elements in stars: forty years of progress. Rev Mod Phys : Wang T et al () Measurement of the total neutron cross-section and resonance parameters of molybdenum using pulsed neutrons generated by an electron linac. Nucl Instrum Methods Phys Res Sect B ():– Weisskopf VF, Ewing DH () On the yield of nuclear reactions with heavy elements. Phys Rev ():–; erratum (): Wigner EP, Eisenbud L () Higher angular momenta and long range interaction in resonance reactions. Phys Rev (): – Wisshak K, Guber K, Kappeler F et al () The Karlsruhe -pi barium fluoride detector. Nucl Instrum Methods Phys Res Sect A ():–  Wisshak K, Kappeler F () Neutron-capture cross-section ratios of Pu-, Pu-, U-, and Au- in energy-range from  to  keV. Nucl Sci Eng ():– Wisshak K, Kappeler F () Neutron-capture cross-section ratios of Pu- and Pu versus Au- in the energy-range from  to  keV. Nucl Sci Eng (): – Wisshak K, Voss F, Arlandini C et al () Stellar neutron capture on Ta-m. I. cross section measurement between  keV and  keV. Phys Rev C (): Woods R, Mckibben JL, Henkel RL () LosAlamos -stage Van de Graaff facility. Nucl Instrum Methods (–):– Younes W, Britt HC () Neutron-induced fission cross sections simulated from (t,pf ) results. Phys Rev C (): Zerkin VV, McLane V, Herman MW et al () EXFOR-CINDA-ENDF: migration of databases to give higher-quality nuclear data services. AIP Conf Proc ():  Zhuravlev BV, Demenkov VG, Lychagin AA et al () A measuring complex for time-of-fight spectrometry of fast neutrons. Instrum Exp Tech ():– Zwillinger D () CRC standard mathematical tables and formulae. CRC Press, Boca Raton   Evaluated Nuclear Data Pavel Obložinský ⋅ Michal Herman ⋅ Said F. Mughabghab National Nuclear Data Center, Brookhaven National Laboratory, Upton, NY, USA oblozinsky@bnl.gov mwherman@bnl.gov mugabgab@bnl.gov  . . .. .. .. . . .. .. .. .. .. . .. .. .. Evaluation Methodology for Neutron Data . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Basic Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Thermal and Resolved Resonance Region . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Thermal Energy Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Westcott Factors and Resonance Integrals . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Resolved Resonance Energy Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Unresolved Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Fast Neutron Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Optical Model and Direct Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Compound Nucleus Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Width Fluctuation Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Preequilibrium Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Light Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Fission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Fission Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Prompt Fission Neutron Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Peculiarities of Fission Cross Section Evaluation . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .                   . .. .. . .. .. . .. .. . . .. .. . . Neutron Data for Actinides . . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . .  U Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  U, Unresolved Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  U, Fast Neutron Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  U Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  U, Resolved and Unresolved Resonance Region . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  U, Fast Neutron Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  Pu Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  Pu, Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  Pu, Fast Neutron Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  Th Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Minor Actinides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .  U Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . ,,,,,, U Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Thermal Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Nubars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .                 Dan Gabriel Cacuci (ed.), Handbook of Nuclear Engineering, DOI ./----_, © Springer Science+Business Media LLC    Evaluated Nuclear Data . .. .. . .. Delayed Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fission-Product Delayed Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  U Thermal ν̄ d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fission Energy Release. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       . . .. .. . .. .. .. .. Neutron Data for Other Materials . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Light Nuclei . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluations of Major Structural Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Evaluations for ENDF/B-VII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fission Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Priority Fission Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complete Isotopic Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Case of  Zr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remaining Fission Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .            . .. .. . . .. .. .. . .. .. .. Covariances for Neutron Data . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Evaluation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fast Neutron Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Case: Gd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Major Actinides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,, U Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Pu Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Th Covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Covariance Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Fidelity Covariance Library. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SCALE- Covariance Library. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AFCI Covariance Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .               . . . .. .. .. . . . . . Validation of Neutron Data .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Criticality Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fast U and Pu Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal U and Pu Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  U Solution Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U Fuel Rod Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pu Solution and MOX Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions from Criticality Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Delayed Neutron Testing, β e f f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reaction Rates in Critical Assemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shielding and Pulsed-Sphere Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing of Thermal Values and Resonance Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . .              . . .. .. Other Nuclear Data of Interest . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Fission Yields .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H O and D O .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O in UO and U in UO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      Evaluated Nuclear Data  .. .. . .. H in ZrH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Other Modified Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Decay Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Decay Heat Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .      . .. .. .. . . .. .. . . . Evaluated Nuclear Data Libraries .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Overview of Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . General Purpose Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Special Purpose Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Derived Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . ENDF- Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . ENDF/B-VII. (USA, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Overview of the ENDF/B-VII. Library . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Processing and Data Verification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . JEFF-. (Europe, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . JENDL-. (Japan, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Web Access to Nuclear Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .             References . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . .     Evaluated Nuclear Data Abstract: This chapter describes the current status of evaluated nuclear data for nuclear technology applications. We start with evaluation procedures for neutron-induced reactions focusing on incident energies from the thermal energy up to  MeV, though higher energies are also mentioned. This is followed by examining the status of evaluated neutron data for actinides that play a dominant role in most of the applications, followed by coolants/moderators, structural materials, and fission products. We then discuss neutron covariance data that characterize uncertainties and correlations. We explain how modern nuclear evaluated data libraries are validated against an extensive set of integral benchmark experiments. Afterward, we briefly examine other data of importance for nuclear technology, including fission yields, thermal neutron scattering, and decay data. A description of three major evaluated nuclear data libraries is provided, including the latest version of the US library ENDF/B-VII., European JEFF-., and Japanese JENDL-.. A brief introduction is made to current web retrieval systems that allow easy access to a vast amount of up-to-date evaluated nuclear data for nuclear technology applications.  Evaluation Methodology for Neutron Data The evaluated (recommended) neutron cross-section data represent the backbone of data needed for nuclear technology applications. The incident energies of interest cover an extremely broad energy range of  orders of magnitude. Thus, for fission and fusion reactor systems one needs neutrons from − to  × + eV ( MeV), to be extended up to about  MeV for accelerator-driven systems. If higher incident energies are needed, one resorts to on-fly calculations (not covered here) rather than to evaluated nuclear data libraries. Such a large range of incident neutron energies represents an enormous challenge for developers of evaluated nuclear data libraries. This is coupled to another challenge that stems from the fact that nuclear technology applications need data for about  atomic nuclei, covering the atomic mass range of A = –, from hydrogen to the actinides. There is no simple way to describe the physics of neutrons interacting with atomic nuclei throughout this vast range of energies and different types of nuclei. Rather, nuclear physics uses different approaches and many different models to describe underlying physics. The role of experimental data in the evaluation process is absolutely crucial, with the understanding that measured data must be combined with physics-based models to fill in the gaps and to gain confidence in the judgment as to what is the best reflection of nature. The goals for the evaluations are to comply as closely as possible with experimental microscopic (differential) data, and at the same time to accurately match results from simple benchmark (integral) experiments. The evaluation is a complex process, requiring a detailed knowledge of nuclear reaction physics, experimental databases, nuclear modeling, considerable experience, and attention to details. Once the whole set of isotopes (materials) is evaluated and a library is assembled, a validation of the entire library is performed against hundreds of benchmark experiments. The description of evaluation methodology given below reflects the state-of-the-art methods used in the development of the latest US-evaluated data library, ENDF/B-VII.. This library, released in December , is described in detail in the extensive paper by Chadwick et al. (). The evaluated data are stored in the internationally adopted ENDF- format (Herman and Trkov ). Evaluated Nuclear Data .  Basic Ingredients Basic ingredients of the evaluation process for neutron cross-section data include the EXFOR database of microscopic experimental data, Atlas of Neutron Resonances, and nuclear reaction model codes. Consequently, the evaluation methodology consists of three parts: • Careful analysis of microscopic experimental data. (“Microscopic” is the term used by nuclear data physicists to describe properties of individual nuclei and their interactions, which should be distinguished from properties of large-scale ensembles of nuclei. Thus, microscopic cross section is the interaction probability of one neutron incident on a single target nucleus.) • The low energy region (thermal energy, resolved resonances, and unresolved resonances) is treated by methods developed to analyze neutron resonances. • The fast neutron region is evaluated using methods based on nuclear reaction model calculations and experimental data. Nuclear theory and modeling has played a central role in developing complete crosssection evaluations, by which we mean representations that cover all incident projectile energies, outgoing particle and photon energies, as well as angular distributions. Nuclear reaction theory codes provide a powerful tool to interpolate and extrapolate from the measured data, and naturally incorporate constraints such as unitarity, and energy and momentum conservation. A number of reaction physics codes have been developed that support this work: • Statistical, preequilibrium, direct, and fission models, for use in modeling medium and heavy nucleus reactions, notably Los Alamos National Laboratory (LANL) code GNASH (Young and Arthur, ; Young et al., , ) and code EMPIRE (Brookhaven National Laboratory, Herman et al., b, ), often used in conjunction with coupledchannels optical model code ECIS (CEA Saclay, Raynal, ). • R-matrix codes for light nucleus reactions, and for lower incident energy reactions on heavier targets, notably the Los Alamos code energy-dependent analysis (EDA) (Hale, ) and the Oak Ridge code SAMMY (Larson, ). • Atlas code system (Oh et al., ) for analyzing neutron resonances in terms of multilevel Breit–Wigner (MLBW) formalism by Mughabghab at Brookhaven to produce a comprehensive evaluation of resonance parameters, thermal cross sections, and average resonance parameters for the Atlas of Neutron Resonances (Mughabghab, ). > Figure  qualitatively summarizes nuclear reaction models and related codes as used in the ENDF/B-VII. evaluations for various combinations of mass number and incident energy. In light nuclei, the excited states are generally sparse and well isolated. This feature necessitates the use of special few-body techniques that are feasible due to a limited number of nucleons in the system. We have used the explicit R-matrix theory, implemented in the Los Alamos code EDA, for evaluations of nuclides up to the atomic mass A ≈  (with a few exceptions). This approach, although formally strict, relies on experimental input. In > Fig.  the few-body regime is depicted as a vertical rectangle at the left of the picture. Note, that in this case the same methodology is applied throughout the whole energy range. Increasing the number of nucleons in the target makes usage of few-body models impractical. On the other hand, the large number of excited states facilitates approaches that,  Evaluated Nuclear Data Preequilibrium  DWBA/CC Compound Width fluct.  ⊡ Figure  Schematic representation depicting the use of various evaluation techniques and related codes (in brackets) depending on target mass and incident energy. Arrows to the right of the figure indicate major reaction mechanisms in the fast neutron region and their energy range of applicability to a certain extent, are built upon statistical assumptions. This “statistical regime” appears in > Fig.  to the right of A∼. We have to deal with three distinct energy regions for these nuclei: • Resolved resonance region (including thermal neutrons) • Unresolved resonance region (URR) • Fast neutron region Since the density of neutron resonances increases with A, the upper limit of the resolved resonance region decreases when moving to heavier nuclei. A neighboring region is known as the URR in which overlapping resonances usually produce quite smooth cross sections. Each of these three regions needs different techniques and different reaction modeling. Except for very light nuclei, there is no theory capable of predicting individual resonances. Therefore, realistic evaluations require experimental data for neutron resonances. In ENDF/B-VII., the Reich–Moore approach derived from the R-matrix theory, as implemented in the Oak Ridge code SAMMY, was utilized for the important actinides. For about  fission product nuclei, the MLBW formalism and statistical methods from the Atlas of Neutron Resonances (Mughabghab, ) were used at BNL. The URR is a transitional region that could be treated with the methods from the resolved region as well as in the terms of the models used in the fast neutron region. The fast neutron region involves a whole suite of nuclear reaction models with a strong statistical component resulting from the averaging over many resonances. The Hauser–Feshbach Evaluated Nuclear Data  formulation of the compound nucleus is a key model for any evaluation in the fast neutron region, although in the low energy range it must be corrected to account for the width fluctuation effects. At incident energies above some  MeV, preequilibrium emission has to be taken into account and one implements a variety of semiclassical and quantum-mechanical models. While most of the nuclear reaction models used for the evaluations are predominantly phenomenological, their usage involves a huge number of input parameters. The development of the ENDF/B-VII. library largely benefited from the reference input parameter library (RIPL) (Ignatyuk et al., ; Young et al., ; Capote et al., ), an international project coordinated by the International Atomic Energy Agency (IAEA), Vienna. . Thermal and Resolved Resonance Region Usually, the first step in neutron resonance evaluation is to inspect the well-known compendium of data produced over the years by Mughabghab, BNL, traditionally known as BNL-. Its fifth and latest edition has been published in  as Atlas of Neutron Resonances: Resonance Parameters and Thermal Cross Sections (Mughabghab, ), representing a considerable update to the  (Mughabghab et al., ) and  editions (Mughabghab, ). Often, one is satisfied with these data and adopts them as they are. Indeed, these latest thermal values and resonance parameters provided a basis for more than  new evaluations included in ENDF/B-VII.. In many other cases, however, one performs additional evaluation by applying sophisticated R-matrix analysis to most recent experiments using the Oak Ridge code SAMMY (Larson, ). .. Thermal Energy Region Accurate knowledge of the thermal neutron capture and fission cross sections are of paramount importance for many applications and considerable experimental as well as evaluation effort was expended in obtaining precise and consistent constants at a neutron energy of . eV (velocity v = , m/s). The parameters under consideration are the absorption (σabs ), radiative capture (σγ ) and fission (σ f ) cross sections, the neutron yield data (ν̄, η), as well as Westcott factors gw . Some of these quantities are interrelated as η = ν̄ σf ν̄ = , σabs  + α () where ν̄ = ν̄ p + ν̄ d is the average number of neutrons emitted per neutron-induced fission obtained by the sum of prompt and delayed values, σabs = σγ + σ f and α= σγ . σf () When the scattering cross section (σ s ) is known, the absorption cross section can be determined absolutely to a high degree of accuracy from a measurement of the total cross section as σabs = σtot − σs . The capture cross section for a single resonance is usually represented by the Breit– Wigner formalism. (“Capture” is the short-hand term used by nuclear physicists to describe    Evaluated Nuclear Data 1,000 n ⊡ Figure  Neutron capture cross sections for  Cs in the thermal and resolved resonance energy region computed from Atlas are compared with experimental data. The calculated cross section is Dopplerbroadened to  K; the experimental resolution is not included. Two bound levels were invoked in order to fit the thermal constants radiative capture, i.e., (n, γ) cross sections. This differs from what nuclear engineers might consider “capture,” which is given by the sum of neutron removal cross sections, i.e., (n, γ) + (n, p) + (n, α) + ⋯ ). In the case of several s-wave resonances, the thermal capture cross section at E = . eV is given by σγ (E) = g J Γnj Γγ j . ×  A +   . ) ∑  ( √  A E j Γ j + (E − E  j ) () In this relation, Γnj , Γγ j , and Γj are the neutron scattering, radiative, and total width of the resonance j, respectively; E j is the resonance energy, A is the atomic mass number of the target nucleus, and g J is the statistical spin weight factor defined as gJ = J +  , (I + ) () where J is the resonance spin and I is the target nucleus spin. As an example, > Fig.  shows the evaluated capture cross sections for  Cs in the thermal along with the low-energy resonance region compared with the available experimental data. Similarly, the fission cross section can be described as a sum over positive and negative energy resonance contributions. In the framework of the Breit–Wigner formalism, the fission cross section can be obtained from Evaluated Nuclear Data σ f (E) = g J Γnj Γfj . ×  A +   , ) ∑  ( √  A E j Γj + (E − E  j )  () where Γ j (E) = Γnj (E) + Γγ j + Γfj . () The formalism for neutron elastic scattering is more complicated. Thus, the elastic cross section for a single resonance can be expressed by the sum of three terms σn (E) = π g J Γn Γ cos(ϕ l ) + (E − E  )Γn sin(ϕ l ) l +   sin ϕ l + π  , k k (E − E  ) +  Γ  () where k is the neutron wave number, ϕ l are the phase shifts determined by k, and the potential scattering radius R ′ . The first term describes potential scattering, σpot , which is nearly constant as a function of energy. The second term stands for the symmetric resonance cross section. The third term, containing (E − E )Γn sin(ϕ l ) in the numerator, describes interference between potential (hard-sphere) and resonance scattering, which is negative at E < E  and positive at E > E  . Note that Eq. () does not include resonance–resonance interference term. In order to obtain the thermal scattering cross section, one should resort to an extended version of the above expression, such as provided by the MLBW formalism. We note that in the low-energy approximation simplified expressions can be obtained, including that for potential scattering σpot ≈ πR ′ . () The neutron scattering can also be expressed in terms of spin-dependent free nuclear scattering lengths, a + and a − , associated with spin states I + / and I − /, as ′ a± = R + ∑ j λ j Γnj , (E − E  j ) − iΓj () where λ j = /k is de Broglie’s wavelength divided by π. We note that a ± contain imaginary components and the summation is carried out over all s-wave resonances with the same spin. The total coherent scattering length for nonzero spin target nuclei is then the sum of the spin-dependent coherent scattering widths, a+ and a − , weighted by the spin statistical factor, g+ = (I + )/(I + ) and g− = I/(I + ), a = g+ a+ + g− a− . () The total scattering cross section can then be expressed as   σs = π (g+ a+ + g− a − ) . () If the results of the calculated cross sections do not agree with measurements within the uncertainty limits, then one or two negative energy (bound) levels are invoked.    Evaluated Nuclear Data The potential scattering length or radius, R ′ , is an important parameter, which is required in the calculation of scattering and total cross sections. It can be expressed as ′ ∞ R = R( − R ), () where R is the channel or interaction radius, and R ∞ is related to the distant s-wave resonance contribution. We note that R ′ can be determined to a high degree of accuracy from the measured coherent scattering amplitude by () when the resonance data are complete. .. Westcott Factors and Resonance Integrals In general, in the thermal energy region, capture cross sections follow the /v law, where v is the neutron velocity. Deviations from this behavior are due to the proximity of the first resonance to the thermal energy of . eV, notable examples being  Cd,  Sm, and  Gd. Westcott factors, ideally equal to unity, can be used as a suitable measure of the validity of this law. They are defined as the ratio of the Maxwellian averaged cross section, σ, to the thermal cross section, σ , gw = ∞  v   σ  = e −v /vT σ(v) dv, ∫  / σ  v σ   π v T () where v = , m/s and v T is the most probable velocity for Maxwellian spectrum at temperature T. Resonance integrals represent useful quantities that characterize cross sections in the thermal and resonance region. For a particular reaction σx (E), such as total, elastic scattering, capture, and fission (x = tot, s, γ, f ), in a /E spectrum these are defined as Ix = ∫ ∞ . eV σ x (E) dE , E () where the low energy is determined by the cadmium cutoff energy usually set to . eV, while the upper energy is sometimes set to  keV (JEFF, ). It is important to note that both the thermal energies and resonances contribute to the resonance integrals. Often Westcott factors and resonance integrals are readily available in tabulated form, an example being JEFF-. library (JEFF, ). They can also be conveniently obtained from the web using retrieval systems such as Sigma (Pritychenko and Sonzogni, ) developed and maintained by the National Nuclear Data Center (NNDC) at Brookhaven. .. Resolved Resonance Energy Region Neutron time-of-flight techniques that employ accelerator facilities as neutron sources are used to perform high-resolution cross-section measurements in the resonance region. Then, the measured data are analyzed by a state-of-the-art tool such as SAMMY (Larson, ). This code combines multichannel multilevel R-matrix formalism with corrections for experimental conditions to fit experimental data using generalized least-squares fitting procedures. Evaluated Nuclear Data  Resolved resonances are described by the R-matrix collision theory, which is exact, and the resulting formalism is fairly transparent, though the expressions look rather formidable. In practical applications, several approximations are widely used. The most precise is Reich– Moore followed by the MLBW, while the least precise is single-level Breit–Wigner (SLBW). In ENDF/B-VII. library, Reich–Moore is mostly used for actinides, MLBW is adopted for majority of other materials, while SLBW was essentially abandoned and its use is restricted to the URR. R-matrix channels are characterized by the two particles with spin i and I, the orbital angular momentum l, the channel spin s (where s = i ± I), and the total spin J (where ⃗J = ⃗s + ⃗l) and parity π. Those channels having the same J and π (the only two quantum numbers that are conserved) are collected in the same spin group. Resonances (which appear generally as peaks in the cross sections) are assigned to particular spin groups depending on their individual characteristics; initial assignments may be changed as knowledge is gained during the evaluation process. The goal of the evaluation is to determine those values for the resonance energy (peak position), channel widths, and spin for each of the resonances that provide the best fit to the measured data. In general, partial cross sections can be obtained from a collision matrix U ab , which connects entrance channels a with exit channels b. The formalism, applied to neutron reactions, implies a = n and σnb = π g J ∣δ nb − U nb ∣ , k () where k is the neutron wave number and δ nb is the Kronecker delta symbol. These partial cross sections must be summed over the appropriate entrance and exit channels to yield observable cross sections. The statistical factor g J is the probability of getting the correct angular momentum J from the spins of collision partners, and π/k  relates probability and cross section. In the Reich–Moore formalism as implemented in ENDF-, the only reactions requiring explicit channel definitions are total, elastic scattering and fission; capture is obtained by subtraction (although it is possible to obtain it directly from the collision matrix elements). Neutron channels are labeled by quantum numbers, l, s, and J. The channel spin s is the vector sum of the target spin I and the neutron spin i = /, and takes on the range of values ∣I − /∣ to I + /. The total angular momentum J is the vector sum of l and s, and runs from ∣l − s∣ to l + s. The fission channels f  and f  do not correspond to individual two-body fission product breakup, but to Bohr-channels in deformation space, which is why two are adequate for describing many neutron-induced fission cross sections. If one sums over all incident channels n and exit channels b, and invokes unitarity, the resulting total cross section can be expressed in terms of the diagonal matrix elements as σtot (E) = π ∑ ∑ g J Re[ − U lsJ,l′ s′ J ′ ] . k  lsJ l ′ s ′ J ′ () The elastic cross section is obtained by summing the incident neutron channels over all possible lsJ values and the exit neutron channels over those quantities l ′ s ′ J ′ that have the same ranges as lsJ. The conservation of total angular momentum requires that J ′ = J; usually additional, simplifying conservation rules are imposed, namely, l ′ = l and s ′ = s. The sixfold summation then reduces to the familiar form    Evaluated Nuclear Data σnn (E) = π  ∑ g J ∣ − U lsJ,lsJ ∣ . k  lsJ () The absorption (nonelastic) cross section is obtained by subtraction σabs (E) = σtot (E) − σnn (E). () Fission is obtained from the collision matrix by summing () over all incident lsJ values and over the two exit fission channels, b = f  and b = f , σ f (E) = π lsJ  lsJ  ∑ g J (∣U nf  ∣ + ∣U nf  ∣ ) . k  lsJ () The level-matrix form of the collision matrix is given as J U nb = e −i(ϕ n +ϕ b ) {[( - K)− ]nb − δ nb } , () where − ( - K)nb = δ nb − / / Γnr Γbr i . ∑  r E r − E − i Γγr / () Here, ϕ b is zero for fission, ϕ n = ϕ l , and the summation is over those resonances r that have partial widths in both of the channels n and b; E r is the resonance energy; Γγr is the “eliminated” radiation width; Γnr and Γbr are the partial widths for the rth resonance in channels n and b. The shift factor has been set equal to zero in the above equations (E r′ → E r ); hence they are strictly correct only for s-wave resonances. Originally, the ENDF Reich–Moore format was used for low-energy resonances in fissile materials, which are s-waves. However, it is believed that the “no-shift” formulae can be safely applied to higher l-values also, since the difference in shape between a shifted resonance and one that is not shifted at the same energy has no practical significance. One of the tasks of the evaluator is to assign the orbital momentum, l, for resonances where this has not been done experimentally. In the Atlas of Neutron Resonances this was done by the Bayesian approach that assigns these values probabilistically. The first investigators to apply Bayes’ conditional probability for the determination of parities of  U resonances were Bollinger and Thomas (). Subsequently, Perkins and Gyullassy () and Oh et al. () extensively applied this procedure in the evaluation of resonance parameters. For a resonance with a neutron width weighted by the spin statistical factor, g J Γn , the probability that this resonance is p-wave is given according to Bayes’ theorem of conditional probability by − P(p∣g J Γn ) = ( + P(g J Γn ∣s) ⟨D  ⟩ ) , P(g J Γn ∣p) ⟨D  ⟩ () where ⟨D  ⟩ / ⟨D  ⟩ are the level-spacing ratio, and P(g J Γn ∣s) is the probability that the neutron width is g J Γn if the resonance is s-wave and similarly for p-waves. The Bayesian equation can be solved by taking into account the Porter and Thomas () distribution and taking into account (J + ) degeneration of nuclear levels. Evaluated Nuclear Data  i –T n ⊡ Figure  Porter–Thomas distribution of reduced neutron widths, gΓn , for s-wave resonances of  Cs in the energy region below , eV > Figures  and >  illustrate the Porter–Thomas analysis as applied to the s- and p-wave resonances of  Cs. From this procedure also the average level spacings and strength functions for the s- and p-waves are determined, see () and (). . Unresolved Resonance Region In the URR, the situation is different than in the resolved resonance energy region. The experimental resolution is larger than the width of the resonances and individual resonance parameters can no more be extracted from cross-section fitting. The formalism used for crosssection treatment in URR is therefore based on average values of physical quantities obtained in the resolved resonance range. The values for statistical quantities are determined from the resolved energy region and used as starting values for the unresolved evaluation. The theoretical basis for URR description is the Lane–Lynn approach (Lane and Lynn, ), which for capture gives ⟨σγ ⟩ = ∑ ⟨σγ ⟩Jl = Jl F(α Jl ) π  / , ∑ ⟨Γγ ⟩ l S l Vl E n g J k  Jl ⟨ΓJ ⟩ ()   Evaluated Nuclear Data 133Cs Experimental data Porter–Thomas fit Number of resonances  (gΓ1n)1/2 (eV1/2) ⊡ Figure  Porter–Thomas distribution of reduced neutron widths, gΓn , for p-wave resonances of Cs in the energy region below  eV where the p-wave resonances are detected where the averaged quantities are given in brackets ⟨ ⟩, Vl is the penetrability factor divided by kR; F(α Jl ) is the fluctuation factor, α J l being the ratio of mean radiative and neutron widths; summation is carried over partial waves l and spins J. The URR is treated within the SLBW formalism, which requires the following parameters: the average level spacing, D l , the strength functions, S l , the average radiative widths, Γγ l , and R ′ . After the determination of l values for all resonances, the reduced neutron widths are analyzed in terms of the Porter–Thomas distribution (Porter and Thomas, ) if the number of measured resonances is large enough for a statistical sample. Instead of working with the Porter–Thomas distribution, it is often much simpler to analyze the resonance parameter data with the cumulative Porter–Thomas distribution. Since resonances with small neutron widths are usually missed in measurements, it is necessary to exclude resonances whose reduced widths are smaller than a certain magnitude. By setting a cutoff value, that is, a minimum magnitude of reduced neutron width, the effect of missed small resonances on the resulting average parameters is reduced significantly. The result is N(y) = N r ( − erf(y)), () Evaluated Nuclear Data  where erf is the error function, N r is the corrected total number of resonances, and N(y) is the total number of resonances with reduced neutron width larger than a specified value y, y = (Γnl / ⟨Γnl ⟩) / , () where Γnl is the reduced neutron width for orbital angular momentum l and ⟨Γnl ⟩ is its average value. The two parameters N r and ⟨gΓnl ⟩ are determined through the fitting procedure. The resulting average level spacing D l and neutron strength function S l in a determined energy interval ΔE are then calculated by Dl = Sl = ΔE , Nr −  ⟨gΓnl ⟩ (l + )D l () . () The average radiative widths of neutron resonances are determined from measurements in the resolved energy region by calculating the weighted as well as unweighted values. For nuclei with unmeasured radiative widths, the systematics of s-, p-, and d-wave radiative widths as a function of atomic mass number are used (Mughabghab, ). > Figure  shows the evaluated capture cross sections in the unresolved energy resonance region, compared with the available experimental data for  Cs. The URR is extended up to the first excited level, which is  keV for  Cs. At higher energies (fast neutrons), the evaluations were done by the code EMPIRE. We note an excellent match of cross sections in the boundary of the two energy regions. We note that in the code SAMMY, the unresolved resonance formalism is based on the methodology adopted by the statistical model code FITACS developed by Fröhner (). Values of the average parameters are found from fitting the calculated cross sections to experimental cross sections. The set of parameters that best reproduces the data cannot be reported directly to ENDF/B because the ENDF- format uses a less rigorous SLBW representation. SAMMY/FITACS parameters must therefore be converted into average widths before insertion into ENDF/B library. . Fast Neutron Region Fast neutron region is defined as incident energies above the URR (materials with Z > ) or above resolved resonances that vary between hundreds keV–MeV for light nuclei and eV–keV for actinides. The upper end of the fast neutron region is in general  MeV, though in about % cases in ENDF/B-VII. this has been extended to – MeV. In this energy range, several distinct nuclear reaction models are used to describe the interaction of fast neutrons with atomic nuclei. .. Optical Model and Direct Reactions Spherical optical model is usually used to calculate transmission coefficients for all ejectiles involved in a reaction. In the case of spherical nuclei, the same calculations also determined    Evaluated Nuclear Data n ⊡ Figure  Neutron capture cross section for  Cs in the unresolved resonance energy region extended up to the first excited level. The evaluation at higher energies was performed by EMPIRE. Evaluation adopted by ENDF/B-VII. is compared with experimental data reaction (absorption) cross sections. For deformed nuclei, the incident channel is treated in terms of coupled-channels rather than the spherical optical model. In the latter case, proper coupling also provided cross sections for inelastic scattering to collective levels and related angular distributions of scattered neutrons. In certain cases we also included direct scattering to the collective levels embedded in the continuum. Generally, we chose optical model potentials from a vast selection available in the RIPL library (Capote et al., ), but in the course of ENDF/B-VII. development the original RIPL potentials were often adjusted to improve agreement with recent experimental data or to match the cross section obtained in the unresolved resonance evaluation, and in some cases totally new potentials were constructed. .. Compound Nucleus Decay The statistical model provides the basic underpinning for the whole evaluation procedure. The decay of the compound nucleus (CN) is modeled by the Hauser–Feshbach equations, using transmission coefficients and level densities to represent the relative probabilities of decay in the various open channels. Evaluated Nuclear Data  Schematically, the cross section for a reaction (a, b) that proceeds through the compound nucleus mechanism can be written as σ a,b = σ a Γb . ∑c Γc () The summation over compound nucleus spin J and parity π, and integration over excitation energy E is implicit in (). The decay width Γc is given by Γc = E−B c  ρ c (E ′ )Tc (E − B c − E ′ ) dE ′ , ∑ πρ CN (E) c′ ∫ () where B c is the binding energy of particle c in the compound nucleus, ρ is the level density, and Tc (є) stands for the transmission coefficient for particle c having channel energy є = E − B c − E ′ . Again, for simplicity, we drop explicit reference to the spin and parity in () and the summation extends over all open channels c ′ . For low incident energies, () needs to be corrected for width fluctuation corrections. Since the evaluations extend at least up to  MeV, sequential multiparticle emission had to be included in the Hauser–Feshbach calculations, which in practice implies an energy convolution of multiple integrals of the type of (). In order to account for the competition between γ-emission and emission of particles along the deexcitation chain, our calculations always involve a full modeling of the γ-cascade that conserves angular momentum. The formalism for γ-rays transitions is based on the giant dipole resonance (GDR) model known as the Brink–Axel hypothesis (Alex, ; Brink, , unpublished). GDR parameters are taken from the experimental compilation and/or systematics contained in the RIPL library. We note, that our calculations account for GDR splitting due to nuclear deformation. In GNASH, the γ-ray transmission coefficients are obtained from the γ-ray strength function formalism of Kopecky and Uhl (). EMPIRE allows for a suite of γ-ray strength functions. Typically, we used Mughabghab and Dunford’s prescription known as GFL (Mughabghab and Dunford, ) or Plujko’s modified Lorentzian referred to as MLO (Plujko and Herman, ). In both codes, the γ-ray strength functions can be, and often are, normalized to experimental information on πΓγ /D  or adjusted to reproduce capture cross sections. Nuclear level densities along with optical model transmission coefficients are the two most important ingredients of the statistical model. In GNASH , the description of the level densities in the continuum follows the Ignatyuk form of the Gilbert–Cameron formalism, including a washing out of shell effects with increasing excitation energy. Most of the evaluations performed with EMPIRE employed level densities that are specific to the EMPIRE code. The formalism uses the superfluid model below a critical excitation energy and the Fermi gas model at energies above it. Collective enhancements due to nuclear vibration and rotation are taken into account in the nonadiabatic approximation, that is, they are washed out when excitation energy increases. Differently from other formulations, EMPIRE-specific level densities explicitly account for the rotation-induced deformation of the nucleus and determine spin distributions by subtracting rotational energy from the energy available for intrinsic excitations.    .. Evaluated Nuclear Data Width Fluctuation Correction At low incident energies, the statistical approximation that entrance and exit channels are independent (Bohr independence hypothesis) is not valid anymore due to correlations between entrance and exit channels. The Hauser–Feshbach equations have to be modified in order to include the so-called width fluctuation correction factors accounting for the coupling between the incident and outgoing waves in the elastic channel. The GNASH code does not calculate these correction factors but rather imports them as the result from an auxiliary code (usually COMNUC, Dunford, , which uses the Moldauer model). EMPIRE, by default, uses internal implementation of the HRTW approach (Hofmann et al., ) that can be summarized with the following equation: − HRT W σab = Va Vb (∑ Vc ) [ + δ ab (Wa − )] . () c This formula is, essentially, equivalent to the Hauser–Feshbach expression () but the elastic channel is enhanced by the factor Wa . In () the quantities Vc replace optical model transmission coefficients that appear in the original Hauser–Feshbach formula. .. Preequilibrium Models The probability that a system composed of an incident neutron and a target nucleus decays before thermal equilibrium is attained becomes significant at incident energies above  MeV. In any preequilibrium model, the excited nuclear system (composite nucleus) follows a series of ever more complicated configurations, where more and more particle-hole (p-h) states are excited. In each stage, a possible emission of a particle competes with the creation of an intrinsic particle-hole pair that brings the system toward the equilibrium stage. Particle emission from the early stages is characterized by a harder spectrum and forward peaked angular distributions. The exciton model is a semiclassical formulation of the preequilibrium emission that is used in GNASH and EMPIRE. The core of the model is the so-called master-equation that governs time dependence of occupation probabilities, Pn , for various p-h stages ħ dPn = ∑ Λ n,m Pm − Γn Pn , dt m () where the total decay width of the stage n is given in terms of the partial transition widths Λ l ,n and partial width Γe,n for the emission of particle e by Γn = ∑ Λ l ,n + ∑ Γe,n . l () e Due to the two-body nature of the nuclear force, intrinsic transitions occur only between neighboring stages, and the transition matrix Λ is tri-diagonal, the off-diagonal terms accounting for backward and forward transitions. In GNASH, the preequilibrium phase is addressed through the semiclassical exciton model in combination with the Kalbach angular-distribution systematics (Kalbach, ). These systematics provide a reasonably reliable representation of the experimental database. Evaluated Nuclear Data  EMPIRE implements a suite of preequilibrium codes including two versions of the exciton model (PCROSS and DEGAS, Běták and Obložinský, ), and the Monte-Carlo approach DDHMS (Blann, ; Blann and Chadwick, , ) in addition to the quantummechanical multistep direct (MSD) and multistep compound models. A new Monte Carlo preequilibrium model allows unlimited emission of preequilibrium neutrons and protons, and is therefore well suited for the study of high-energy reactions up to a few hundreds of MeV. The model of choice in EMPIRE is the statistical MSD theory of preequilibrium scattering to the continuum originally proposed by Tamura, Udagawa, and Lenske (TUL) (Tamura et al., ). The evolution of the projectile-target system from small to large energy losses in the open channel space is described in the MSD theory with a combination of direct reaction (DR), microscopic nuclear structure, and statistical methods. The modeling of multistep compound (MSC) processes in EMPIRE follows the approach of Nishioka et al. (NVWY) (). The formal structure of the NVWY formula resembles the matrix representation of the master-equation typical for classical preequilibrium models. However, the NVWY formalism is strictly derived from basic principles. Microscopic quantities that constitute ingredients of the NVWY formula were linked to the macroscopic, experimentally known, quantities in Herman et al. (), which was an essential step allowing for practical application of the theory. .. Light Nuclei In the case of light nuclei (from hydrogen to oxygen, A = –, which mostly serve as coolants and moderators), the statistical approach cannot be applied and the above methodology should be replaced by the R-matrix approach. In the USA, this approach is pursued by Los Alamos and virtually all light nuclei evaluations in ENDF/B-VII. were performed by the EDA code developed over the years by Hale (, ), which is based on R-matrix formalism in its most general form. R-matrix theory is a general framework for describing nuclear reactions that are particularly well suited for including resonances. It is the mathematically rigorous phenomenological description of what is actually seen in an experiment. This is not a model of neutron–nucleus interaction, rather it parametrizes measurements in terms of observable quantities. This theory properly describes multichannel nuclear reactions and builds in all the fundamental conservation laws, symmetries, and analytic properties of nuclear interactions. The experimental cross-section data from all relevant reactions, including neutron and charged particles, are taken into account and fitted simultaneously. This allows obtaining a single set of multichannel, multilevel R-matrix parameters that describe all the desired neutron-induced cross sections for light nucleus under consideration. . Fission Nuclear fission remains the most complex topic in applied nuclear physics. Since its discovery, it has remained an active field of research, and from the evaluation point of view it poses one of the most difficult problems.    .. Evaluated Nuclear Data Fission Modeling The status of fission modeling and related parametrization relevant to evaluation is summarized in the extensive paper dedicated to the RIPL by Capote et al. (). The main concepts of nuclear fission theory are based essentially on the liquid-drop model (Bohr and Wheeler, ; Frenkel, ). According to this model, when a nucleus is being deformed, the competition between the surface tension of a nuclear liquid drop and the Coulomb repulsion related to the nuclear charge leads to the formation of an energy barrier, which prevents spontaneous decay of the nucleus by fission. The penetrability of this barrier depends on its height and width and is a dominant factor in determining the fission cross section. Decrease in height and/or width results in an exponential increase in barrier penetrability, which leads to increased fission. For most of the practical calculations, one-dimensional fission barrier is considered. The present knowledge indicates that the pre-actinides have single-hump barriers, while the actinides have double- or triple-humped barriers. Usually, the barriers are parametrized as a function of the deformation (β) by inverted parabolas,  B i (β) = B fi − μħ  ω i (β − β i ) ,  i = , N, () where N is the number of humps, the energies B fi represent maxima of the deformation potential, β i are the deformations corresponding to these maxima (saddle points), the harmonic oscillator frequencies ω i define the curvature of the parabolas, and μ is the inertial mass parameter approximated usually by a semiempirical expression. > Figure  illustrates the relationship among the above-mentioned quantities in a typical case of a double-humped fission barrier (N=). The quasi-stationary states in the second well (the class II states) are also depicted in the plot. B Eci (Jp) ei (KJp) Transition states U Ei (KJp) bi ai Bfi Class II states b ⊡ Figure  Energy of a double-humped fission barrier in function of the deformation β along with associated barrier parameters: Bfi is the height of the fundamental fission barrier i; є i (KJπ) is the energy of the transition state i; Eci (Jπ) is the cutoff energy, above which the continuum starts for a barrier i. Fission barriers associated with each discrete transition state are shown Evaluated Nuclear Data  Above each barrier hump there is a spectrum of excited levels commonly referred to as transition states. Close to the top of the hump these levels are well separated and can be treated individually while at higher excitation energies (above E ci ) the concept of level densities must be invoked, as indicated in > Fig.  with the shaded regions. The discrete transition states for all barriers i (i = ,  for a double-humped barrier) are obtained by building rotational levels on top of vibrational or non-collective levels that serve as a base (bandheads). They are characterized by a set of quantum numbers (angular momentum J, parity π, and angular momentum projection on the nuclear symmetry axis K) with the excitation energies E i (K Jπ) = B fi + є i (K Jπ) = B fi + є i (Kπ) + ħ [J(J + ) − K(K + )], ℑ i () where є i (Kπ) are the bandhead energies and ħ  /ℑ i are the inertial parameters. A parabolic barrier with height E i (KJπ) and curvature ħω i is associated with each transition state. The transmission coefficients through each hump are expressed in first-order approximation in terms of the momentum integrals for the humps K i (U) = ± ∣∫ bi ai √ μ[U − B fi (β)]/ħ  d β∣ , i = , , () where a i and b i are the points indicated in > Fig.  and U is the excitation energy in the fissioning nucleus. The + sign is taken when the excitation energy U is lower than the hump under consideration and the − sign when it is higher. In the case of a single parabolic barrier, () yields the well-known Hill–Wheeler transmission coefficient (Bjornholm and Lynn, a) TiHW (U) =  .  + exp[−(π/ħω i )(U − B fi )] () The total fission transmission coefficient for a given excitation energy U, spin J, and parity π is determined by summing the penetrabilities through barriers associated with all allowed transition levels, that is, Ti (U Jπ) = ∑ Ti (U Kπ) + ∫ K≤J ∞ E ci ρ i (εJπ) dε .  + exp [− π ω i (U − B fi − ε)] ħ () The sum runs over all the discrete transition levels having the same spin J and parity π as those of the decaying compound nucleus, and the integration runs over the continuum of the transition levels described by the level densities ρ i (εJπ). Usually the wells are considered deep enough so that the transmission coefficient can be averaged over the intermediate structures. For a doublehumped barrier the fission coefficient becomes T f (U Jπ) = T (U Jπ)T(U Jπ) , T (U Jπ) + T (U Jπ) + Tγ II (U Jπ) () where T and T are the penetrabilities of the inner and the outer humps, respectively, calculated according to (), and Tγ II is the probability for gamma decay in the second well. Finally, the relation used in the statistical model for the fission cross section is σ a, f (E) = ∑ σ a (E Jπ)Pf (E Jπ), Jπ ()    Evaluated Nuclear Data where σ a (E Jπ) is the population of the fissioning nucleus in the state E Jπ, and P f (E Jπ) represents the fission probability computed for a specific representation of fission barrier. The fission probability is usually defined as the ratio of the fission coefficient T f to the sum of all the transmission coefficients including the competing channels ∑d Td , P f (E Jπ) = T f (E Jπ) . T f (E Jπ) + ∑d Td (E Jπ) () Similar definitions apply for other decay probabilities. In GNASH, fission probabilities are calculated from the quantum-mechanical transmission coefficient through a simple double-humped fission barrier, using uncoupled oscillators for the representation of the barriers. The barrier penetrabilities are computed using the Hill–Wheeler formula for inverted parabolas. An additional parameter is used to account for level density enhancement due to asymmetry at saddle points (Young et al., ). Version . (Lodi) of the EMPIRE code introduced an advanced fission formalism that is applicable to multi-chance fission induced by light particles and photons. It uses an optical model for fission, that is, allows for an absorption through the imaginary potential in the second well, to calculate transmission through coupled single-, double-, and triple-humped fission barriers. Such calculations can start from sub-barrier excitation energies. In the case of a double-humped barrier, the expression is generalized to account for multimodal fission. For light actinides, a triple-humped fission barrier with a shallow tertiary well, which accommodates undamped vibrational states, is employed. This fission model can provide a good description of experimental data, including gross vibrational resonant structure at sub-barrier energies. As an example of the complexity of the fission channel modeling in EMPIRE, we show  Th fission cross sections in > Fig. . Fine details can be seen in the insert, revealing complex resonance-like structure of fission in the threshold energy region. These fine details are fully described by the model, with the caveat that the fission model parametrization was obtained from careful fits to data. .. Prompt Fission Neutron Spectra One of the most intriguing aspects of evaluation of actinides are prompt neutron fission spectra. The Los Alamos (Madland–Nix) model of the prompt fission neutron spectrum and average prompt neutron multiplicity is based upon classical nuclear evaporation theory and utilizes an isospin-dependent optical potential for the inverse process of compound nucleus formation in neutron-rich fission fragments (Madland and Nix, ). This model, in its exact energydependent formulation, has been used to calculate the prompt fission neutron spectrum matrix for the n +  U, n +  U, and n +  Pu systems, and these appear in ENDF/B-VII. with the tabulated distribution (LF=) law. > Figure  shows the average prompt fission neutron emission energy as a function of incident energy for , U and  Pu for both the new ENDF/B-VII. evaluations and the old ENDF/B-VI evaluations. Evaluated Nuclear Data 232Th(n,f) 1 Cross section (b)  0.15 10 –2 0.10 10 –4 0.05 0 1.0 10 –6 –1 10 1.5 2.0 2.5 3.0 1 10 Incident neutron energy (MeV) ⊡ Figure  Average secondary neutron energy (MeV) Neutron-induced fission cross section on  Th compared with experimental data from EXFOR. Insert shows fine details of the resonance-like structure of these fission cross sections 2.5 U235 ENDF/B-VI ENDF/B-VII U238 ENDF/B-V ENDF/B-VII Pu239 ENDF/B-VI ENDF/B-VII 2.4 2.3 2.2 2.1 2 1.9 1.8 0 5 10 15 Incident neutron energy (MeV) 20 ⊡ Figure  First moments (average energies) of , U and  Pu prompt fission neutron spectra from ENDF/BVII. calculated with the Los Alamos model (Madland and Nix, ) in comparison with those of ENDF/B-VI    .. Evaluated Nuclear Data Peculiarities of Fission Cross Section Evaluation It is notoriously difficult to describe and parametrize the fission process. In particular, fission is extremely sensitive to fission barriers that are very difficult to predict and their actual values depend on other parameters such as nuclear level densities. In view of this, in most cases fission cross sections are directly adopted from experiments. Then, the modeling is performed and used to evaluate all other reaction channels. This is the approach applied for years by Los Alamos, using the code GNASH. Most recently it has been used in the unprecedented evaluation of a complete set of ten uranium isotopes ,,,,,,,,, U as well as  Pu by Young et al. (), from keV energies to  MeV. These evaluations can be seen as the core of the ENDF/B-VII. library, with more details provided in the subsequent part of this chapter. Recent advances in fission modeling, gradually implemented into the codes such as EMPIRE, allow cautious bypassing traditional mantra of using purely experimental data for fission cross-section evaluations. An example would be  Th evaluation adopted by ENDF/BVII.. In that evaluation, the fission channel was based on modeling with parametrization derived from fits to data, see > Fig. .  Neutron Data for Actinides In general, actinides are the most important materials (isotopes) in most of the nuclear technology applications. The “big three” actinides,  U,  U, and  Pu, usually play the dominant role. These three plus  Th constitute the major actinides; the remaining about  actinides including other isotopes of U and Pu as well as Np, Am, and Cm fall into the category of minor actinides. It should be emphasized that major actinides are evaluated extremely carefully and attention paid to details is exceptionally high. The nuclear data community spent considerable amount of time on these evaluations (Young et al., ). What is shown below is a glimpse on progress made between  release of the US library ENDF/B-VII. and  release of the ENDF/BVII. library (Chadwick et al., ). .  U Evaluation The ENDF/B-VII. evaluation in the URR was performed by ORNL. This was complemented with the evaluation in the fast neutron region performed by LANL. The remaining data were taken over from ENDF/B-VI.. ..  U, Unresolved Resonance Region The SAMMY code has been used to perform the unresolved resonance evaluation of the  U cross sections from . up to  keV (Leal et al., ). SAMMY generates average resonance parameters based on a statistical model analysis of the experimental average cross sections. The primary use of the average resonance parameters is to reproduce the fluctuations in the cross sections for the purposes of energy self-shielding calculations. Evaluated Nuclear Data  Cross section (b) 235U(n, l) 1,000 Incident neutron energy (KeV) ⊡ Figure  Evaluated  U(n, γ) capture cross section compared with data and with the JENDL-. evaluation A good representation of the average cross section was achieved with the new evaluation as shown for the  U radiative capture in > Fig. . The thermal ν̄ value for  U , which was taken over from ENDF/B-VI., is ν̄ = .. This value is slightly higher than that from the neutron standards, ., but within experimental uncertainties in order to optimize agreement with the critical assembly benchmarks. ..  U, Fast Neutron Region The new  U evaluation builds upon the previous ENDF/B-VI. file, with a number of improvements from Los Alamos. They include improved fission cross sections from the new standards, prompt ν̄ based on a covariance analysis of experimental data, (n, n), (n, n) cross sections based on new data, new prompt fission spectra taken from Madland, new delayed neutron time-dependent data, and improved inelastic scattering at  MeV and below. The previous  U ENDF/B-VI. evaluation has performed reasonably well in integral validation tests based on simulations of critical assemblies. The principal deficiency the ENDF/BVII. developers wanted to remove was an underprediction of reactivity. For instance, the calculated keff for Godiva, a fast critical assembly based upon highly enriched uranium (HEU) in a spherical configuration, was ., compared to experiment of ..   Evaluated Nuclear Data 2.2 235 U(n,f) 2.0 Cross section (b)  1.8 1.6 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII ENDF/B-VII Std (exp) 1.4 1.2 1.0 0 5 10 15 Incident neutron energy (MeV) 20 ⊡ Figure  Evaluated fission cross section compared with measured data, as represented by a covariance analysis of experimental data (referred to as ENDF/B-VII. Standard). Our new evaluation follows the Standard evaluation of the experimental data. Other evaluations from joint evaluated fission and fusion (JEFF) and JENDL are also shown The  U fission cross section is shown in > Fig. , with comparison to the previous ENDF/B-VI. evaluation, and to the latest joint evaluated fission and fusion (JEFF) and JENDL evaluations. This new result comes from the recent international standards project, and the evaluation follows the statistical analysis of the measured data. This evaluation is .–.% higher than the previous ENDF/B-VI Standard in the – MeV region, and significantly higher above  MeV. The impact of the higher fission cross section in the fast region (a few MeV) is particularly important, having the effect of increasing the criticality of fast systems. The Nuclear Energy Agency (NEA Paris) project studied the fission prompt neutron spectrum for  U. The final report (Weigmann et al., ) noted that significant uncertainties still exist in the prompt spectrum at thermal energies. Because of this uncertainty, adopted were Madland’s new data for all energies except thermal, where the previous ENDF/B-VI evaluation was preserved. In > Fig. , we show the prompt fission neutron emission spectrum, compared with measurements by Boykov et al. () for . MeV neutrons on  U and plotted as a ratio to the σ c = constant approximation to the Los Alamos model (Madland and Nix, ). It is evident that the present ENDF/B-VII. agrees better with the data by Boykov et al. The new  U(n, n) cross section comes from a GNASH code theory prediction, baselined against the measured data. A comparison with experimental data, and with ENDF/B-VI., JEFF-., and JENDL-. is given in > Fig. . The previous  U evaluation was known to poorly model Livermore pulsed sphere data that measure the downscattering of  MeV neutrons, in the region corresponding to inelastic scattering (the – MeV excitation energy region in  U). The angle-integrated spectrum for Evaluated Nuclear Data  Ratio to constant sc calculation 1.5 n (2.9MeV) + 235U 1.3 1.1 0.9 Experiment Present calculation Enitratio Unitratio 0.7 0.5 100 101 Laboratory emission neutron energy (MeV) ⊡ Figure  Prompt fission spectrum for . MeV neutrons incident on  U shown as a ratio to the σc = constant approximation to the Los Alamos model. The data are from Boykov et al. () 1.0 235U(n,2n) Cross section (b) 0.8 0.6 Becker, 1998 Frehaut, 1980 Mathur, 1969 Mathur, 1972 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII 0.4 0.2 0.0 5 10 15 Incident neutron energy (MeV) 20 ⊡ Figure  Evaluated  U(n, n) cross section compared with data and with previous evaluations  MeV is shown in > Fig. , and the oscillatory structure between  and  MeV emission energy is due to the new inelastic scattering to collective states. This is the first time that preequilibrium and DWBA mechanisms for inelastic scattering have been included high into the continuum for evaluated actinide databases. In ENDF/B-VII., this approach was followed for ,,, U,  Pu,  Th, and , Pa.   Evaluated Nuclear Data 101 235U+n Neutron emission E n =14 MeV Cross section (b/MeV)  100 10–1 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII 10–2 10–3 0 2 4 6 8 10 12 Emission neutron energy (MeV) 14 16 ⊡ Figure  Evaluated  U(n, xn) neutron production energy-spectrum compared with previous evaluations. No measured data exist, though our calculations were guided by measured data for  U .  U Evaluation In natural uranium,  U is the dominant isotope, with .% abundance and a half-life of . ×  years. Although it is fast fissioner, not suitable for thermal systems, its high abundance stipulates that it must be evaluated very carefully. ..  U, Resolved and Unresolved Resonance Region Numerous criticality studies, involving low-enriched thermal benchmarks demonstrated a systematic keff underprediction of about −.% (− pcm) or more with ENDF/B-VI.. International activity was formed to solve this problem. First, the  U capture cross sections were investigated using specific integral experiments sensitive to the capture resonance integrals: • Correlation between keff and  U capture fraction • Measurements of  U spectral indices and effective capture resonance integral • Postirradiation experiments that measure the  Pu isotopic ratio as a function of burn-up These tests (Courcelle, ) supported a slight reduction of the effective resonance integral between . and %. A new analysis of the  U cross section in the resolved-resonance range was performed at ORNL in collaboration with the CEA (Derrien et al., ). The SAMMY (Larson, ) analysis of the lowest s-wave resonances below  eV led to resonance parameters slightly different from those of ENDF/B-VI. as shown in > Table . The  U(n, γ) thermal cross section, recently recommended by Trkov et al. (), σ = . ± .b, was adopted. The scattering cross section at thermal energy was also Evaluated Nuclear Data  ⊡ Table  Resonance parameters of the  U s-wave resonances in ENDF/B-VII. and ENDF/B-VI. Energy (eV) . ENDF/B-VII. R′ = . fm ENDF/B-VI. R′ = . fm Γγ (MeV) Γn (MeV) Γγ (MeV) . . . Γn (MeV) . . . . . . . . . . . . . . . . . . . . . . . . . . Although the differences look small, they have a positive impact on performance revisited and the effective scattering radius Reff as well as the parameters of the external levels have been carefully assessed. > Figure  shows an example of the SAMMY fit of capture measurements in the keV energy range. As suggested by integral experiments, this new evaluation proposes a slight decrease of the effective capture resonance integral by about .%, compared to ENDF/B-VI.. One expected consequence of this new evaluation is an increase of the calculated multiplication factor for low-enriched lattices from about . to .% (– pcm), depending on the moderation ratio. The combination of the new LANL  U inelastic data in the fast neutron region with the ORNL resonance parameter set gave a satisfactory correction of the reactivity underprediction. ..  U, Fast Neutron Region The new ENDF/B-VII. evaluation is based upon evaluations of experimental data and use of GNASH and ECIS nuclear model calculations to predict cross sections and spectra. Prior to the present work, there were some long-standing deficiencies, as evident in critical assembly integral data testing. First, there was the reflector bias – the phenomenon whereby fast critical assemblies showed a reactivity swing in the calculated keff in going from a bare critical assembly (e.g., Godiva (HEU) or Jezebel ( Pu)) to  U-reflected critical assembly (e.g., Flattop-, or Flattop-Pu), whereas measurements showed keff =  for both assemblies. Second, thermal critical assemblies involving  U have showed a calculated underreactivity for ENDF/B-VI.. Third, some intermediate energy critical assemblies involving large quantities of  U, such as Big-, were modeled very poorly using ENDF/B-VI. data. The nuclear data improvements made for ENDF/B-VII. largely removed these deficiencies. Similar methods in the fast neutron region applied at the CEA/Bruyères-le-Châtel lead to similar improvements in the JEFF-. library (López Jiménez et al., ).   Evaluated Nuclear Data 5 σg (b) 4 238U(n,g ) 3 2 σg (b) 1 0 5 4 3 2 1 0 5 4 σg (b)  3 2 1 0 5.5 5.6 5.7 5.8 Incident neutron energy (keV) 5.9 6.0 ⊡ Figure  Experimental capture data on  U (one sample measured by De Saussure and two samples by Macklin) compared to the results of the SAMMY fit in the range .–. keV The fission cross section was taken from the new recommendations of the IAEA Standards group, based on a Bayesian analysis of measured data. As can be seen in > Fig. , the fission cross section differs from the previous ENDF/B-VI. cross section in some important ways, being ≈.% larger in the – MeV region, and –% larger in the – MeV region. Above  MeV, the principal reason for the change is newer and more precise measurements from various laboratories, which were not available for ENDF/B-VI. The prompt fission spectrum in ENDF/B-VII. for  U came from a new analysis by Madland using the Los Alamos model. The average energies are compared with Los Alamos model predictions in > Fig. , and the agreement with experimental data is seen to be good. Nuclear reaction modeling with the GNASH and ECIS codes played an important role for improving the treatment of inelastic scattering to discrete levels and to the continuum. This work impacts both the scattering in the fast region, as well as at  MeV and below. In the former case – inelastic scattering in the fast (a few MeV) region – our improved data for inelastic scattering result in significant improvements in the critical assembly validation tests, not just for fast critical assemblies, but also for more moderated and thermal assemblies (the LEU-COMPTHERM series). An example of the secondary neutron emission spectrum at . MeV incident energy on  U is shown in > Fig.  for an emission angle of ○ . It is evident that the new ENDF/BVII. evaluation provides a much more accurate representation of the secondary spectrum, and its angular distribution, than the earlier ENDF/B-VI. evaluation. Evaluated Nuclear Data  1.5 Cross section (b) 238U(n,f ) 1.0 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII ENDF/B-VII Std (exp) 0.5 0.0 0 5 10 15 Incident neutron energy (MeV) 20 ⊡ Figure  Evaluated  U fission cross section based on a covariance analysis of the experimental data from the Standards project (labeled Std) 2.5 n +238U prompt fission neutron spectrum matrix Average energy <E> (MeV) 2.4 2.3 2.2 2.1 2.0 1.9 1.8 Los Alamos model Previous exp. FIGARO exp. 1.7 1.6 1.5 0 2 4 6 8 10 12 14 Incident neutron energy (MeV) 16 18 ⊡ Figure  First moment (average energies) of the n+ U prompt fission neutron spectrum matrix calculated with the Los Alamos model shown together with those extracted from earlier experiments and the more recent CEA/Los Alamos FIGARO measurements (Ethvignot et al., ) Our evaluated neutron capture cross section is shown in > Fig. , and is compared with the result from the Standards project (which represents a Bayesian analysis of a large amount of experimental data). It should be noted that in the s–s keV region, the evaluated cross section lies below the bulk of the measurements that one might find in the CSISRS (EXFOR) experimental database. This is intentional and represents the conclusions of evaluators who   Evaluated Nuclear Data 100 238U+n Cross section (b/sr-MeV) Neutron emission En = 6.1 MeV, q =45° 10 –1 10 – 2 10 – 3 0.0 Baba, 1989 Los Alamos, 2004 ENDF/B-VI JEF-3.0 JENDL-3.3 2.0 4.0 6.0 Emission neutron energy (MeV) 8.0 ⊡ Figure  Evaluated  U(n, xn) neutron production energy-spectrum, compared with data, and with different evaluations 100 238U(n,g) Cross section (b)  10–1 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII ENDF/B-VII Std (exp) 10–2 10 –1 100 Incident neutron energy (MeV) 101 ⊡ Figure  Evaluated  U(n, γ) neutron capture cross section, compared with data (labeled Std), and with previous evaluations. The standards evaluation resulted in uncertainties less than % over the En = − –. MeV region Evaluated Nuclear Data  2.0 238 U(n,2n) Cross section (b) 1.5 1.0 0.5 0.0 5 Knight, 1958 Frehaut, 1980a Frehaut, 1980b Veeser, 1978 Karius, 1979 Raics, 1980 Barr, 1966 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII 10 15 20 Incident neutron energy (MeV) 25 ⊡ Figure  Evaluated  U(n, n) cross section compared with data and with previous evaluations have studied the various measurements and concluded that the lower measurements are most accurate. See, for instance, the NEA WPEC Subgroup- report (Kanda and Baba, ). The neutron capture cross section of  U can be tested in an integral way by comparing the production of  U in a critical assembly for various neutron spectra in different critical assemblies, ranging from soft spectra to hard spectra. The results show that the evaluation reproduces integral capture rates reasonably well. Like radiative capture, cross sections such as (n, n) and (n, n) are also important for production–depletion studies of uranium isotope inventories and transmutation. Our new evaluation of the (n, n) cross section is shown in > Fig.  and is compared with ENDF/B-VI. and with measured data. . ..  Pu Evaluation  Pu, Resonance Region The evaluation by Derrien and Nakagawa of the resonance region was taken over from the ENDF/B-VI. library without any change. ..  Pu, Fast Neutron Region The upgrades made to the  Pu evaluation included improved description of  Pu (n, n), adoption of fission cross section from Standards, new analysis of the prompt fission spectrum,   Evaluated Nuclear Data 2.6 239 Pu(n,f ) 2.4 Cross section (b)  2.2 2.0 1.8 1.6 1.4 0.0 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII ENDF/B-VII Std (exp) 5.0 10.0 15.0 Incident neutron energy (MeV) 20.0 ⊡ Figure  Evaluated  Pu fission cross section compared with measured data, as represented by a covariance analysis of experimental data (referred to as ENDF/B-VII. Standard). The new evaluation follows the Standard evaluation of the experimental data. Other evaluations from JEFF and JENDL are also shown new delayed neutron time-dependent data, ν̄ modifications, and improved inelastic scattering at  MeV and below. The earlier  Pu ENDF/B-VI. evaluation exhibited an under-reactivity, with the simulated Jezebel keff being ≈ .. The new evaluation is more reactive, mainly because of the higher fission cross section in the fast region, with keff ≈ .. The new fission cross section is shown in > Fig.  and is compared with the older ENDF/BVI. evaluation. Because the earlier  U ENDF/B-VI. standard fission cross section was too low in the fast neutron energy region, and has now been increased in ENDF/B-VII., this leads to an increased  Pu fission cross section in this energy region too, since the plutonium fission cross section is strongly dependent on  Pu/ U fission ratio measurements. The prompt fission neutron spectrum, as a function of incident neutron energy, was reevaluated using the Madland–Nix approach. An example of the prompt fission spectrum is shown in > Fig. , for . MeV neutrons incident on  Pu, compared with the data by Staples et al. (). The new (n, n) cross section was based upon a Livermore–Los Alamos collaboration, involving GEANIE gamma-ray measurements of the prompt gamma rays in  Pu, together with GNASH code theory predictions of unmeasured contributions to the cross section. Prior to this work, precision activation measurements had been made near  MeV by Lougheed et al. (). Other measurements based on measuring the two secondary neutrons were thought to be problematic and were therefore discounted in the evaluation (> Fig. ). Evaluated Nuclear Data  100 239 Neutron energy spectrum (1/MeV) n (1.5 MeV) + Pu 10 –1 10 –2 10 –3 10 –4 Experiment Los Alamos model 10 –5 100 101 Laboratory emission neutron energy (MeV) ⊡ Figure  Prompt fission neutron spectrum for . MeV neutrons incident on  Pu. The data of Staples et al. () are shown together with the least-squares adjustment to the Los Alamos model 600 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII Other Data Lougheed, 2000 239 Pu(n,2n) Cross section (mb) 500 400 Frehaut ,1985 GEANIE-GNASH 300 200 100 0 5 8 10 12 15 18 20 Incident neutron energy (MeV) ⊡ Figure  Evaluated  Pu(n, n) cross section compared with data, and with previous evaluations. The evaluation was based upon the GEANIE-GNASH data and the Lougheed et al.  MeV data (Lougheed et al., ). The ENDF/B-VII. evaluation (red line) is also referred to as the “GEANIE-project” evaluation   Evaluated Nuclear Data 1 10 239 Pu+n Neutron emission 10 Cross section (b/MeV)  10 En = 14 MeV 0 –1 10 –2 10 –30 JENDL-3.3 JEFF-3.0 ENDF/B-VI.8 ENDF/B-VII 2 4 6 8 10 12 14 16 Emission neutron energy (MeV) ⊡ Figure  Evaluated  Pu(n, xn) neutron production energy-spectrum compared with previous evaluations. No fundamental experimental data exist for this reaction, although Livermore pulsed data for transmission do exist As was the case for  U, the previous  Pu evaluation did not include enough inelastic scattering into the continuum for  MeV neutron energy and below, leading to poor performance in simulations of Livermore pulsed spheres in the – MeV excitation energy region. We inferred collective excitation strength from DR analyses of  U data by Baba et al. and assumed similar strengths for  Pu. > Figure  shows the new angle-integrated neutron spectrum data compared to the ENDF/B-VI. evaluation (no measurements exist). This procedure led to a much improved MCNP modeling of the pulsed-sphere data. .  Th Evaluation Recent developments in innovative fuel cycle concepts and accelerator-driven systems for the transmutation of nuclear waste have created a new interest in nuclear data for light actinides, with fission being crucially important for the design of new reactor systems. Additionally, there is strong scientific interest in the “thorium anomaly” (Bjornholm and Lynn, b), which implies that in the thorium region the second-order shell effects split the outer fission barrier giving the so-called triple-humped structure. The evaluation of  Th was completed in  (Trkov, , ). The resonance parameters were obtained by Leal and Derrien () from a sequential Bayes analysis with the SAMMY code of the experimental database including Olsen neutron transmission at Evaluated Nuclear Data  ORELA (Olsen and Ingle, ), not yet published capture data by Schillebeeckx (GELINA), and Gunsing (n-TOF) in the energy range – keV. Unresolved resonance parameters (– keV) were derived by Sirakov et al. (,  Th: evaluation of the average resonance parameters and their covariances in the URR from  to  keV, private communication). Evaluation in the fast energy region (Capote et al. , evaluation of fast neutron-induced reactions on  Th and , Pa up to  MeV, private communication) was fully based on nuclear model calculations using the EMPIRE-. code (Herman et al., a,b; Sin et al., ). A crucial point was the selection of the proper coupled-channel optical model potential. The direct interaction cross sections and transmission coefficients for the incident channel on  Th were obtained from the dispersive coupled-channel potential of Soukhovitskii et al. () (RIPL ). Hauser–Feshbach (Hauser and Feshbach, ) and HRTW (Hofmann et al., ) versions of the statistical model were used for the compound nucleus cross section calculations. Both approaches include fission decay probabilities deduced in the optical model for fission (Sin et al., ) and account for the multiple-particle emission and the full γ-ray cascade. A new model to describe fission on light actinides, which takes into account transmission through a triple-humped fission barrier with absorption, was developed (Sin et al., ) and applied for the first time to fission cross section evaluations. This formalism is capable of interpreting complex structure in the light actinide fission cross section in a wide energy range. The agreement with experimental fission cross sections is impressive as can be seen in > Fig.  discussed earlier. The complex resonance structure in the first-chance neutron-induced fission cross section of  Th has been very well reproduced. Prompt fission neutron spectra and ν̄ values were calculated using a new PFNS module of the EMPIRE code. The calculated ν̄ values were normalized to BROND- values (Ignatyuk et al., ), which are based on an extensive experimental database and contain covariance information. The EMPIRE calculations were merged with the resonance data, including resonance covariance file and the delayed neutron data from the BROND- file (Ignatyuk et al., ). Since the evaluation extends up to  MeV, exclusive spectra are only given for the first three emissions, such as (n, n) and (n, np), while all the remaining channels are lumped into MT = . The validation of the thorium file was carried out by Trkov and Capote (), showing improvement over previous evaluations. . Minor Actinides Minor actinides are defined broadly as fissionable nuclei beyond the four major actinides. They include minor isotopes of U and Pu as well as isotopes of Np, Am, and Cm and ultimately also all heavier actinides. For example, advanced reactor systems are interested in  minor actinides, ,, U, ,,, Pu,  Np, ,m, Am, and ,,, Cm. ..  U Evaluation The latest evaluation for  U (Young et al., ) was specifically performed for ENDF/B-VII. library. The fission cross section is taken from a covariance statistical analysis of all experimental data, including  U/ U fission ratio measurements converted using the ENDF/B-VII.   Evaluated Nuclear Data 2.6 233 U(n,f ) Cross section 2.4 Cross section (b)  2.2 2.0 Exp data (Cov Anal.) 1.8 ENDF/B-VII (LANL) ENDF/B-VI.8 JEFF-3.0 1.6 JENDL-3.3 1.4 0 5 10 Incident neutron energy (Mev) 15 20 ⊡ Figure  Evaluated fission cross section that follows the measured data (shown as a covariance analysis of the experimental data). Other evaluations from JEFF and JENDL are also included standard  U cross section, as shown in > Fig. . The somewhat higher  U fission cross section in the fission spectrum region produces better agreement with fast critical benchmark experiments. .. ,,,,,, U Evaluations These evaluations were done by the well-established Los Alamos group led by Young et al. (). Depending upon the isotope, varying amounts of measured data are available. In some cases, the experimental database is extremely sparse. For example, for  U, there are no direct measurements of the fission cross section at monoenergetic incident neutron energies, and there are no capture measurements. However, for  U and  U, indirect information does exist on the fission cross section in the few-MeV region, using surrogate (t,p) “Decay Ratio” (DR) experiments from Los Alamos, which have recently been reanalyzed by Younes and Britt () at Livermore and from a more recent Lawrence Livermore National Laboratory (LLNL) experiment by Bernstein et al. (Plettner et al., ). These data allow an assessment of the equivalent neutron-induced fission cross section, and Younes and Britt have shown that such surrogate approaches can be accurate to better than %. In the case of  U, a measurement has been made of the fission cross section in a fast fission spectrum within a Flattop (fast) critical assembly, at two locations – the center region and the tamper region (where the spectrum is softer). This kind of measurement also provides indirect information on the fission cross section. Evaluated Nuclear Data  ⊡ Table  f and Thermal (. eV) constants obtained from the standards evaluation, gw abs gw are the Westcott factors Quantity  σnf (b) .± .% .± .% .± .% .± .% (.) (.) (.) (.) .± .% .± .% .± .% .± .% (.) (.) (.) (.) .± .% .± .% .±.% .± .% (.) (.) (.) (.) .±.% .± .% .±.% .± .% (.) (.) (.) (.) .±.% .± .% .±.% .± .% (.) (.) (.) (.) . ±.% .± .% .±.% .± .% (.) (.) (.) (.) σnγ (b) σnn (b) gfw gabs w ν̄ tot U  U  Pu  Pu The neutron sublibrary values are given in brackets. The nubar obtained from the standards evaluation process for  Cf is ν̄ tot = . ± .%, comprised of ν̄ p =. and ν̄ d =.. In ENDF/B-VI., ν̄ tot was ., comprised of ν̄ p =. and ν̄ d =. . Thermal Constants It is useful to summarize the thermal constants in the ENDF/B-VII. neutron sublibrary for important materials and compare them with the values given in the neutron cross section standards sublibrary. This is done in > Table . One can see that there are differences between the two sublibraries, though these are generally very small and within ≈ . standard deviation. The only item shown in this table that is considered a standard is the thermal  U(n, f ) cross section. . Nubars The average number of neutrons per fission, also known as fission neutron multiplicity, represent quantities of exceptional importance. These quantities are evaluated with utmost care and high precision of data has been achieved. The total nubar (denoted as ν̄ or ν̄tot ) is obtained as a sum of prompt and delayed nubars, ν̄ tot = ν̄ p + ν̄ d . () For  U, the energy dependence of the prompt ν̄ p is shown in > Fig. . This new ENDF/B-VII. evaluation follows covariance analysis of the experimental data, generally within uncertainties, and includes renormalization of the measured values to the latest standard value for  Cf.   Evaluated Nuclear Data · 235 U+n Prompt nubar 5.20 4.80 · 4.40 Nubar (n/f )  4.00 3.60 Exp data (Cov Anal.) 3.20 ENDF/B-VII.0 ENDF/B-VI.8 JEFF-3.0 JENDL-3.3 2.80 2.40 0 2 4 6 8 10 12 14 16 18 20 Incident neutron energy (MeV) ⊡ Figure  Evaluated  U prompt fission neutron multiplicity, ν̄ p , compared with measured data, as represented by a covariance analysis of experimental data. Other evaluations from JEFF (Europe) and JENDL (Japan) are also shown For  U, the energy dependence of prompt fission neutron multiplicity, the ENDF/B-VII. data are identical to ENDF/B-VI, except the energy range was extended from  to  MeV. The ENDF/B-VI data are based on an evaluation by Frehaut (). For the results and comparison see > Fig. . The  Pu evaluated prompt fission nubar is shown in > Fig. , compared with statistical covariance analysis of all measured data (again re-normalized to the latest californium standard). In the fast region, our evaluation follows the upper uncertainty bars of the statistical analysis of the experimental data, allowing us to optimize the integral performance in criticality benchmarks for the fast Jezebel  Pu spherical assembly. . Delayed Neutrons Delayed neutrons originate from the radioactive decay of nuclei produced in fission and hence they are different for each fissioning system. .. Fission-Product Delayed Neutrons Delayed neutrons, also referred to as temporal fission-product delayed neutrons, are stored in ENDF- formatted files as MF=, MT=. Related experiments typically report data as a series Evaluated Nuclear Data Bao (1975) Savin (1972) Malynovskyj (1983) Frehaut (1980) Asplund (1964) 4.8 4.4 238 U+n prompt nubar Leroy (1960) Fieldhouse (1966) ENDF/B-VII.0 ENDF/B-VI.8 JEFF-3.0 JENDL-3.3 4.0 Nubar (n/f )  3.6 3.2 2.8 2.4 0 4 10 6 8 Incident neutron energy (MeV) 2 14 12 16 ⊡ Figure  Evaluated  U prompt fission neutron multiplicity based on a covariance analysis of the experimental data 239 Pu+n prompt nubar 5.6 Nubar (n /f ) 5.2 4.8 4.4 4.0 Exp data (Cov Anal.) ENDF/B-VII.0 ENDF/B-VI.8 JEFF-3.0 JENDL-3.3 3.6 3.2 2.8 0 2 4 6 8 10 12 14 Incident neutron energy (MeV) 16 18 20 ⊡ Figure  Evaluated  Pu prompt fission neutron multiplicity, ν̄ p compared with measured data, as represented by a covariance analysis of experimental data. Other evaluations from JEFF and JENDL are also shown    Evaluated Nuclear Data of exponential terms. An experiment includes measurements characteristically made for a set of irradiation, cooling, and counting periods. Integrally detected delayed neutrons, adjusted for efficiencies and assigned uncertainties, are fit for maximum likelihood with an exponential series. The most common function used has been a series of six exponential terms emulating the sum of contributions of six uncoupled delayed-neutron precursors or precursor groups of differing time constants – hence the use of “six-group fits” in common parlance. This series is generally given in terms of ν̄ d – the total number of delayed neutrons per fission – times the normalized sum of six exponential terms giving the temporal production at time t following a fission event. The delayed neutron yields in ENDF/B-VII. were carried over from ENDF/B-VI., with the exception of the modifications to  U thermal ν̄ d as described in the next subsection. Therefore, these yields are based on experimental data. The six-group parameters describing the time dependence of the delayed neutrons are discussed in more detail below. But we note that an explicit incident energy dependence is not given in the ENDF file as was also the case in the previous ENDF/B-VI. evaluation. The six-group values in the ENDF file correspond to fast neutron incident energies. CINDER’ calculations of a single fission pulse were replaced with a series of calculations for a variety of irradiation periods followed by decay times to  s, defining delayed neutron production in terms of irradiation and cooling times improving fits at very short and very long cooling times (Wilson and England, ). Subsequent improvements in Pn and half-life data were obtained using evaluated measured data of Pfeiffer et al. () and NUBASE (Audi et al., ). Use of the earlier systematics of Kratz and Herrmann was then replaced by results obtained with our own model. A new CINDER’ data library, including all delayed neutron data developed, now includes  delayed neutron precursors, with  precursors in the fission-product range  < A < . Use of the fission product yields (FPY) data (England and Rider, ) results in the production of – of these precursors yielded in the  fission systems. These data have been used to produce new temporal delayed neutron fits for all  fission systems. Fits for some systems are included in this release of ENDF/B-VII.; spectra, where present, are taken from the ENDF/BVI. files using the new group abundances. For illustration, in > Fig.  we show delayed neutron fraction emitted as function of the time following a  U thermal fission pulse. As shown in the inset, differences between ENDF/BVII. and the other evaluations are smaller than % for times larger than  s. ..  U Thermal ν̄ d When Keepin () measured delayed nubar (ν̄d ) for  U, he found a difference between thermal and fast values: .±. and .±., respectively. Since second-chance fission was above the energy of his measurements, he assumed it was an experimental error, and recommended the cleaner fast data for kinetics applications, including thermal. This posed a problem for thermal reactor designers: use the more accurate fast value, or the more relevant thermal data. Mostly, they opted for the latter. Experiments continued to send mixed signals. Conant and Palmedo () and Tuttle () supported a difference between fast and thermal values, and thermal reactor kinetics Delayed neutron fraction Evaluated Nuclear Data f  p Time after fission burst (s) ⊡ Figure  Delayed neutron fraction as a function of time following a  U thermal fission pulse for ENDF/BVII., ENDF/B-VI., JEFF ., Brady and England (), and Keepin (). The inset shows the ratio of the delayed neutron fractions for the other evaluations to the ENDF/B-VII. calculations failed to show a problem with the lower value. Fast measurements and summation calculations tended to raise the . value even higher. Two recent developments provided a plausible resolution to this problem: . Fission theory allowed an energy-variation of delayed nubar in the resonance region (Ohsawa and Oyama, ; Ohsawa and Hambsch, ). The change in  U delayed nubar is a series of small dips, one at each resonance, but for engineering purposes, only the average value over the thermal region is important. As the energy increases, the fluctuations decrease and the value approaches the higher fast value. . Analysis of beta-effective measurements supported the view that thermal delayed nubar is about % lower than the fast value (Nakajima, ; Sakurai and Okajima, ; van der Marck, ). The ENDF/B-VII.  U delayed nubar file is not a reevaluation of the data, but a minimum adjustment to ENDF/B-VI, which reflects current usage and recognizes the thermal-fast difference. An appropriate time to revisit this issue will be when the ANS-. Standard is finalized. The delayed value at thermal energy (ν̄d = .) was taken from JENDL-.. It then ramps linearly to . at  keV. JENDL ramps to ., but . minimizes the change to ENDF/B-VI. Above  keV, the ENDF/B-VI data are unchanged. To avoid disturbing thermal criticality benchmark results, which depend on total nubar, the thermal prompt value was changed to keep total nubar the same, ν̄tot = .–..    . Evaluated Nuclear Data Fission Energy Release The ENDF/B-VII. library includes new information for the energy released in fission for the major actinides, , U and  Pu. A recent study by Madland () found a new representation for the prompt fission product energy EFR (einc ), prompt neutron energy ENP (einc ), and prompt photon energy E GP (einc ) functions. Their sum, the average total prompt fission energy deposition, is given by ⟨E d (einc )⟩ = E FR (einc ) + E NP (einc ) + E GP (einc ). () This expression is based upon published experimental measurements and application of the Los Alamos model (Madland and Nix, ), and it shows that, to first order, these quantities can be represented by linear or quadratic polynomials in the incident neutron energy einc ,  E i (einc ) = c  + c  einc + c  einc , () where E i is one of E FR , E NP , or E GP . The recommended coefficients for , U and  Pu are provided in > Table . The average total prompt energy deposition ⟨E d ⟩ obtained using these coefficients in () and () is shown in > Fig. . Madland’s recommended c  values for E FR have been adopted in the new ENDF/BVII. files for , U and  Pu. .. Nuclear Heating Nuclear heating is an important quantity in any nuclear system. It is a topic that should be explored in relation to the energy release presented above and its handling by the processing code NJOY. In general, heating as a function of energy, H(einc ), may be given in terms of kinetic ⊡ Table  Madland’s recommended energy release polynomial coefficients, in MeV Nuclide Parameter c c c  U EFR ENP EGP . −. . . +. . . +. .  U EFR ENP EGP . −. . +. . +.  Pu EFR ENP EGP . . . See () for explanation . . . −. . . . +. −. Energy deposition <Ed > (MeV) Evaluated Nuclear Data Ed en Ed en en Ed  en en Incident neutron energy, en (MeV) ⊡ Figure  Average total prompt fission energy deposition as a function of the incident neutron energy. See () for explanation energy released in materials (KERMA) coefficients (the International Commission on Radiation Units and Measurements in its document ICRU- (ICRU, ) recommends using the name “KERMA coefficient” instead of “KERMA factor”), kij (einc ), as H(einc ) = ∑ ∑ ρ i k ij (einc )Φ(einc ), i () j where ρ i is the number density of the ith material, k ij (einc ) is the KERMA coefficient for the ith material and jth reaction at energy einc , and Φ(einc ) is the scalar flux. A rigorous calculation of the KERMA coefficient for each reaction requires knowledge of the total kinetic energy carried away by all secondary particles following that reaction, data that frequently are not available in evaluated files. An alternative technique, known as the energy balance method (Muir, ), is used by NJOY. KERMA coefficient calculations by this method require knowledge of the incident particle energy, the reaction Q-value, and other terms. The prompt fission reaction Q-value required for prompt fission KERMA including the energy-dependent prompt fission Q-value can be calculated as Q(einc ) = E R − . ×  [ν̄(einc ) − ν̄()] + .einc − E B − EGD , () where E R is the total energy minus neutron energy, E B is the total energy released by delayed betas, and E GD is the total energy of delayed photons.    Evaluated Nuclear Data ⊡ Table  Prompt fission Q-values in MeV obtained with ENDF/B-VII. dataa Nuclide Incident energy einc ENDF/B VII. Madland NJOY (old) NJOY ()  U . eV . MeV . MeV . . . . . . . . . . . .  U . eV . MeV . MeV . . . . . . . . . . . .  Pu . eV . MeV . MeV . . . . . . . . . . . . a Given for the sum of prompt fission products, prompt neutrons, and prompt gammas To get total energy deposition, add the incident energy to total Q-values tabulated here Results based upon new ENDF/B-VII. are shown in > Table . We note that the prompt fission Q-value calculated with the traditional ENDF formulas are now in much better agreement with Madland’s calculations.  Neutron Data for Other Materials In addition to actinides, there are three other categories of materials of interest for nuclear technology applications. These are light nuclei that often serve as moderators and coolants, structural materials, and fission products. . Light Nuclei Several light-element evaluations were contributed to ENDF/B-VII., based on R-matrix analysis done at Los Alamos using the EDA code. Among the neutron-induced evaluations were those for  H,  H,  Li,  Be, and  B. For the light-element standards, R-matrix results for Li(n, α) and  B(n, α) were contributed to the standards process, which combined the results of two different R-matrix analyses with ratio data using generalized least-squares. Differences persisted between the two R-matrix analyses even with the same data sets that are not completely understood, but probably are related to different treatments of systematic errors in the experimental data. Below we summarize upgrades that have been made for ENDF/B-VII.. Where no changes have been made compared to ENDF/B-VI. (e.g., for n +  H), we do not discuss reactions on these isotopes. H: The hydrogen evaluation came from an analysis of the N − N system that includes data for p + p and n + p scattering, as well as data for the reaction  H(n, γ) H in the forward (capture) and reverse (photodisintegration) directions. The R-matrix parametrization, which is  Evaluated Nuclear Data  completely relativistic, uses charge-independent constraints to relate the data in the p+ p system to those in the n + p system. It also uses a new treatment of photon channels in R-matrix theory that is more consistent with identifying the vector potential with a photon “wavefunction.” In the last stages of the analysis, the thermal capture cross section was forced to a value of . mb (as in ENDF/B-VI.), rather than the “best” experimental value of . ±. mb (Cokinos and Melkonian, ), since criticality data testing of aqueous thermal systems showed a slight preference for the lower value. Also, the latest measurement (Schoen et al., ) of the coherent n + p scattering length was used, resulting in close agreement with that value, and with an earlier measurement of the thermal scattering cross section (Houk, ), but not with a later, more precise value (Dilg, ). This analysis also improved a problem with the n + p angular distribution in ENDF/B-VI. near  MeV by including new measurements (Boukharaba et al., ; Buerkle and Mertens, ) and making corrections to some of the earlier data that had strongly influenced the previous evaluation.  H: The n+ H evaluation resulted from a charge-symmetric reflection of the parameters from a p+ He analysis that was done some time ago. This prediction (Hale et al., ) resulted in good agreement with n + t scattering lengths and total cross sections that were newly measured at the time, and which gave a substantially higher total cross section at low energies than did the ENDF/B-VI evaluation. At higher energies, the differences were not so large, and the angular distributions also remained similar to those of the earlier evaluation.  Be: The n+ Be evaluation was based on a preliminary analysis of the  Be system that did a single-channel fit only to the total cross section data at energies up to about  MeV. A more complete analysis should take into account the multichannel partitioning of the total cross section, especially into the (n, n) channels. An adequate representation of these multibody final states will probably require changes in the EDA code. For ENDF/B-VII., the elastic (and total) cross section was modified to utilize the new EDA analysis, which accurately parametrizes the measured total elastic data, while the previous ENDF/B-VI. angular distributions were carried over. Data testing of the file (including only the changes in the total cross sections) appeared to give better results for beryllium-reflecting assemblies, and so it was decided to include this preliminary version in the ENDF/B-VII. release.  O: The evaluated cross section of the  O(n, α  ) reaction in the laboratory neutron energy region between . and . MeV was reduced by % at LANL. The  O(n, α) cross section was changed accordingly and the elastic cross sections were adjusted to conserve unitarity. This reduction was based upon more recent measurements. We note that this led to a small increase in the calculated criticality of LCT assemblies. . Structural Materials Structural materials play a prominent role in nuclear applications and hence neutron reaction data are evaluated very carefully. In the ENDF/B-VII., the main evaluation effort was concentrated on the major actinides and the fission products (Z = –) that together cover more than half of the ENDF/B-VII. neutron sublibrary. Outside of these two groups, only few materials were fully or partially evaluated for ENDF/B-VII., as described below.    .. Evaluated Nuclear Data Evaluations of Major Structural Materials Structural materials fall into a category of priority materials in all major evaluated data libraries. The list is dominated by Cr, Fe, and Ni, the most important isotopes being major structural materials  Cr (natural abundance .%),  Fe (.%), and  Ni (.%), followed by less abundant isotopes , Cr, , Fe,  Ni, etc. In the USA, considerable attention to evaluations of structural materials has been devoted in the past. These evaluations have been performed by the highly experienced team at ORNL up to  MeV in the s, in particular in reference to the celebrated ENDF/B-V library. We note that ORNL supplied complete evaluations in the entire energy range, combining the capabilities in the thermal and resonance region (code SAMMY) with the then advanced nuclear reaction modeling code TNG in the fast neutron region. An example would be the  update for iron by Fu et al. (). Since then, virtually no updates below  MeV have been made. In view of the data need for accelerator driven systems in the s, the evaluations of structural materials have been extended to  MeV by Los Alamos and incorporated into the ENDF/B-VI library. Then, these have been adopted without any change by the latest version of the ENDF/B-VII. library, which was released in . .. New Evaluations for ENDF/B-VII. V: Cross sections for the (n, np) reaction were revised at BNL (Rochman et al., ) by adjusting the EMPIRE-. calculations to reproduce two indirect measurements by Grimes et al. () and Kokoo et al. () at . and . MeV, respectively. This resulted in the substantial reduction (about  mb at the maximum) of the (n, np) cross section Similarly, the (n, t) reaction was revised to reproduce experimental results of Woelfle et al. (). The inelastic scattering to the continuum was adjusted accordingly to preserve the original total cross section. nat , Ir: These are two entirely new evaluations performed jointly by T- (LANL) and the NNDC (BNL) in view of recent GEANIE data on γ-rays following neutron irradiation. The resolved and unresolved resonance parameters are based on the analysis presented in Mughabghab (). New GNASH model calculations were performed for the γ-rays measured by the GEANIE detector, and related (n, xn) reactions cross sections were deduced (Kawano et al., ). We also include an evaluation of the  Ir(n, n ′ ) reaction to the isomer. The remaining cross sections and energy-angle correlated spectra were calculated with the EMPIRE code. The results were validated against integral reaction rates.  Pb: A new T- (LANL) analysis with the GNASH code was performed over the incident neutron energy range from . to . MeV. The Koning–Delaroche optical model potential (Koning and Delaroche, ) from the RIPL- database was used to calculate neutron and proton transmission coefficients for calculations of the cross sections. Minor adjustments were made to several inelastic cross sections to improve agreement with experimental data. Additionally, continuum cross sections and energy-angle correlated spectra were obtained from the GNASH calculations for (n, n′ ), (n, p), (n, d), (n, t), and (n, α) reactions. Elastic scattering angular distributions were also calculated with the Koning–Delaroche potential and incorporated in the evaluation at neutron energies below  MeV. Evaluated Nuclear Data  This new  Pb evaluation led to a significant improvement in the lead-reflected critical assembly data. This is especially true for fast assemblies, but some problems remained for thermal assemblies. . Fission Products Fission products represent the largest category of nuclei (materials) in the evaluated nuclear data libraries. Defined broadly as materials with Z = –, in the latest US library ENDF/BVII. they supply  nuclei. We follow this definition with the understanding that it covers also several other important materials such as structural Mo and Zr, and absorbers Cd and Gd. Many fission product evaluations in ENDF/B had not been revised for nearly – years. Not surprisingly an analysis performed by Wright (ORNL) and MacFarlane (LANL) in  revealed considerable deficiencies in ENDF/B-VI (Wright and MacFarlane, ). In this situation, fission product evaluations in ENDF/B-VI. were completely abandoned and ENDF/B-VII. adopted new or recently developed evaluations. For a set of  materials, including  materials considered to be of priority, entirely new evaluations were performed using the Atlas-EMPIRE evaluation procedure (Herman et al., ) including those by Kim et al. (, ). For the remaining  materials, evaluations were adopted from the recently developed International Fission Product Library of Neutron Cross Section Evaluations completed in December  and described in the  report (Obložinský et al., b). .. Priority Fission Products New evaluations were performed for materials considered to be priority fission products. The list includes  materials, •  Mo,  Tc,  Ru,  Rh,  Pd,  Ag,  Xe,  Cs,  Pr,  Eu, , Nd, ,,,, Sm, Gd. , This selection (Oh et al., ) was based on the analysis by DeHart, ORNL, which was performed in  (DeHart, ). It was motivated by the need to improve existing evaluations for materials of importance for a number of applications, including criticality safety, burnup credit for spent fuel transportation, disposal criticality analysis, and design of advanced fuels. .. Complete Isotopic Chains As a part of modern approach to evaluation, complete isotopic chains were evaluated for several fission products, including Ge, Nd, Sm, Gd, and Dy. A simultaneous evaluation of the complete isotopic chain for a given element became possible thanks to tremendous progress in the development of evaluation tools in recent years. This includes highly integrated evaluation code systems such as EMPIRE, coupled to experimental database EXFOR and to the library of input parameters RIPL. Complete isotopic chains for Ge,   Evaluated Nuclear Data Nd-isotopes (n,tot) 10 Cross section (b)  8 142 Nd 143 Nd 144 Nd 145 Nd 146 Nd 147 Nd 148 Nd 150 Nd 6 4 1997 Wisshak, 142Nd 1973 Djumin, 142Nd 1997 Wisshak, 143Nd 1973 Djumin, 143Nd 1997 Wisshak, 144Nd 1980 Shamu, 144Nd 1973 Djumin, 144Nd 1983 Poenitz 1971 Foster Jr 1967 Haugsnes 1965 Seth 1958 Conner 1954 Okazaki 10 –1 1997 Wisshak, 145Nd 1973 Djumin, 145Nd 1997 Wisshak, 146Nd 1973 Djumin, 146Nd 1997 Wisshak, 148Nd 1973 Djumin, 148Nd 1973 Djumin, 150Nd 1 Incident neutron energy (MeV) 10 ⊡ Figure  Total cross sections for Nd isotopes. Note the consistency among different isotopes resulting from the simultaneous evaluation of the full chain of neodymium isotopes Nd, Sm, Gd, and Dy, totaling  materials were evaluated for ENDF/B-VII.. As an example, we discuss isotopes of Nd. A set of neodymium evaluations include two priority fission products, , Nd, another five stable isotopes, and the radioactive  Nd. Neodymium is one of the most reactive rare-earth metals. It is important in nuclear reactor engineering as a fission product that absorbs neutrons in a reactor core. A new evaluation was performed by Kim et al. (). In > Fig. , we show total neutron cross sections of all Nd isotopes in comparison with available data measured on isotopic samples as well as on elemental samples. Of special interest to radiochemical applications is the radioactive  Nd for which no data exist in the fast neutron region. A good fit to available data on other stable isotopes gives confidence that predictions for  Nd cross sections are sound. This is illustrated in > Figs.  and >  where we show (n, n) and neutron capture cross sections, respectively, for all Nd isotopes. .. Specific Case of  Zr Zirconium is an important material for nuclear reactors since, owing to its corrosion-resistance and low absorption cross section for thermal neutrons, it is used in fuel rods cladding. Benchmark testing performed at Bettis and KAPL showed an undesirable drop in the reactivity when the beta version of ENDF/B-VII. was used. Taking into account the importance of zirconium in reactor calculations, BNL undertook an entirely new evaluation of the fast neutron region in  Zr using the EMPIRE code. A good description of the total cross section of  Zr confirmed the higher elastic scattering cross section, see > Fig. . The new file met expectations when validated against integral measurements at KAPL. Evaluated Nuclear Data 2.5 142 Nd 143 Nd 144 Nd 145 Nd 146 Nd 147 Nd 148 Nd 150 Cross section (b) 2.0  Nd-isotopes (n,2n) 1997 Kasugai 142 Nd 1983 Gmuca 142 Nd 1980 Frehaut 142 Nd 1978 Sothras 142 Nd 1977 Kumabe 142 Nd 1974 Lakshman142 Nd 142 Nd 1974 Qaim 1970 Bormann 142 Nd 142 Nd 1968 Dilg 1966 Grissom 142 Nd 1980 Frehaut 144 Nd 1980 Frehaut 146 Nd 1983 Gmuca 148 Nd 1980 Frehaut 148 Nd 1977 Kumabe 148 Nd 148 Nd 1974 Qaim 148 Nd 1971 Bari 148 Nd 1969 Rama 1997 Kasugai 150 Nd 1984 An Jong 150 Nd 1983 Gmuca 150 Nd 1980 Frehaut 150 Nd 1977 Kumabe 150 Nd 1974 Lakshman150 Nd 150 Nd 1974 Qaim 150 Nd 1971 Bari 1967 Menon 150 Nd Nd 1.5 1.0 0.5 0 5 10 15 20 Incident neutron energy (MeV) 25 ⊡ Figure  (n, n) cross sections for Nd isotopes. Good fit to the available data justifies prediction of cross sections for the radioactive  Nd 108 142 Nd 143 Nd 144 Nd 145 Nd 146 Nd 147 Nd 148 Nd 150 Nd-isotopes (n,g ) 106 Nd 150 ×10 7 Cross section (b) 148 ×10 6 104 147 146 145 102 ×10 4 144 ×10 3 143 1 ×10 2 ×10 Nd-142 10–2 10– 4 1997 Wisshak 142Nd 1997 Wisshak 143Nd 1985 Bokhovko 143Nd 1997 Wisshak 144Nd 1977 Kononov 144Nd 1997 Wisshak 145Nd 1985 Bokhovko 145Nd 1997 Wisshak 146Nd 1993 Trofimov 146Nd 1999 Afzal 1997 Wisshak 1993 Trofimov 1977 Kononov 1993 Trofimov 1977 Kononov ×105 148Nd 148Nd 148Nd 148Nd 150Nd 150Nd 1 10 –1 Incident neutron energy (MeV) 10 ⊡ Figure  Neutron capture cross sections for Nd isotopes. Good fit to the available data endorses prediction of cross sections for the radioactive  Nd   Evaluated Nuclear Data 90-Zr(n,elastic) 10 Cross section (b)  8 6 Yamamuro ENDF/B-VIIb2 ENDF/B-VII.0 Igarasi65 Wilmore-Hodgson Igarasi66 Smith 1977 Mc Daniel 1976 Stooksberry 1974 Mcdaniel 1973 Wilson 4 2 0.2 0.5 2 1 Incident neutron energy (MeV) 5 ⊡ Figure  Comparison of  Zr elastic cross sections calculated with various optical model potentials. The preliminary ENDF/B-VII evaluation (denoted ENDF/B-VIIb) yields the lowest cross sections. The ENDF/B-VII. evaluation is considerably higher as suggested by the integral experiments .. Remaining Fission Products Recognizing a need to modernize the fission product evaluations, an international project was conducted during – to select the best evaluations from the available evaluated nuclear data libraries. Evaluated nuclear data libraries of five major efforts were considered, namely, the USA (ENDF/B-VI. and preliminary ENDF/B-VII), Japan (JENDL-., released in ), Europe (JEFF-., released in ), Russia (BROND-., released in ), and China (CENDL-., made available for this project in ). As a result, the International Fission Product Library of Neutron Cross Section Evaluations (IFPL) was created for  materials (Obložinský et al., b). Afterward, IFPL was adopted in full by the ENDF/B-VII. library, see > Table  for a summary.  Covariances for Neutron Data A covariance matrix specifies uncertainties and usually energy–energy correlations of data (cross sections, ν̄, etc.) that are required to assess uncertainties of design and operational parameters in nuclear technology applications. Covariances are obtained from the analysis of experimental data and they are stored as variances and correlations in the basic nuclear data libraries. Early procedures for generating nuclear data covariances were widely discussed in the s and s (Smith, ). Accordingly, many of the presently existing covariance data were developed about  years ago for the ENDF/B-V library (Kinsey, ; Magurno et al., ). This earlier activity languished during the s due to limited interest by users and constrained resources. Evaluated Nuclear Data  ⊡ Table  Summary of  fission product evaluations included in the ENDF/B-VII. library Library (data source) Full file Resonance region Fast region ENDF/B-VI., released in  New evals for ENDF/B-VII. JEFF-., released in  JENDL-., released in  CENDL-., released in  BROND-., released in          −  − −  − −   −    Total number of materials Full files were taken over from single libraries (data sources) for  materials; the remaining  files were put together from two different data sources ⊡ Table  Summary of methods used for the ENDF/B-VII. covariance evaluations Energy region Evaluation method Material Resolved resonances Direct SAMMY Retroactive SAMMY Atlas-KALMAN  Experimental Atlas-KALMAN EMPIRE-KALMAN  ,,,,,,, Gd,  Y,  Ir EMPIRE-KALMAN EMPIRE-GANDR ,,,,,,, Gd,  Y,  Tc, , Ir Unresolved resonances Fast neutrons Th ,,,,,,,   Gd Y,  Tc, , Ir Th Tc,  Ir  Th Distinguished are three energy regions: resolved resonances, unresolved resonances, and fast neutron region More recently, intensive interest in the design of a new generation of nuclear power reactors, as well as in criticality safety and national security applications has stimulated a revival in the demand for covariances as demonstrated at the major Covariance Workshop held in  (Obložinský, ). . Evaluation Methodology New covariance data in the ENDF/B-VII. neutron sublibrary were produced for  materials using the evaluation techniques summarized in > Table .    .. Evaluated Nuclear Data Resonance Region Covariances in the resonance region can be produced by three different methods. The most sophisticated approach is based on the code SAMMY (Larson, ), which uses generalized least-squares fits to experimental data. The intermediate Atlas method propagates resonance parameter uncertainties (Mughabghab, ) to cross-section covariances. The simplest and most transparent method uses uncertainties of thermal cross sections and resonance integrals as estimate of covariances (Williams, ). SAMMY Covariance Method This method is normally applied to actual experimental data as the integral part of simultaneous evaluation of both resonance parameters and their covariances (files MF and MF in ENDF- format terminology, Herman and Trkov, ). However, out of practical necessity an alternative retroactive procedure is often used. One proceeds in three steps: First, one either starts with actual experimental data or these are artificially (“retroactively”) generated using R-matrix theory and known resonance parameters. In the latter case, usually transmission, capture, and fission is calculated assuming realistic experimental conditions. Then, realistic statistical uncertainties are assigned to each data point, and realistic values are assumed for data-reduction parameters such as normalization and background. Afterward, initial covariance matrix is established. Let D represent the experimental data and V the covariance matrix for experimental/retroactive data. Values for V (both on- and offdiagonal elements) are derived from the statistical uncertainties of the individual data points, v i , and from the uncertainties of the data-reduction parameters, in the usual fashion Vij = ν i δ ij + ∑ g ik Δ  r k g jk . () k In this equation, Δ r k represents the uncertainty on the kth data-reduction parameter r k , and g ik is the partial derivative of the cross section at energy E i with respect to r k . Then, the covariance matrix Vij describes all the known experimental uncertainties. Finally, the generalized least-squares equations are used to determine the set of resonance parameters, P ′ , and associated resonance parameter covariance matrix, M ′ , that fit these data. If P is the original set of resonance parameters (for which we wish to determine the covariance matrix), and T is the theoretical curve generated from those parameters, then, in matrix notation, the least-squares equations are P ′ = P + M + G t V − (D − T) and M ′ = (G t V − G)− . () Here, G is the set of partial derivatives of the theoretical values T with respect to the resonance parameters P; G is sometimes called the sensitivity matrix. The solutions of () provide the new parameter values P ′ and the associated resonance parameter covariance matrix M ′ , fitting all directly measured/retroactive data simultaneously and using the full off-diagonal data covariance matrix for each data set. Atlas Covariance Method This method combines the wealth of data given in the Atlas of Neutron Resonances (Mughabghab, ) with the filtering code KALMAN (Kawano et al., ; Kosako and Yanano, ). Atlas Evaluated Nuclear Data  provides values and uncertainties for neutron resonance parameters and also integral quantities such as capture thermal cross sections, resonance integrals, and -keV Maxwellian averages. The procedure consists of two major steps: • One starts with the resonance parameters and their uncertainties and uses MLBW formalism to compute cross sections along with their sensitivities. • Uncertainties of resonance parameters are propagated to cross sections with the code KALMAN. Uncertainties of thermal values are obtained by suitable adjustment of resonance parameter uncertainties, if necessary, inferring anticorrelation with bound (negative energy) resonances. An alternative approach would be to take resonance parameter uncertainties, put them into file MF, and leave the job of propagation of these uncertainties into cross-section covariances to well-established processing codes. Low-Fidelity Covariance Method This simple, yet extremely transparent and useful method, provides an estimate of covariance data using known uncertainties of thermal cross sections and integral quantities. It was proposed by Williams () and later used extensively in the “Low-fidelity Covariance Project” by ANL-BNL-LANL-ORNL collaboration that was completed in  (Little et al., ). .. Fast Neutron Region EMPIRE-KALMAN Covariance Method EMPIRE-KALMAN methodology can serve as an example of recently developed covariance methods in the fast neutron region. The KALMAN filter techniques are based on minimum variance estimation and naturally combine covariances of model parameters, of experimental data, and of cross sections. This universality is a major advantage of the method. The key ingredient of the method is the sensitivity matrix, which represents complex nuclear reaction calculations. If we denote the combination of nuclear reaction models as an operator M̂ that transforms the vector of model parameters p into a vector of cross sections σ(p) for a specific reaction channel, then the sensitivity matrix S can be interpreted as the linear term in the expansion of the operator M̂, M̂p = σ(p) and M̂(p + δp) = σ(p) + Sδp + ⋯, () where M̂ is the operator rather than a matrix. In practice, the elements s i, j of the sensitivity matrix are calculated numerically as partial derivatives of the cross sections σ at the energy E i with respect to the parameter p j , s i, j = ∂σ(E i , p) . ∂p j () In case of covariance determination, the initial values of the parameters, p , are already optimized, that is, when used in the model calculations they provide the evaluated cross sections.    Evaluated Nuclear Data Their covariance matrix P is assumed to be diagonal while the uncertainties of the parameters are estimated using systematics, independent measurements, or educated guesses. The modelbased covariance matrix (prior) for the cross sections, C , can be obtained through a simple error propagation formula, C = SP ST , () where superscript T indicates a transposed matrix. The experimental data, if available, are included through a sequential update of the parameter vector p and the related covariance matrix P as exp pn+ = pn + Pn ST Qn+ (σ n+ − σ(pn )) , () T Pn+ = Pn − Pn S Qn+ SPn . Here, exp − Qn+ = (Cn + Cn+ ) , () where n = , , , ... and n +  denotes update related to the sequential inclusion of the (n + )th experimental data set. In particular, the subscript  ≡ + denotes updating model prior (n = ) with the first experiment. Vector pn+ contains the improved values of the parameters starting exp from the vector pn , and P n+ is the updated covariance matrix of the parameters pn+ . The Cn+ is the cross-section covariance matrix for the (n + )th experiment. The updated (posterior) covariance matrix for the cross sections is obtained by replacing P with Pn+ in (), Cn+ = SPn+ ST . () The updating procedure described above is often called Bayesian, although ()–() can be derived without any reference to the Bayes theorem as shown in Muir (). exp The experimental covariance matrix, Cn , is usually non-diagonal, due to the correlations among various energy points E i . Assuming that systematic experimental uncertainties are fully exp correlated, the matrix elements are expressed through the statistical, Δsta σ n , and systematic, sys exp Δ σ n , experimental uncertainties. This yields exp  exp exp c n (i, i) = (Δsta σ n (E i )) + (Δsta σ n (E i ))  () and, for i ≠ k, exp sys exp sys exp c n (i, k) = Δ σ n (E i ) × Δ σ n (E k ) . () The quality and consistency of the evaluated cross sections can be assessed by scalar quantity N T − exp exp exp χ  = ∑ (σ n − σ(p N )) (Cn ) (σ n − σ(p N )) , n= ()  Evaluated Nuclear Data where p N is the final set of model parameters corresponding to the inclusion of N experiments. A value of χ  per degree of freedom exceeding unity indicates underestimation of the evaluated uncertainties. It is a fairly common practice to multiply such uncertainties by a square root of χ  per degree of freedom to address this issue. . Sample Case: Gd Covariance evaluation of Gd isotopes was produced as a sample case for ENDF/B-VII.. There are seven stable Gd isotopes, ,,,,,, Gd and the radioactive  Gd. All covariances were produced by SAMMY retroactive method in the resolved resonance region and EMPIREKALMAN at higher energies.  > Figure  shows uncertainties for Gd capture cross sections at low energies. The thermal cross section and its uncertainty in the Atlas of Neutron Resonances (Mughabghab, ) are very well reproduced. The particular feature that  Gd shares with  Gd is a very close vicinity of the first positive resonance to the thermal energy. Therefore, the thermal cross section is determined by the first resonance rather than /v dependence typical for other nuclei. > Figure  also demonstrates anti-correlation between uncertainties and cross sections in the resonance region. The highest uncertainties are being found between the resonances, that is, at dips of cross sections. This feature is clearly visible although it is, to some extent, obscured by the groupwise representation that lumps close resonances together. 1e+05 Cross sections Gd(n,g ) 1e+04 First resonance at 0.0268 eV 1,000 100 15 10 Mughabghab 2006 (thermal energy 0.0253 eV) 10 Cross section (b) Relative uncertainty (%) 155 Relative uncertainty 5 0.001 0.01 0.1 1 Incident neutron energy (eV) 10 100 ⊡ Figure  Relative uncertainties for  Gd(n, γ) obtained with the retroactive SAMMY method plotted along with the cross sections to show anti-correlation between the two quantities. The experimental cross section (, b) and its uncertainty (.%) at the thermal energy (Mughabghab, ) are well reproduced   Evaluated Nuclear Data Relative uncertainty (%)  Incident neutron energy (MeV) ⊡ Figure  Relative uncertainties in the unresolved resonance and fast neutron range for the total, elastic and capture cross sections on  Gd obtained with the EMPIRE-KALMAN method The covariances for the unresolved resonance and fast neutron regions were produced with the EMPIRE-KALMAN method. In > Fig.  we show relative uncertainties for  Gd(n,tot),  Gd(n,elastic), and  Gd(n, γ) cross sections for incident neutron energies above  keV.  > Figure  shows the correlation matrix for the Gd(n, γ) cross section. This matrix reveals complicated structures with strong correlations aligned within a relatively narrow band along the diagonal. . Major Actinides Covariances for major actinides play a crucial role in many applications. There was insufficient time to complete new covariance evaluations for these important actinides prior to the release of the ENDF/B-VII. library in . This work was completed in , but it has not yet been officially approved by Cross Section Evaluation Working Group (CSEWG) (status at the end of ). One of the issues that has been resolved was conversion of huge multimillion line long resonance parameter covariance matrices (MF files) into cross-section covariances (MF files). Such a conversion reduced the size of the files considerably, though the  U and  Pu covariance files still remain very large (about  and  MB, respectively). Evaluated Nuclear Data  157 Gd(n,g ) Incident neutron energy (MeV) 10 100 80 1 60 40 20 0.1 0 0.01 0.01 0.1 1 Incident neutron energy (MeV) 10 ⊡ Figure  Correlation matrix for the  Gd neutron capture cross sections in the fast neutron region obtained with the EMPIRE-KALMAN method .. ,, U Covariances The evaluation of ,, U covariances was performed by ORNL-LANL collaboration. ORNL covered the resonance region and LANL supplied covariances in the fast neutron region. Due to the huge size of the resonance parameter covariances (MF), the files were converted into cross-section covariances (MF). Still, the size of the largest  U file is considerable, in excess of  MB. The preliminary version of these evaluations are available in the recently reestablished ENDF/A library, which contains candidate evaluations for the next release of ENDF/B-VII library. It is expected that all these files will be included into ENDF/B-VII. release. As an example of these evaluations,  U(n, f ) covariances are shown in > Fig. , see also discussion later in this chapter under the advanced fuel cycle initiative (AFCI) covariance library. ..  Pu Covariances The evaluation of  Pu covariances was also performed by ORNL-LANL collaboration. ORNL covered the resonance region and LANL supplied covariances in the fast neutron region. The file was also converted into cross-section covariances (MF), its reduced size is  MB. The preliminary version of this evaluation is available in the ENDF/A library and it is expected that the file will be included into ENDF/B-VII. release.    Evaluated Nuclear Data Ds/s vs. E for 235 U(n,f ) 10 1 Ordinate scales are % relative standard deviation and barns. 10 Abscissa scales are energy (eV). 0 10 4 3 2 10 7 10 6 1 10 4 10 5 10 10 3 0 10 0 10 1 10 2 10 10 –1 10 10 –1 10 –2 10 s vs. E for 235 U(n,f ) 10 –2 10 –1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 Correlation matrix 1.0 0.8 0.6 0.4 0.2 0.0 –1.0 – 0.8 – 0.6 – 0.4 – 0.2 0.0 ⊡ Figure  Covariances for  U(n, f ) taken from ENDF/A (November ), which collects candidate evaluations for the next release of ENDF/B-VII. Data are given in -energy group representation adopted by the advanced fuel cycle initiative (AFCI) covariance library: top – cross-section uncertainties, right – cross sections, middle – energy–energy correlations ..  Th Covariances The covariance evaluation for  Th includes the resolved resonance, unresolved resonance, and the fast neutron regions as well as ν̄. In the resolved resonance region, a Reich–Moore evaluation was performed (Derrien et al., ) in the energy range up to  keV using the code Evaluated Nuclear Data Incident neutron energy (eV) 232 10  Th(n,g ) 3 100 80 60 1 10 40 20 0 10 –1 10–3 1 10 10 10–1 Incident neutron energy (eV) 3 ⊡ Figure  Correlation matrix for  Th neutron radiative capture cross sections in the thermal and resolved resonance region SAMMY. The correlation matrix for the radiative capture cross section is shown in > Fig. . In the URR, the experimental method was used (Sirakov et al. ,  Th: evaluation of the average resonance parameters and their covariances in the URR from  to  keV, private communication). Cross-section covariance data in the fast neutron region were generated by the Monte Carlo technique (Smith, ) using the EMPIRE code . In the Monte Carlo approach, a large collection of nuclear parameter sets (normally more than ,) is generated by randomly varying these parameters with respect to chosen central values. These parameter sets are then used to calculate a corresponding large collection of nuclear model derived values for selected physical quantities, such as cross sections and angular distributions. These results are subjected to a statistical analysis to generate covariance information. The GANDR code system (Trkov, ) updates these nuclear model covariance results by merging them with the uncertainty information for available experimental data using the generalized least-squares technique. Covariances for ν̄ were obtained from the unpublished evaluation performed by Ignatyuk (Obninsk) for the Russian library BROND- (this library has not been released yet). This evaluation was based on the analysis of experimental data. . Covariance Libraries Despite the fact that only a limited amount of covariance evaluations have been produced to date (end of ), three covariance libraries were created in the USA as briefly described below. It should be understood that each of these libraries represents an approximate solution and has, therefore, inherent limitation. This stems from the fact that the development of quality covariances is a formidable task that requires considerable resources. This challenge should    Evaluated Nuclear Data be addressed by future releases of major evaluated libraries. Accordingly, it is expected that covariances will be part of the next release of the ENDF/B-VII library. .. Low-Fidelity Covariance Library The development of this library was funded by the US Nuclear Criticality Safety Program. The library provides the first complete, yet simple, estimate of neutron covariances for  materials listed in ENDF/B-VII.; for details see Little et al. (). Covariances cover main reaction channels, elastic scattering, inelastic scattering, radiative capture, and fission (cross sections and nubars) over the energy range from − eV to  MeV. Various approximations were utilized depending on the mass of the target, the neutron energy range, and the neutron reaction. The resulting covariances are not an official part of ENDF/B-VII., but they are available for testing in nuclear applications. In general, at low energies simple estimates were made following the approach proposed by Williams (). In the thermal region, defined by the cutoff energy . eV, experimental uncertainties of thermal cross sections were uniformly adopted. In the epithermal region, from . eV to  keV, the uncertainties of resonance integrals were used and uniformly applied. This led to simple and often reasonable estimates of cross section uncertainties. An example is shown in > Fig. . In the fast neutron region, the model-based estimates of covariances were produced for  materials from  F to  Bi (Pigni et al., ). To this end, EMPIRE code was employed n – – – ⊡ Figure  Low-fidelity uncertainties for  Th(n, γ) cross sections, labeled as integral quantities, are compared with ENDF/B-VII. values Evaluated Nuclear Data  ⊡ Table  Sources of covariance data in the SCALE- covariance library Source Material ENDF/B-VII. ,, ENDF/A ,, ENDF/B-VI.  Gd,  Th,  Tc, , Ir U,  Pu Na, ,, Si,  Sc,  V, ,,, Cr,  Mn, ,,, Fe, ,,,, Ni, , Cu,  Y,  Nb, nat In, , Re,  Au, ,, Pb,  Bi,  Am JENDL-. , LANL new  Low-fidelity About  materials, mostly fission products and minor actinides Pu  H, Li,  B and parametrization from the latest version of the RIPL library (Capote et al., ) was adopted along with global estimates of related model parameters. Parameter uncertainties were propagated into cross-section uncertainties. Light nuclei, A < , were treated differently. Recent R-matrix evaluations were adopted for three materials; for the remaining materials, simple estimates were supplied by looking into experimental data in the entire energy region. Actinides in the fast neutron region were again treated differently. Latest full scale evaluations by ORNL-LANL were used for major actinides, while simple model-based estimates using EMPIRE were used for minor actinides. .. SCALE- Covariance Library This is the covariance library included in the well-known ORNL reactor licensing code SCALE (). The library was produced by selecting covariances from a variety of sources as summarized in > Table  (Williams and Rearden, ). It should be noted that inherent limitation of selecting covariances from various sources is inconsistency with basic cross sections, which may or may not be negligible. .. AFCI Covariance Library This library was developed for AFCI funded by the US DOE Nuclear Energy. It should be suitable for nuclear data adjustment needed for fast reactor applications, such as advanced burner reactor (ABR). The list of materials contains  materials, including • Twelve light nuclei • Seventy-eight structural materials and fission products • Twenty actinides Covariances are produced by BNL-LANL collaboration (Obložinský et al., a). Major reaction channels are covered and covariance data are supplied in -energy group representation.    Evaluated Nuclear Data It is important to note that these covariances are tested by highly experienced reactor users in Argonne and Idaho National Laboratory. It is expected that a fairly robust version of the AFCI covariance library will be available by the end of . Then, it should serve as the basis for producing ENDF- formatted files suitable for inclusion into the next release of ENDF/B-VII. We note that whenever possible, AFCI library adopts new covariance evaluations produced for ENDF/B-VII. As an example, we refer the reader to > Fig.  where we have shown covariances for  U(n, f ) in AFCI -energy group representation.  Validation of Neutron Data Integral data testing of evaluated cross sections plays an essential role for validation purposes. The importance is twofold: First, since many of the integral experiments are very well understood (especially the critical assembly experiments), they provide a strong test of the accuracy of the underlying nuclear data used to model the assemblies, and can point to deficiencies that need to be resolved. Second, such integral data testing can be viewed as a form of “acceptance testing” prior to these data being used in various applications. Many applications, ranging from reactor technologies to defense applications, have a high standard required of a nuclear database before it is adopted for use. Critical assemblies, while involving many different nuclear reaction processes, can still be thought of as “single-effect” phenomena that probe the neutronics and nuclear data (but not other phenomena), and therefore an important acceptance test is that a sophisticated radiation transport simulation of the assembly should reproduce the measured keff to a high degree. Of necessity, the testing of an evaluated data library must be performed after the evaluation process. Ideally though, this testing is not performed as an afterthought, but more as an integral part of the evaluation process. It has been demonstrated many times that a close link between the data evaluators and the data users can provide valuable feedback to the evaluation process – quantifying the sensitivity of performance parameters to specific changes in nuclear data. We note that ENDF/B-VII. benchmarking was largely facilitated by the fact that for the first time benchmark model descriptions were available from the International Criticality Safety Benchmark Evaluation Project (ICSBEP) handbook. This criticality safety handbook contains benchmark descriptions of almost , critical assembly configurations (compared to the tens of benchmark descriptions contained in the ENDF- Benchmark Specifications used previously). Furthermore, this rich collection of benchmark descriptions spans the range of fuel types, compositions, spectra, geometries, etc. However, an additional feature of these benchmarks is the evaluation of the benchmark uncertainties, that is, estimates of the total experimental uncertainties combined with any additional modeling uncertainties (Briggs et al., ). As a result, the most diverse and robust aspect of the ENDF/B-VII. validation effort was the analysis of hundreds of criticality configurations compared with their benchmark eigenvalues and uncertainties. . Criticality Testing In reference to ENDF/B-VII., C/E values for keff (for the sake of clarity we use the term “C/E value for keff ” rather than the term “normalized eigenvalue;” here, C/E stands for the ratio of  Evaluated Nuclear Data ⊡ Table  The number of benchmarks per main International Criticality Safety Benchmark Evaluation Project (ICSBEP) category for compound, metal, and solution systems with thermal, intermediate, fast, and mixed neutron spectrum used in ENDF/B-VII. validation (van der Marck, ) COMP MET SOL Ther Inter Fast Mix Ther Inter Fast Mix ther Total Low-enriched U Intermediate-enriched U   High-enriched U Mixed U Total     Low-energy Pu                                     calculated to experimental values, and keff means the effective multiplication factor defined as the ratio of the average number of neutrons produced to the average number of neutrons absorbed per unit time) have been calculated for hundreds of critical benchmarks using continuous energy Monte Carlo programs including MCNP (versions c or ), RCP, RACER, and VIM. These calculations generally use benchmark models derived from the ICSBEP Handbook. Benchmark evaluations in this handbook are revised and extended on an annual basis. Unless otherwise noted, benchmark models derived from the  or  editions of the handbook were used in the calculations described below. Since the C/E values for keff have all been obtained using continuous energy Monte Carlo calculations there is a stochastic uncertainty associated with each C/E value for keff . The magnitude of this uncertainty is very small, typically less than  pcm (.%, pcm is derived from Italian “per cento mille,” meaning per hundred thousands; it is a unit of reactivity, where  pcm = . Δk/k, i.e.,  pcm is a .% discrepancy). A paper by van der Marck () presents independent European data testing of ENDF/BVII., using MCNPc with data processed by NJOY, and also shows extensive neutron transmission benchmark comparisons. > Table  provides a summary of  benchmark criticality experiments that were simulated and compared with measurements. > Table  summarizes the average value of C/E- (the average deviation of Calculation/Experiment from unity) for these benchmarks. We show for comparison in italics the values for the previous ENDF/B-VI.. While it is important to study the individual benchmark results for a more thorough understanding, it is still very useful to observe the overall averaged behavior shown in > Table : • The low-enriched U (LEU) compound benchmarks are modeled much more accurately (owing to improved  U, as well as  O and  H). • The intermediate-enriched U (IEU) benchmarks are modeled more accurately. • The Pu and HEU fast benchmarks are modeled more accurately.    Evaluated Nuclear Data ⊡ Table  The average value of C/E − in pcm ( pcm = .%) for ENDF/B-VII. per main ICSBEP benchmark category COMP Ther LEU IEU U Fast Mix Ther Inter Fast Mix ther  −  −     − −  ,  −     ,  − − −       −    − PU  Inter SOL − HEU MIX MET , ,     ,     −a  − − − a This becomes − pcm versus − pcm if we restrict ourselves to the well-understood UMF- and UMF- assemblies Shown in italics are the values for the ENDF/B-VI. library • The  U thermal benchmarks are modeled more accurately. Although the  U fast benchmarks simulations appear to have become worse, this is perhaps more due to deficiencies in modeling of beryllium for two of the assemblies studied – for bare  U (Jezebel- and Flattop-) the new ENDF/B-VII. are clearly much better. • Lower energy Pu (PU) benchmarks were modeled poorly in ENDF/B-VI. and continue to be modeled poorly in the new library. . Fast U and Pu Benchmarks Fast U and Pu benchmarks were given considerable attention in the validation of the ENDF/BVII. library. Shown below are selected examples for several benchmark categories. Bare, and  U reflected, assemblies: A large number of well-known Los Alamos fast benchmark experiments have been incorporated into the ICSBEP Handbook and are routinely calculated to test new cross-section data. Unmoderated enriched  U benchmarks include Godiva (HEU-MET-FAST- or HMF) (“HEU-MET-FAST-” is the identifier assigned in the ICSBEP Handbook for this assembly. It is comprised of four parts which, respectively, classify the assembly by fissile materials (PU, HEU, LEU), fuel form (METal, SOLution, COMPound), and energy-spectrum (FAST, INTERmediate, THERMal, or MIXED) and benchmark Evaluated Nuclear Data  1.015 Open squares - ENDF/B-VI.8 Closed squares - ENDF/B-VII Error bars - ICSBEP estimated 1 sigma experimental C/E value for k eff 1.010 1.005 1.000 0.995 0.990 HMF1 (Godiva) HMF28 (Flattop – 25) IMF7s (Big–10) PMF1 (Jezebel) PMF2 (Jezebel – 240) PMF6 (Flattop /Pu) PMF8c (Thor) UMF1 (Jezebel – 23) UMF6 (Flattop – 23) 0.985 Benchmark ⊡ Figure  Los Alamos National Laboratory (LANL) highly enriched uranium (HEU), Pu and  U unmoderated benchmark C/E values for keff calculated with ENDF/B-VI. and ENDF/B-VII. cross-section data number (-NNN). It is also common to use a shorthand form for this identifier, such as HMF for HEU-MET-FAST-), Flattop-, and Big- assemblies. Results of MCNP keff calculations with ENDF/B-VI. and ENDF/B-VII. cross sections for this suite of benchmarks are displayed in > Fig. . The improved accuracy in calculated keff for these systems with the new ENDF/B-VII. cross sections is readily apparent. Assemblies with various reflectors: A number of additional HEU benchmarks, either bare or with one of a variety of reflector materials including water, polyethylene, aluminum, steel, lead, and uranium have also been calculated with MCNP and both ENDF/B-VI. and ENDF/BVII. cross sections. The calculated values for keff are illustrated in > Fig. . Once again, significant improvement in the calculated keff is observed with the ENDF/B-VII. cross sections. Pb-reflected assemblies: Two reflector elements of particular historical interest are lead and beryllium. Often there are multiple evaluations that contain similar materials, in particular the same core with differing reflectors, thereby facilitating testing of cross-section data for individual reflector materials. Such is the situation for lead, displayed in > Fig. , with calculated keff for a variety of benchmarks. The significant improvements in these lead-reflected calculated keff reflects improvements made in the new ENDF/B-VII.  Pb evaluation, which was based on modern calculational methods together with careful attention to accurately predicting cross-section measurements, and by adopting the JEFF-. evaluations for ,, Pb, which (together with the JEFF-. file for  Pb) have reduced this bias by a similar amount in JEFF-. benchmarking. The situation is not so clear for thermally moderated systems (LCT) as shown in > Fig. . This figure also shows the keff calculations for the HMF benchmark, which consists of either a spherical or cylindrical HEU core with a lead reflector. In any event, the C/E values for keff are significantly different from unity in most cases regardless of cross-section data set.   Evaluated Nuclear Data 1.015 Open squares - ENDF/B-VI.8 Closed squares - ENDF/B-VII Error bars - ICSBEP estimated 1 sigma experimental 1.010 C /E value for keff 1.005 1.000 0.995 0.990 HMF8 (bare) HMF18 (bare) HMF51.x (bare) HMF4 (water) HMF11 (poly) HMF22 (Al) HMF13 (steel) HMF21 (steel) HMF27 (Pb) HMF14 (U) 0.985 HMF12 (Al) Benchmark ⊡ Figure  HEU-MET-FAST benchmark C/E values for keff calculated with ENDF/B-VI. and ENDF/B-VII. crosssection data 1.040 1.035 1.030 1.025 C/E value for keff  Open squares - ENDF/B-VI.8 Closed squares - ENDF/B-VII Error bars - ICSBEP estimated 1 sigma experimental 1.020 1.015 1.010 1.005 1.000 0.995 0.990 0.985 HMF18 HMF27 PMF22 PMF35 LCT2 LCT10 HMF57 0.980 Benchmark ⊡ Figure  Bare and lead-reflected C/E values for keff calculated with ENDF/B-VI. and ENDF/B-VII. crosssection data for several HMF, PMF, and LCT benchmarks Evaluated Nuclear Data  1.015 Open squares - ENDF/B-VI.8 Closed squares - ENDF/B-VII Error bars - ICSBEP estimated 1 sigma experimental C/E value for keff 1.010 1.005 1.000 0.995 0.990 0.985 0.0 5.0 10.0 15.0 20.0 25.0 Beryllium reflector thickness (cm) ⊡ Figure  HEU-MET-FAST- benchmark C/E values for keff with ENDF/B-VI. and ENDF/B-VII. cross-section data as a function of the beryllium reflector thickness Be-reflected assemblies: C/E values for keff of beryllium-reflected benchmarks are shown in  and > . These comparisons are useful to assess the changes made in the  Be cross sections for ENDF/B-VII.. > Figs. Zero power reactor assemblies: The keff calculations by the code VIM for a suite of  Argonne zero power reactor (ZPR) or zero power physics reactor (ZPPR) benchmarks are presented in > Fig. . These benchmarks come from various areas of the ICSBEP Handbook. These benchmarks exhibit large variation in calculated keff , with the smallest keff being biased several tenth of a percent below unity while the maximum positive C/E keff bias is in excess of %. Calculated keff with ENDF/B-VII. cross sections are generally an improvement over those obtained with ENDF/B-VI., but significant deviations from unity remain. . Thermal U and Pu Benchmarks Thermal benchmarks are of considerable interest to reactor applications. Shown below are selected examples for  U solution benchmarks, fuel rod U benchmarks, and Pu solution, as well as MOX benchmarks. ..  U Solution Benchmarks Thermal, highly enriched  U homogeneous solution benchmarks were used to test the accuracy of low energy ENDF/B cross-section data sets for many years. The new ENDF/B-VII. library, like the old ENDF/B-VI. library, performs well for these assemblies. C/E values for keff have been calculated for a suite of critical assemblies from  HEU-SOL-THERM (HST)   Evaluated Nuclear Data 1.015 C/E value for keff 1.010 Open square - ENDF/B-VI.8 Closed squares - ENDF/B-VII Error bars - ICSBEP estimated 1 sigma experimental 1.005 1.000 0.995 0.990 0.985 0.0 5.0 10.0 15.0 Beryllium reflector thickness (cm) 20.0 25.0 ⊡ Figure  HEU-MET-FAST- benchmark C/E values for keff for ENDF/B-VI. and ENDF/B-VII. cross-section data as a function of the beryllium reflector thickness. The poorer agreement using ENDF/B-VII. appears to be in contradiction to the results shown in > Fig.  1.04 Open squares - ENDF/B-VI.8 Closed squares - ENDF/B-VII C/E value for keff 1.03 1.02 1.01 1 R6 ZP /6A RZP 6/7 R ZP -9/1 R ZP -9/2 R ZP -9/3 RZP 9/4 R ZP -9/5 RZP 9/6 R ZP -9/7 R ZP -9/8 ZP R-9/ 9 ZP R-6/ P 10 ZP R-21 P A ZP R-21 P B ZP R-21 P C ZP R-21 PR D ZP -21 PR E ZP -21F R ZP -3/2 ZP ZPR R-3 3 R 6 /4 ZP -9/34 /9 (U 1 P ( 9 ZP R-20 U/Fe) PR C ) ZP -20 L10 P 5 D ZP R-20 L13 PR D 6 -20 L12 EL 9 16 0 0.99 ZP  ⊡ Figure  C/E values for keff for  zero power reactor (ZPR) and zero power physics reactor (ZPPR) benchmarks from Argonne. The ENDF/B-VII results are for the beta version, but one does not expect significant changes for ENDF/B-VII. Evaluated Nuclear Data  1.025 Error bars - ICSBEP estimated 1 sigma experimental Normalized eigenvalue 1.020 1.015 1.010 1.005 1.000 0.995 0.990 0.985 0.980 0.975 0.0 HST1 HST12 HST42 E7 2 Fit HST10 HST13 HST43 HST9 HST11 HST32 HST50 0.1 0.2 0.3 0.4 0.5 Above-thermal leakage fraction (ATLF) 0.6 ⊡ Figure  HEU-SOL-THERM (HST) benchmark C/E values for keff with ENDF/B-VII. cross sections or LEU-SOL-THERM (LST) benchmark evaluations. These benchmarks have most commonly been correlated versus above-thermal leakage fraction (ATLF), for example, k c al c (AT LF) = b  + b  *ATLF, where ATLF is the net leakage out of the solution of neutrons whose energies exceed . eV. Smaller systems with large ATLF test the higher energy cross sections, the  U fission spectrum, elastic scattering angular distributions, and, for reflected systems, the slowing down and reflection of above-thermal neutrons back into the fissile solution. An important goal in developing the new ENDF/B-VII. library was to improve the data files while at the same time retain the good performance seen with ENDF/B-VI. (in the homogeneous solution benchmark category). As shown in > Fig. , this goal has been attained. This result is nontrivial, since we have made changes in the ENDF/B-VII. library for  O (the n, α cross section was significantly reduced) and for hydrogen (a new standard cross section, as well as an updated scattering kernel). .. U Fuel Rod Benchmarks The  U cross-section data in ENDF/B-VII. have led to major improvements in the ability to accurately calculate thermal LEU benchmark C/E values for keff . Calculated C/E for keff for arrays of low-enriched UO fuel rods have historically been biased with previous data libraries including ENDF/B-VI., frequently falling –, pcm below unity. These C/E values have also varied systematically when correlated against parameters such as rod pitch, average fission energy, unit cell H/U ratio, or  U absorption fraction. Some of these characteristics are illustrated in > Figs.  and > , which illustrate calculated keff obtained with MCNP and either ENDF/B-VI. and/or ENDF/B-VII. cross sections. Results using the new ENDF/B-VII. cross sections are significantly more accurate as shown in > Fig. , which illustrates calculated keff for the LCT benchmark with both ENDF/B-VI.   Evaluated Nuclear Data 1.015 Normalized eigenvalue 1.010 1.005 LCT1 (2.35 w/o, USA - PNL) LCT2 (4.31 w/o, USA - PNL) LCT6 (2.6 w/o, Japan) LCT7 (4.74 w/o, France - Valduc) LCT22 (9.8 w/o, Russia - Kurchatov) LCT24 (9.8 w/o, Russia - Kurchatov) LCT39 (4.74 w/o, France - Valduc) 1.000 0.995 0.990 0.985 Benchmark experiment ⊡ Figure  LEU-COMP-THERM benchmark C/E values for keff with the old ENDF/B-VI. cross sections 1.015 1.010 Normalized eigenvalue  1.005 Opened squares - ENDF/B-VI.8 Closed squares - ENDF/B-VII Error bars - average experimental uncertainty for the four square pitch configurations, per the ICSBEP Handbook. ENDF/B-VII results include both rectangular and triangular pitch configurations. 1.000 0.995 0.990 0.985 0.0 2.0 4.0 6.0 8.0 10.0 Volume (moderator) / volume (UO 2 ) 12.0 ⊡ Figure  LEU-COMP-THERM- benchmark C/E values for keff for the ENDF/B-VI. and ENDF/B-VII. cross sections and ENDF/B-VII. cross sections. The ENDF/B-VI. C/E for keff trend and bias have both been eliminated with the ENDF/B-VII. cross-section data set. A summary of all water moderator and reflected LCT k eff reveals that previously identified deficiencies have been largely eliminated. A total of  LEU-COMP-THERM benchmarks have Evaluated Nuclear Data  been calculated with the ENDF/B-VII. cross-section data set. The average calculated keff is . with a population standard deviation of .. This standard deviation represents a significant decrease over that obtained with ENDF/B-VI. cross sections and is further evidence for the reduction or elimination in C/E keff trends, such as versus H/U ratio (> Fig. ), with the ENDF/B-VII. cross sections. The ENDF/B-VII. cross section changes that are responsible for the improved C/E keff are due primarily to () ORNL and CEA  U revisions in the resonance range for  U, and () the new Los Alamos analysis of  U inelastic scattering in the fast region. The contribution to the increased calculated criticality of these two revisions are of about the same magnitude. Two additional cross section changes also contributed to increase the calculated keff of these assemblies: the reduced  O(n, α) cross section, and a revised scattering kernel for hydrogen bound in water. .. Pu Solution and MOX Benchmarks While excellent calculated k eff results continue to be obtained for thermal uranium solution critical assemblies, the same cannot be said for plutonium solution (PU-SOL-THERM, or PST) assemblies. MCNP C/E values for keff , calculated with ENDF/B-VII. cross sections are plotted versus ATLF and versus H/Pu ratio in > Figs.  and > , respectively. There are obvious variations in these C/E values for k eff when plotted versus ATLF or H/Pu ratio, but it is not obvious what changes in the plutonium cross-section evaluation that could also be supported by the underlying microscopic experimental cross-section data would mitigate these trends. The results for a MOX benchmark, six critical configurations from MIX-COMP-THERM, show less variation than the solutions, possibly because there are fewer of them. The fuel pins contain  wt.% MOX, and the plutonium contains %  Pu. The benchmarks include a 1.025 1.020 Normalized eigenvalue 1.015 1.010 1.005 1.000 0.995 0.990 0.985 PST1 PST2 PST3 PST4 PST5 PST6 PST9 0.980 PST11 PST12 PST18 PST22 PST28 PST32 PST7 0.975 0.0 0.1 0.2 0.3 0.4 Above-thermal leakage fraction (ATLF) 0.5 PST10 0.6 ⊡ Figure  PU-SOL-THERM (PST) benchmark C/E values for keff with ENDF/B-VII. cross sections as a function of the above-thermal leakage fraction (ATLF)   Evaluated Nuclear Data 1.025 1.020 1.015 Normalized eigenvalue  1.010 1.005 1.000 0.995 0.990 0.985 0.980 0.975 0 PST1 PST2 PST3 PST4 PST5 PST6 PST7 PST11 PST12 PST18 PST22 PST28 PST32 PST9 500 1,000 1,500 2,000 2,500 PST10 3,000 H/Pu (in solution) ⊡ Figure  PST benchmark C/E values for keff with ENDF/B-VII. cross sections as a function of the H/Pu ratio highly (several hundred ppm) borated case and a slightly (a few ppm) borated case for each of the three lattice pitches. The highly borated cases contain many more fuel pins than the slightly borated cases and exhibit a small positive bias, but all three fall within a band of about a quarter of a percent in reactivity. . Conclusions from Criticality Testing Hundreds of criticality benchmarks from the ICSBEP Handbook have been calculated to test the accuracy of the ENDF/B-VII. cross-section library. Significant improvement in C/E values for keff has been observed in many cases, including bare and reflected fast uranium and plutonium systems and in particular for arrays of low-enriched fuel rod lattices. The C/E values for keff for bare HEU and Pu assemblies are larger compared to those obtained with ENDF/B-VI. data, and now agree very well with the measurements. The reflector bias for the  U reflected Flattop assemblies has been largely eliminated. Furthermore, major improvements have been obtained in the calculations for intermediate energy assemblies such as Big- and, to a lesser extent, the Argonne ZPR assemblies. Homogeneous uranium solution systems have been calculated accurately with the last several versions of ENDF/B-VI cross sections, and these accurate results are retained with the ENDF/BVII. cross-section library. Many fast-reflected systems are more accurately calculated with the ENDF/B-VII. cross-section library, but disturbing discrepancies remain, particularly in leadand beryllium-reflected systems (although certain reflector-bias improvements were obtained using the new data for these isotopes). A significant accomplishment has been excellent C/E for keff for the LCT assemblies, where the historical underprediction of criticality has been removed. This advance has come from improved  U evaluation (both in the resonance region and in the fast region), together with Evaluated Nuclear Data  revisions to the  O(n, α) cross section and the hydrogen bound in water scattering kernel. Plutonium solution systems are not calculated as well as uranium solutions, with C/E values for keff typically being several tenths of a percent greater than unity. There is a .% spread in these C/E values for keff , but there does not appear to be a trend as a function of  Pu abundance. Although advances have been made at Los Alamos to the  Pu cross sections in the fast region, there has been no recent work on  Pu at lower energies. Clearly such efforts are needed in the future. The performance of the new ENDF/B-VII. evaluations for  U and  Th is much improved in both fast and thermal critical assemblies; an analysis of the Np-U composite fast benchmark suggests important improvements have been made in  Np fission cross-section evaluation. . Delayed Neutron Testing, βeff Delayed neutron data can be tested against measurements of the effective delayed neutron fraction βeff in critical configurations. Unlike the situation for keff , only a handful of measurements of βeff have been reported in the open literature with sufficiently detailed information. In van der Marck (), more than twenty measurements are listed, including two thermal spectrum cores and five fast spectrum cores: TCA: A light water-moderated low-enriched UO core in the Tank-type Critical Assembly (Nakajima, ). IPEN/MB-: A core consisting of  ×  UO (.% enriched) fuel rods inside a light waterfilled tank (dos Santos et al., ). Masurca: Measurements of βeff by several international groups in two unmoderated cores in Masurca (R had ∼% enriched uranium, ZONA had both plutonium and depleted uranium), surrounded by a –% UO -Na mixture blanket and by steel shielding (Okajima et al., ). FCA: Measurements of βeff in three unmoderated cores in the Fast Critical Assembly (HEU; plutonium and natural uranium; plutonium), surrounded by two blanket regions, one with depleted uranium oxide and sodium, the other with only depleted uranium metal (Okajima et al., ). The calculation of βeff for these systems was done using a version of MCNP-C with an extra option added to it as described in Klein Meulekamp and van der Marck (). This method was used earlier to test delayed neutron data from JEFF-. and JENDL-. (van der Marck et al., ). The results based on ENDF/B-VII. are given in > Table , as well as the results based on those other libraries. One can see that the calculated βeff now agrees better with experiment as compared to the old ENDF/B-VI. (specially for the more thermal systems). . Reaction Rates in Critical Assemblies Many different critical assemblies have been developed over the years: Godiva is a bare sphere of HEU; Jezebel is a bare sphere of plutonium; Jezebel  is a bare sphere of  U. The Flattop experiments involved spherical cores of HEU or plutonium surrounded by  U reflector material to make the composite systems critical. These different systems all produce neutron spectra within them that are “fast,” that is, the neutrons are predominantly of energies in the  keV to   Evaluated Nuclear Data ⊡ Table  C/E values for βeff of several critical systems, using ENDF/B-VII. and other libraries C/E βeff System ENDF/B-VII. TCA . ± . IPEN/MB- . ± . Masurca R . ± . Masurca ZONA ENDF/B-VI. JEFF-. JENDL-. . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . FCA XIX- . ± . . ± . . ± . . ± . FCA XIX- . ± . . ± . . ± . . ± . FCA XIX- . ± . . ± . . ± . . ± . The uncertainties in the C/E values are statistical uncertainties from the calculations only 0.6 241 242g Am(n,g) Am/ 239 Pu(n,f ) 0.5 0.4 0.3 Ratio  0.2 0.1 0.1 U238f /U235f (spectrum hardness) ⊡ Figure  The integral  Am neutron capture rate (divided by the  Pu fission rate) as a function of spectral index for different critical assembly locations. In this case, the measurements, which detect the  Cm, are divided by . to account for the fraction of g Am that beta decays to  Cm few MeV region, but the exact spectra vary from system to system. The neutron spectrum gets softer as one moves out from the center of the assembly, thereby giving additional information about the quality of the cross-section data in different energy regimes. Neutron capture of  Am is shown in > Fig.  for different critical assembly spectra. We see good agreement between the calculations and the measurements, except that for the Evaluated Nuclear Data  hardest-spectrum system (Jezebel) the measurement appears to be underpredicted by up to %. This tells us that the  Am capture cross section to the ground state may be too low in the current evaluation in the ≈.– MeV region. . Shielding and Pulsed-Sphere Testing In a paper (van der Marck, ), Steven van der Marck presents extensive data testing results for neutron transmission (shielding) benchmarks. We show some illustrative examples from that paper, focusing on validation benchmarks that test  MeV evaluations that have changed between ENDF/B-VI. and ENDF/B-VII. (e.g., , U,  Pu, Pb, Li, and Be). These comparisons test the accuracy of the secondary emission spectra of neutrons following nuclear reactions. > Figure  shows an example from the above paper for the fusion neutronics source (FNS) benchmark corresponding to  MeV neutrons transmitted through  cm lead at an angle of .○ . The agreement between simulation and the FNS data is seen to be good, it shows an improvement on the earlier ENDF/B-VI. data and provides support for the accuracy of the new  Pb evaluation. Numerous high-energy pulsed-sphere experiments (Wong et al., ; Webster et al., ) have been performed in which small, medium, and large spheres of  different materials were pulsed with a burst of high-energy ( MeV) neutrons at LLNLs insulated core transformer (ICT) accelerator facility. Measured time-dependent neutron fluxes at collimated detectors located at a distance of – m provide a benchmark by which various neutron transport codes and cross-section libraries may be evaluated. The results can be seen in > Fig. . The peak on the left-hand side corresponds to the transmission of the  MeV source neutrons; the broad peak further right (lower energies) corresponds to the neutrons created through compound nucleus and fission mechanisms. Flux/lethargy [1/(cm2 source-neutron)] 0.01 FNS, material: Pb, thickness: 20.3 cm, angle: 42.8 0.001 Experiment ENDF/B-VI.8 ENDF/B-VII.0 0.0001 1e–05 1e–06 1e–07 0.1 ⊡ Figure  1 Neutron energy (MeV) 10 Simulation of  MeV neutron transmission through  cm Pb at .○ (van der Marck, )   Evaluated Nuclear Data 235 U, 0.7 mfp 1.0 Measured Normalized flux/source neutron/Bin  ENDF/B-VII.6 ENDF/B-VII.0 0.1 0.01 0.001 10 –4 20 30 40 50 Neutron flight time (shakes) ⊡ Figure  Comparison of the simulated results using ENDF/B-VI. and ENDF/B-VII. data for the . mfp  U sphere. The experiment used a NE- detector biased at . MeV and located . m along the ○ flightpath. See the footnote for the definition of shake. Note the improved simulation predictions in the minimum region (En ≈ – MeV), where preequilibrium and direct inelastic scattering are present Numerous improvements (Marchetti and Hedstrom, ; Frankle, a,b; Bucholz and Frankle, ) to the simulations have been made since the early implementations of these benchmarks. Simulations were performed by comparing the measured data with calculated results using ENDF/B-VI. or ENDF/B-VII. data with MCNP for the smallest spheres of  U,  U, and  Pu. With improvements in the modeling of the pulsed sphere experiments, problems with down-scattering from -MeV to the – MeV energy region had been noted especially for  U and  Pu. Recent efforts by LANL to improve the evaluated data for inelastic scattering at these higher incident neutron energies have been incorporated into the ENDF/B-VII. evaluations. As shown in > Fig. , the ENDF/B-VII. library improves the treatment of inelastic scattering for  U and  Pu showing much better agreement with the measured data. The improvements in modeling these integral transmission data experiments in the minimum region around  shakes (a shake is an informal unit of time used in nuclear science, − s; the word comes from the expression “two shakes of a lamb’s tail” to mean a very short time Evaluated Nuclear Data  interval) can be directly related to the cross-section improvements in the fundamental data for an incident energy of  and – MeV emission energies, (see > Fig. ). . Testing of Thermal Values and Resonance Integrals Important quantities at low neutron energies are thermal capture cross sections and capture resonance integrals. These quantities can be extracted from the ENDF/B-VII. files and compared with the data in the recently published Atlas of Neutron Resonances (Mughabghab, ). Ratios of capture cross sections at thermal energies are shown in > Fig. . Overall, there is a fairly good agreement between the values in ENDF/B-VII. and the Atlas, although in several instances there are notable discrepancies. The thermal region in  Pa was revised for ENDF/BVII. by Wright (ORNL) leading to the thermal capture nearly  times bigger than the one reported in the Atlas. The origin of this discrepancy is not clear and should be addressed in the future. Ratios for capture resonances integrals can be seen in > Fig. . Several comments should be made. Resonance integrals for  Xe,  Nd,  Gd, and  U: these ratios deviate from unity since there are inconsistencies between resonance parameters and resonance integral measurements reported in Mughabghab () and the evaluators adopted resonance parameters rather than the experimental integrals. In the extreme case of m Ho, the experiment is deemed doubtful due to the cadmium cutoff because of the low energy resonance at . eV. In  Dy, the resolved resonance range is very limited (up to  eV) and extrapolation of the URR to such low energies might not be reliable (in particular, the exact position of the lower boundary might play Ratio of ENDF/B-VII to Atlas Thermal capture cross sections Z ⊡ Figure  Thermal neutron capture cross sections in ENDF/B-VII. compared to the Atlas of Neutron Resonances (Mughabghab, )   Evaluated Nuclear Data Ratio of ENDF/B-VII to Atlas  Z ⊡ Figure  Neutron capture resonance integrals in ENDF/B-VII. compared to the Atlas of Neutron Resonances (Mughabghab, ) a significant role). The remaining outliers ( P,  Ca,  Co,  Hf,  Hg,  Pu) are real discrepancies. These are old evaluations that will have to be updated in future releases of the ENDF/B library.  Other Nuclear Data of Interest Other evaluated nuclear data of interest to nuclear technology applications, primarily to fission reactor applications, are fission yields (also termed FPY), thermal neutron scattering, and radioactive decay data. . Fission Yields The fission yields from the  LANL evaluation by England and Rides () were adopted by ENDF/B-VII.. In this evaluation, fission yield measurements reported in the literature and calculated charge distributions were used to produce a recommended set of yields for the fission products. Independent yields were taken from a calculated charge distribution model. A Gaussian charge distribution was calculated by using the most probable charge and Gaussian width. The weighted average experimental independent yields, the weighted average experimental cumulative yields, the weighted average experimental cumulative yields, and the calculated cumulative yields were combined statistically to form a recommended value. Evaluated Nuclear Data  There are two fission product yield sublibraries in ENDF/B-VII.. Both of them have been taken over from ENDF/B-VII. without any change: • The neutron-induced fission yields sublibrary contains data for  actinides. Incident neutron interacts with the target material that undergoes fission, which gives rise to extensive number of fission products. There are  such target materials, from  Th to  Fm; neutron incident energies include the thermal energy of . eV,  keV, and  MeV. While for some materials, such as  U, fission yields are given for all three energies, for many other materials, yields are given only for one or two energies. • Spontaneous fission yields sublibrary contains nine materials,  U, ,, Cm, , CF,  Es, and , Fm. Each of these material undergoes spontaneous fission, which again gives rise to an extensive set of fission products. Fission yields can be conveniently retrieved and plotted by Sigma web interface (Pritychenko and Sonzogni, ), which was developed and continues to be maintained by the NNDC, BNL. . Thermal Neutron Scattering In ENDF/B-VII., this sublibrary contains  evaluations. As described below, seven were reevaluated or updated due to the combined efforts of MacFarlane, Los Alamos, and by Mattes and Keinert, IKE Stuttgart (Mattes and Keinert, ). The remaining evaluations were taken over from the ENDF/B-VI. library. New thermal neutron scattering evaluations were generated by the LEAPR module of the NJOY code (MacFarlane, ). The physical model has been improved over the one used at General Atomics in  to produce the original ENDF/B-III evaluations (Koppel and Houston, ). The alpha and beta grids have been extended to allow for larger incident energies and to properly represent the features of S(α, β) for the various integrations required. The physical constants have been updated for ENDF/B-VII. to match the current hydrogen and oxygen evaluations. The changes include additional alpha and beta points, interpolating the rotational energy distributions and translational energies onto the new temperature grid, and slightly reducing the rotational energies to improve the energy region between . and . eV. .. H O and D O H O: This evaluation was generated by Mattes and Keinert (). Water is represented by freely moving H O molecule clusters with some temperature dependence to the clustering effect. Each molecule can undergo torsional harmonic oscillations (hindered rotations) with a broad spectrum of distributed modes. The excitation spectra were improved over the older ENDF model, and they are given with a temperature variation. In addition, there are two internal modes of vibration at  and  meV. The stretching mode was reduced from the older ENDF value of  meV to account for the liquid state. Scattering by the oxygen atoms is not included in the tabulated scattering law data. It should be taken into account by adding the scattering for free oxygen of mass . We note that the new H O thermal scattering kernel in ENDF/B-VII. led to a slight increase in calculated criticality of LEU-COMP-THERM critical assemblies.    Evaluated Nuclear Data D O: This was based on the IKE-IAEA-JEFF-. evaluation done by Mattes and Keinert (). Changes made for ENDF/B-VII include using a more ENDF-like temperature grid and an extension of the α and β grids to improve results for higher incident energies. .. O in UO and U in UO Uranium dioxide has a structure similar to fluoride, CaF . A lattice dynamical model was developed by Dolling, Cowley, and Woods to fit dispersion curve measurements. In additional to short-range core–core forces, the model includes shell–core, shell–shell, and long-range Coulomb interactions. Weighted frequency distributions were calculated from a dynamical matrix based on this model. The O in UO part is kept separate from the U in O part, and one-fourth of the coherent elastic cross section from the original General Atomics evaluation was included. The various constants were updated to agree with the ENDF/B-VII. evaluation of oxygen. .. H in ZrH The lattice dynamics of ZrH were computed from a central force model. The slightly tetragonal lattice of ZrH was approximated by a face-centered cubic lattice. Four force constants (μ, γ, ν, and δ) were introduced describing, respectively, the interaction of a zirconium atom with its nearest neighbors ( H atoms) and its next nearest neighbors ( Zr atoms), and the interaction of a hydrogen atom with its next nearest neighbors ( H atoms) and its third nearest atoms ( H atoms). Eigenvalues and eigenvectors of the dynamical matrix were calculated and a phonon frequency spectrum was obtained by means of a root sampling technique. Weighted frequency spectra for hydrogen in ZrH were then obtained by appropriate use of the dynamical matrix eigenvectors. The final values of the four force constants were obtained by fitting both specific heat and neutron data. The position of an optical peak observed by neutron scattering techniques to be centered roughly around . eV determines the constant μ, while the overall width and shape of this peak determine ν and δ, respectively. Existing neutron data are not sufficiently precise to confirm the structure predicted in the optical peak by the central force model. Specific heat data were used to determine the force constant γ, which primarily determines the upper limit on the phonon energies associated with acoustic modes. .. Other Modified Materials Liquid methane at  K used the model of Agrawal and Yip as implemented by Picton, modified to include a diffusive component. Solid methane at  K used the model of Picton based on the spectrum of Harker and Brugger. Liquid para and ortho hydrogen at  K were computed with LEAPR. The scattering law is based on the model of Keinert and Sax (), which includes spin correlations from the Young and Koppel () model, diffusion and local hindered motions from an effective translational scattering law based on a frequency distribution, and intermolecular coherence after Vineyard (). Evaluated Nuclear Data  Data for aluminum are provided for temperatures of , , ., , , and  K using frequency distribution of Stedman et al. ().  Fe was modeled using iron frequency distribution of Brockhouse et al. (). . Decay Data The decay data part of the ENDF/B-VII. library was produced by Sonzogni (NNDC, BNL) in . This new sublibrary contains , materials and is mostly derived from the Evaluated Nuclear Structure Data File (ENSDF, ) and the  edition of the Nuclear Wallet Cards (Tuli, ). Each material corresponds to the ground state or an isomeric level of a given nucleus. The library provides information for stable and unstable nuclei, from the neutron to  Rg (Z = ). For sections of the library corresponding to unstable levels, the half-life, decay modes, and energy released during the decay is presented. For stable levels, the only information given is the spin and parity of the level. The energy released can be given with varying degrees of detail. The most basic information includes mean electromagnetic energy (EEM), mean light particle energy (ELP), and mean heavy particle energy (EHP). For materials whose decay scheme is well known, that is, satisfying that the sum of the average energies for each radiation type is very close to the effective Q-value, the ENSDF database was used and discrete radiation information was provided. In contrast, for materials with unknown or poorly known decay schemes, the Nuclear Wallet Cards database was used. In this case, a simple rule was used to obtain the mean energies. If for instance the level in question undergoes beta decay, it was assumed that EEM and ELP corresponds each to a third of decay Q-value, while the neutrinos take the remaining third. For β-delayed particle emission, it was assumed that the neutrinos carried away a quarter of the available energy and that leptons, baryons, and photons took a quarter each. The measurement of decay characteristics of fission products becomes increasingly difficult as the fission products are further from the valley of stability. Typically, as the β-decay Q-value increases, more weak gamma rays are produced, which are difficult to place or simply escape detection. To address this issue, a series of measurements using a  Cf source and a total absorption gamma spectrometer (TAGS) were performed at INL, Idaho (Greenwood et al., ). Using these data, EEM and ELP values were obtained for  materials, which can improve the decay heat predicting power of the library (Hagura et al., ). To obtain the EEM and ELP values from TAGS experiments, the evaluator followed the prescription developed by Hagura et al. (), where it is assumed that the decay from excited levels proceeds only by gamma emission, that is, conversion electrons are neglected. As a result, the EEM values is really an upper limit and the ELP a lower one. The effect of electron conversion is expected to be small, less than % of EEM. The use of TAGS data in decay data libraries is one of the issues under study by WPEC Subgroup  (Yoshida et al., ). Additionally, the following features were included: • Internal conversion coefficients were calculated for all gamma rays of known multipolarity using the code BRICC (Band et al., ).  I, and  Cs average β-energies for second forbidden nonunique transitions were calculated using the code SPEBETA (Cassette, ). • For  Cl,  Fe,  Tc,   Evaluated Nuclear Data 235 U decay heat decay neat (MeV/fission) Data: Tobias 1989 Full: TAGS included Dashed: No TAGS p Time  , Time after fission burst (s) ⊡ Figure  Decay heat per fission for a  U sample as a function of time. Shown is the total decay heat and its two components (light particles, electromagnetic) • Theoretical β-decay half-lives and β-delayed neutron emission probabilities (Pn ) using the Kratz–Hermann systematics (Pfeiffer et al., ) were used for some neutron-rich nuclides that were produced in the fission of  U and  Pu with limited experimental T/ or Pn information. .. Decay Heat Calculations A plot of the decay heat following a fission event of  U can be seen in > Fig. . The total decay heat is separated into two components, electromagnetic and light particles. The former includes gamma and X-rays, while the latter includes electrons from β-decay as well conversion and Auger electrons. A heavy particle component, including neutrons and alphas is negligible. The data come from the  compilation of Tobias and Mills (, private communication). The effect of the TAGS data is clearly visible. Without including it, many unmeasured weak gamma rays would be missing due to incomplete decay schemes, resulting in artificially high values of electron and neutrino mean energies as well as artificially low values of mean gamma energies. The JEFF-. decay data library was released in  (correction to TAGS not yet introduced) and shares a similar spirit and scope with the ENDF/B-VII. decay data library. One Evaluated Nuclear Data  U decay heat , 10,000 ⊡ Figure  Decay heat per fission for a  U sample as a function of time using the ENDF/B-VII. and the JEFF-. decay data without total absorption gamma spectrometer (TAGS) data possible way of comparing both libraries would be to plot decay heats without TAGS data for  U. This is shown in > Fig.  and, as expected, both libraries give very similar results under this condition.  Evaluated Nuclear Data Libraries There are numerous evaluated nuclear data libraries available from various nuclear data centers. National interests and different applications are the two principal factors causing this variety. Countries with strong nuclear programs, such as the USA, the European Union, Japan, Russia, and China, develop their own general purpose libraries to maintain evaluation expertise and ensure technological independence. On the other hand, various applications of nuclear technology require special purpose libraries that satisfy particular needs of a given application. These derived libraries add another class to that mentioned above. One should also take into account various versions (releases) of the major libraries. Frequent sharing of evaluations among different libraries, often with some modifications, makes this picture even more complicated.    . Evaluated Nuclear Data Overview of Libraries A brief overview of evaluated nuclear data libraries should assist users to make the right choice for their application. It should be understood that there is internal dynamics in data development. Therefore, users should always consult webpages of the most prominent data centers to make sure that the library they are interested in is the latest version available. .. General Purpose Libraries General purpose libraries are not limited to any specific application and they are meant to satisfy a broad class of users. In practice, though, they often started as libraries for reactor applications. Evaluations in a general purpose library are usually most complete in terms of physical quantities and nuclear reactions. They have to be suitable for transport calculations and as such have to fulfill quite strict requirements regarding completeness and consistency. Thus, neutron evaluations have to cover thermal, resolved, and unresolved resonance as well as fast neutron ranges extending at least up to  MeV, contain all major reaction channels, provide cross sections and possibly angular distributions, energy-angle correlated cross sections, and photon production data. Internal consistency implies that individual cross sections must sum up to the total cross section and the integrals of emission spectra correspond to the respective reaction cross sections. Typically, the general purpose libraries are extensively validated against integral measurements. Sometimes, results of these integral measurements are incorporated into a library. This procedure introduces implicit correlations between various reactions and materials causing such a library to become an entity rather than a simple collection of individual evaluations. Major general purpose libraries are maintained by the following countries: . . . . USA – ENDF/B-VII., released in ; new version is expected in . Europe – JEFF-., released in ; new version is expected in . Japan – JENDL-., released in ; JENDL- is expected in . Russia – BROND-., released in ; BROND- has not been completed yet, it is partly available in the selected evaluations of ROSFOND, which was released in . . China – CENDL-, released in ; CENDL- was developed but not internationally released; development of CENDL- is underway. Over the first decade of the twenty-first century three evaluated data libraries (ENDF/B, JEFF, and JENDL) have been continuously updated and improved. These libraries will be briefly summarized later in this section. .. Special Purpose Libraries The special purpose libraries address particular exigencies of certain applications. Typical examples of such libraries are: • • • • International Reactor Dosimetry File, IRDF (Griffin and Paviotti-Corcuera, ) European Activation File, EAF (Forrest et al., ) Standards neutron cross section library (Carlson et al., ), and Fusion Evaluated Nuclear Data Library, FENDL (Lopez Aldama and Trkov, ) Evaluated Nuclear Data  Evaluations in these libraries are not as comprehensive as those in the general purpose libraries but excel in certain features that would be impractical or too costly to be implemented in the general purpose files. For example, activation library does not have to cover the full energy range, does not require spectra and angular distributions but does need to provide cross sections for the reactions leading to the radioactive products. Often, these are metastable states that are rarely considered in the general purpose libraries. There is no internal consistency requirement, but the amount of materials in the activation library is usually far larger than in the general purpose libraries. The dosimetry libraries are similar to the activation ones but cover very limited number of well-known reactions that are used for the determination of neutron spectra. Cross-section covariances are critical for spectra deconvolution and are mandatory in dosimetry libraries. On top of this pyramid are libraries of standards that include cross sections and covariances for an even smaller number of reactions that are known to a very high accuracy. The evaluation of standards is particularly thorough and is predominantly based on a detailed analysis of precise experimental data. Since standards are used as reference in many measurements they are a potential source of correlations among seemingly independent experiments. Therefore, cross-correlations among standards cross sections are required in the standards library. These libraries provide by far most accurate and reliable data but their coverage is very fragmentary. Another type of special purpose libraries are those that are created as a selection of evaluations from various major libraries to better serve particular application. The prominent example is FENDL (Lopez Aldama and Trkov, ), the international library compiled under auspices of the IAEA in support of the fusion program. Some examples of special purpose libraries are: IRDF-: The International Reactor Dosimetry File (Griffin and Paviotti-Corcuera, ) is a standardized, updated, and benchmarked evaluated cross section library of neutron dosimetry reactions with uncertainty information for use in lifetime management assessments of nuclear power reactors and other neutron metrology applications such as boron neutron capture therapy, therapeutic use of medical isotopes, nuclear physics measurements, and reactor safety applications. It contains damage cross sections, decay data, standard spectra, and dosimetry cross sections in ENDF- pointwise and groupwise representation. The development of IRDF- was coordinated by IAEA during –. INDL/TSL: An improved set of thermal neutron scattering law data prepared for ten elements/compounds in – by Mattes and Keinert under auspices of the IAEA. IAEA-Standards, : The most respected international library of neutron cross-section standards (Carlson et al., ). It contains data for nine reactions including covariances. The library relies on a very careful evaluation of the selected set of most precise and reliable experiments. ENDF/B-VII. adjusted its neutron sublibrary to these cross sections and the whole set is available in its standards sublibrary. EAF-: The European Activation File (Forrest et al., ) is the most extensive library of neutron activation cross sections. It contains   excitation functions on  different targets from  H to  Fm stored in the extended ENDF- format (EAF format). The ENDF formatted version is included as JEFF-./A sublibrary in the general purpose library JEFF-.. EAF-: An extension of the European Activation File to proton- and deuteron-induced reactions (Forrest et al., ) in addition to the traditional neutron-induced data (Forrest et al., ). The deuteron-induced library contains , reactions, while the proton-induced    Evaluated Nuclear Data library contains , reactions. The library makes extensive use of model calculations with the TALYS code (Koning et al., ). JENDL/AC-: The JENDL Actinoid File  is a consistent set of new evaluations for  actinides (≤ Z≤) released in  with the intention of being included in JENDL-. MINSK: The library of original evaluations for  actinides developed by Maslov et al. (Minsk, Belarus) between  and . It contains data for isotopes of Th, Pa, U, Np, Pu, Am, and Cm. MENDL-: The neutron reaction data library for nuclear activation and transmutation at intermediate energies developed by Shubin et al. (IPPE Obninsk) around –. It contains production cross sections for the formation of radioactive product nuclides for incident neutrons with energies up to  MeV. The  nuclides included cover the range from  Al to  Po with half-lives larger than one day. ENDF/HE-VI: The high-energy library developed by S. Perlstein (BNL) and T. Fukahori (JAERI) in the s containing neutron and proton data for  C,  Fe,  Pb, and  Bi up to , MeV. .. Derived Libraries Derived libraries are obtained from the libraries discussed above by processing them with a dedicated computer code such as NJOY (MacFarlane and Muir, ). In most cases, this processing is carried out to reconstruct cross sections in the resonance region, perform their Doppler broadening at a given temperature (pointwise representation), and to provide averages over certain energy intervals (groupwise representation). Derived libraries are generally needed for transport calculations (e.g., ACE libraries used in the Monte Carlo MCNP code (X-MCNP-Team, )). The derived libraries may also be adjusted to reproduce particular set of integral experiments. In most cases, such adjustment is performed on the groupwise library and targets very well-defined applications such as sodium cooled fast reactors. The performance of the adjusted library is superior when it is used for the intended application but it might be poor for other applications. This is a consequence of the selection of integral experiments and adoption of energy-group structure that are tailored for the intended application. . ENDF- Format Use of the ENDF- format is common for most of the evaluated nuclear data libraries. Only some of the activation and derived libraries deviate from this standard. Otherwise, all major libraries are using ENDF- format that has been accepted internationally. This unification had a great impact on the worldwide cooperation, greatly facilitated by exchanging files between the national libraries and easy comparison of the data. The ENDF format has been developed by the CSEWG and it is maintained by the NNDC. The work started in , the first version was released in , and then in , , , , and  along with the subsequent releases of the US ENDF library. The current Evaluated Nuclear Data  version, ENDF- (Herman and Trkov, ), has been used for both the ENDF/B-VI and ENDF/B-VII library implying that a new version of the format has not been developed for the ENDF/B-VII. We note that to differentiate it from the library that is denoted with Roman numerals (say, ENDF/B-VI), the ENDF format is always denoted with the Arabic numeral (ENDF-). For historic reasons, the ENDF- format uses -character records conforming to the oldfashioned versions of FORTRAN. It is organized in a strict hierarchical structure. Any library is a collection of material evaluations from a recognized evaluation group. It is divided into sublibraries that distinguish between different incident particles and types of data, namely, neutron induced-reactions, proton-induced reactions, thermal scattering data, fission yields, decay data, etc. The sublibraries contain the data for different materials identified by MAT numbers. Each material evaluation contains data blocks referred to as “Files” and denoted with MF numbers. File : MF= is the descriptive part of the numerical file with details of evaluation, it also contains ν̄ values. File : MF= contains neutron resonance parameters. Neither thermal constants, nor cross sections in the resonance region are provided, these are reconstructed from resonance parameters by processing codes. File : MF= contains cross sections. The minimum required energy range for neutron reactions is from the threshold or from âĹŠ eV up to  MeV, but higher energies are allowed. There is a section for each important reaction or sum of reactions. The reaction MT-numbers for these sections are chosen based on the emitted particles. Files –: Energy and angle distributions for emitted neutrons and other particles or nuclei. File  is used for simple two-body reactions (elastic, discrete inelastic). Files  and  are used for simple continuum reactions, which are nearly isotropic and emit only one important particle. File  is used for more complex reactions that require energy-angle correlation that are important for heating or damage, or that have several important products that must be tallied. Files –: If any of the reaction products are radioactive, they should be described further in File . This file indicates how the production cross section is to be determined (from Files , , , or ) and gives minimal information on the further decay of the product. Additional decay information can be retrieved from the decay data sublibrary when required. Branching ratios (or relative yields) for the production of different isomeric states of a radionuclide may be given in File . Alternatively, radionuclide isomer-production cross sections can be given in File . Files –: For compatibility with earlier versions, photon production and photon distributions can be described using File  (photon production yields), File  (photon production cross sections), File  (photon angular distributions), and File  (photon energy distributions). File  is preferred over File  when strong resonances are present (capture, fission). Files –: Covariance data are given in Files –, with ν̄ covariances in File , resonance parameter covariances in File , and cross-section covariances in File . A concise list of basic definitions and constants used in the ENDF- format is given in Table . For a detailed description we refer to the extensive manual (Herman and Trkov, ). >    Evaluated Nuclear Data ⊡ Table  ENDF- format: Selected definitions and constants. See Herman and Trkov () for more details File Section MF= MF= General information MT= Description of the evaluation MT= Average number of neutrons per fission, ν̄ (ν̄ = ν̄d + ν̄p ) MT= Average number of delayed neutrons per fission, ν̄ d MT= Average number of prompt neutrons per fission, ν̄ p MT= Energy release in fission for incident neutrons MT= β-Delayed photon spectra MF= MF= Resonance parameters MT= Resolved resonance parameters, flag LRU= MT= Unresolved resonance parameters, flag LRU= MF= MF= Quantity Reaction cross sections MT= Total cross sections MT= Elastic cross sections MT= Sum of all inelastic cross sections MT= Sum of cross sections for all reaction channels not given explicitly under other MT numbers MT= (n,n) cross sections MT= (n,n) cross sections MT= (n,xf ) total fission cross sections MT= (n,f ) first chance fission cross sections MT= (n,nf ) second chance fission cross sections MT= (n,nf ) third chance fission cross sections MT= (n,n’α) cross sections MT= (n,n’p) cross sections MT= (n,n’ ) cross sections (inelastic scattering to the st excited level) MT= (n,n’ ) cross sections (inelastic scattering to the nd excited level) MT= (n,n’cont ) cross sections (inelastic scattering to continuum) MT= (n,γ) cross sections MT= (n,p) cross sections MT= (n,t) cross sections MT= (n,α) cross sections Evaluated Nuclear Data  ⊡ Table  (continued) File Section Quantity MT= (n,p ) cross sections for the (n,p) reaction leaving residual nucleus in the st excited level MT= (n,p ) cross sections for the (n,p) reaction leaving residual nucleus in the ground state MT= (n,α ) cross sections for the (n,α) reaction leaving residual nucleus in the st excited level MF= MF= Angular distributions of emitted particles expressed as normalized probability distributions MT= Angular distributions for elastic scattering MT= Angular distributions for inelastic scattering to the st excited level MF= MF= Energy distributions (spectra) of emitted particles expressed as normalized probability distributions MT= MF= MF= Energy-angle distributions of emitted particles (for a given reaction should contain subsections for all reaction products including γ’s and recoils) MT= Energy-angle distributions of products for all reactions lumped into MT= (reactions are identified by the residual nuclei) MT= Energy-angle distributions of continuum neutrons (only those neutrons that were not followed by any other particle emission are counted) MT= Energy-angle distributions of neutrons, protons, residual nuclei, and photons emitted in the (n,n’p) reaction MF= MF= Thermal neutron scattering on moderating materials MT= Elastic thermal neutron scattering MT= Inelastic thermal neutron scattering MF= MF= MF= Spectra of emitted neutrons, photons, and recoils for the (n,n) reaction (neutron spectra contain both cascading neutrons) Decay data and fission-product yields MT= Independent fission product yields MT= Cumulative fission product yields MT= Radioactive decay data Multiplicities for production of radioactive nuclei (activation/isomeric cross sections expressed as a fraction of the respective cross sections in MF=)    Evaluated Nuclear Data ⊡ Table  (continued) File Section Quantity MF= Absolute cross sections for production of radioactive nuclei (similar to MF= but without reference to MF=) MF= Multiplicities for photon production and branching ratios for γ transitions between discrete levels (respective cross sections in MF= must be used for absolute values) MF= Absolute photon spectra and photon production cross sections (similar to MF= but without reference to MF=) MF= Angular distributions for discrete and continuum photons MF= Energy spectra for continuum photons (normalized distributions to be multiplied by the respective cross sections in MF=) MF= Electromagnetic interaction cross sections (such as total, coherent and incoherent (Compton) elastic scattering for photons and elastic scattering, brehm- sstrahlung and ionization for electrons) MF= MT= Total cross section for incident photons MT= Photon coherent scattering cross section MT= Pair production cross section MT= Photoelectric absorption cross section MT= Electro-atomic brehmsstrahlung cross section MF= Spectra and angular distributions of photons and electrons emitted in inter–action of photons or electrons with atoms MF= Atomic form factors or scattering functions for angular distribution of photons MF= Atomic relaxation data (emission of X-rays and electrons from ionized atoms) MF= Covariances for nubar (ν̄) MF= Covariances for resonance parameters MF= MF= Covariances for cross sections MT= Covariances for total cross sections MT= Covariances for elastic cross sections MT=– Covariances for cross sections of lumped channels MF= Covariances for angular distributions of emitted particles MF= Covariances for energy spectra of emitted particles MF= Covariances for activation cross sections Evaluated Nuclear Data  ⊡ Table  (continued) NLIB Library Full name  ENDF/B US Evaluated Nuclear Data File  ENDF/A US Evaluated Nuclear Data File for preliminary or incomplete evaluations  JEFF Joint European Evaluated File  EFF European Fusion File (now in JEFF)  ENDF/HE US High Energy File  CENDL Chinese Evaluated Nuclear Data Library  JENDL Japanese Evaluated Nuclear Data Library  IFPL NEA International Fission Product Library  FENDL IAEA Fusion Evaluated Nuclear Data Library  IRDF IAEA International Reactor Dosimetry File  FENDL/A FENDL Activation File  BROND Russian Biblioteka (library) of Recommended Neutron Data Symbol Definition Recommended value mn neutron mass .   amu me electron mass .  ×− amu mp proton mass .   amu md deuteron mass .   amu mt triton mass .   amu mh  .   amu mα α-particle mass .   amu amu atomic mass unit .  × eV e elementary charge .   ×− C h Planck’s constant .   ×− eV s h/π Planck’s const./π .   ×− eV s k Boltzmann’s constant .  ×− eV/K c Speed of light    m/s NA Avogadro’s number .   × /mol He mass    Evaluated Nuclear Data ⊡ Table  Major releases of the ENDF/B library of the USA. The library is maintained by the Cross Section Evaluation Working Group (CSEWG), established in  . ENDF/B I II III IV V VI VII Year        ENDF/B-VII. (USA, ) The ENDF/B-VII. library, released by the US CSEWG in December , contains data primarily for reactions with incident neutrons, protons, and photons on almost  isotopes, based on experimental data and theory predictions. The new library plays an important role in nuclear technology applications, including transport simulations supporting national security, nonproliferation, advanced reactor and fuel cycle concepts, criticality safety, fusion, medicine, space applications, nuclear astrophysics, and nuclear physics facility design. Major releases of the US ENDF/B library are summarized in > Table . After an initial -year release cycle, CSEWG moved to ever longer release cycles. Recent releases occurred at widely spaced intervals: ENDF/B-V was released in , ENDF/B-VI in , followed by this ENDF/B-VII. release in . However, interim releases have occurred more frequently and ENDF/B-VI had upgrades embodied in eight releases, the last one occurring in October  and referred to as ENDF/B-VI. (CSEWG-Collaboration, ). .. Overview of the ENDF/B-VII. Library The ENDF/B-VII. library contains  sublibraries, summarized in > Table , according to the identification number NSUB. The number of materials (isotopes or elements) are given for both the new (VII.) and previous (VI.) versions of the ENDF/B library. Although the ENDF/B library is widely known for evaluated neutron cross sections, it also contains a considerable amount of non-neutron data. Out of the total of  sublibraries, there are two new sublibraries; seven sublibraries were considerably updated and extended, and the remaining five sublibraries were taken over from ENDF/B-VI. without any change: . The photonuclear sublibrary is new; it contains evaluated cross sections for  materials (all isotopes) mostly up to  MeV. The sublibrary has been supplied by LANL and it is largely based on the IAEA-coordinated collaboration completed in  (Chadwick et al., ). . The photo-atomic sublibrary has been taken over from ENDF/B-VI.. It contains data for photons from  eV up to  GeV interacting with atoms for  materials (all elements). The sublibrary has been supplied by LLNL. . The decay data sublibrary has been completely reevaluated and considerably extended by the NNDC, BNL. . The spontaneous fission yields were taken over from ENDF/B-VI.. The data were supplied by LANL. Evaluated Nuclear Data  ⊡ Table  Contents of the ENDF/B-VII. library, with ENDF/B-VI. shown for comparison No. NSUB Sublibrary name Short name ENDF/B- VII. materials ENDF/B-VI. materials   Photonuclear g  −   Photo-atomic Photo     Radioactive decay Decay ,     Spontaneous fission yields s/fpy     Atomic relaxation ard     Neutron n     Neutron fission yields n/fpy     Thermal scattering tsl     Standards Std     Electro-atomic e     Proton p     Deuteron d     Triton t     he    He NSUB stands for the sublibrary number, given in the last two columns are the number of materials (isotopes or elements) . The atomic relaxation sublibrary was taken over from ENDF/B-VI.. It contains data for  materials (all elements) supplied by LLNL. . The neutron reaction sublibrary represents the heart of the ENDF/B-VII. library. The sublibrary has been considerably updated and extended; it contains  materials, including  isotopic evaluations and  elemental ones (C, V, and Zn). These evaluations can be considered to be complete (the only exception is  Es that contains (n, γ) dosimetry cross sections) since they contain data needed in neutronics calculations. Important improvements were made to the actinides by LANL, often in collaboration with ORNL. Evaluations in the fission product range (Z = –) have been entirely changed. Of the  materials, about two-thirds of the evaluations are based upon recent important contributions from US evaluators. The remaining evaluations were adopted from other sources (mostly the JENDL-. library). LLNL provided β-delayed γ-ray data for  U and  Pu for the first time in ENDF/B. . Neutron fission yields were taken over from ENDF/B-VI.. The data were supplied by LANL. . The thermal neutron scattering sublibrary contains thermal scattering-law data, largely supplied by LANL, with several important updates and extensions (Mattes and Keinert, ). . The neutron cross-section standards sublibrary is new. Although standards traditionally constituted part of the ENDF/B library, in the past these data were stored on a tape. As the    . . . . . Evaluated Nuclear Data concept of tapes has been abandoned in ENDF/B-VII., the new sublibrary (short name std, sublibrary number NSUB = ) has been introduced. Out of eight standards materials, six were newly evaluated, while the  He(n,p) and nat C(n,n) standards were taken over from ENDF/B-VI.. The standard cross sections were adopted by the neutron reaction sublibrary except for the thermal cross section for  U(n, f ) where a slight difference occurs to satisfy thermal data testing. These new evaluations come from the international collaboration coordinated by the IAEA (Carlson et al., ). The electro-atomic sublibrary was taken over from ENDF/B-VI.. It contains data for  materials (all elements) supplied by LLNL. The proton-induced reactions were supplied by LANL, the data being mostly up to  MeV. There are several updates and several new evaluations. The deuteron-induced reactions were supplied by LANL. This sublibrary contains five evaluations. The triton-induced reactions were supplied by LANL. This sublibrary contains three evaluations. Reactions induced with  He were supplied by LANL. This sublibrary contains two evaluations. .. Processing and Data Verification The ENDF/B-VII. library was issued in its basic format defined by the ENDF- Formats Manual (Herman and Trkov, ). For practical applications, the library must be processed so that basic data are converted into formats suitable as input for applied codes such as the Monte Carlo transport code, MCNP, and the reactor licensing code, SCALE (SCALE, ). Recommended processing codes: • Los Alamos code NJOY- (MacFarlane and Muir ; MacFarlane and Kahler, , “NJOY-.”, www.t.lanl.gov/codes/njoy/index.html ( october ), to be obtained from RSICC (RSICC, ) or NEA Data Bank (), with patches available at the LANL T- webpage (). • Two codes are available for processing of covariance data, ERRORJ (Kosako, ) – since recently a part of the NJOY package, and PUFF (Wiarda and Dunn, ) – a module of the Oak Ridge processing code AMPX (Dunn and Greene, ). Data verification was performed by the NNDC, BNL: • Checking the library by ENDF- utility codes (CHECKR, FIZCON, PSYCHE) (Dunford, ) for possible formatting problems and inconsistencies in physics. • Processing of photonuclear, neutron, thermal scattering, and proton sublibraries by NJOY to ensure that a processed library suitable for neutronics calculations can be produced. • Use of the processed files by the Monte Carlo codes MCNP (X-MCNP-Team, ) and MCNPX () in simple neutronics test to ensure that neutronics calculations can be performed. • Processing of covariance data to ensure that multigroup data for applied calculations can be produced. Evaluated Nuclear Data  Data validation is a complex process described earlier in this chapter. CSEWG used continuous energy Monte Carlo transport codes and validation focused on the neutron reaction and thermal neutron scattering sublibraries. These are the best validated sublibraries. The neutron standards sublibrary contains a special category of data where the highest quality was achieved. The photonuclear sublibrary was subject to partial validation, and the decay data sublibrary went through some limited testing. The remaining nine sublibraries were not included in the ENDF/B-VII. validation process. . JEFF-. (Europe, ) The JEFF project is a collaborative effort among the European member countries of the NEA Data Bank. The initial objective was to improve performance for existing reactors and fuel cycles. More recently, the goal is to provide users with a more extensive set of data for a wider range of applications, including innovative reactor concept (Gen-IV), transmutation of radioactive waste, fusion, and medical applications. These data include neutron- and protoninduced reactions, radioactive decay, fission yields, thermal scattering law, and photo-atomic interactions. The JEFF-. version of the library was released in May , for summary description see JEFF Report  (JEFF, ). The library combines the efforts of the JEFF and EFF/EAF (European Fusion File/European Activation File) working groups. The neutron general purpose sublibrary contains  materials from  H to  Fm. The activation sublibrary is based on the EAF- and contains cross sections for neutron reactions on  targets; radioactive decay sublibrary contains three  isotopes of which only  are stable; proton sublibrary covers  materials from  Ca to  Bi; thermal scattering law sublibrary includes nine materials; neutron-induced-fission-yield sublibrary covers  isotopes from  Th to  Cm, and spontaneous-fission-yield sublibrary contains , Cm and  Cf. The JEFF-. library was upgraded mainly because of underprediction of the reactivity for LEU systems relevant to light water reactors. The reactivity issue was linked to the  U cross sections and the improved evaluation was assembled as a result of the broad international effort. Transport calculations proved that the predictions of this reactivity was appreciably improved. New evaluations or major revisions were performed for Ti isotopes (IRK Vienna); Ca, Sc, Fe, Ge, Pb, and Bi isotopes (NRG Petten);  Rh, , I, and Hf isotopes; and ,, U and  Am (CEA). For other isotopes, more recent evaluations from other libraries were adopted. Revised thermal scattering data have been produced for all important moderator and structural materials. The JEFF project put considerable effort to validation of the library, which was done particularly carefully from the point of view of nuclear reactor applications. The overall performance of the library is excellent. . JENDL-. (Japan, ) JENDL-. is the Japanese evaluated library that was released in , see the summary paper by Shibata et al. (b). The library is largely based on evaluations that originated in Japan, thus representing probably the most extensive source of independent evaluations, just after the US effort.    Evaluated Nuclear Data The objective of the JENDL effort is to supply Japanese evaluated data for fast breeder reactors, thermal reactors, fusion neutronics, and shielding calculations, as well as other applications. The JENDL-. library contains data for  materials, from − to  MeV. Major issues in the previous version of the library, JENDL-., were addressed: overestimation of criticality values for thermal fission reactors was improved by the modification of fission cross sections and fission neutron spectra for  U; incorrect energy distributions of secondary neutrons from important heavy materials were replaced by the statistical model results; inconsistency between elemental and isotopic evaluations were removed for medium-heavy nuclides. JENDL-. also contains covariances for  most important materials. Of them,  materials have been originally developed for JENDL-. covariance file, made available in March  (Shibata et al., a), and adopted shortly afterward with minor modifications by JENDL-.. Three additional materials were produced for JENDL-., while the dosimetry material  Mn was taken over. The list of resulting  materials includes actinides, structural materials, and light nuclides, which are of interest primarily for fast reactor applications: •  H, , B,  O,  Na,  Ti, V,  Cr,  Mn,  Fe,  Co, , Ni,  Zr, ,, U, ,, Pu. A new version of the library, JENDL-, is under development, with a release expected in . . Web Access to Nuclear Data Several major webpages offer evaluated nuclear data. These are regularly maintained by the four well-established data centers: . . . . NNDC, USA, www.nndc.bnl.gov IAEA, Nuclear Data Section (IAEA-NDS), Vienna, www-nds.iaea.org Nuclear Energy Agency, Data Bank (NEA-DB), Paris, www.nea/fr/html/dbdata Nuclear Data Center, Japan Atomic Energy Agency (NDC-JAEA), Japan, wwwndc.tokaisc.jaea.go.jp The NNDC has probably the largest portfolio of data with web retrieval capabilities, including both nuclear structure (ENSDF, NuDat, Chart of Nuclides) and nuclear reactions (EXFOR, ENDF, Sigma). For the readers of the present Handbook of most interest would be Sigma nuclear reaction retrieval and plotting system, which was developed by the NNDC to satisfy needs of both professional users and those without knowledge of complex ENDF- formatting system. Sigma uses the latest web technologies to provide browsing and search tools as well as interactive graphics. It can be easily accessed at www.nndc.bn.gov/sigma and includes the following capabilities: . Retrieval, browsing, search . Plotting • Cross sections • Angular distributions, energy spectra • Covariances (MF) • Fission yields . Computations (ratios, integrals, weighting) Evaluated Nuclear Data  . Thermal values and resonance integrals Sigma offers data from the following nuclear reaction libraries: • • • • • • • ENDF/B-VII. (USA, ) JEFF-. (Europe, ) JENDL-. (Japan, ) ROSFOND (Russia, ) ENDF/B-VI. (USA, ) ENDF/A (USA, selected files only) EXFOR (NRDC network, experimental data, latest version) The IAEA Nuclear Data Service (Vienna) offers major nuclear reaction data libraries as well as a number of smaller libraries and specialized results of the IAEA-coordinated data projects. Its signature nuclear reaction retrieval system is ENDF, which is also offered by the NNDC as an alternative to Sigma. The NEA Data Bank (Paris) data services are more restricted, focusing on nuclear reaction data and nuclear reaction computer codes. Its signature nuclear reaction data retrieval and plotting system is Java-based JANIS. The Nuclear Data Center of the Japan Atomic Energy Agency (Tokaimura) offers data services with the focus on data included in the Japanese Evaluated Nuclear Data Library, JENDL. Acknowledgments It would not have been possible to write this chapter without results produced by the dedicated work of numerous scientists and colleagues over years, both in the USA and abroad. We are particularly grateful to members of CSEWG and many other colleagues who contributed to ENDF/B-VII. and other nuclear data libraries. We owe special thanks to the authors of the “Big Paper” on ENDF/B-VII. (Chadwick et al., ), which supplied most of the figures and served as the basis for preparing this chapter. References Audi G, Bersillon O, Blachot J, Wapstra A () The NUBASE evaluation of nuclear and decay properties. Nucl Phys A : Axel P () Electric dipole ground state transition width strength function and  MeV photon interaction. Phys Rev : Band I, Trzhaskovskaya M, Nestor C Jr, Tikkanen P, Raman S () Dirac-Fock internal conversion coefficients. At Data Nucl Data Tables : Běták E, Obložinský P () Code PEGAS: preequilibrium exciton model with spin conservation and gamma emission. Technical report INDC(SLK)-, IAEA/Slovak Academy of Sciences (Note: DEGAS is an extended version of the code PEGAS using two-parameteric p-h level densities.) Bjornholm S, Lynn J (a) The double-humped fission barrier. Rev Mod Phys :– Bjornholm S, Lynn JE (b) The double-humped fission barrier. Rev Mod Phys : Blann M () New precompound decay model. Phys Rev C : Blann M, Chadwick MB () New precompound model: angular distributions. Phys Rev C : Blann M, Chadwick MB () Precompound Monte-Carlo model for cluster induced reactions. Phys Rev C : Bohr N, Wheeler J () The mechanism of nuclear fission. Phys Rev :– Bollinger LM, Thomas GE () p-Wave resonances of  U. Phys Rev :    Evaluated Nuclear Data Boukharaba N et al () Measurement of the n-p elastic scattering angular distribution at E n =  MeV. Phys Rev C : Boykov G et al () Prompt neutron fission spectra of  U. Sov J Nucl Phys : Brady M, England T () Delayed neutron data and group parameters for  fissioning systems. Nucl Sci Eng :– Briggs JB et al () International handbook of evaluated criticality safety benchmark experiments. Technical report NEA/NSC/DOC() /I, Nuclear Energy Agency, Paris, France Brockhouse BN, Abou-Helal HE, Hallman ED () Lattice vibrations in iron at ○ K. Solid State Commun : Bucholz J, Frankle S () Improving the LLNL pulsed-sphere experiments database and MCNP models. Trans Am Nucl Soc :– Buerkle W, Mertens G () Measurement of the neutron-proton differential cross section at . MeV. Few Body Syst : Capote R, Herman M, Obložinský P et al () RIPL – reference input parameter library for calculation of nuclear reactions and nuclear data evaluations. Nucl Data Sheets : Carlson A, Pronyaev V, Smith D et al () International evaluation of neutron cross section standards. Nucl Data Sheets :– Cassette P () SPEBETA programme de calcul du spectre en energie des electros emis par des radionucleides emetterus beta. Technical report CEA, Technical Note LPRI///J, CEA Saclay Chadwick M, Obložinský P, Blokhin A et al () Handbook on photonuclear data for applications: cross-sections and spectra. Technical report IAEA-TECDOC-, International Atomic Energy Agency Chadwick M, Obložinský P, Herman M, Greene N, McKnight R, Smith D, Young P, MacFarlane R, Hale G, Frankle S, Kahler A, Kawano T, Little R, Madland D, Moller P, Mosteller R, Page P, Talou P, Trellue H, White M, Wilson W, Arcilla R, Dunford C, Mughabghab S, Pritychenko B, Rochman D, Sonzogni A, Lubitz C, Trumbull T, Weinman J, Brown D, Cullen D, Heinrichs D, McNabb D, Derrien H, Dunn M, Larson N, Leal L, Carlson A, Block R, Briggs J, Cheng E, Huria H, Zerkle M, Kozier K, Courcelle A, Pronyaev V, van der Marck S () ENDF/B-VII.: next generation evaluated nuclear data library for nuclear science and technology. Nucl Data Sheets ():– Cokinos D, Melkonian E () Measurement of the  m/sec neutron-proton capture cross section. Phys Rev C : Conant J, Palmedo P () Delayed neutron data. Nucl Sci Eng : Courcelle A () First conclusions of the WPEC Subgroup-. In: Haight R, Chadwick M, Kawano T, Talou P (eds) Proceedings of the international conference on nuclear data for science and technology, American Institute of Physics, New York, Santa Fe, Sept –Oct , , pp – CSEWG-Collaboration () Evaluated nuclear data file ENDF/B-VI.. http://www.nndc.bnl. gov/endf, Mar ,  DeHart MD () Sensitivity and parametric evaluations of significant aspects of burnup credit for PWR spent fuel packages. Technical report ORNL/TM-, ORNL Derrien H, Courcelle A, Leal L, Larson NM, Santamarina A () Evaluation of  U resonance parameters from  to  keV. In: Haight R, Chadwick M, Kawano T, Talou P (eds) Proceedings of the international conference on nuclear data for science and technology, American Institute of Physics, New York, Santa Fe, Sept –Oct , , pp – Derrien H, Leal LC, Larson NM () Evaluation of  Th neutron resonance parameters in the energy range  to  keV. Technical report ORNL/TM-/, ORNL Dilg W () Measurement of the neutronproton total cross section at  eV. Phys Rev C : dos Santos A, Diniz R, Fanaro L, Jerez R, de Andrade e Silva G, Yamaguchi M () The experimental determination of the effective delayed neutron parameters of the IPEN/MB- reactor. In: PHYSOR, Chicago, April – Dunford C () Compound nuclear analysis programs COMNUC and CASCADE. Technical report T---, Atomics International Dunford C () ENDF utility codes, version .. http://www.nndc.bnl.gov / nndcscr / endf / endfutil-./, Dec ,  Dunn M, Greene N () AMPX-: a cross section processing system for generating nuclear data for criticality safety applications. Technical report, Transactions of the American Nuclear Society, ORNL England T, Rider B () Evaluation and compilation of fission product yields. Technical report ENDF-, Los Alamos National Laboratory, Los Alamos England T, Rider B () Nuclear modeling of the  Pu(n, xn) excitation function. Technical report LA-UR-- ENDF-, Los Alamos National Laboratory, Los Alamos Evaluated Nuclear Data ENSDF () ENSDF, Evaluated Nuclear Structure Data File. http://www.nndc.bnl.gov/ensdf,  Dec  Ethvignot T et al () Prompt-fission-neutron average energy for  U(n, f ) from threshold to  MeV. Phys Lett B : Forrest R, Kopecky J, Sublet J-C () The European activation file: EAF- cross section library. Technical report FUS , United Kingdom AEA Forrest R, Kopecky J, Sublet J-C () The European activation file: EAF- neutron-induced cross section library. Technical report FUS , United Kingdom AEA Frankle S (a) Possible impact of additional collimators on the LLNL pulsed-sphere experiments. Technical report X-:SCF-- and LA-UR-, Los Alamos National Laboratory, Los Alamos Frankle S (b) LLNL pulsed-sphere measurements and detector response functions. Technical report X-:SCF-- and LA-UR--, Los Alamos National Laboratory, Los Alamos Frehaut J () Coherent evaluation of nu-bar (prompt) for , U and  Pu. Technical report JEFDOC-, NEA, JEFF Project Frenkel Y () On the splitting of heavy nuclei by slow neutrons. Phys Rev :– Fröhner F, Goel B, Fischer U () FITACS computer code. Technical report ANL-, Kernforschungszentrum Karslruhe, p  (Note: Presented at Specialists’ Meeting on Fast Neutron Capture Cross Sections, ANL.) Fu C, Hetrick D () Update of ENDF/B-V Mod- iron: neutron-producing reaction cross sections and energy-angle correlations. ORNL/TM-, ENDF-, ORNL Greenwood R, Helmer R, Putnam M, Watts K () Measurement of β-decay intensity distributions of several fission-product isotopes using a total absorption γ-ray spectrometer. Nucl Instr Meth Phys Res A : Griffin PJ, Paviotti-Corcuera R () Summary report of the final technical meeting on international reactor dosimetry file: IRDF-. Technical report INDC(NDS)-, IAEA, Vienna, Austria Grimes S et al () Charged particle-producing reactions of -MeV neutrons on  V and  Nb. Phys Rev C : Hagura N, Yoshida T, Tachibana T () Reconsideration of the theoretical supplementation of decay data in fission-product decay heat summation calculations. J Nucl Sci Technol :– Hale GM () Use of R-matrix methods in light element evaluations. In: Dunford C (ed) Proceedings of the international symposium  on nuclear data evaluation methodology, Brookhaven National Laboratory, World Scientific, Singapore, pp – Hale G () Covariances from light-element R-matrix analyses. Nucl Data Sheets : Hale GM et al () Neutron-triton cross sections and scattering lengths obtained from p-  He scattering. Phys Rev C : Hauser W, Feshbach H () The inelastic scattering of neutrons. Phys Rev :– Herman M, Trkov A () ENDF- formats manual: data formats and procedures for the evaluated nuclear data file ENDF/B-VI and ENDF/B-VII. Technical report BNL--, Brookhaven National Laboratory Herman M, Reffo G, Weidenmüller HA () Multistep-compound contribution to precompound reaction cross section. Nucl Phys A : Herman M, Obložinský P, Capote R, Sin M, Trkov A, Ventura A, Zerkin V (a) Recent developments of the nuclear reaction model code EMPIRE. In: Haight R, Chadwick M, Kawano T, Talou P (eds) Proceedings of the international conference on nuclear data for science and technology, American Institute of Physics, New York, Santa Fe, Sept –Oct , , p  Herman M, Capote R, Carlson B, Obložinský P, Sin M, Trkov A, Zerkin V (b) EMPIRE nuclear reaction model code, version . (Lodi). http://www.nndc.bnl.gov/empire/, Dec ,  Herman M, Capote R, Carlson B, Obložinský P, Sin M, Trkov A, Wienke H, Zerkin V () EMPIRE: nuclear reaction model code system for data evaluation. Nucl Data Sheets ():– Hofmann HM, Richert J, Tepel JW, Weidenmüller HA () Direct reactions and Hauser-Feshbach theory. Ann Phys : Houk TL () Neutron-proton scattering cross section at a few electron volts and charge independence. Phys Rev C : ICRU () ICRU-Report-. In: Nuclear data for neutron and proton radiotherapy and for radiation protection, International Communications on Radiation Units and Measurements, Bethesda Ignatyuk A, Obložinský P, Chadwick M et al () Handbook for calculations of nuclear reaction data: reference input parameter library (RIPL). Technical report TECDOC-, IAEA, Vienna Ignatyuk A, Lunev V, Shubin Y, Gai E, Titarenko N () Evaluation of n +  Th cross sections for the energy range up to  MeV. Provided    Evaluated Nuclear Data to the IAEA CRP on Th-U cycle, see Report INDC(NDS)-, pp – JEFF () The JEFF-. nuclear data library. Technical report JEFF Report , OECD Nuclear Energy Agency Kalbach C () Systematics of continuum angular distributions: extensions to higher energies. Phys Rev C :– Kanda Y, Baba M () WPEC-SG report:  U capture and inelastic cross sections. Technical report NEA/WPEC-, NEA, Paris Kawano T, Ohsawa T, Shibata K, Nakashima H () Evaluation of covariance for fission neutron spectra. Technical report -, JAERI Kawano T et al () Production of isomers by neutron-induced inelastic scattering on  Ir and influence of spin distribution in the preequilibrium process. Nucl Instr Meth : Keepin GR () Physics of neutron kinetics. Addison-Wesley, New York Keinert J, Sax J () Investigation of neutron scattering dynamics in liquid hydrogen and deuterium for cold neutron sources. Kerntechnik : Kim H-I, Lee Y-O, Herman M, Mughabghab SF, Obložinský P, Rochman D () Evaluation of neutron induced reactions for  fission products. Technical report BNL---IR, Brookhaven National Laboratory Kim H-I, Herman M, Mughabghab SF, Obložinský P, Rochman D, Lee Y-O () Evaluation of neutron cross sections for a complete set of Nd isotopes. Nucl Sci Eng : Kinsey R () ENDF-: ENDF/B summary documentation. Technical report BNL-NCS-, National Nuclear Data Center, BNL (Note: rd Edition, ENDF/B-V Library.) Klein Meulekamp R, van der Marck S () Reevaluation of the effective delayed neutron fraction measured by the substitution technique for a light water moderated low-enriched uranium core. Nucl Sci Eng :– Kokoo O, Murata I, Takahashi A () Measurements of double-differential cross sections of charged-particle emission reactions for several structural elements of fusion power reactors by .-MeV incident neutrons. Nucl Sci Eng : Koning AJ, Delaroche JP () Local and global nucleon optical models from  keV to  MeV. Nucl Phys A : Koning A, Hilaire S, Duijvestijn M () TALYS: comprehensive nuclear reaction modeling. In: Haight R, Chadwick M, Kawano T, Talou P (eds) Proceedings of the international conference on nuclear data for science and technology, American Institute of Physics, New York, Santa Fe, Sept –Oct , , pp – Kopecky J, Uhl M () Test of gamma-ray strength functions in nuclear-reaction model-calculations. Phys Rev C :– Koppel J, Houston D () Reference manual for ENDF thermal neutron scattering data, report GA- revised and reissued as ENDF- by the National Nuclear Data Center, General Atomic, July  Kosako K () Covariance data processing code: ERRORJ. In: Katakura J (ed) The specialists’ meeting of reactor group constants, JAERI, Tokai, Japan, – February , JAERI-Conf -, p  Kosako K, Yanano N () Preparation of a covariance processing system for the evaluated nuclear data file, JENDL. Technical report JNC TJ-, -, JAERI Lane A, Lynn, J () Fast neutron capture below  MeV: the cross sections for  U and  Th. Proc Phys Soc (London) A : LANL () Webpage, T- Information Service, Los Alamos National Laboratory, Los Alamos. http://www.t.lanl.gov, Dec ,  Larson NM () Updated users’ guide for SAMMY: multilevel R-matrix fits to neutron data using Bayes’ equations. Technical report ORNL/TM/R, Document ENDF-, ORNL Leal L, Derrien H () Evaluation of the resonance parameters for  Th in the energy range  to  keV. Technical report, ORNL Leal LC, Derrien H, Larson NM, Courcelle A () An unresolved resonance evaluation for  U. In: PHYSOR conference, Chicago, – April  Little R, Kawano T, Hale G et al () Low-fidelity covariance project. Nucl Data Sheets : Lopez Aldama D, Trkov A () FENDL-. update of an evaluated nuclear data library for fusion applications. Technical report INDC(NDS)-, IAEA, Vienna, Austria López Jiménez M, Morillon B, Romain P () Overview of recent Bruyères-le-Châtel actinide evaluations. In: Haight R, Chadwick M, Kawano T, Talou P (eds) Proceedings of the international conference on nuclear data for science and technology, American Institute of Physics, New York, Santa Fe, Sept –Oct , , pp – Lougheed RW et al ()  Pu and  Am (n,n) cross section measurements near  MeV. Technical report UCRL-ID-, Lawrence Livermore National Laboratory MacFarlane RE () New thermal neutron scattering files for ENDF/B-VI release . Technical Evaluated Nuclear Data report LA--MS (), Los Alamos National Laboratory, Los Alamos MacFarlane RE, Muir DW () The NJOY nuclear data processing system, version . Technical report LA--M, Los Alamos National Laboratory, Los Alamos MacFarlane RE, Kahler AC () NJOY-., www.t.lanl.gov/codes/njoy/index.html ( October ) Madland D () Total prompt energy release in the neutron-induced fission of  U,  U, and  Pu. Nucl Phys A : Madland DG, Nix JR () New calculation of prompt fission neutron-spectra and average prompt neutron multiplicities. Nucl Sci Eng :– Magurno B, Kinsey R, Scheffel F () Guidebook for the ENDF/B-V nuclear data files. Technical report EPRI NP-, BNL-NCS-, ENDF-, Brookhaven National Laboratory and Electric Power Research Institute Marchetti A, Hedstrom G () New Monte Carlo simulations of the LLNL pulsed-sphere experiments. Technical report UCRL-ID-, Lawrence Livermore National Laboratory Mattes M, Keinert J () Thermal neutron scattering data for the moderator materials H  O, D  O, and ZrH x in ENDF- format and as ACE library for MCNP(X) codes. Report INDC(NDS)-, IAEA MCNPX () MCNPX transport code for charged particles. http://www.mcnpx.lanl.gov Mughabghab SF () Neutron cross sections: Z=–, vol B. Academic, New York Mughabghab SF () Atlas of neutron resonances: thermal cross sections and resonance parameters. Elsevier, Amsterdam Mughabghab SF, Dunford CL () A dipolequadrupole interaction term in E photon transitions. Phys Lett B : Mughabghab SF, Divadeenam M, Holden NE () Neutron cross sections: Z=–, vol A. Academic, New York Muir D () Gamma rays, Q-values and kerma factors. Technical report LA--MS, Los Alamos National Laboratory, Los Alamos Muir DW () Evaluation of correlated data using partitioned least squares: a minimum-variance derivation. Nucl Sci Eng :– Nakajima K () Re-evaluation of the effective delayed neutron fraction measured by the substitution technique for a light water moderated low-enriched uranium core. J Nucl Sci Technol :– NEA () Webpage, Nuclear Energy Agency Data Bank. http://www.nea.fr/html/databank/  Nishioka H, Verbaarschot JJM, Weidenmüller HA, Yoshida S () Statistical theory of precompound reactions: multistep compound process. Ann Phys : Obložinský P (ed) () Workshop on neutron cross section covariances. In: Nuclear data sheets, vol , Port Jefferson, – June , pp – Obložinský P, Mattoon C, Herman M, Mughabghab S, Pigni M, Talou P, Hale G, Kahler A, Kawano T, Little R, Young P (a) Progress on nuclear data covariances: AFCI-. covariance library. Technical report unpublished, Brookhaven National Laboratory and Los Alamos National Laboratory Obložinský P et al (b) Evaluated data library for the bulk of fission products. Technical report NEA/WPEC-, OECD Nuclear Energy Agency, Paris Oh S, Chang J, Mughabghab SF () Neutron cross section evaluations of fission products below the fast neutron region. Technical report BNLNCS-, Brookhaven National Laboratory Ohsawa T, Hambsch F-J () An interpretation of energy-dependence of delayed neutron yields in the resonance region for  U and  Pu. Nucl Sci Eng : Ohsawa T, Oyama T () Possible fluctuations in delayed neutron yields in the resonance region of U-. In: Katakura J (ed) Proceedings of the specialists’ meeting on delayed neutron nuclear data, JAERI, American Institute of Physics, Tokaimura, Japan, p  Okajima S, Sakurai T, Lebrat J, Averlant V, Martini M () Summary on international benchmark experiments for effective delayed neutron fraction. Prog Nucl Energy :– Olsen D, Ingle R () Measurement of neutron transmission spectra through  Th from  MeV to  keV. Report ORNL/TM-(ENDF-), ORNL Perkins ST, Gyulassy GE () An integrated system for production of neutronics and photonics calculational constants, vol . Technical report UCRL-, University of California Pfeiffer B, Kratz K-L, Moller P () Status of delayed-neutron data: half-lives and neutron emission probabilities. Prog Nucl Energy : – Pigni M, Herman M, Obložinský P () Extensive set of cross section covariance estimates in the fast neutron region. Nucl Sci Eng : – Plettner C, Ai H, Beausang CW et al () Estimation of (n,f ) cross sections by measuring reaction probability ratios. Phys Rev C ():    Evaluated Nuclear Data Plujko V, Herman M () Handbook for calculations of nuclear reaction data, RIPL, ch. : gamma-ray strength functions. No. TECDOC-, IAEA, Vienna, p  Porter CE, Thomas RG () Fluctuations of nuclear reaction widths. Phys Rev :– Pritychenko B, Sonzogni A () Sigma: web retrieval interface for nuclear reaction data. Nucl Data Sheets : Raynal J () Notes on ECIS. CEA-N-, Commissariat à l’Energie Atomique Rochman D, Herman M, Obložinský P () New evaluation of  V(n, np+pn) and (n, t) cross sections for the ENDF/B-VII library. Fusion Eng Des :– RSICC () Webpage, Radiation Safety Information Computational Center. http://www-rsicc. ornl.gov Sakurai T, Okajima S () Adjustment of delayed neutron yields in JENDL-.. J Nucl Sci Technol :– SCALE () SCALE: a modular code system for performing standardized computer analysis for licensing evaluation. NUREG/CR-, Rev.  (ORNL/NUREG/CSD-/R), vols I, II, and III, May  (Available from the Radiation Safety Information Computational Center at ORNL as CCC-.) Schoen K et al () Precision neutron interferometric measurements and updated evaluations of the n-p and n-d coherent neutron scattering lengths. Phys Rev C : Shibata K, Hasegawa A, Iwamoto O et al (a) JENDL-. covariance file. J Nucl Sci Technol ():– Shibata K, Kawano T, Nakagawa T et al (b) Japanese evaluated nuclear data library version  revision-: JENDL-.. J Nucl Sci Technol :– Sin M, Capote R, Herman M, Obložinský P, Ventura A, Trkov A () Improvement of the fission channel in the EMPIRE code. In: Haight R, Chadwick M, Kawano T, Talou P (eds)Proceedings of the international conference on nuclear data for science and technology, American Institute of Physics, New York, Santa Fe, Sept –Oct , , p  Sin M, Capote R, Ventura A, Herman M, Obložinský P () Fission of light actinides:  Th(n,f ) and  Pa(n,f ) reactions. Phys Rev C ():  Smith DL () Probability, statistics, and data uncertainties in nuclear science and technology. American Nuclear Society, LaGrange Park Smith D () Covariance matrices for nuclear cross sections derived from nuclear model calculations. Report ANL/NDM- November, Argonne National Laboratory Soukhovitskii ES, Capote R, Quesada JM, Chiba S () Dispersive coupled-channel analysis of nucleon scattering from  Th up to  MeV. Phys Rev C : Staples P et al () Prompt fission neutron energy spectra induced by fast neutrons. Nucl Phys A :– Stedman R, Almqvist L, Nilsson G () Phononfrequency distributions and heat capacities of aluminum and lead. Phys Rev :– Tamura T, Udagawa T, Lenske H () Multistep direct reaction analysis of continuum spectra in reactions induced by light ions. Phys Rev C : Trkov A () nd IAEA research co-ordination meeting on evaluated nuclear data for thorium-uranium fuel cycle. Technical report INDC(NDS)-, IAEA, Vienna Trkov A () Summary report of a technical meeting on covariances of nuclear reaction data: GANDR project. INDC(NDS)-, IAEA, Vienna Trkov A () Evaluated nuclear data for ThU fuel cycle summary report of the third research coordination meeting. Technical report INDC(NDS)-, IAEA, Vienna Trkov A, Capote R () Validation of  Th evaluated nuclear data through benchmark experiments. In: Proceedings of the international conference on nuclear energy for new Europe , Portorož, Slovenia, – Sept  Trkov A et al () Revisiting the  U thermal capture cross-section and gamma emission probabilities from  Np decay. Nucl Sci Eng :– Tuli J () Nuclear wallet cards. Electronic version. http://www.nndc.bnl.gov/wallet Tuttle R () Delayed neutron data. Nucl Sci Eng : van der Marck S () Benchmarking ENDF/B-VII beta. CSEWG  meeting report. http:// www.nndc.bnl.gov/csewg/. Dec ,  van der Marck SC () Benchmarking ENDF/B-VII.. Nucl Data Sheets :– van der Marck S, Klein Meulekamp R, Hogenbirk A, Koning A () Benchmark results for delayed neutron data, AIP Conf. Proc. . American Institute of Physics. New York, Santa Fe, Sept –Oct , , pp – Vineyard GH () Frequency factors and isotope effects in solid state rate processes. J Phys Chem Solids (–):– Webster W et al () Measurements of the neutron emission spectra from spheres of N, O, W, Evaluated Nuclear Data U-, U-, and Pu-, pulsed by  MeV neutrons. Technical report UCID- Weigmann H, Hambsch J, Mannhart W, Baba M, Tingjin L, Kornilov N, Madland D, Staples P () Fission neutron spectra of  U. Report NEA/WPEC-, OECD Wiarda D, Dunn M () PUFF-IV: code system to generate multigroup covariance matrices from ENDF/B-VI uncertainty files. Radiation Safety Information Computational Center (RSICC) Code Package PSR-, ORNL Williams M () Generation of approximate covariance data. ORNL memo, August  Williams M, Rearden B () SCALE- sensitivity/uncertainty methods and covariance data. Nucl Data Sheets : Wilson W, England T () Delayed neutron study using ENDF/B-VI basic nuclear data. Prog Nucl Energy :– Woelfle R et al () Neutron activation cross section measurements. Radiochimica Acta : Wong C et al () Livermore pulsed sphere program: program summary through July . Technical report UCRL-, Rev I, and Addendum (), Lawrence Livermore National Laboratory Wright RQ, MacFarlane RE () Review of ENDF/B-VI fission-product cross sections. Technical report ORNL/TM-, ORNL, Oak Ridge X-MCNP-Team () MCNP – a general Monte Carlo N-particle transport code, version , volume I: overview and theory. Technical report LA-UR--, Los Alamos National Laboratory  Yoshida T et al () Validation of fission product decay data for decay heat calculations . WPEC Subgroup-. http://www.nea.fr/ html/science/projects/SG, Dec   Younes W, Britt HC () Neutron-induced fission cross sections simulated from (t,pf ) results. Phys Rev C (): Young PG, Arthur ED () GNASH: a preequilibrium statistical nuclear model code for calculations of cross sections and emission spectra. Technical report LA-, Los Alamos National Laboratory, Los Alamos Young JA, Koppel JU () Slow neutron scattering by molecular hydrogen and deuterium. Phys Rev :A–A Young PG, Arthur ED, Chadwick MB () Comprehensive nuclear model calculations: introduction to the theory and use of the GNASH code. Technical report LA--MS, Los Alamos National Laboratory, Los Alamos Young PG, Arthur ED, Chadwick MB () Comprehensive nuclear model calculations: theory and use of the GNASH code. In: Gandini A, Reffo G (eds) Proceedings of the IAEA workshop on nuclear reaction data and nuclear reactors, World Scientific Publishing, Singapore, April –May  , pp – Young P, Herman M, Obložinský P et al () Handbook for calculations of nuclear reaction data: reference input parameter Library-. TECDOC-, IAEA, Vienna Young P, Chadwick M, MacFarlane R, Talou P, Kawano T, Madland D, Wilson W, Wilkerson C () Evaluation of neutron reactions for ENDF/B-VII: − U and  Pu. Nucl Data Sheets ():–   Neutron Slowing Down and Thermalization Robert E. MacFarlane Nuclear and Particle Physics, Astrophysics and Cosmology, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA ryxm@lanl.gov  . . . . . . . . . . . . . . . . . . Thermal Neutron Scattering . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Chemical Binding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Coherent and Incoherent Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Quantum Mechanical Scattering Function . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Intermediate Scattering Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Scattering Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Phonon Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Short Collision Time Approximation . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Diffusive Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Discrete Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Incoherent Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Coherent Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Example of Thermal Scattering in Graphite . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Example of Thermal Scattering in Water. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Example for Thermal Scattering in Heavy Water . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Example for Thermal Scattering in Zirconium Hydride . . . . . . . .. . . . . . . . . . . . . . . Using the ENDF/B Thermal Scattering Evaluations . . . . . . . . . . . . .. . . . . . . . . . . . . . .                     . . . .. .. .. .. .. .. .. .. .. . Neutron Thermalization . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Monte Carlo Simulations of Neutron Thermalization . . . . . . . . . .. . . . . . . . . . . . . . . Discrete Ordinates Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . S N Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Transport Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Fission Source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Eigenvalue Iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . S N Data Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Example for HST- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Preparing S N Cross-Section Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Example for an Infinite Pin-Cell Lattice. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Monte Carlo vs. Multigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Collision Probability Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .               Dan Gabriel Cacuci (ed.), Handbook of Nuclear Engineering, DOI ./----_, © Springer Science+Business Media LLC    Neutron Slowing Down and Thermalization . Size Effects in Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . . . . . .. .. .. .. .. .. .. . . . . Steady-State Slowing Down .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Introduction. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slowing-Down Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectra for Elastic Downscatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectra for Inelastic Downscatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Level Breit–Wigner Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multi-Level Breit–Wigner Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reich–Moore Representation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reich–Moore-Limited Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance Reconstruction and Doppler Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance Slowing Down. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flux Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermediate Resonance Self-Shielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unresolved Resonance Range Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                   . . . . . Time and Space in Slowing Down . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Introduction. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Dependence of the Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Dependence of the Spatial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eigenvalues and Eigenfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytic Age Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        Concluding Remarks and Outlook. .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  References . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. .  Neutron Slowing Down and Thermalization  Abstract: The theory behind the generation of thermal cross sections is presented, concentrating on the phonon expansion method. Examples are given for graphite, water, heavy water, and zirconium hydride. The graphite example demonstrates incoherent inelastic scattering and coherent elastic scattering for crystalline solids. The water example demonstrates incoherent inelastic scattering for liquids with diffusive translations. Heavy water adds a treatment for intermolecular coherence. Zirconium hydride shows the effects of the “Einstein oscillations” of the hydrogen atoms in a cage of zirconium atoms, and it also demonstrates incoherent elastic scattering. Neutron thermalization is introduced using Monte Carlo simulations of several systems, followed by multigroup discrete-ordinates and collision-probability methods. Size effects in thermalization are demonstrated. Steady-state slowing down is discussed by illustrating typical cross-section data, and showing slowing down by elastic scattering, inelastic scattering, and resonance cross sections in the narrow resonance approximation. Intermediate resonance selfshielding effects are introduced using the NJOY flux calculator and the WIMS implementation. The effects of time and space on slowing down are demonstrated using Monte Carlo simulations, and the theoretical basis is summarized.  Thermal Neutron Scattering . Introduction At high energies, the wavelengths of neutrons are small, and it is reasonable to treat scattering as classical collisions between particles. At thermal energies, however, the wavelengths of neutrons approach the size of molecules and the spacing of crystalline lattices. Scattering becomes a quantum mechanical problem. The theory for this was worked out in the late s and is described in detail by Williams (). This theory was reduced to practice for the US Evaluated Nuclear Data File (ENDF) in the s, mainly by researchers at General Atomic (Koppel and Houston ). This theoretical basis remains largely valid today except for the improvements in scope and detail allowed by modern computing machines (MacFarlane ; Mattes and Keinert ). In this section, we will summarize the theoretical basis for the thermal scattering evaluations in ENDF/B-VII (Tuli et al. ), and show examples for graphite, water, heavy water, and zirconium hydride as they are currently given in ENDF format (Herman ). . Chemical Binding The simplest system of targets is a free gas of protons moving in thermal equilibrium with a temperature T. From basic statistical mechanics, the energies of these protons will be given by the Maxwell–Boltzmann distribution M(E) = e−E/kT () where k is Boltzmann’s constant. These protons are free to recoil, so a neutron can lose energy by colliding with a proton, but because the protons are moving, the neutron can also gain energy in a scattering event. It is assumed that the density of neutrons is small with respect to the density of scatterers; thus, the distribution of neutron energies does not affect the thermal distribution    Neutron Slowing Down and Thermalization of the scatterers. In this one case, the thermal scattering can be obtained by both quantum and classical methods. Now if we bind the protons into a free gas of water molecules, the scattering process gets more interesting. Collisions of a neutron with a water molecule can result in a gain or loss of energy by reducing or increasing the velocity of the molecule (translational modes), and they can interact to make the molecule slow down or speed up its rotations (rotational modes) or vibrations (vibrational modes). The thermal energy kT for the gas of water molecules is divided among the various possible translational, rotational, and vibrational modes according to their characteristic energies by the Maxwell–Boltzmann distribution. The chemical binding of the proton into the water molecule reduces its ability to change the energy of the neutron upon scattering. Now, to get more realistic, we condense this gas of water molecules into a liquid. The vibrational modes are not changed very much by this condensation. The rotational modes become hindered rotations. You can visualize little V-shaped molecules moving back and forth like torsional oscillators. The translational modes act like the diffusion of clusters of molecules with some internal order moving through a matrix of other clusters. All of these modes are considered to be quantized. The good quantum numbers are energy and momentum, and the energies of the modes are distributed according to the Maxwell–Boltzmann distribution. The good quantum numbers for the incident and scattered neutron are also energy and momentum (or wave vector). When the neutron scatters in the liquid, it changes its energy and momentum. The difference is associated with a change in the energy and momentum of various modes of excitation in the liquid. In a solid, the possible modes of excitation are various wave-like vibrations of the crystal lattice. In quantum language, these excitations are called “phonons.” Phonons are characterized by particular energies and momenta (or wave vectors). The equilibrium distribution of phonon energies is described by the Maxwell–Boltzmann distribution. When a neutron scatters through a crystalline lattice, it can exchange energy and momentum by creating or annihilating phonons in such a way as to conserve energy and momentum. This can lead to a change of energy and direction (wave vector) for the neutron. There is another thing that can happen. Because the neutron can be characterized as a wave, it can scatter from the lattice in the same way that light scatters from a diffraction grating. This is called “coherent elastic scattering,” and it results in a change of direction for the neutron with no change in energy. . Coherent and Incoherent Scattering When the neutron wave impinges on the crystalline lattice, there is a secondary wave produced from each scattering center. If the resulting waves are “coherent,” they can combine constructively and destructively to produce maxima and minima in various directions. This is similar to the behavior of a diffraction grating for light. For a material of randomly ordered crystallites, this would result in rings of neutron intensity in the forward direction. However, the diffraction can also be “incoherent.” A primary example of this is scattering from bound protons. Protons have spin, like the neutron, and the spins can be randomly arranged in a hydrogenous scatter. This randomness breaks the coherence and destroys the diffraction pattern. Inelastic scattering also tends to reduce the coherence in the scattering, and we normally assume that inelastic thermal scattering can be treated as incoherent. In general, hydrogenous materials Neutron Slowing Down and Thermalization  scatter incoherently. Nuclei with spin zero (many important materials) scatter coherently for elastic scattering (no energy change), but incoherently for inelastic scattering. . The Quantum Mechanical Scattering Function This process has been analyzed in terms of a position and time correlation function. Let G(r, t) be the probability that a particle will be at the origin at time t =  and at position r at time t (s for “self ” particle). Using the incoherent approximation and assuming a liquid or a powdered crystalline material, the scattering function can be written as ′   σ inc (E → E , μ) = (a inc + a coh ) k′ i(κ⋅r−εt/ħ) dr ∫ dt e G s (r, t) , k ∫ () where E and E ′ are the incident and secondary energies, μ is the cosine of the scattering angle, a inc and a coh are the characteristic scattering lengths for the material, k and k ′ are the incident and outgoing wave numbers (momenta), κ is the change in wave vector in scattering, and ε is the energy transfer in scattering. . The Intermediate Scattering Function This expression is often written in terms of the “intermediate scattering function” χ, or σ inc (E → E ′ , μ) = (a inc + a coh ) k′ dt e i εt/ħ) χ s (κ, t) , k ∫ () where the coherent scattering is treated with the incoherent approximation. For most of the cases of interest in thermal scattering, it turns out that χ takes on a Gaussian form in momentum transfer κ χ(κ, t) = e−γ(t) , () where  γ(t) = κ ∫ ∞ −∞ P(ε) ( − e −i εtħ )e −ε/kT dε , () and where P(ε) = ρ(ε) . ε sinh(ε/kt) () The quantity ρ(ε) is the energy spectrum of excitations in the system. In this context, the Gaussian approximation is basically equivalent to treating the internal excitations of the system as quantized harmonic oscillators. The excitation energy spectrum is often written as ρ(ω), with ε = ħω and called a “frequency distribution.”    . Neutron Slowing Down and Thermalization Detailed Balance The thermal scattering cross section must obey a very important property called the condition of “detailed balance” or “reciprocity.” From very basic principles like time reversal invariance, it can be shown that for thermal equilibrium, the down-scattering must be balanced by the up-scattering. Thus, EM(E, T)σ(E → E ′ , μ) = E ′ M(E ′, T)σ(E ′ → E, μ) . () This condition guarantees that the thermal neutron flux spectrum in an infinite non-absorbing medium at temperature T will take on a Maxwell–Boltzmann flux shape appropriate to the temperature T; that is, the flux shape will be ϕ(E) = Ee −E/kT . () Absorption and leakage from the medium can modify the spectrum from this ideal shape. . The Scattering Law It has proved to be convenient to write the thermal neutron scattering cross section in terms of the “scattering law.” First, the momentum transfer and the energy transfer are written in terms of two new variables √ ħ  κ  E ′ + E − μ E ′ E α= = , () kT AkT and ε E′ − E β= = , () kT kT where A is the ratio of the target mass to the neutron mass. Then we write σ σ inc (E → E , μ) = b kT ′ √ E′ Ŝ(α, β) , E () where σb = π (a coh + a inc ) is the characteristic bound scattering cross section for the material. Note that all the nuclear terms involving the cross section, the incident wave, and the scattered wave have been separated from the chemical-binding effects, all of which have been consolidated in Ŝ. Specifically, Ŝ does not depend on incident energy E or the scattered energy E ′ , but only on the amount of energy transferred between the scattered neutron and the material. This will be true as long as the atomic motions are small enough for the oscillations to remain harmonic. Anharmonicity and actual disruption of molecules or displacements of atoms from their normal lattice positions could eventually cause this theory to break down for large energy transfers at high incident energies. Using this form of the incoherent scattering cross section and the principle of detail balance, we see that β Ŝ(α, β) = e Ŝ(α, −β) . () The scattering law must also obey two so-called moments theorems or sum rules: the zeroth moment theorem or sum rule () ∫ Ŝ(α, β)d β =  , Neutron Slowing Down and Thermalization  and the first moment theorem or sum rule ∫ Ŝ(α, β)βd β = −α . () The latter guarantees that the thermal cross section will approach the correct limit at high incident energies, namely, the free cross section σ f , where σb = ( A+  ) σf . A () In practice, the symmetric form of the scattering law is often used S(α, β) = e β/ Ŝ(α, β) . () Then, the detail balance condition becomes S(α, β) = S(α, −β) , () and the scattering law is symmetric in β (energy transfer). Thermal neutron scattering laws in ENDF format are normally given in terms of S, but this can lead to numerical difficulties. The asymmetric Ŝ function can be represented by normal numbers (say − to ) for all β. But the symmetric S can be smaller than Ŝ by factors like e−β/ ≈ e− ≈ − . Processing codes have to be careful to allow for this huge range of values. The problem can be even more difficult for cold moderators where numbers ranging from  to − can be required in evaluated data files. As discussed above, the chemical binding for protons in water can be decomposed into translational, rotational, and vibrational modes. More formally, we can write ρ(ε) = ∑ w i ρ i (ε) , () i where the following possibilities are allowed ρ j (β) = δ(β j ) discrete oscillator () ρ j (β) = ρ s (β) solid-type spectrum () ρ j (β) = ρ t (β) translational spectrum () The solid-type spectrum must vary as ε  , as ε goes to zero, and it must integrate to w s , the weight for the solid-type law. The translational spectrum is sometimes represented as a free gas for liquids, but more realistically, it can be represented using a diffusion-type spectrum represented with the approximation of Egelstaff and Schofield that will be discussed later. In either case, the spectrum must integrate to w t, the translational weight. The sum of all the partial weights must equal . This partition of the energy distribution leads to a recursive formula for the scattering law: (K) () Ŝ(α, β) = Ŝ (α, β) ,    Neutron Slowing Down and Thermalization where Ŝ (J) ′ (α, β) = ∫ Ŝ J (α, β ) Ŝ (J−) ′ ′ (α, β−β ) d β . () As an example of the use of this recursive procedure, consider a case where the desired frequency spectrum is a combination of ρ s and two discrete oscillators. First, calculate Ŝ ()= Ŝ  using ρ s . Then calculate Ŝ  using ρ(β  ), the distribution for the first discrete oscillator, and convolve Ŝ  with Ŝ () to obtain Ŝ () , the composite scattering law for the first two partial distributions. Repeat the process with the second discrete oscillator to obtain Ŝ () , which is equal to Ŝ(α, β) for the full distribution. One simple example of the translational component of the scattering law represents the free gas with a weight of w t : (w i α − β)  Ŝ(α, −β) = √ exp [− ], w t α πw i α () −β Ŝ t (α, β) = e Ŝ t (α, −β), () and with β positive. Note that for large α and β, the main contribution to this comes from the region w t α ≈ β. The shape is a Gaussian, and as α and β increase (high incident energies), the Gaussian gradually goes over to a delta function, and we recover the normal classical elastic scattering behavior. A number of approximate methods and full-up computer methods have been used over the years to compute the scattering law for realistic cases. Notable among these was the GASKET method, (Koppel et al. ) which was used by General Atomic to prepare the initial ENDF/B thermal scattering evaluations. Here we will limit our discussion to the phonon expansion method as implemented in the LEAPR module of the NJOY Nuclear Data Processing System (MacFarlane and Muir ), which has been used to prepare the more recent thermal scattering evaluations available in modern neutron cross-section compilations. LEAPR is a descendent of the British codes LEAP and ADDELT (Butland ). . The Phonon Expansion Consider first γ s (t), the Gaussian function for solid-type excitation spectra. Expanding the time-dependent part of the exponential gives ∞ e−γ s (t) = −α λs ∑ n= n ∞  [α∫ Ps (β) e−β/ e−i β t d β ] , n! −∞ () where λ s is the Debye–Waller coefficient λs = ∫ ∞ −∞ Ps (β) e −β/ dβ . () Neutron Slowing Down and Thermalization  The scattering law becomes ∞ Ŝ s (α, β) = e−α λ s ∑ n=  n α n! n ∞ ∞ ′ ′  × e i β t [∫ Ps (β ′ ) e−β / e−i β t d β ′ ] dt . ∫ π −∞ −∞ () For convenience, define the quantity in the second line of this equation to be λ sn Tn (β). Then clearly, ∞  −α λ n () Ŝ s (α, β) = e s ∑ [α λ s ] Tn (β) , n! n= where T (β) = and T (β) = ∫ ∞ −∞ ∞  e i β t dt = δ(β) , π ∫−∞ () ′ ∞ Ps (β) e−β/ Ps (β ′ ) e−β /  i(β−β ′ )t ′ { ∫ e dt} d β = . λs π −∞ λs In general, Tn (β) = ∫ ∞ −∞ T (β ′ ) Tn− (β−β ′ ) d β ′ . () () The T functions obey the relationship Tn (β) = e−β Tn (−β). In addition, each of the Tn functions obeys the following normalization condition: ∫ ∞ −∞ Tn (β) d β =  . () In LEAPR, the T functions are computed up to some specified maximum value, typically , and these results are used to compute Ŝ(α, β). What does the phonon expansion mean physically? Consider a neutron scattering in water. If the energy loss of the neutron is large, there is no way that energy can be absorbed by only kicking the mass  molecule into additional motion. That would violate conservation of momentum. But it is possible to transfer part of the energy to translations and the rest to exciting rotations and vibrations. In other words, a multiplicity of phonons is excited. For small energy changes in scattering, only a few phonons may be required, but large transfers may require up to . For water in equilibrium at temperature T, intermolecular collisions will excite a spectrum of these  different modes in accordance with their characteristic energies following the Maxwell–Boltzmann distribution. This very complicated target motion can then deliver energy and momentum to a slow neutron, thus scattering it up to higher energies. It is assumed that the changes in the occupation of target modes caused by the neutrons are very small when compared to the thermal excitation of the target modes – this linearizes the scattering problem. . The Short Collision Time Approximation Very large energy transfers may even require more than  phonons to be created or annihilated. In this case, a limiting form of the phonon expansion is available called the short collision time (SCT) approximation, which can be written    Neutron Slowing Down and Thermalization (w s α − β)  exp [− ], Ŝ s (α, −β) = √ w s αT s /T πw s αT s /T () Ŝ s (α, β) = e−β Ŝ s (α, −β) , () and where β is positive and where the effective temperature is given by Ts = ∞ T β  Ps (β) e−β d β . ∫ w s −∞ () As above, w s is the weight for the solid-type spectrum. The effective temperature can be substantially higher than the ambient temperature; for example, the effective temperature is about , K at room temperature for hydrogen bound in water. . Diffusive Translations As discussed above, translational effects in liquids like water are often represented using a diffusion term. The “effective width model” of Egelstaff and Schofield provides a tractable approximation for the diffusive motion: Ŝ t (α, β) = cw t α c  w t α−β/ e π √ √ √ c  + . K  { c  + . β  + c  w t α  } , √ β  + c  w t α  and ρ(β) = w t √ c √  c + . sinh(β/) K  { c  + . β} . πβ () () In these equations, K  (x) is a modified Bessel function of the second kind, and the translational weight w t and the diffusion constant c are chosen to try to represent experiment. The scattering law for a combination of solid-type modes and diffusion can be computed as follows: Ŝ(α, β) = Ŝ t (α, β) e−α λ s + ∫ ∞ −∞ Ŝ t (α, β ′) Ŝ s (α, β−β ′ ) d β ′ . () The first term in this equation comes from the delta function in (), the “zero phonon” term, which is not included in the normal calculation of Ŝ. The effective temperature for a combination of solid-type and diffusive modes is given by Ts = w t T + ws T s . w t + ws () The “effective width” refers to the width of the pseudo-elastic peak seen in neutron scattering experiments. Crudely speaking, one can think of clusters of water molecules with some internal order diffusing through a matrix of other clusters. Larger clusters are harder to set into motion Neutron Slowing Down and Thermalization  with neutron collisions, and the translational effects on neutron scattering are reduced from the free molecule value. Although this diffusive model helps to achieve better agreement with experiment at modest thermal energies, it leaves something to be desired at very low thermal energies. However, because of the effects of detail balance, the actual value of the cross section at the lowest neutron energies has little effect on the shape of the equilibrium flux, so this failure of the diffusive effect has normally been accepted. Some experiments seem to show that there is a distribution of cluster sizes in real liquids, and taking better account if this distribution might improve the agreement with experiment. . Discrete Oscillators Polyatomic molecules normally support a number of vibrational modes that can be represented as discrete oscillators. The distribution function for one oscillator is given by w i δ(β i ), where w i is the fractional weight for mode i, and β i is the energy-transfer parameter computed from the mode’s vibrational frequency. The corresponding scattering law is given by Ŝ i (α, β) = e ∞ −α λ i αw i −nβ / ]e i β i sinh(β i /) ∑ δ(β − nβ i ) I n [ n=−∞ ∞ = ∑ A in (α) δ(β − nβ i ) , () n=−∞ where coth(β i /) . () βi The combination of a solid-type mode (s) with discrete oscillators () and () would give λi = wi Ŝ () (α, β) = Ŝ s (α, β) , Ŝ () (α, β) = ∫ ∞ −∞ ∞ () Ŝ  (α, β ′) Ŝ () (α, β−β′ ) d β ′ = ∑ A n (α) Ŝ () (α, β−nβ  ) , () n=−∞ Ŝ () (α, β) = ∫ ∞ −∞ ∞ ′ Ŝ  (α, β ) Ŝ () ′ (α, β−β ) d β ′ ∞ = ∑ A m (α) ∑ A n (α) Ŝ () (α, β−nβ −mβ  ) . m=−∞ () n=−∞ and so on through Ŝ  (α, β), Ŝ () (α, β), etc., until all the discrete oscillators have been included. The Debye–Waller λ for the combined modes is computed using N λ = λs + ∑ λ i . () i= The effective temperature for the combined modes is given by N T s = w t T + ws T s + ∑ w i i= βi βi coth ( ) T .   ()    . Neutron Slowing Down and Thermalization Incoherent Elastic Scattering In hydrogenous solids (such as polyethylene), there is an elastic (no energy loss) component of scattering arising from the zero-phonon term of the phonon expansion. This is called the “incoherent elastic” term. Clearly, S iel (α, β) = e−α λ δ(β) . () The corresponding differential scattering cross section is σb −W E(−μ) , e  () σb  − e−W E { }.  W E () σ(E, μ) = and the integrated cross section is σ(E) = In these equations, the Debye–Waller coefficient is given by W= λ , AkT () where λ is computed from the input frequency spectrum as modified by the presence of discrete oscillators (if any) any as shown above. . Coherent Elastic Scattering In solids consisting of coherent scatterers (e.g., graphite) the zero-phonon term leads to interference scattering from the various planes of atoms of the crystals making up the solid. Once again, there is no energy loss. The neutrons only change direction. This is called “coherent elastic scattering.” The differential scattering cross section is given by σcoh (E, μ) = σc −W E i δ(μ − μ i ) , ∑ fi e E E i <E () μ i =  − E i /E , () where and the integrated cross section is given by σcoh = σc −W E i . ∑ fi e E E i <E () In these equations, σc is the effective bound coherent scattering cross section for the material, W is the effective Debye–Waller coefficient, E i are the so-called Bragg Edges, and the f i are related to the crystallographic structure factors. Neutron Slowing Down and Thermalization  It can be seen from () that the cross section is zero below the first Bragg edge (typically about – meV). It then jumps sharply to a value determined by f  and the Debye–Waller term. At higher energies, the cross section drops off as /E until E = E  . It then takes another jump and continues falling off like /E. The sizes of the jumps gradually become smaller, and at high energies, there is nothing left but an asymptotic /E decrease (typically above – eV). The calculation of the E i and f i depends on a knowledge of the crystal structure of the scattering material. The Bragg edges are given by Ei = ħ  τ i , m () where τ i is the length of the vectors of one particular “shell” of the reciprocal lattice, and m is the neutron mass. The f i factors for a material containing a single atomic species are given by fi = πħ   ∑ ∣F(τ)∣ , mN V τ i () where the sum extends over all reciprocal lattice vectors of the given length, and the crystallographic structure factor is given by   N   ∣F(τ)∣ = ∑ e iπϕ j  ,   j=   () where N is the number of atoms in the unit cell, ϕ j = τ⃗ ⋅ ρ⃗ j are the phases for the atoms, and the ρ⃗ j are their positions. The situation is a little more difficult for a material with more than one atomic species, such as BeO, and an approximation has to be made for the effective Debye–Waller factor for the cell. As an example of this process, consider graphite. It has an hexagonal lattice described by the constants a and c. The reciprocal lattice vector lengths are given by ( τ   ) =  (ℓ  + ℓ  + ℓ  ℓ  ) +  ℓ  , π a c () where ℓ  , ℓ  , and ℓ  run over all the positive and negative integers, including zero. The volume of the unit cell is √ () V = a  c/ . For graphite, there are four atoms in the unit cell at positions (Wycoff )         (, , ), (− , , ) , (− , − , ) , (− , , ) .         These positions give the following phases: ϕ =  , () ϕ  = (−ℓ  + ℓ  )/ , () ϕ  = −(/)ℓ  − (/)ℓ  + (/)ℓ  , () ϕ  = −(/)ℓ  + (/)ℓ  + (/)ℓ  . ()   Neutron Slowing Down and Thermalization 15 10 Rho  5 0 0 50 100 150 Energy (eV) 200 250 *10–3 ⊡ Figure  The phonon frequency spectrum ρ(ε) used for graphite The form factor for graphite becomes  ∣F∣ = {  +  cos[π(ℓ  − ℓ  )/]  sin  [π(ℓ  − ℓ  )/] ℓ  even . ℓ  odd () Similar methods can be used to obtain the lattice vectors and structure factors for the HCP structure (beryllium), the FCC structure (aluminum), or the BCC structure (iron). These options are built into the LEAPR module of the NJOY processing system. . Example of Thermal Scattering in Graphite As an example of the application of the theory presented above, we consider graphite, the moderator used in the very first critical reactor. To obtain the phonon excitation spectrum, the binding in graphite is modeled using four force constants: a nearest neighbor central force that binds the hexagonal planes together, a bond-bending force in an hexagonal plane, a bondstretching force between nearest neighbors in the plane, and a restoring force against bending of the hexagonal plane. These force constants are then used to compute the phonon energies in various directions in the crystal, the “phonon dispersion curves.” These results are analyzed to get the phonon density of states; that is, the number of phonons that can exist with energies in a given range. The phonon excitation curve used for the ENDF/B-VII evaluation of thermal scattering in graphite is shown in > Fig. . Note the ε  dependence at low energy transfers. This frequency spectrum can be used with the phonon expansion method to generate the Ŝ scattering law using NJOY’s LEAPR module. Some results are shown in > Figs.  and > . Neutron Slowing Down and Thermalization  S(α, β) 10– 1 10– 2 β =0 β =2 10– 3 10– 4 10– 2 β =4 10– 1 β = 6.44 100 101 α ⊡ Figure  Ŝ(α, β) vs. α for several values of β in graphite at . K 100 a = 0 .252 a = 1.01 a = 3.08 S-hat(α , −β) 10–1 a = 7.52 a = 20 a = 40 10– 2 10– 3 10– 4 100 101 β 102 ⊡ Figure  Ŝ(α, β) vs. β for several values of α in graphite at . K. Note how the shape approaches a smooth Gaussian as α increases Note that as α increases, the curves get smoother and begin to look like Gaussians, as predicted by the SCT extension to the phonon expansion. In > Fig. , we show the incoherent part of the inelastic scattering from graphite at two temperatures, and in > Fig. , we show the coherent elastic contribution to the cross section displaying the low-energy cutoff and the Bragg edges.   Neutron Slowing Down and Thermalization Sigma-in (barns) 102 101 100 10–1 10– 5 10– 4 10– 3 10– 2 10 –1 Energy (eV) 100 101 ⊡ Figure  Incoherent inelastic cross section of graphite at . K (solid curve) and , K (dashed curve). Note how the cross section approaches the free-atom value at high incident energies 101 Sigma-el (barns)  100 10 –1 10 –3 10 –2 10 –1 Energy (eV) 10 0 101 ⊡ Figure  Coherent elastic cross section of graphite at . K (solid curve) and , K (dashed curve) showing the Bragg edges and the asymptotic /E decrease at high energies There are also thermal effects on the angular distributions for scattering. > Figure  shows the average cosine for scattering for both coherent elastic and incoherent inelastic scattering. Note how the scattering cosine for elastic scattering varies from a backward direction at the Bragg edge toward the forward direction as the energy increases above the edge. The inelastic μ̄ is less than the normal high-energy value of .. Neutron Slowing Down and Thermalization  1.0 Mubar 0.5 0.0 –0.5 –1.0 10–11 10–10 10–9 10–8 10–7 Energy (MeV) 10–6 10–5 ⊡ Figure  Average scattering cosine for inelastic (solid curve) and elastic (dashed curve) in graphite at room temperature In > Fig. , we show the neutron emission curves for the incoherent inelastic component of the scattering at room temperature (. K) featuring the low-energy range. For the low-energy incident neutrons, these curves show upscatter structure with peaks coming from the excitation peaks in the graphite phonon spectrum that are excited at thermal equilibrium at room temperature. For the higher incident energies, the emission spectra show features in the downscatter resulting from exciting phonons corresponding to the peaks in the phonon spectrum. At the highest incident energy, the scattering begins to look more and more like the rectangle-shaped characteristic of high-energy elastic scattering. Finally, > Fig.  gives a perspective view of the energy distribution for inelastic scattering in graphite at room temperature. . Example of Thermal Scattering in Water The current ENDF/B-VII evaluation for the thermal scattering law for H bound in H O is based on recent work done under IAEA auspices (Mattes and Keinert ) with some slight modifications. It is assumed that scattering from the oxygen atom can be treated as a free gas of mass . The hydrogen binding has a component describing hindered rotations of the water molecules using the phonon frequency distribution show in > Fig.  with a weight of .. The translational modes are treated as the diffusion of water clusters through the liquid with a weight of .. Finally, there are two discrete oscillator frequencies with energies of . and . eV and weights of . and ., respectively. The scattering law computed from this model using the LEAPR module of NJOY is shown in the next few figures. In > Fig. , the peak at low α and β comes from the diffusive translations.   Neutron Slowing Down and Thermalization 103 0.0005 eV 102 0.2907 eV 0.95 eV Prob/eV 101 3.12 eV 100 10 –1 10 – 2 10 – 3 10 – 3 10 – 2 10 –1 Energy (eV) 100 101 ⊡ Figure  Spectra from thermal inelastic scattering for graphite at room temperature showing several incident energies 0 1 10 10 0 10 – 1 10 –1 – 10 ) V (e 10 0 4 –2 10 – y rg ne E 10 – 3 4 10 ne 10 rg – 2 y (e V ) 10 E Prob/eV  ⊡ Figure  Perspective view of the spectra from thermal scattering on graphite at . K Neutron Slowing Down and Thermalization  8 rho 6 4 2 0 0 50 100 150 *10– 3 Energy (eV) ⊡ Figure  The phonon frequency spectrum ρ(ε) used for H in H O 100 α =0.05 α =0.20 α =0.61 α = 2.5 α =15 S-hat(α, −β ) 10–1 α = 39 10 – 2 10 – 3 10 – 4 10 –1 100 101 102 β ⊡ Figure  Ŝ(α, −β) for H in H O at room temperature plotted versus β for various values of α For high α, the curves begin to take on the Gaussian shape predicted by the SCT approximation. In between, the peaks in the curves come from the effects of the discrete oscillators. As discussed above, there are peaks at the main oscillator energies and at various “harmonics,” that is, sums and differences of multiples of the basic energies. > Table  shows the oscillator energies possible for α = .   Neutron Slowing Down and Thermalization ⊡ Table  Discrete oscillator β values and weights for α =  for H in H O Beta Weight . .E- −. .E- −. .E- −. .E- −. .E- −. .E- . .E- −. .E- −. .E- −. .E- −. .E- 102 101 100 S(α, β)  10 –1 10– 2 10– 3 10– 4 10–3 10–2 10–1 α 100 101 ⊡ Figure  Ŝ(α, −β) vs. α for a number of β values In > Fig.  the high-energy cutoff of the energy distribution for the rotational modes is visible, as well as the effect of the discrete oscillators. Note the singularity at low α and β where the slope changes sign. This is an effect characteristic of the translational modes in liquids. The neutron emission spectra for incoherent inelastic scattering that results from processing this scattering law is shown in > Fig.  for several incident energies. For very low incident energies, the neutron gains energy from the rotational modes excited at thermal equilibrium. Neutron Slowing Down and Thermalization  10 3 0.0005 eV 0.0253 eV 10 2 0.2907 eV 0.95 eV 101 Prob/eV 3.12 eV 100 10–1 10–2 10–3 10–3 10–2 10–1 100 101 Energy (eV) ⊡ Figure  Incoherent inelastic spectra for several incident energies for H in H O 1.0 Prob/eV 0.8 0.6 0.4 0.2 0.0 2.5 2.6 2.7 2.8 2.9 3.0 Energy (eV) 3.1 3.2 3.3 ⊡ Figure  Detailed view of a neutron emission spectrum for inelastic scattering for H in H O For higher energies, it is more probable that the neutron will lose energy, and the effects of exciting translational, rotational, and vibrational modes are visible. The downscatter behavior is shown in more detail in > Fig. . The sharp peak at E ′ = E is the quasi-elastic peak coming from the diffusive translations. The next lower hump is from rotational modes, and the other peaks are from vibrational modes. The integrated cross section is shown in > Figs.  and >  for two temperatures. As the incident energy increases, the cross section begins to approach the free-atom value, as predicted   Neutron Slowing Down and Thermalization Prob/eV 103 102 600 K 293.6 K 101 10–4 10–3 10–2 Energy (eV) 10–1 100 ⊡ Figure  The incoherent inelastic cross section for H in H O at two temperatures 21.0 293.6 K 20.8 Sigma-in (barns)  600 K 20.6 20.4 Free gas Static 20.2 3 4 5 6 7 Energy (eV) 8 9 10 ⊡ Figure  The incoherent inelastic cross section for H in H O for higher incident energies showing the static limit (scattering from atoms at rest) and the free-gas cross section by the theory. In practice, multigroup codes would normally change from the thermal value to the target-at-rest value at some particular break-point energy chosen so that error caused by the additional discontinuity at that energy group was not too significant. Monte Carlo codes normally change from the thermal cross section to the free-gas cross section at some break-point Neutron Slowing Down and Thermalization  ⊡ Table  Effective temperatures for the short collision time approximation for H in H O Temp. (K) Eff. Temp. (K) . ,  ,  ,  ,  ,  ,  ,  , 1.0 Bound Mubar 0.8 Static 0.6 0.4 0.2 0.0 10– 4 10– 3 10– 2 10– 1 Energy (eV) 100 101 ⊡ Figure  The average scattering cosine for H in H O compared to the static value for scattering from atoms at rest. The effect of the binding of H in H O is to make the scattering more isotropic at thermal energies energy. There again, it is hoped that the break-point can be made high enough to minimize the adverse effects of the discontinuity. Another approach that has been used in practice is to shift from the thermal cross section to an SCT cross section, that is, a free-gas cross section at a higher temperature than the ambient value. > Table  shows the effective SCT temperatures for H in H O. This approach gives a fairly good integrated cross section versus energy above the thermal cutoff of the scattering law calculation, and it gives a good downscatter spectrum, but the upscatter is too large. The angular behavior of thermal scattering is also of interest. For H bound in H O, the hydrogen atom is not as free to recoil as the free atom. This makes it look like it has a higher effective mass, and it causes the scattering to be more isotropic on the average. See > Fig. .   Neutron Slowing Down and Thermalization 1 10 0 10 – 1 10 2 0 – 0. 0.5 10 – 0.0 sin e Co 3 5 Se c.e ne rg y –1 10 –1. 10 – Prob/cosine  1.0 ⊡ Figure  A perspective view of an angle-energy distribution for H in H O However, as > Fig.  demonstrates, there are still interesting anisotropies seen, especially near E ′ = E, where translational effects are important. > Figure  shows a perspective view of the thermal scattering for H in H O. As the description of the evaluation for H in H O demonstrates, ENDF scattering law data are not obtained directly from experimental measurement. That would require more complete differential data than are currently available. Instead, they are modeled based on various kinds of input data ranging from neutron scattering measurements to optical results. The model results are then compared to the available experimental data to see how good a job was done with the evaluation. Examples of the comparison of the modeled thermal cross section for water with experiment are shown in > Figs.  and > . Additional comparisons with differential data are shown in the report on the IAEA evaluation (Mattes and Keinert ). The results are fairly good, except around – meV and below  meV. The problem at the lowest energies comes from the failure of the diffusive model. As discussed above, because of the principle of detailed balance, the value of the cross section in this region does not have much influence on the computed flux in a water-moderated system. Therefore, this discrepancy can be accepted for nuclear engineering calculations. . Example for Thermal Scattering in Heavy Water The structure of heavy water is similar to that of water, except that the weights on the ends of the arms of the vee are twice as heavy. This shifts the scale of the rotational and vibrational modes.  10 0 Neutron Slowing Down and Thermalization 10 –1 1 10 10 –2 –3 10 –2 10 – 10 0 –3 1 10 ) eV y( rg ne E 10 ne rg y( eV ) 0 10 E Prob/eV 2 10 ⊡ Figure  A perspective view of the isotropic part of the incoherent inelastic scattering from H in H O. The variations in the size of the quasi-elastic peak are artifacts of the plotting program Drista M (1967) report from misc. OECD countries to EANDC, No. 63. Heinloth K (1961) Zeitschrift fuer Physik 163, 218. σ (barns) Smith RR (1953) private communication. Walton W (1960) private communication. Springer VT (1961) J. Nucleonik (3), 110. Simpson OD (1964) J. Nuclear Instruments and Methods 30, 293. 102 Russell JL Jr. (1966) General Atomic report 7581. ENDF/B-VII 10–4 10–3 10–2 10–1 Energy (eV) ⊡ Figure  Comparison of the ENDF/B-VII thermal cross section for water at lower incident energies with experimental results for the CSISRS compilation at the the National Nuclear Data Center of the Brookhaven National Laboratory   Neutron Slowing Down and Thermalization 80 Drista M (1967) report from misc. OECD countries to EANDC, No. 63. Heinloth K (1961) Zeitschrift fuer Physik 163, 218. Smith RR (1953) private communication. 70 σ (barns)  Walton W (1960) private communication. Springer VT (1961) J. Nucleonik (3), 110. Simpson OD (1964) J. Nuclear Instruments and Methods 30, 293. Russell JL Jr. (1966) General Atomic report 7581. ENDF/B-VII 60 50 40 10–1 100 Energy (eV) 101 ⊡ Figure  Comparison of the ENDF/B-VII thermal cross section for water at higher incident energies with experimental results for the CSISRS compilation at the the National Nuclear Data Center of the Brookhaven National Laboratory The work of Mattes and Keinert () resulted in the temperature-dependent frequency distribution shown in > Fig. . The two oscillator frequencies are . and . eV (they were . and . for water – the scaling is the square root of the mass ratio)). The new feature illustrated by this example is intermolecular coherence. As discussed above, the random orientation of proton spins in water allows us to use an incoherent approximation for the calculation. But the spin of the deuterons in D O is zero and this simplification does not apply. Even though heavy water is a liquid, there is still some persistent structure. There tends to be certain characteristic values of the spacing between the molecules, and this is sufficient to preserve some degree of coherence in the scattering. This effect is made more objective by using a “static structure factor,” such as the ones shown in > Fig. . These were generated assuming a hard core for small distances and a Lennard-Jones potential for larger distances. The scattering function is then computed using the Skold approximation (Skold ): S(α, β) = ( − f ) S inc (α, β) + f s inc (α ′ , β) S(κ) , where f = and () σ coh , σ inc + σ coh () α . S(κ) () ′ α = Neutron Slowing Down and Thermalization  8 ρ(ε) 6 4 2 0 0 20 40 60 80 100 120 140 160 180 *10–3 ε (eV) ⊡ Figure  S(κ) Frequency distribution for D in heavy water. The solid line is for a temperature of . K, and the dashed line is for a temperature of  K κ ⊡ Figure  Static structure factor for D in heavy water. The sold curve is for a temperature of . K, and the dashed curve is for a temperature of  K The static structure factor for room temperature shows strong structure, but as the temperature increases, the correlations in spacing get smoothed out and the coherence is suppressed. This is evident in the computed cross sections shown in > Fig. . The dip below  meV in the room temperature curve comes from the coherence – it has basically disappeared at  K. The curve at the higher temperature looks more like the water cross section.   Neutron Slowing Down and Thermalization 102 Cross section (barns)  101 100 10–5 10–4 10–3 10–2 Energy (eV) 10–1 100 ⊡ Figure  Incoherent inelastic scattering for D in D O. The solid curve is for a temperature of . K, and the dashed curve is for a temperature of  K > Figure  compares the calculated cross section with experimental data for heavy water extracted from the CSISRS compilation at the NNDC. The match is not perfect, but it is clear that the new evaluation partially accounts for the dip below  meV due to intramolecular coherence. . Example for Thermal Scattering in Zirconium Hydride Zirconium hydride, ZrHx , has variable stoichiometry with x near . Therefore, it is necessary to treat it as two separate materials, namely, H bound in ZrH, and Zr bound in ZrH. We will only consider the first of those here. The ENDF/B-VII evaluation follows the GA model (Koppel and Houston ) with a few small changes. It used a force-constant model to generate the frequency distribution. Because there are two atoms in the unit cell, there are both low-energy acoustic modes and high-energy optical modes. These two parts of the distribution are shown separately in > Figs.  and > . The optical modes form an isolated peak near . eV. This comes about because the hydrogen atoms sit in a cage of surrounding zirconium atoms (the variable stoichiometry results from some of these cages being occupied and some being empty). The cage provides a potential well for the hydrogen atom to vibrate in. In quantum language, this is an “Einstein oscillator,” and the spectrum of evenly spaced states in this well lead to interesting oscillations in S(α, β), the integrated cross section (> Fig. ), and the spectra of emitted neutrons (> Fig. ). > Figure  shows the incoherent elastic cross section. . > Using the ENDF/B Thermal Scattering Evaluations Table  gives a summary of the thermal scattering evaluations available in ENDF/B-VII. Cross section (barns) Neutron Slowing Down and Thermalization  Calculated Kropf F (1974) private communication. Rainwater LJ (1948) Phys. Rev. 73, 733. Meyers VW (1953) private communication. 101 10–4 10–3 10–2 10–1 Energy (eV) 100 101 ⊡ Figure  Comparison of the calculated cross section for heavy water to experiment. The dip due to coherence below  meV is partially accounted for by the evaluation. The experimental data are from the CSISRS compilation maintained by the NNDC at BNL. See the compilation for the references 400 *10–3 ρ (ε) 300 200 100 0 0 5 10 15 20 ε (eV) 25 30 35 *10–3 ⊡ Figure  Frequency distribution for H in ZrH for the acoustic modes Note the “Secondary scatterer” column. For H in H O, the evaluation only describes the H scattering. The effect of oxygen is to be added on using free-gas scattering. On the other hand, the benzine evaluation (C H ) includes both C and H scattering. Care must be taken to not add on free C scattering in the thermal range.   Neutron Slowing Down and Thermalization 70 60 50 ρ (ε) 40 30 20 10 0 110 120 130 140 ε (eV) 150 160 170 *10–3 ⊡ Figure  Frequency distribution for H in ZrH for the optical modes. The function is zero from the top of the acoustic modes to the start of the optical modes 102 Cross section (barns)  101 100 10–5 10–4 10–3 10–2 Energy (eV) 10–1 100 ⊡ Figure  Incoherent inelastic scattering cross section for H in ZrH. The solid curve is for a temperature of . K, and the dashed curve is for a temperature of  K Neutron Slowing Down and Thermalization  Cross section (barns) 102 101 100 10–5 10–4 10–3 10–2 Energy (eV) 10–1 100 ⊡ Figure  Incoherent elastic scattering cross section for H in ZrH. The solid curve is for a temperature of . K, and the dashed curve is for a temperature of  K 102 probability/eV 101 100 10–1 10–2 10–3 10–4 10–3 10–2 10–1 Energy (eV) 100 ⊡ Figure  Neutron emission spectra for incoherent inelastic scattering from H in ZrH at energies of . and . eV. The effect of the Einstein oscillator is seen as peaks in the upscatter for the low-energy curve and as strong oscillations in the downscatter for the high-energy curve    Neutron Slowing Down and Thermalization ⊡ Table  Summary of the thermal scattering evaluations available in ENDF/B-VII Evaluation name Secondary scatterer Material number Temperatures (K) H in H O Free O  ., , , , , , , ,  H in CH Free C  ,  Benzine None  , , , , , , , , H in ZrH None  , , , , , , ,, , D in D O Free O  ., , , , , , ,  Be in metal None  , , , , , , ,, , Be in BeO None  ., , , , , , ,, , Graphite None  , , , , , ,, ,, ,, , O in BeO None  ., , , , , , ,, , O in UO None  , , , , , , ,, , Al in metal None  , , ., , ,  Fe in metal None  , , ., , ,  Zr in ZrH None  , , , , , , ,, , U in UO None  ., ,  , , , ,, , Para H None   Ortho H None   Para D None   Ortho D None   Liquid CH None   Solid CH None   A number of these evaluations were prepared for use at neutron scattering centers where moderators are cooled down to make long wavelength neutrons. They are not needed for reactor calculations. The ENDF/B thermal scattering evaluations are not ready to be used directly in practical reactor calculations. The have to be converted into cross sections in appropriate formats by a nuclear data processing code such as NJOY (MacFarlane and Muir ). In NJOY, this is done using the THERMR module. It reads in an ENDF thermal scattering evaluation and produces cross sections versus energy and scattering distributions giving incident energy, secondary energy, probability, and a set of discrete emission cosines for incoherent inelastic scattering. For crystalline coherent scattering, it just produces a cross section with Bragg edges. The angular distribution can be deduced from that in subsequent codes. For incoherent elastic scattering, it produces a cross section and a set of emission cosines. The output from the THERMR module can be passed to GROUPR to be formatted for multigroup codes or to ACER to be formatted for the MCNP continuous energy Monte Carlo code. Neutron Slowing Down and Thermalization  Neutron Thermalization . Introduction  In nuclear reactors, neutrons are born at million electron-volts (MeV) energies, and they slow down (are “moderated”) by elastic and inelastic collisions with the materials in the reactor until they reach the thermal range below a few electron-volts (eV). In this range, in addition to losing energy in collisions, they can also gain energy by collisions with atoms and molecules in thermal motion. After some time, the distribution of the neutrons will come into equilibrium with the thermal motion of the atoms or molecules of the material and show a Maxwellian-like shape. Fission reactions caused by this distribution of neutrons will lead to the production of more neutrons at high energies, continuing the thermalization process. In addition to causing fission, the thermal neutrons can suffer absorption losses, and neutrons can be lost by leaking out of the system. These absorptions and leakages can affect the shape of the equilibrium neutron spectrum. In this section, we will consider the process of neutron thermalization in the region below a few eV, how to compute the equilibrium neutron spectrum, and the various influences on the shape of the equilibrium spectrum. We will first demonstrate this by direct Monte Carlo simulation of the thermalization process, and then move on to methods based on solving the transport equation for thermal neutrons (the Boltzmann equation) using multigroup techniques. . Monte Carlo Simulations of Neutron Thermalization The Monte Carlo method is based on following the histories of many particles using a random selection of the reactions that the neutrons go through between their production in a fission event, and their eventual death by absorption or leakage from the system. There are a number of Monte Carlo codes available that perform well for thermal neutrons. The examples in this section use MCNP (). As a first example, let us consider a cylindrical steel tank partly filled with a solution of highly enriched uranium nitrate in water. This particular case is called HIGH-ENRICHEDSOLUTION-THERMAL-- (or shortly HST-) in the International Criticality System Benchmark Experiment Program (ICSBEP) handbook (Briggs et al. ). Using cross sections from ENDF/B-VII (Tuli et al. ) and running this assembly for  million histories gives a predicted multiplication k eff of . with a estimated standard deviation of . as compared to an experimental prediction of . ± .. What this means is that the production rate of neutrons by fission is almost exactly balanced by the loss rate of neutrons to absorption and leakage from the system, or keff = ν̄σ f ϕ , (σ f + σnγ )ϕ + L () where these quantities stand for integrals over energy and space, ν̄ is the number of neutrons produced per fission, σ f is the fission cross section, σnγ is the capture cross section, L is the leakage rate from the system, and ϕ is the neutron flux. In practice, this equation is a little too simple. The reactions (n, n) and (n, n) also produce neutron multiplication, and there are also other less-common reactions that produce neutron multiplication. MCNP and many of   Neutron Slowing Down and Thermalization the multigroup transport codes handle this by reducing the absorption to compensate for the increased production k eff = ν̄σ f ϕ . (σ f + σnγ − σ nn − σ nn )ϕ + L () MCNP normalizes itself so that the denominator of this equation is one. Some other transport codes use the full production in the numerator, getting k eff = (ν̄ + σnn + σ nn )ϕ . (σ f + σ nγ + σnn + σnn )ϕ + L () These two different definitions are equivalent when k eff is not too far from unity. The MCNP result for the HST- case gives L = .; therefore, the leakage is fairly small for this case. It is strongly thermalized. > Figure  shows the computed flux (solid curve) compared to a Maxwellian at room temperature and the effective contribution due to slowing down from higher energies (dotted curves). The numbers plotted in this figure are the hits in MCNP tally bins defined logarithmically with  bins per decade; therefore, they are proportional to “flux per unit lethargy.” The term “lethargy” is often used in reactor physics. It is defined as  MeV lethargy = exp [ ]. () E It increases as the neutron slows down. The flux per unit lethargy is equivalent to E ∗ ϕ(E). Because the epithermal slowing down in a system like this gives a shape close to /E, the flux per unit lethargy plots are approximately flat in the epithermal region. 10–6 10–7 10–8 E*flux(E)  10–9 10–10 10–11 10–12 10 –13 10 –14 10 –11 10 –10 10–9 10–8 E(eV) 10–7 10–6 10–5 ⊡ Figure  Calculated flux for the HST- critical assembly (solid curve). The dotted curves show the theoretical Maxwellian flux and nominal /E source due to slowing down from higher energies Neutron Slowing Down and Thermalization  Flux per unit lethargy 10–1 10–2 10–3 10–4 10–9 10–8 10–7 10–6 10–5 10–4 10–3 10–2 10–1 Energy (MeV) 100 101 ⊡ Figure  Calculated flux for the LCT- critical assembly. The average over the lattice region is shown For a more realistic example, we consider the assembly called LCT-. This one has a ×  square lattice of .%-enriched uranium oxide rods with aluminum cladding placed in a tank of water. The water outside the lattice acts as a reflector. The calculated k eff using ENDF/B-VII cross sections is . ± . as compared to the evaluated model k eff of . ± .. > Figure  shows the flux in the lattice region (not including the water reflector). The region around each fuel rod can be defined as a “cell,” where each square cell contains a fuel pin, the aluminum cladding, and the associated water. With MCNP, we can calculate (or “tally”) the flux in the fuel pin and the flux in the associated water separately as averages over the entire lattice. The result is shown in > Fig. . At high energies, the flux in the pin is higher than the flux in the water because of the fission neutron source there. At the middle energies, we see strong dips in the fuel flux from absorption resonances, but the water flux is smoother. The /E shape we saw in the previous example is modified somewhat because of the losses to absorption in the fuel. At thermal energies, we see typical Maxwellian flux shapes from thermalization, but the thermal flux in the fuel is depressed because of the absorption by capture and fission. > Figure  shows an expanded view of the thermal region. The solid line is the water flux, and the dotted lines are the theoretical Maxwellian and slowing-down shapes. It is clear that the water flux matches the expected Maxwellian shape quite well. However, the fuel flux (dashed curve) has been depressed by the absorption from capture and fission, and its shape is been “hardened” somewhat. > Figure  shows an expanded view of the resonance region. The slowing down in the water shows a slightly harder shape than /E because of the losses to absorption in the fuel, and also because some effect of the strong absorption resonances in the fuel shows up as dips in the water flux. The absorption dips in the fuel are very large and lead to the “self-shielding” effects that will be discussed below. In order to get a better idea of the details of thermalization in a lattice like LCT-, we consider an infinite lattice of LCT pin cells. With MCNP, this is represented as a single pin   Neutron Slowing Down and Thermalization Flux per unit lethargy 10–1 10–2 10–3 10–9 10–8 10–7 10–6 10–5 10–4 10–3 10–2 10–1 Energy (MeV) 100 101 ⊡ Figure  Calculated flux for the LCT- critical assembly showing water flux (solid) and fuel flux (dashed) 10–2 Flux per unit lethargy  10–3 10–4 10–5 10–10 10–9 10–8 10–7 Energy (MeV) 10–6 ⊡ Figure  Expanded view of the thermal region flux for the LCT- critical assembly showing water flux (solid), fuel flux (dashed), and the theoretical Maxwellian and slowing down shapes (dotted) of fuel and clad surrounded by water in a square cross section with reflecting surfaces and extended vertically with reflecting ends. Of course, without the leakage, this cell is supercritical with k∞ = .. > Figure  shows the net current out of the pin (actually at the surface of the clad) into the water moderator. Note that the current is positive at high energies where the fission neutron Neutron Slowing Down and Thermalization  Flux per unit lethargy 10–2 10–3 10–6 10–5 10–4 Energy (MeV) 10–3 10–2 ⊡ Figure  Expanded view of the resonance region flux for the LCT- critical assembly. The solid curve shows the flux in the cell water, and the dashed curve shows the flux in the fuel pin Current per unit lethargy 20 *10–3 15 10 5 0 –5 –10 –15 –20 10–9 10–8 10–7 10–6 10–5 10–4 10–3 10–2 10–1 100 Energy (MeV) 101 ⊡ Figure  Net current out of pin into water for an infinite lattice of LCT pin cells showing fission neutrons entering the water at high energies, slowing down, and reentering the pin at low energies are born. Many of the fission neutrons are leaking from the fuel pin into the moderator. They then slow down quickly to thermal energies and renter the fuel pin (negative current) to cause new fissions or to be killed by capture. In the epithermal range, there are occasional negative dips in the current where neutrons are being drawn out of the moderator by strong absorption   Neutron Slowing Down and Thermalization 10–1 Flux per unit lethargy  0.0 0.2 0.4 0.6 Radius (cm) 0.8 1.0 ⊡ Figure  Flux vs. radius in the fission peak around  MeV (solid) and the thermal dip around . eV (dashed). The vertical dotted lines show the location of the clad resonances in the fuel (especially around . eV). At higher energies, you can see the effects of the aluminum and oxygen resonances. > Figure  shows the radial shape of the flux in the fission peak around  MeV and in the thermal dip around . eV. The fission source is slightly peaked in the center and decreases as you move out into the moderator due to the leakage into the moderator. The thermal shape is depressed in the center of the pin due to absorption from fission and capture. More dramatically, > Fig.  shows the strong depression in the center of the pin due to absorption in the . eV capture resonance of U-. The mean free path of the neutrons near the center of the resonance is very small, and they do not penetrate very far into the pin. Only a small fraction of the U- in the pin actually contributes to the absorption, and this is the real source of the dip in the flux seen in > Fig. . The U- in the center of the pin is “self-shielded” from the neutrons in the moderator. Energy self-shielding and spatial self-shielding are seen to closely related. This example helps to explain why a heterogeneous arrangement of fuel pins in a moderator is used in thermal reactors. Having the neutrons slow down in a moderator outside the fuel pins allows most of the neutrons to avoid being absorbed in the strong resonances of U-, thereby allowing them to reach thermal energies where they can continue the fission chain reaction. The HST example could reach criticality using a homogeneous arrangement for fuel and moderator because the fuel was highly enriched in U-, making the U- absorptions less of a problem. The LCT fuel is a low-enriched material (.% U-) and U- absorption must be well managed to design an effective reactor. . Discrete Ordinates Methods The examples above used simulations of the real physical process going on during thermalization, following the neutron as it bounced around the assembly losing and gaining energy until Neutron Slowing Down and Thermalization  Flux per unit lethargy 10–2 10–3 0.0 0.2 0.4 0.6 Radius (cm) 0.8 1.0 ⊡ Figure  Flux vs. radius in the . eV resonance (solid) and below the resonance at  eV (dashed). The vertical dotted lines show the location of the clad it came into equilibrium and was lost by absorption or escape from the system. An alternate approach is to solve the Boltzmann transport equation, which describes the balance of neutrons in a region of space and energy. There are a number of highly developed computer codes that solve this equation using the multigroup “discrete ordinates,” or S N , method. An example of this is PARTISN (Alcouffe et al. ) from Los Alamos. SN Theory .. The S N transport codes solve the equation (Bell and Glasstone ) μ N ∂ N ϕ g (μ, x) + σ gS N (x) ϕ g (μ, x) = ∑ Pℓ (μ) ∑ σ ℓSg←g ′ (x) ϕ l g ′ + S g (μ, x), ∂x g′ ℓ= () where one-dimensional plane geometry has been used for simplicity, μ is the scattering cosine, x is position, ϕ(μ, x) is the angular flux for group g, ϕ l g is the Legendre flux for group g, Pℓ (μ) is a Legendre polynomial, and S g (μ, x) is the external and fission source into group g. The cross sections in () must be defined to make ϕ g as close as possible to the solution of the Boltzmann equation. As shown in the reference, the multigroup Boltzmann equation can be written in the PN form: μ N N ∂ P PN ψ g (μ, x) + ∑ Pℓ (μ) σ ℓtNg (x) ψ ℓ g = ∑ Pℓ (μ) ∑ σ ℓ g←g ′ (x) ψ ℓ g ′ + S g (μ, x), ∂x g′ ℓ= ℓ= ()    Neutron Slowing Down and Thermalization where the PN cross sections are given by the following group averages: σ ℓtPNg = ∫ σ t (E) Wℓ (E) dE g , ∫ Wℓ (E) dE () g and PN σℓ g←g ′ = ′ ′ ′ ∫ ′ dE ∫ dE σ ℓ (E →E) Wℓ (E ) g g ′ ′ ∫ ′ dE Wℓ (E ) . () g ′ In these formulas, σ t (E) and σ ℓ (E →E) are the basic energy-dependent total and scattering cross sections, and W ℓ (E) is a weighting flux that should be chosen to be as similar to ψ as possible. So “there’s the rub.” To solve for the unknown flux, we have to have a good estimate of the flux to start with! This is not a serious problem for thermalization. As we saw in > Fig. , the thermalized flux is well represented by a Maxwellian flux shape with a /E slowing-down contribution. This will almost always be the case for a practical thermal reactor. To keep good neutron economy, it is important not to lose too many neutrons to unproductive leakage or capture, which tends to preserve the well-thermalized flux shape shown in the picture. We can use the Maxwellian flux shape with a slowing-down contribution for the weighting function Wℓ (E). It is still necessary to divide the thermal range into a number of energy groups to account for deviations from the simple Maxwell plus slowing-down shape, such as the hardening seen in > Fig. , and especially around the break between the two functions or where strong resonances may occur (e.g., the  eV resonance of Pu-). The venerable THERMOSLASER structure used  groups between . and . eV with velocity spacing over most of the range and extra coverage around  eV. The widely use WIMS structure uses  thermal groups between − and  eV. Because of the increased capabilities of modern computers, many newer systems use even more thermal groups. .. Transport Corrections When () is compared with (), it is evident that the SN equations require S P ′ N N σℓ g←g′ = σ ℓ g←g′ for g ≠ g, () N N = σℓPg←g − σℓtPNg + σ gS N , σ ℓSg←g () and where σ gS N is not determined. The choice of σ gS N gives rise to a “transport approximation” and various recipes are in use. It is convenient to write N N σℓSg←g = σℓPg←g − (σ ℓtPNg − σotPNg ) − Δ Ng , () σ gS N = σotPNg − Δ Ng . () and Neutron Slowing Down and Thermalization  The term in parentheses corrects for the anisotropy in the total reaction rate term of the Boltzmann equation, and Δ Ng can be chosen to minimize the effects of truncating the Legendre expansion at ℓ = N. Some recipes for doing this follow: Consistent-P approximation: N Δ g = , () Inconsistent-P approximation: N P P N Δ g = σotNg − σ N+,t g, () Diagonal transport approximation: N P P P N N Δ g = σotNg − σ N+,t g + σ N+,g←g , () Bell–Hansen–Sandmeier (BHS) or extended transport approximation: N P PN PN Δ g = σotNg − σ N+,t g + ∑ σ N+,g ′ ←g , () g′ and Inflow transport approximation: P N ∑ σ N+,g←g ′ ϕ N+,g ′ PN PN Δ Ng = σot g − σ N+,t g + g′ ϕ N+,g . () The first two approximations are most appropriate when the scattering orders above N are small. The inconsistent option removes most of the delta-function of forward scattering introduced by the correction for the anisotropy in the total scattering rate and should normally be more convergent than the consistent option. For libraries produced with an ℓ-independent flux guess and in the absence of self-shielding, the difference between “consistent” and “inconsistent” vanishes. The diagonal and BHS recipes make an attempt to correct for anisotropy in the scattering matrix and are especially effective for the forward-peaked scattering normally seen for high neutron energies. The BHS form is most often used, but the diagonal option can be substituted when BHS produces negative values, which is often the case in the thermal range. The inflow recipe makes the N+ term of the PN expansion vanish, but it requires a good knowledge of the N+ flux moment from some previous calculation. Inflow reduces to BHS for systems in equilibrium by detail balance (i.e., the thermal region). The diffusion approximation obtained using the inflow formula is equivalent to a P transport solution. These corrections require data from the (N+)-th Legendre moments of the cross sections to prepare a corrected N-table set. .. Fission Source The source of fission neutrons into a group is given by S g = ∑ σ f g←g ′ ϕ g ′ , g′ ()    Neutron Slowing Down and Thermalization where σ f g←g ′ is the fission matrix computed from the energy-dependent fission spectra given in the ENDF-format evaluation. However, most existing transport codes do not use this matrix form because the upscatter is expensive to handle and a reasonably accurate alternative exists. Except for relatively high neutron energies, the spectrum of fission neutrons is only weakly dependent on initial energy. Therefore, the fission source can be written as S g = χ g ∑ ν̄ g ′ σ f g ′ ϕ g′ , () g′ where ν̄ g is the fission neutron yield, σ f g is the fission cross sections, and χ g is the average fission spectrum, which can be defined by ∑σ f g←g ′ ϕ g′ χg = g′ , ∑ν̄ g ′ σ f g ′ ϕ g′ () g′ where the fission neutron production rate can also be written as ∑ ν̄ g ′ σ f g ′ ϕ g′ = ∑ ∑ σ f g←g ′ ϕ g ′ . g′ g () g′ Clearly, χ g as given by () depends on the flux in the system of interest. The dependence is weak except for high incident energies, and a rough guess for ϕ g usually gives an accurate spectrum. When this is not the case, a sequence of calculations can be made, using the flux from each step to improve the χ g for the next step. The matrix as described above represents the prompt part of fission only. Steady-state (SS) fission is obtained using two auxiliary pieces of data: delayed ν̄ and delayed χ. Therefore, SS D ν̄ g ′ σ f g ′ = ∑ σ f g←g ′ + ν̄ g ′ σ f g ′ , () g and D SS χg = g′ g′ D ′ ∑ ∑σ f g←g ′ ϕ o′ + ∑ν̄ g ′ σ f g ′ ϕ g g .. D ∑σ f g←g ′ ϕ g ′ + χ g ∑ν̄ g ′ σ f g ′ ϕ g ′ g′ . () g′ The Eigenvalue Iteration In order to solve (), it is converted into an eigenvalue problem by multiplying the fission source term by a factor k. Starting with an initial estimate for k (such as k = ), the flux is computed. This is called the “inner iteration.” This flux is then used to compute a new value for k, and a new flux is computed. This is called the “outer iteration.” It is continued until a sufficiently converged value for k is obtained. The outer iteration is needed to adjust the fission source and the thermal upscatter source. Referring back to the discussion of () and (), we see that this method produces the value of k eff defined for fission productions. Neutron Slowing Down and Thermalization ..  SN Data Requirements The data required for an S N flux solution using () have been shown to be σ g , σℓ g←g ′ , ν̄ g σ f g , and χ g , where the superscript S N has been omitted for simplicity. The SN codes also traditionally use a particle-balance cross section (often loosely called “absorption”) defined by σ a g = σ g − ∑ σg ′ ←g . () g′ This quantity can also be computed by adding all the absorption reactions [(n, γ), fission, (n, p), (n, α), etc.] and subtracting (n, n), twice (n, n), and so on. The two methods are formally equivalent, except that small numerical differences due to cross-section processing lead to unreasonable values for σ a g , as computed from (), when σ a g is small in relation to σ g . In such cases, σ a g can be replaced by the value from the direct calculation, and the σ g position of the transport table is adjusted accordingly. Note that σ a g can be negative if more particles are produced by (n, xn) reactions than are absorbed. Note that σ a is the quantity that appears in the denominator of the fission-based definition for k eff (see ()). When the flux calculation is complete, it is often necessary to compute some response such as heating, radiation damage, gas production, photon production, or dose to tissue. Therefore, S N codes allow for reading several response-function edit cross sections, σ E g . The original S N codes read χ as a special array, and the cross sections were arranged into “transport tables” by “position” as shown in > Table . Note that the positions containing scattering data give all the source groups that scatter into the same final or “sink” group. Even the newer codes retain many features of this structure. We call this “DTF format” in honor of the pioneering discrete ordinates code (Lathrop ). A transport table like this is required for each group, Legendre order, and material. The tables may be “material-ordered” or “group-ordered.” Material ordering is the natural result of preparing a library from evaluated nuclear data. The tables are written onto a library with the outermost loop being material, then Legendre order, and then group as the inner loop. However, group ordering is the way the cross sections are needed in the codes. The S N equations are solved by sweeping down from group  to the lower energy groups, so cross sections for all materials and orders are needed for each group in sequence. The library is ordered with group as the outermost loop, then Legendre order, and then material. .. Example for HST- An S N model for HST- can be constructed using two-dimensional r–z geometry, including the vessel and the solution level inside the vessel. Effective cross sections can be constructed (see below) including the appropriate thermal scattering data for H in H O and self-shielded cross sections in the resonance range. When PARTISN is used to run the problem, it finds a k eff value of . as compared to . from MCNP, which is satisfactory agreement. This calculation was done with  groups; P , S ,  radial intervals; and  axial intervals. > Figure  shows the flux at an energy of . eV (or kT, the maximum of the flux per    Neutron Slowing Down and Thermalization ⊡ Table  Transport table terminology Contents for Group g Position  ⋮ NED ............ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ σEg Response edits ............ ....... . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . NED+ σag NED+ ν−g σf g σg NED+ ............ NED+ ⋮ NED+NUP+ ............ ............ ....... . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ σg←g′ Upscatter (g′ > g) ............ ....... . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . σg←g′ In-group (g′ = g) ............ ....... . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ σg←g′ Downscatter (g′ < g) NED+NUP+ ............ NED+NUP+ ⋮ NTABL Standard edits NED = number of extra response edits (NED ≥ ) NUP = maximum number of upscatter groups ( ≤ NUP ≤ NGROUP) NTABL = table length (NED +  + NUP ≤ NTABL ≤ NED +  + NUP + NGROUP) NGROUP = number of energy groups IPTOT = NED +  = position of total cross section unit lethargy) vs. the radial coordinate r and the axial coordinate z. The cosine shape of the axial distribution is typical; it is called the “fundamental mode” shape. > Figure  shows a view of the HST- flux per unit lethargy vs. energy and radius at the axial midpoint of the solution region. Just as for the MCNP result shown in > Fig. , we see the fission source at high energies, the /E slowing down region, and the thermal peak at low energies. > Table  compares the reaction-rate balances for PARTISN and MCNP. At this point, it might be well to remind ourselves that a k calculation does not determine the actual values of the flux or reaction rates; a reactor can be critical at a wide range of operating power values. The normalization in a table like this is arbitrary. The default normalization for both PARTISN and MCNP is to make the effective loss rate (i.e., the denominator of ()) equal to one. The entry “(n, xn)” in the table is the amount subtracted from capture + fission to get the effective absorption. Specifically, it is (n, n) + (n, n) + . . . . As a counterexample, the TART Monte Carlo code (Cullen ) normalizes itself so that the total production νσ f + (n, n) + (n, n) + . . . is one (see the numerator of ()). Neutron Slowing Down and Thermalization  1.5 20 0 –6 16 0 18 0 1.0 0 z( 120 cm 1 ) 40 Flux *10 60 40 40 80 ) cm r( 60 20 20 0 0 80 10 0.5 ⊡ Figure  Flux for HST- vs. r and z at . eV as computed by SN methods –5 10 5 10 –7 ) V rg ne 10 – 80 3 ) m (c 60 – r 40 1 E 20 10 0 10 1 y (e 10 –6 3 10 10 Flux 7 10 ⊡ Figure  Flux per unit lethargy for HST- vs. energy and radius at the axial midpoint as computed by SN methods .. Preparing SN Cross-Section Data Normal multigroup cross-section data need some extra processing to be used for a calculation like this HST- example. The microscopic cross sections need to be assembled into macroscopic cross sections that include the shielding effects on the resonance features of the actinides    Neutron Slowing Down and Thermalization ⊡ Table  Reaction balances for HST- Reaction rate PARTISN MCNP νσf . Leakage . . Absorption . . Losses . . Fission . (n,xn) . in the solution. In addition, the normal low-energy scattering cross section for H- has to be modified to include the H in H O thermal scattering, and the other materials need to be modified to include thermal free-gas scattering. One way to do this additional processing is with the TRANSX code (MacFarlane ). TRANSX works with multigroup libraries produced by NJOY in MATXS format. The libraries provide tables of self-shielded microscopic cross sections vs. background cross section and temperature (more on these kinds of self-shielded cross sections will be found below). TRANSX determines the background cross section using the mixture and geometry information provided by the user and interpolates for the effective cross sections. The libraries contain normal target-at-rest down scatter at low energies, and they also contain tables of the bound and free thermal-scattering cross sections with upscatter. TRANSX will replace the target-at-rest values with the proper thermal data as specified by the user. The code will also prepare the fission source in the χ and νσ f representation, using a user-supplied weighting flux to get the good χ vector. This feature can be used to iterate for an improved χ for cases where the library weighting function does not match the computed flux very well. This problem is common for fast critical assemblies. Other features provided by TRANSX include group collapse, cell homogenization, coupled sets, adjoint cross sections, and data for the multigroup mode of MCNP. Output can be prepared in formats adaptable to a number of popular multigroup transport codes. .. Example for an Infinite Pin-Cell Lattice With S N codes, the normal way to simulate an infinite pin-cell lattice is to convert the square cells with reflective boundaries into cylindrical shells with white boundary conditions. This allows us to use a one-dimensional calculation. That runs very fast with PARTISN, giving k ∞ = . (as compared to the MCNP value of .). The result of that calculation is region fluxes for  groups in the fuel, clad, and moderator. These fluxes can be used in a second TRANSX run to perform a cell homogenization and group collapse. A simplified result of such a calculation is shown in > Table . These macroscopic effective cross sections can then be used in a large-scale calculation for the entire reactor using transport, diffusion, or collisionprobability methods. The TRANSX self-shielding calculation for the infinite lattice also included a Dancoff correction. When a neutron escapes from the fuel pin in a lattice, there is a possibility that it will reach another fuel pin before scattering from the moderator material. In that case, it would be as Neutron Slowing Down and Thermalization  ⊡ Table  Two-group macroscopic cross sections for a pin cell in an infinite lattice Reaction Group  . eV– MeV σ cm− Group  e- eV–. eV σ cm− nf .- .- ng .- .- abs .- .- nusigf .- .- Total .- .- if it did not escape from the fuel pin at all. There is a reduction in the effective escape probability. TRANSX handles that effect with several possible Dancoff correction options, for example, pins in a square lattice. Another improvement to this calculation is possible using PARTISN. It is possible to simulate the cosine “fundamental mode” axial shape seen in > Fig.  by imposing a “buckling.” This simulates axial leakage from the top and bottom of the lattice of cells, leaving the effective radius or x–y extent of the entire core large (zero leakage). .. Monte Carlo vs. Multigroup From these examples, we can begin to appreciate the different advantages and disadvantages of the Monte Carlo and multigroup methods. The Monte Carlo method uses good physics for slowing down and resonance reactions, and it allows for a very complete geometry representation. However, it is a statistical calculation, and it is not so good for looking at small differences or for getting smooth detailed distributions like those shown in > Fig. . The Monte Carlo calculations are also relatively expensive. On the other hand, the multigroup methods are relatively fast, and they are useful for looking for the effects of small differences in materials or geometry. They are good for getting smooth distributions of things like flux or nuclear heating over the geometry. However, they work best for simple geometries (requiring some approximation of the actual arrangements). The self-shielding effects from the resonances are not handled as accurately as with Monte Carlo. In the end, it has been the speed advantages of the multigroup methods that have kept them prominent in reactor core calculations. This is especially important when time-dependent fuel burnup and fission-product accumulation are considered – in these cases many time steps are required, making the Monte Carlo approach even more expensive. . Collision Probability Methods Collision probability methods are based on the integral form of the transport equation. As used above, the transport equation was of integro-differential form. As shown in Bell and Glasstone (), standard mathematical methods can be used to convert that form of the equation    Neutron Slowing Down and Thermalization into integral form. For our purposes, let us limit ourselves to the time-independent case with isotropic scattering and source. Then ϕ(r, E) = ∫ exp[−τ(E, r′ → r)] ′ ′ ′ ′ ′ ′ × [∫ σ(r , E → E)ϕ(r , E ) dE + Q(r , E)] dV , π∣r − r′ ∣ () where τ(E, r′ → r) is the optical path length from r′ to r (i.e., the number of mean free paths, or the integral of σ(r, E) along the path from r′ to r). Clearly, the quantity in square brackets is the source into energy E and volume element dV ′ around r′ due to scattering from other energies and external contributions. The factor in the first line is the attenuation of that while traveling from r′ to r. In practice, the energy range is divided into a number of energy groups within which the cross section can be taken as constant, and the geometry is divided into a number of subregions in which the flux and source can be taken as constant. The result is ϕ g,i ⎡ ⎤ ⎢ ⎥  ⎢ ′ ′ = ∑ Vj Pg, j→i ⎢∑ σ g →g, j ϕ g , j + Q g, j ⎥ ⎥, Vi σ g,i j ⎢ g′ ⎥ ⎣ ⎦ () where Pg, j→i is the probability that a neutron born in group g and subregion j will suffer its next collision in subregion i. This is the collision probability matrix Pg, j→i = σ g,i exp[−τ(E g , r′ → r)] dV dV ′ . ∫ ∫ V j Vi V j π∣r − r′ ∣ () The limitation to isotropic scattering can be alleviated by using a transport correction similar to the ones described above in connection with the SN method. Collision probability methods have been widely used for core physics calculations. An early example was the THERMOS code (Honek ), which could compute CP matrices for plate and pin-cell geometries in the thermal region. For a pin cell, the fuel, clad, and moderator regions were subdivided into a number of parts in order to handle the spatial self-shielding well. The resulting flux was then used to homogenize the cell and collapse to a coarser group structure. These numbers could then be used by simpler methods, such as diffusion, to complete the calculation for the whole reactor. In comparing CP and S N methods, we see that the CP approach can handle very complex geometries, while the S N method works best when it can work with simpler geometries, such as slab, sphere, or cylindrical r–z. However, the CP methods couples every cell in the calculation to every other cell. With N subregions and G groups, the problem grows like N  G. This makes the CP method impractical for representing a reactor in all its awful detail. However, it has been used for quite complex fuel bundles in the past. To handle larger problems, it makes sense to use the CP method on a fuel element or cluster, homogenize, and collapse. Then another CP calculation can be made using the effective fuel element cross sections for the core region and modeling the radial and axial components as a CP matrix. . Size Effects in Thermalization Full thermalization requires multiple collisions between the neutrons and the material to bring the neutrons into equilibrium at the temperature of the material. This implies that the system Neutron Slowing Down and Thermalization  ⊡ Table  Mean free paths near  meV (− eV) for several moderators Moderator material Mean free path near  meV Water . cm Heavy water . cm Graphite  cm 10 –5 E*flux 10 –6 10 –7 10 –8 10 –9 10 –10 10 –10 10 –9 10 –8 10 –7 10 –6 10 –5 Energy (MeV) ⊡ Figure  Effect of size on thermalization in graphite. The flux in spheres with radii of , , , and  cm are shown (some curves were shifted vertically for clarity) must be large with respect to the mean free path of the neutrons in the material. > Table  shows the mean free paths at energies around  meV (− MeV) for three common moderators. > Figure  shows fluxes computed using MCNP for spheres of graphite of various sizes. At a radius of  cm, the flux is nearly thermalized, showing the typical Maxwellian shape. However, for a radius of  cm, the flux shows features coming from the cross-section shape. The Bragg edges previously seen in > Fig.  are evident. It is apparent that systems with sizes of ten mean free paths or more will produce a Maxwellian flux shape independent of the details of the shape of the scattering cross section. This is a consequence of the detail balance as discussed in > Sect. .. It is also the reasoning behind the statement at the end of > Sect. . claiming that the problem that theory has in matching the experimental cross section below  meV (see > Fig. ) is not too important in practice. The part of any capture or fission reaction rate that comes from the millivolt region will not depend on the detailed shape of the scattering cross section there. In heavy water, the ENDF/B-VII evaluation shows a dip in the scattering cross section below about  meV coming from intermolecular coherence (see > Fig. ). This effect was not included in earlier ENDF/B versions. The original GA evaluators (Koppel and Houston )    Neutron Slowing Down and Thermalization said, “... integral quantities like the thermal neutron spectra can actually be predicted quite accurately with an incoherent model.” This is born out by MCNP calculations of the flux in spheres of heavy water, which show a Maxwellian shape at low energies even at relatively small sizes. The effect of the coherence dip is not visible.  Steady-State Slowing Down . Introduction If a nuclear assembly is driven by a steady-state source, the neutrons will slow down and become thermalized. In this section, we will summarize the elastic and inelastic cross sections that are responsible for the slowing down, study the spectra of the downscattered neutrons, and discuss how resonance structure affects slowing down. . Slowing-Down Cross Sections The important materials for neutron slowing down are the ones that occur in large concentrations in nuclear systems. We will concentrate on H-, H-, O-, C-nat, and U- in this section. The simplest materials are H- and H-. They only support elastic scattering and (n, γ) capture. > Figure  shows those cross sections as obtained from the tabulated data in the ENDF/B-VII evaluations for the two isotopes. The elastic cross sections for both materials are basically constant over much of their energy range, and the capture cross sections are small with a /v energy dependence (mostly). This makes materials containing H- or H- very useful as neutron moderators. The angular distributions in the center-of-mass (CM) system are nearly isotropic for the lower energies, gradually becoming somewhat backwardly peaked at the higher energies. In the ENDF/B-VII file for H-, the angular distribution in the CM is represented using an expansion in Legendre polynomials f (E, ω) = ∑ ℓ ℓ +  a ℓ (E)Pℓ (ω) ,  () where ω is the scattering cosine in the CM system and ℓ is the Legendre order varying from zero to some stated maximum. The angular distribution for H- is given as a set of tabulations of f (E, ω) vs. ω. The center-of-mass motion causes the angular distributions in the laboratory system to be forward-peaked. At the lower energies, the average cosine for elastic scattering goes like /A, where A is the ratio of the scatterer mass to the neutron mass. Thus, the nominal μ̄ is . for H- and . for H-. Here, we use the symbol μ for the scattering cosine in the laboratory system. For O-, the elastic cross section has the additional complication of resonant behavior at the higher energies, as shown in > Fig. . The (n, γ) capture cross section for O- is extremely small, and it is not plotted in the figure. However, at higher energies, an (n, α) channel opens, and this absorption cross section, although small, does have some effect on criticality in water systems. O- also has the additional complication of inelastic scattering, as shown in > Fig. . Only the first four inelastic levels are shown. The deep dip in the elastic scattering is a crosssection “window.” This kind of feature can lead to additional leakage because the mean free path Neutron Slowing Down and Thermalization  10 2 Cross section (barns) 10 1 10 0 10 –1 10 –2 10 –3 10 –4 10 –5 10 –6 100 101 102 103 104 105 106 107 Energy (eV) ⊡ Figure  Elastic and capture cross sections for H- (solid) and H- (dashed) from ENDF/B-VII Cross section (barns) 10 1 10 0 10 –1 10 –2 10 –3 102 103 104 105 106 107 Energy (eV) ⊡ Figure  Elastic (solid) and (n, α) (dashed) cross sections for O- from ENDF/B-VII is very large in the window. The CM angular distribution for elastic scattering is fairly isotropic up to the  keV region. Because of the higher atomic mass, the laboratory scattering angular distribution is fairly isotropic with a μ̄ of .. The total cross section for U- is shown in > Fig. . For heavy materials like this, the resonance behavior becomes pronounced. The range from − eV to  keV is the “resolved resonance region” (RRR), the range from  to  keV is the “unresolved resonance region” (URR), and the range above  keV is the “fast region.”   Neutron Slowing Down and Thermalization 400 Cross section (barns) *10 –3 300 200 100 0 107 Energy (eV) ⊡ Figure  The first four levels of inelastic scattering for O- from ENDF/B-VII (in increasing order of excitation energy, solid, dash, chain-dash, dotted 105 Cross section (barns)  104 103 102 101 100 10 –2 10 –1 100 101 102 103 104 105 106 107 Energy (eV) ⊡ Figure  The total cross section of U- at room temperature > Figure  shows the cross sections for U- in the fast range. (The (n, n) reaction, which is open above . MeV, was omitted.) The inelastic reaction includes  discrete inelastic channels and a continuum inelastic channel. Note that U- is a threshold fissioner – there is only a small amount of subthreshold fission below the effective threshold near  keV. Also note that the (n, γ) capture cross section is very small in this range. The angular distributions for elastic scattering are given using the Legendre representation, and they show significant Neutron Slowing Down and Thermalization  Cross section (barns) 5 4 3 2 1 0 0 5 10 Energy (eV) 15 20 *106 ⊡ Figure  The fast region for U- showing elastic (solid), inelastic (dashed), fission (chain-dash), (n, n) (chain-dot), and (n, γ) capture (dotted) structure. This structure has proved to be important for getting the proper leakage out of uranium bodies in critical assembly benchmarks. The angular distributions for the discrete inelastic levels are also given using the Legendre representation and show significant structure. The continuum inelastic and (n, n) reactions are represented as coupled energy–angle distributions. An angular distribution specification is given for each E → E ′ scattering. At lower energies, these distributions are fairly isotropic. The excited compound system resulting from the initial collision has had time to come to equilibrium; thus, the emitted neutron has “forgotten” the direction of the incident neutron. At higher energies, we begin to see preequilibrium emissions, and the angular distributions become more and more forward peaked. In the unresolved resonance region, a tabulated cross section is given to represent the average behavior. However, there are resonance effects there that lead to self-shielding of the cross sections. This will be discussed in more detail below. The resonance effects are described by giving average parameters and distribution rules for the unresolved resonances. The average parameters include spacing and resonance widths. These data can be used to produce selfshielded cross sections for multigroup methods or probability tables for Monte Carlo methods. The resolved resonance region can be extremely complex. > Figure  shows the elastic and capture cross sections between  and  eV. The . eV capture resonance is very important for thermal reactors. We have already seen its effect in > Fig. . . Spectra for Elastic Downscatter Because the neutron wavelength is very small at the higher energies, neutron elastic scattering can be treated by classical “billiard ball” kinematics. If ω is the cosine of the scattering angle in   Neutron Slowing Down and Thermalization 10 4 10 3 Cross section (barns)  10 2 10 1 10 0 10 –1 10 –2 100 101 Energy (eV) 102 ⊡ Figure  The elastic (solid) and capture (dashed) cross sections of U- for room temperature at the lower end of the resolved resonance range the CM system, E is the incident energy, and A is the ratio of the target mass to the neutron mass, the secondary energy E ′ and laboratory scattering cosine μ are given by E′ = E (A +  + Aω) , (A + ) and μ= √ ()  + Aω . () +  + Aω Clearly, the maximum fractional energy change in elastic scattering (commonly called α) occurs when ω = − and is given by A−  α=[ () ] . A+ For H- with A = ., α is almost zero, and a neutron can lose almost all of its energy in an elastic scattering event. If the CM scattering distribution is isotropic, the energy distribution for elastically scattered neutrons is given by A ′ f (E, E ) =  , ( − α)E () for E ′ between αE and E, and zero otherwise. We can simulate the behavior of elastic downscatter by running MCNP on a “broomstick.” We take a very long, very thin cylinder of a material, start neutrons into one end, and look at the spectrum of neutrons escaping from the surface integrated over all angles. Because the cylinder is very thin, these neutrons will have undergone only one scattering event, and the distribution we see will be just σ(E → E ′ ) = σ(E) f (E, E ′). > Figure  shows an example of elastic downscatter spectra for D O. At the lowest energy, we see the rectangular shapes predicted by () with the effects of both H- and O- visible. The rectangle for O- is narrower than that for H- because of its larger mass (larger A). For the higher incident energies, the effects of anisotropy in the scattering cross section show up. Neutron Slowing Down and Thermalization  3.0 E*spect (E-prime) 2.5 2.0 1.5 1.0 0.5 0.0 10 –3 10 –2 10 –1 100 101 E-prime (MeV) ⊡ Figure  Elastic slowing-down spectra for D O at ., ., and  MeV . Spectra for Inelastic Downscatter At high incident energies, inelastic neutron scattering becomes possible. For the lighter isotopes, the first inelastic threshold is in the million electron volts range, but for heavy isotopes, it can come down to tens of kiloelectron volts or lower. We can extend the kinematics formulas to include inelastic scattering (and even charged-particle emission) as follows: ′ E = A′ E  (β +  + βω) , (A + ) and μ= √ where β=(  + βω β  +  + βω () , A+ Q A(A +  − A′ ) [ + ]) A′ A E () / . () The quantity A′ is the ratio of mass of the emitted particle to the mass of the incident particle; thus it is equal to one for neutron inelastic scattering. The quantity (A + )(−Q)/A is the threshold energy for the inelastic level with excitation energy Q. The quantity β acts sort of like an effective mass ratio. When E is close to the inelastic threshold, β is small, and a scattered neutron can lose a large part of its energy, just as elastic scattering from a light target can cause the neutron to lose a large part of its energy. > Figure  illustrates this for U- just above the threshold for the first inelastic level at . keV. As the neutron energy increases in the inelastic region, the levels get closer and closer together. At some point, it becomes reasonable to treat them continuously. > Figure  shows the continuum inelastic spectra for incident energies near . and  MeV. The dotted line shows   Neutron Slowing Down and Thermalization 1,000 xsec per unit energy *10 –6 800 600 400 200 0 2 4 6 8 10 *10 –3 E-prime (MeV) ⊡ Figure  Inelastic slowing-down spectra for the first inelastic level for U- at incident energies of ., ., . and . MeV. The threshold for this level is . MeV Spectrum per unit energy  10 –6 10 –7 10 –8 102 103 104 E-prime (eV) 105 106 ⊡ Figure  Inelastic slowing-down spectra for the continuum inelastic reaction for U- at incident energies of . and  MeV. The dotted line goes as sqrt(E) the expected shape for low energies, namely sqrt(E). From theory, we expect that the lowenergy shape for the CM distribution will go as sqrt(E CM ). The Jacobian for transforming from the CM to the laboratory frame is sqrt(E lab /E CM ), so both the CM and the lab distributions must take on the sqrt shape. Actual ENDF/B evaluations often use fairly coarse histogram representations for curves like this (they come from nuclear model calculations). We have smoothed the low-energy sqrt(E) shape of these curves for this plot. Neutron Slowing Down and Thermalization .  Resonance Cross Sections In the ENDF format (Herman ), resolved resonance range cross sections are represented using resonance parameters based on four different methodologies: Single-Level Breit–Wigner (SLBW), Multi-Level Breit–Wigner (MLBW), Reich–Moore (RM), and Reich–Moore Limited (RML). Additional representations were used in the past, but they are no longer represented in the ENDF/B-VII library (Tuli et al. ). .. Single-Level Breit–Wigner Representation The SLBW resonance shapes are given by σ n = σ p + ∑ ∑ σ mr {[cos ϕ ℓ − ( − ℓ r σ f = ∑ ∑ σ mr ℓ r σγ = ∑ ∑ σ mr ℓ r Γnr )] ψ(θ, x) + sin ϕ ℓ χ(θ, x)}, Γr () Γf r ψ(θ, x), Γr () Γγr ψ(θ, x), Γr () and σp = ∑ ℓ π (ℓ + ) sin θ ℓ , k () where σ n , σ f , σγ , and σ p are the neutron (elastic), fission, radiative capture, and potential scattering components of the cross section arising from the given resonances. There can be “background” cross sections given that must be added to these values to account for competitive reactions such as inelastic scattering or to correct for the inadequacies of the single-level representation with regard to multilevel effects or missed resonances. The sums extend over all the ℓ values and all the resolved resonances r with a particular value of ℓ. Each resonance is characterized by its total, neutron, fission, and capture widths (Γ, Γn , Γf , Γγ ), by its J value (AJ in the file), and by its maximum value σ mr /Γr σ mr = π Γnr gJ , k Γr () where g J is the spin statistical factor J +  , () I +  and I is the target spin, and k is the neutron wave number, which depends on incident energy E and the atomic weight ratio to the neutron for the isotope A as follows: gJ = k = (. × − ) A √ E. A+ () There are two different characteristic lengths that appear in the ENDF resonance formulas: first, there is the “scattering radius” â, and second, there is the “channel radius” a, which is given by a = . A/ + .. ()    Neutron Slowing Down and Thermalization In some cases, a is set equal to â in calculating penetrabilities and shift factors (see below). The neutron width in the equations for the SLBW cross sections is energy-dependent due to the penetration factors Pℓ ; that is, Pℓ (E) Γnr Γnr (E) = , () Pℓ (∣E r ∣) where P = ρ, ()  P = ρ ,  + ρ () P = ρ ,  + ρ  + ρ  () P = ρ , + ρ  + ρ   + ρ  and P =  + ρ  ρ , + ρ  + ρ  + ρ  () () where E r is the resonance energy and ρ = ka depends on the channel radius or the scattering radius. The phase shifts are given by ϕ  = ρ̂, () ϕ  = ρ̂ − tan− ρ̂, ()  ρ̂ ,  − ρ̂  ()  ρ̂ − ρ̂  ,  −  ρ̂  ()  ρ̂ −  ρ̂  ,  −  ρ̂  + ρ̂  () ϕ  = ρ̂ − tan − ϕ  = ρ̂ − tan− and ϕ  = ρ̂ − tan− where ρ̂ = k â depends on the scattering radius. The final components of the cross section are the actual line shape functions ψ and χ. At zero temperature,  ,  + x x χ= ,  + x  (E − E r′ ) , x= Γr ψ= and E r′ = E r + S ℓ (∣E r ∣) − S ℓ (E) Γnr (∣E r ∣), (Pℓ (∣E r ∣) () () () () Neutron Slowing Down and Thermalization  in terms of the shift factors S  = , ()  S = − ,  + ρ () S = −  + ρ  ,  + ρ  + ρ  () S = −  + ρ  + ρ  ,  + ρ  + ρ  + ρ  ()  + ρ  + ρ  + ρ  .  + ρ  + ρ  + ρ  + ρ  () and S = − To go to higher temperatures, define θ=√ Γr , () kT E A where k is the Boltzmann constant and T is the absolute temperature. The line shapes ψ and χ are now given by √ π θx θ ψ= θ Re W ( , ) , ()    and √ π θx θ θ Im W ( , ) , () χ=    in terms of the complex probability function. ∞ e−t i dt, ∫ π −∞ z − t   W(x, y) = e−z erfc(−iz) = () where z = x+iy. The ψ χ method is not as accurate as kernel broadening (see below) because the backgrounds (which are sometimes quite complex) are not broadened, and terms important for energies less than about kT/A are neglected; however, the ψ χ method is less expensive than kernel broadening. The SLBW formalism is deprecated for new evaluations, and there are only a few holdovers remaining in ENDF/B-VII. It has the problem of sometimes producing negative cross sections from the χ interference term. However, it is the basis for the unresolved range methodology, and it is still important to understand it. .. Multi-Level Breit–Wigner Representation The MLBW representation is formulated as follows: I+  σ n (E) = l +s   π ℓs J ∑ ∑ ∑ g J ∣ − U nn (E)∣ ,  k ℓ s=∣I−  ∣ J=∣l −s∣  ()    Neutron Slowing Down and Thermalization with ℓJ i ϕ U nn (E) = e ℓ − ∑ r iΓnr , E r′ − E − iΓr / () where the other symbols are the same as those used above. Expanding the complex operations gives ⎧ I+   l +s ⎪ π Γnr  ⎪ ) σ n (E) =  ∑ ∑ ∑ g J ⎨( − cos ϕ ℓ − ∑  k ℓ s=∣I−  ∣ J=∣l −s∣ ⎪ ⎪ r Γr  + x r ⎩  ⎫ ⎪ Γnr x r ⎪ ⎬, + (sin ϕ ℓ + ∑ )  ⎪ Γ  + x r ⎪ r r ⎭ () where the sums over r are limited to resonances in spin sequence ℓ that have the specified value of s and J. Unfortunately, the s dependence of Γ is not known. The file contains only ΓJ = Γs  J + Γs  J . It is assumed that the ΓJ can be used for one of the two values of s, and zero is used for the other. Of course, it is important to include both channel-spin terms in the potential scattering. Therefore, the equation is written in the following form: σn (E) = ⎡ ⎧  ⎪ ⎢ π Γnr  ⎢∑ g J ⎪ ⎨ − ) ( − cos ϕ ∑ ∑ ℓ  ⎪ k ℓ ⎢ ⎢ J ⎪ r Γr  + x r ⎩ ⎣ ⎤ ⎫ ⎪ ⎥ Γnr x r ⎪ ⎥ + (sin ϕ ℓ + ∑ ) ( − cos ϕ ) ⎬ + D ℓ ℓ ⎥,  ⎪ ⎥ ⎪ r Γr  + x r ⎭ ⎦ () where the summation over J now runs from   ∣∣I − ℓ∣ − ∣ → I + ℓ + ,   () and D ℓ gives the additional contribution to the statistical weight resulting from duplicate J values not included in the new J sum; namely, I+  Dℓ = ∑ I+ℓ+  l +s ∑ gJ − s=∣I−  ∣ J=∣l −s∣ = (ℓ + ) − ∑ gj () J=∣∣I−ℓ∣−  ∣ I+ℓ+  ∑ gj. () J=∣∣I−ℓ∣−  ∣ A case where this correction would appear is the ℓ =  term for a spin- nuclide. There will be J values: /, /, and / for channel spin /; and / and / for channel spin /. All five contribute to the potential scattering, but the file will only include resonances for the first three. The fission and capture cross sections are the same as for the single-level option. The ψ χ Doppler-broadening cannot be used with this formulation of the MLBW representation. Neutron Slowing Down and Thermalization  However, there is an alternate formulation that can be used with ψ χ broadening: σn = σ p + ∑ ∑ σ mr {[cos ϕ ℓ − ( − ℓ r + (sin ϕ ℓ + where G rℓ =   and H rℓ = Γnr G rℓ )+ ] ψ(θ, x) Γr Γnr H rℓ ) χ(θ, x)} , Γnr Γr + Γr ′ , ∑ Γnr Γnr ′ (E r − E r′ ) + (Γr + Γr′ ) / ′ r ≠r Jr′ ≠ Jr E r − Er ′ . ∑ Γnr Γnr′ (E r − E r ′ ) + (Γr + Γr ′ ) / ′ r ≠r J r′ ≠ Jr () () () Nominally, this method is slower than the previous one because it contains a double sum over resonances at each energy. However, it turns out that G and H are slowly varying functions of energy, and the calculation can be accelerated by computing them at just a subset of the energies and getting intermediate values by interpolation. It is important to use a large number of r ′ values on each side of r. The GH method works well at higher energies when compared to the more accurate kernel broadening method (more on this below). However, in the electron volts range and below, it compares more poorly with kernel broadening. Since the accelerated GH method is only marginally faster than kernel broadening, it probably should not be used at the lower energies for cases where the details of elastic scattering are important. This still leaves it useful for materials like fission products where absorption is the most important factor. The MLBW representation does not produce the negative cross sections that plague the SLBW method. However, it is more expensive to use. It is used extensively for mid-mass isotopes (such as fission products) in the ENDF/B-VII library. .. Reich–Moore Representation The RM representation is a multilevel formulation with two fission channels; hence, it is useful for both structural and fissionable materials. (The name “Reich” is pronounced like “rich” by the co-originator of this method.) The cross sections are given by π ℓJ ∑ ∑ g J {( − Re U nm ) + d ℓ J [ − cos(ϕ ℓ )]}, k ℓ J π ℓJ  σ n =  ∑ ∑ g J {∣ − U nn ∣ + d ℓ J [ − cos(ϕ ℓ )] , k ℓ J σt = σf = π ℓJ  ∑ ∑ g J ∑ ∣Inc ∣ , k ℓ J c () () () and σγ = σ t − σn − σ f , ()    Neutron Slowing Down and Thermalization where Inc is an element of the inverse of the complex R-matrix and ℓJ U nn = e i ϕ ℓ [ Inn −  ]. () The elements of the R-matrix are given by ℓJ R nc = δ nc − / / i Γnr Γcr . ∑  r E r − E − i Γγr () In these equations, “c” stands for the fission channel, “r” indexes the resonances belonging to spin sequence (ℓ, J), and the other symbols have the same meanings as for SLBW or MLBW. Of course, when fission is not present, σ f can be ignored. The R-matrix reduces to an R-function, and the matrix inversion normally required to get Inn reduces to a simple inversion of a complex number. As in the MLBW case, the summation over J runs from   ∣∣I − ℓ∣ − ∣ → I + ℓ + .   () The term d ℓ J in the expressions for the total and elastic cross sections is used to account for the possibility of an additional contribution to the potential scattering cross section from the second channel spin. There is unity if there is a second J value equal to J, and zero otherwise. This is just a slightly different approach for making the correction discussed in connection with (). Returning to the I = , ℓ =  example given above, d will be one for J = / and J = /, and it will be zero for J = /. The RM representation is used for many of the most important isotopes in ENDF/B-VII, because it is very true to the underlying physics, resulting in good fits to experimental data. For fissile materials, its ability to handle two fission channels provides a better representation of the effects of the double-humped fission barriers seen for isotopes like U-. .. Reich–Moore-Limited Representation The new RML representation is a more general multilevel and multichannel formulation. In addition to the normal elastic, fission, and capture reactions, it allows for inelastic scattering and Coulomb reactions. Furthermore, it allows resonance angular distributions to be calculated. The RML processing in NJOY is based on the SAMMY code (Larson ). The quantities that are conserved during neutron scattering and reactions are the total angular momentum J and its associated parity π, and the RML format lumps all the channels with a given Jπ into a “spin group.” In each spin group, the reaction channels are defined by c = (α, ℓ, s, J), where α stands for the particle pair (masses, charges, spins, parities, and Qvalue), ℓ is the orbital angular momentum with associated parity (−) ℓ , and s is the channel spin (the vector sum of the spins of the two particles of the pair). The ℓ and s values must vector sum to Jπ for the spin group. The channels are divided into incident channels and exit channels. Here, the important input channel is defined by the particle pair neutron + target. There can be several such incident channels in a given spin group. The exit channel particle pair defines Neutron Slowing Down and Thermalization  the reaction taking place. If the exit channel is the same as the incident channel, the reaction is elastic scattering. There can be several exit channels that contribute to a given reaction. The R-matrix in the RM “eliminated width” approximation for a given spin group is given by γ λc γ λc ′ b R cc ′ = ∑ () + R c δ cc ′ , λ E λ − E − iΓλγ / where c and c ′ are incident and exit channel indexes, λ is the resonance index for resonances in this spin group, E λ is a resonance energy, γ λc is a resonance amplitude, and Γλγ is the “eliminated width,” which normally includes all of the radiation width (capture). The channel indexes run over the “particle channels” only, which does not include capture. The quantity R bc is the “background R-matrix.” In order to calculate the contribution of this spin group to the cross sections, we first compute the following quantity: / − − / X cc ′ = Pc L c ∑ Ycc ′′ R c ′′ c ′ Pc ′ , () c ′′ where − Ycc ′′ = L c δ cc ′′ − R cc ′′ , () L c = S c − B c + iPc . () and Here, the Pc and S c are penetrability and shift factors, and the B c are boundary constants. The cross sections can now be written down in terms of the X cc ′ . For elastic scattering σelastic = π  i r  ∑[sin ϕ c ( − X cc ) − X cc sin(ϕ c ) + ∑ ∣X cc ′ ∣ ] , k α Jπ ′ c () r i where X cc ′ is the real part of X cc ′ , X cc ′ is the imaginary part, ϕ c is the phase shift, the sum over Jπ is a sum over spin groups, the sum over c is limited to incident channels in the spin group with particle pair α equal to neutron + target, and the sum over c ′ is limited to exit channels in the spin group with particle pair α. Similarly, the capture cross section becomes σcapture = π i  ∑ ∑ g Jα ∑[X cc − ∑ ∣X cc ′ ∣ ] , k α Jπ c ′ c c () where the sum over Jπ is a sum over spin groups, the sum over c is a sum over incident channels in the spin group with particle pair α equal to neutron + target, and the sum over c ′ includes all channels in the spin group. The cross sections for other reactions (if present) are given by σreaction = π i  ∑ g Jα ∑[X cc − ∑ ∣X cc ′ ∣ ] , k α Jπ c c′ () where the sum over c is limited to channels in the spin group Jπ with particle pair α equal to neutron + target, and the sum over c ′ is limited to channels in the spin group with particle pair α ′. The reaction is defined by α → α ′ . This is one of the strengths of the RML representation. The reaction cross sections can include multiple inelastic levels with full resonance behavior. They   Neutron Slowing Down and Thermalization can also include cross sections for outgoing charged particles, such as (n, α) cross sections, with full resonance behavior. The total cross section can be computed by summing up its parts. For non-Coulomb channels, the penetrabilities P, shift factors S, and phase shift ϕ are the same as those given for the SLBW representation. They are a little more complicated for Coulomb channels. See the SAMMY reference for more details. The RML representation is new to the ENDF format and is not represented by any cases in ENDF/B-VII. There are experimental evaluations for F- and Cl- available. However, the RML approach provides a very faithful representation of resonance physics, and it should see increasing use in the future. NJOY is currently able to process RML evaluations using coding adapted from SAMMY. .. Angular Distributions One of the physics advances available when using the RML format is the calculation of angular distributions from the resonance parameters. A Legendre representation is used: dσ αα ′ = ∑ B Lαα ′ (E)PL (cos β) , dΩ CM L () where the subscript αα ′ indicates the cross section as defined by the particle pairs, PL is the Legendre polynomial of order L, and β is the angle of the outgoing particle with respect to the incoming neutron in the CM system. The coefficients B Lαα ′ (E) are given by a complicated six-level summation that we will omit from this text. > Figure  shows the first few Legendre coefficients for the elastic scattering cross sections as computed by NJOY from the experimental evaluation for F-. 0.5 0.4 0.3 0.2 Coefficient  0.1 –0.0 –0.1 –0.2 –0.3 –0.4 –0.5 104 105 106 Energy (eV) ⊡ Figure  Legendre coefficients of the angular distribution for elastic scattering in F- using the RML resonance representation (P solid, P dashed, P dotted) Neutron Slowing Down and Thermalization ..  Resonance Reconstruction and Doppler Broadening In practice, the formulas presented in the preceding paragraphs must be used to generate cross section vs. energy tables – this is called “resonance reconstruction.” There are a number of computer codes available that carry out this reconstruction. The examples in this chapter were generated using the RECONR module of NJOY (MacFarlane and Muir ). Another system that provides resonance reconstruction and Doppler broadening is PREPRO (Cullen ). Most commonly, the resonances are reconstructed at  K, and they are Doppler broadened to the required temperatures. For this chapter, we used the BROADR module of NJOY. The methods for resonance reconstruction and Doppler broadening are described in detail in Volume I, > Chap. . .. Thermal Constants In thermal-reactor work, people make very effective use of a few standard thermal constants to characterize nuclear cross sections. These parameters include the cross sections at the standard thermal value of . eV (, m/s), the integrals of the cross sections against a Maxwellian distribution for . eV, the g-factors (which express the deviation of the cross section from √ /v; namely, / π times the ratio of the Maxwellian integral to the corresponding thermal cross section), η, α, and K. Here, η is the Maxwellian-weighted average of (ν̄σ f )/(σ f + σc ), α is the average of σ c /σ f , and K is the average of (ν̄ − )σ f − σ c . These quantities are routinely calculated by NJOY when it does Doppler broadening to the temperature corresponding to . eV (. K). > Table  shows values for three important nuclides from ENDF/B-VII. . Resonance Slowing Down Continuous-energy Monte Carlo codes like MCNP () simulate the physics of the slowingdown process. The cross-section tables are used to randomly select the reaction channel following a collision. If elastic scattering is selected, the angular distribution is used to randomly select an emission cosine to get a new particle direction, and this cosine is used to compute the energy of the scattered neutron. This process is very faithful to the physics for some materials and energy ranges, but in practice there are two problems. First, at low energies the motion of the target nuclei corresponding to the temperature of the material becomes important. The Doppler broadening method described in Volume I, > Chap.  preserves the total reaction rate for the neutron at energy E averaged over all angles, but it does not produce a correct shape for the energy spectrum of the scattered neutrons. Current codes use the Doppler-broadened cross section with the target-at-rest downscatter spectrum given by (). Second, the detailed variation of the angular distribution as the energy through a resonance is often not available in current data libraries. Average angular distributions over a range of resonances are more commonly provided. Multigroup methods for calculating neutron slowing down require average cross sections like those defined in () and (). The problem is coming up with good estimates for the neutron flux shape inside the group W(E). One approach to this is based on B  theory. Using    Neutron Slowing Down and Thermalization ⊡ Table  Thermal constants for three important resonance nuclides from ENDF/B-VII Thermal constant U- U- Pu- Thermal fission xsec .E− .E+ .E+ Thermal fission nubar .E+ .E+ .E+ Thermal capture xsec .E+ .E+ .E+ Thermal capture integral .E+ .E+ .E+ Thermal capture g-factor .E+ .E- .E+ Capture resonance integral .E+ .E+ .E+ Thermal fission integral .E− .E+ .E+ Thermal fission g-factor .E+ .E− .E+ Thermal alpha integral .E+ .E− .E− Thermal eta integral .E− .E+ .E+ Thermal K integral −.E+ .E+ .E+ Equivalent K −.E+ .E+ .E+ .E+ .E+ .E+ Fission resonance integral one-dimensional slab geometry for simplicity, the Boltzmann transport equation can be written in the form μ ∞ ∂ Σ s (E ′ → E, μ ′ → μ, x) ϕ(E ′, μ ′ , x) ϕ(E, μ, x) + Σ(E, x) ϕ(E, μ, x) = ∫ ∂x E + S(E, μ, x), () where we are now using the symbol Σ for the macroscopic cross sections. Next, we assume that the flux and the source can be separated into spatial- and energy-angle parts i Bx , () i Bx . () ϕ(E, μ, x) = ϕ(E, μ)e and S(E, μ, x) = S(E, μ)e This assumption certainly is not valid over the entire energy range, but it may be reasonable for the energy range of one group. To get an idea of the physical meaning of this, consider a critical slab. The B parameter is positive, and the real part of the flux has a cosine shape across the slab with its zeros a bit outside of the slab. Here /B becomes a measure of the thickness of the system. For a slab with a source on one face, B is imaginary, and the flux decreases exponentially as you move through the slab. After inserting the separated flux and source into the Boltzmann equation, expanding the flux using Legendre polynomials, and assuming that the scattering cross section is isotropic, we Neutron Slowing Down and Thermalization  get the B  equations: Σ(E)ϕ ℓ (E) = A ℓ ∫ ∞ E Σ s (E ′ → E)ϕ  (E ′ ) + S  (E), () where the A coefficients that we need are given by A = tanh− y , y () A  = A  −  , y () iB . Σ(E) () and where y= Note that small y indicates that the system is large with respect to the mean free path of the neutrons. This leads us to our next approximation. If the system is very large, we can take the limit of small y, getting A =  and A  = y/. The P and P (current) components of the flux become ∞  ϕ (E) = Σ s (E ′ → E)ϕ(E ′ ) + S  (E)] , () [∫ Σ(E) E and ϕ (E) = ∞ B [∫ Σ s (E ′ → E)ϕ(E ′) + S  (E)] .  Σ(E) E () We can now make another important approximation, the “narrow resonance approximation” (NR). If the resonances near E are narrow with respect to the downscatter predicted by Σ s , the contributions to the bracketed term in () and () will come from energies well above the resonances near E and will not contain any structure correlated with those resonances. Therefore, we can replace the bracketed term with a smooth function C(E), such as /E. Let us assume that the material consists of a resonance isotope mixed with a moderator material that has a constant cross section. Then the weighting function for the group cross sections becomes Wℓ (E) = C(E) , [σ + σ t (E)]ℓ+ () where σ t is the microscopic total cross section for the absorber nuclide the “background cross section” σ represents the effect of the other isotopes in the material. As an example, the multigroup total cross section for isotope i and group g becomes ∫ σ tig = ∫ g σ ti (E)C(E) dE i + σ i (E) σg t g C(E) dE i σg + σ ti (E) . () Physically, the term σ ti (E) in the denominators puts a dip in the weighting flux that corresponds to the resonance peak in the cross section in the numerator. The size of this dip is controlled by    Neutron Slowing Down and Thermalization i σg . When the background cross section is large with respect to the heights of the resonances, the dips are very small. This is called “infinite dilution,” because it corresponds to the case of a solution of the absorber and the moderator with very small amounts of the absorber. When the background cross section is smaller, there can be a significant dip in the weighting flux. This dip then cancels out a part of the effect of the resonance in the numerator – the resonance is “self-shielded.” This approach is often called the “Bondarenko Method” after the Russian scientist who originally put it into practice (Bondarenko ). For a mixture of materials, the background cross section for isotope i is given by i σg =  j ∑ ρ j σt g . ρ i j≠i () This formula makes sense for homogeneous systems, but when heterogeneity effects are important, the background cross-section method can be extended as follows. In an infinite system of two regions (fuel and moderator), the neutron balance equations are Vf σ f ϕ f = ( − P f )Vf S f + Pm Vm S m , () Vm σ m ϕ m = P f Vf S f + ( − Pm )Vm S m , () and where V f and Vm are the region volumes, σ f and σ m are the corresponding total macroscopic cross sections, S f and S m are the sources per unit volume in each region, P f is the probability that a neutron born in the fuel will suffer its next collision in the moderator, and Pm is the probability that a neutron born in the moderator will suffer its next collision in the fuel. We then apply the reciprocity theorem, Vf σ f P f = Vm σ m Pm , () and the Wigner rational approximation to the fuel escape probability, Pf = σe , σe + σ f () where σ e is a slowly varying function of energy called the escape cross section, to obtain an equation for the fuel flux in the form (σ f + σ e )ϕ f = σe Sm + Sf . σm () In the limit where the resonances are narrow with respect to both fuel and moderator scattering, the source terms S f and S m take on their asymptotic forms of σ p /E and σ m /E, respectively, and this equation becomes equivalent to the Bondarenko model quoted above with f σ = and C(E) = σe , ρf σe + σ p . ρf E () () Neutron Slowing Down and Thermalization  Note that a large escape cross section (a sample that is small relative to the average distance to collision), corresponds to infinite dilution as discussed above. To illustrate the general case, consider a neutron traveling through a lump of uranium oxide with an energy close to a resonance energy. If the neutron scatters from an oxygen nucleus, it will lose enough energy so that it can not longer react with the uranium resonance. Similarly, if the neutron escapes from the lump, it can no longer react with the uranium resonance. The processes of moderator scattering and escape are equivalent in some way. Comparing () with () gives an “equivalence principle” that says that a lump of particular dimensions and a mixture of particular composition will have the same self-shielded cross sections when the narrow resonance approximation is valid. The effects of material mixing and escape can simply be added to obtain the effective σ for a lump containing admixed moderator material. Therefore, () is extended to read i σg =  j j {σ e + ∑ ρ j σ tg (σg , T)}, ρi j≠i () where the escape cross section for simple convex objects (such as plates, spheres, or cylinders) is given by (V/S)− , where V and S are the volume and surface area of the object, respectively. The quantity V /S is often called the “mean chord length” ℓ̄. For example, the mean chord length for a sphere is equal to the radius, the mean chord length for a cylinder is equal to twice its radius, and the mean chord length for a slab is twice its thickness. Many codes that use the background cross-section method modify the escape cross section as defined above to correct for errors in the Wigner rational approximation (“Bell factor,” “Levine factor”), or to correct for the interaction between different lumps in the moderating region (“Dancoff correction”). These enhancements will not be discussed here. Note that a thin slab is equivalent to a dilute solution – they both will have infinitely dilute cross sections. . Flux Calculations This narrow-resonance approach is quite useful for practical fast reactor problems. However, for nuclear systems sensitive to energies from  to  eV, there are many broad- and intermediatewidth resonances that cannot be self-shielded with sufficient accuracy using the Bondarenko approach. The GROUPR module of NJOY contains a flux calculator that can give some insight for such problems. Consider an infinite homogeneous mixture of two materials and assume isotropic scattering in the center-of-mass system. The integral slowing-down equation becomes σ(E) ϕ(E) = ∫ E/α  E E/α  σs (E ′ ) σs (E ′ ) ′ ′ ′ ′ ϕ(E ) dE + ϕ(E ) dE . ∫ ( − α  )E ′ ( − α  )E ′ E () Furthermore, assume that material  is a pure scatterer with constant cross section and transform to the σ representation. The integral equation becomes [σ + σ t (E)] ϕ(E) = ∫ E/α  E E/α  σ (E ′ ) σ s ϕ(E ′ ) dE ′ + ∫ ϕ(E ′ ) dE ′ . () ′ ( − α  )E ( − α  )E ′ E Finally, assume that the moderator (material ) is light enough so that all the resonances of material  are narrow with respect to scattering from material . This allows the first integral    Neutron Slowing Down and Thermalization to be approximated by its asymptotic form /E. More generally, the integral is assumed to be a smooth function of E given by C(E). In this way, material  can represent a mixture of other materials just as in the Bondarenko method. Fission source and thermal upscatter effects can also be lumped in C(E). The integral equation has now been reduced to [σ + σ t (E)] ϕ(E) = C(E) σ + ∫ E/α E σs (E ′ ) ′ ′ ϕ(E ) dE . ( − α)E ′ () This is the simplest problem that can be solved using the flux calculator. The results still depend on the single parameter σ , and they can be used easily by codes that accept Bondarenko cross sections. For heterogeneous problems, when the narrow-resonance approximation fails, both S f and S m in () will show resonance features. To proceed further with the solution of this equation, it is necessary to eliminate the moderator flux that is implicit in S m . As a sample case, consider a fuel pin immersed in a large region of water. The fission neutrons appear at high energies, escape from the pin, slow down in the moderator (giving a /E flux), and are absorbed by the resonances in the pin. In this limit, any dips in the moderator flux caused by resonances in the fuel are small. On the other hand, in a closely packed lattice, the flux in the moderator is very similar to the flux in the fuel, and resonance dips in the moderator flux become very evident. Intermediate cases can be approximated (MacFarlane ) by assuming ϕ m = ( − β) C(E) + βϕ f , () where β is a heterogeneity parameter given by β= Vf σ e , Vm σ m () Note that β →  gives the isolated rod limit and β →  gives the close-packed lattice limit. This substitution reduces the calculation of the fuel flux to (σ f + σ e ) ϕ f = ( − β) C(E) σ e + S β , () where S β is the source term corresponding to a homogeneous mixture of the fuel isotopes with the isotopes from the moderator region changed by the factor βσ e /σm . If the fuel and moderator each consisted of a single isotope and for isotropic scattering in the center-of-mass system, the integral equation would become [σ + σ t (E)] ϕ f (E) = ( − β) C(E) σ + ∫ +∫ E/α f E E/α m E βσ ϕ f (E ′ ) dE ′ ( − α m )E ′ σs f (E ′ ) ′ ′ ϕ f (E ) dE , ( − α f )E ′ () where σ is σ e divided by the fuel density (units are barns/atom), α m and α f are the maximum fractional energy change in scattering for the two isotopes, and σs f (E ′ ) is the fuel-scattering cross section. This result has a form parallel to that of (), but the solution depends on the two parameters β and σ . For any given data set, β must be chosen in advance. This might not be difficult if Neutron Slowing Down and Thermalization  the data are to be used for one particular system, such as pressurized water reactors. The routine also has the capability to include one more moderator integral with a different α value and a constant cross section. The full equation is [σ + σ t (E)] ϕ f (E) = ( − β) C(E) σ + ∫ +∫ +∫ E/α  E E/α f E E/α  E β( − γ)(σ − σ am ϕ f (E ′ ) dE ′ ( − α  )E ′ σ am + βγ(σ − σ am ϕ f (E ′ ) dE ′ ( − α  )E ′ σs f (E ′ ) ϕ f (E ′ ) dE ′ , ( − α f )E ′ () where σ am is the cross section of the admixed moderator (with energy loss α  ), and γ is the fraction of the admixed moderator that is mixed with the external moderator (which has energy loss α  ). This allows calculations with H O as the moderator and an oxide as the fuel. The flux calculator can thus obtain quite realistic flux shapes for a variety of fuel, admixed moderator, and external moderator combinations. An example comparing the Bondarenko flux with a more realistic computed flux is given in > Fig. . . Intermediate Resonance Self-Shielding In this section, we will describe how the reactor-physics code WIMS (Askew et al. ) treats the problem of self-shielding for intermediate resonances. First, we refer back to (). If the 10 0 Flux (arb units) 10 –1 10 –2 10 –3 10 –4 5.0 5.5 6.0 6.5 7.0 Energy (eV) 7.5 8.0 ⊡ Figure  A comparison of the Bondarenko flux model (dashed) with a realistic computed flux (solid) for a U- oxide pin in water in the region of the . eV resonance    Neutron Slowing Down and Thermalization energy lost by the downscatter represented by the integral is small with respect to the width of a resonance, the integral can be replaced by Σ s (E)ϕ(E), and the equation becomes [Σ(E) − Σ s (E)]ϕ(E) = S  (E) . () The weighting flux would become W (E) = C(E) , σ + σ a (E) () where σ a is the absorption cross section. This is called the “wide resonance” (WR) or the “narrow resonance infinite mass” (NRIM) approximation. For resonances with widths intermediate to the NR and WR limits, we can change the formulas for the weighting flux and background cross section for for isotope i to C(E) i , () Wg (E) = i σP g + σ ai g and σPi g =  j j {σ e + ∑ ρ j λ g σ pg } . ρi j () Note that the sum is now over all j, and we have used the potential scattering cross section rather than the total cross section (which neglects the effect of scattering resonances on the weighting flux). If λ ig = , we get back the narrow resonance result because σ pi adds to σ ai g to return the total cross section (still neglecting resonance scattering). For λ ig = , we get the WR result. With intermediate values of λ, we get the “intermediate resonance” (IR) approximation. The equivalence principle is the same as for narrow resonances; namely, all systems with the same value of IR σPi g will have the same self-shielded cross sections for that isotope and group. WIMS takes the additional step of expressing the self-shielding data in terms of “resonance integrals,” instead of using the self-shielded cross sections produced by GROUPR. That is, σ x (σ ) = σP I x (σ P ) , σ P − I a (σ P ) () I x (σ P ) = σ P σ x (σ ) , σ P + σ a (σ ) () and where σ x (σ ) is a normal cross section as produced by the GROUPR module of NJOY, and x can stand for capture, fission, or nu-fission. In order to clarify the meaning of this pair of equations, consider a homogeneous mixture of U- and hydrogen with concentrations such that there are  barns of hydrogen scattering per atom of uranium. The GROUPR flux calculator can be used to solve for the flux in this mixture, and GROUPR can then calculate the corresponding absorption cross section for U-. Assuming that λ = . and σ p =  for the uranium, the numbers being appropriate for WIMS group , we get σP = . This value of σ P goes into the the WIMS library along with the corresponding I a . At some later time, a WIMS user runs a problem for a homogeneous mixture of U- and hydrogen that matches these specifications. WIMS will compute a value of σP of ., interpolate in the table of resonance integrals, and compute a new absorption cross section that is exactly equal to the accurate computed result from the original GROUPR flux-calculator run. Neutron Slowing Down and Thermalization  This argument can be extended to more complex systems. For example, the assembly calculated using the flux calculator could represent an enriched uranium-oxide fuel pin of a size typical of a user’s reactor system with a water moderator. The computed absorption cross section is converted to a resonance integral and stored with the computed value of σ P . In any later calculation that happens to mimic the same composition and geometry, WIMS will return the accurate calculated absorption cross section. Equivalence theory, with all its approximations, is only used to interpolate and extrapolate around these calculated values. This is a powerful approach, because it allows a user to optimize the library in order to obtain very accurate results for a limited range of systems without having to modify the methods used in the lattice-physics code. Let us call the homogeneous uranium–hydrogen case discussed above “case .” Now, consider a homogeneous mixture of U-, oxygen, and hydrogen. Arrange the ratio of the number densities to the uranium density such that there is  barn/atom of oxygen scattering and  barn/atom of hydrogen scattering. Carry out an accurate flux calculation for the mixture, and call the result “case .” Also do an accurate flux calculation with only hydrogen, but at a density corresponding to  barns/atom. Call this result “case .” The IR lambda value for oxygen is then given by λ= σ a () − σ a () . σ a () − σ a () () Note that λ will be  if the oxygen and hydrogen have exactly the same effect on the absorption cross section. In practice, λ = . for WIMS group  (which contains the large . eV resonance of U-), and λ =  for all the other resonance groups. That is, all the resonances above the . eV resonance are effectively narrow with respect to oxygen scattering. This process can be continued for additional admixed materials from each important range of atomic mass. The result is the table of λ g i values needed as WIMSR input. What are the implications of this discussion? The foremost is the observation that the lambda values for the isotopes are a function of the composition of the mixture that was used for the base calculation. To make the effect of this clear, let us consider two different types of cells: . A homogeneous mixture of U- and hydrogen . A homogeneous mixture of U- and hydrogen A look at the pointwise cross sections in group  shows very different pictures for the two uranium isotopes. The U- cross section has one large, fairly wide resonance at . eV, and the U- cross section has several narrower resonances scattered across the group. If the lambda values are computed for these two different situations, the results in > Table  are obtained. It is clear that the energy dependence of the two lambda sets is quite different. This is because of the difference in the resonance structure between U- and U-. Clearly, the one resonance in group  in U- is effectively wider than the group of resonances in group  for U-. Group  has essentially no resonance character for U-, which reverses the sense of the difference. In groups  and , the U- resonances become more narrow, while the U- resonances stay fairly wide. Finally, in group , the U- resonances begin to get narrower. These results imply that completely different sets of λ values should be used for different fuel/moderator systems, such as U-/water, U-/water, or U-/graphite. In practice, this is rarely done.    Neutron Slowing Down and Thermalization ⊡ Table  IR λ values for several resonance groups and two different reactor systems . WIMS group λ(U) U-@b λ(O) U-@b λ(U) U-@b λ(O) U-@b  . . . .  . . . .  . . . .  . . . .  . . . . Unresolved Resonance Range Methods In the unresolved resonance range, there are still significant fluctuations in the cross sections, but we do not know exactly where they are. However, it is possible to evaluate the average properties of the unresolved resonances. The ENDF format provides the average spacing D and the average widths for the elastic, capture, fission, and competitive reactions (Γ n , Γ γ , Γ f , and Γ x ). All of these can depend on E, J, and ℓ. The resonance spacing is normally assumed to be distributed with the Wigner distribution, and the resonance widths are distributed with chi-square distributions with degrees of freedom μ n , μ γ , μ f , and μ x , for the elastic, capture, fission, and competitive widths, respectively. These quantities are also functions of E, J, and ℓ. The infinitely dilute cross sections in the unresolved resonance region can be computed using these parameters in the SLBW method. σn (E) = σ p + σx (E) = gJ  π   [Γ n R n − Γ n sin ϕ ℓ ] , ∑  k ℓ,J D gJ π  ∑ Γn Γx R x  k ℓ,J D and σp = π  ∑(ℓ + ) sin ϕ ℓ , k ℓ () () () where σ p is the potential scattering cross section, the sin term is the interference correction, x stands for either fission or capture, Γ i and D are the appropriate average widths and spacing for the ℓ,J spin sequence, and R i is the fluctuation integral for the reaction and sequence. These integrals are simply the averages taken over the chi-square distributions specified in the file; for example, Γn Γ f R i = ⟨ Γn Γ f ⟩ Γ Neutron Slowing Down and Thermalization  = ∫ dx n Pμ (x n ) ∫ dx f Pν (x f ) ∫ dx c Pλ (x c ) × Γn (x n ) Γf (x f ) , Γn (x n ) + Γf (x f ) + Γγ + Γc (x c ) () where Pμ (x) is the chi-square distribution for μ degrees of freedom. The integrals are evaluated with the quadrature scheme developed by R. Hwang for the MC - code (Henryson et al. ) giving μ Q i Q νj μ R f = ∑ Wi ∑ Wjν ∑ Wkλ . () μ Γ n Q i + Γ f Q νj + Γγ + Γ c Q kλ i j k μ μ The Wi and Q i are the appropriate quadrature weights and values for μ degrees of freedom, and Γγ is assumed to be constant (many degrees of freedom). The competitive width Γ c is assumed to affect the fluctuations, but a corresponding cross section is not computed (we will discuss this in more detail below). It should be noted that the reduced average neutron width Γn is given in the evaluations, and √ Γ n = Γn E Vℓ (E), () where the penetrabilities for the unresolved region are defined as V = , ρ ,  + ρ () ρ . ρ + ρ  + ρ  () V = and V = () Other parameters are defined as for SLBW. In practice, this simple unresolved resonance representation can not always fit the experimental data with sufficient accuracy. For actinide fission, the double-humped fission barriers seen there lead to “secondary structure” in the cross sections that are not included in the simple theory. In addition, if the break from the resolved range to the unresolved range is not high enough, there can be semi-resolved resonances or clusters of resonances underlying the resonances that do not act like the simple theory predicts. Because of this, the ENDF format has an option to provide evaluated experimental values directly for the infinitely dilute unresolved resonance range cross sections (examples are U- and U-). In these cases, the simple unresolved resonance theory is only used to calculate self-shielding (see below). In other cases, the parameters of the theory are adjusted to try to give reasonable agreement to experiment, even for the features that are not consistent with the simple theory (e.g., Pu-). Both the infinitely dilute cross section and the self-shielding are computed using the simple theory. Both of these procedures leave much to be desired. However, the self-shielding effects of the unresolved energy range are quite small for thermal reactor systems and modest for fast reactor system; therefore, fairly large uncertainties can be accepted in the unresolved resonance range without serious impact on normal calculations. Work is underway in the nuclear-data community to improve this situation.    Neutron Slowing Down and Thermalization There are two methods in common use for calculating self-shielding effects in the unresolved resonance range. For multigroup codes, tables of self-shielded cross sections using the Bondarenko method are normally provided. For continuous-energy Monte Carlo codes like MCNP, probability tables are normally used. The PURR module of NJOY can be used to generate both types of data. This method is based on generating “ladders” of resonances using the statistical properties of the unresolved range. One ladder can be generated appropriate for an energy E by randomly selecting a starting resonance energy for one ℓ, J sequence, and also randomly selecting a set of widths for that resonance using the appropriate average widths and chi-square distribution functions. We can then select the next higher resonance energy by sampling from the Wigner distribution for resonance spacings, and we can choose a new set of widths for that resonance. The process is continued until we have a long ladder of resonances for that ℓ, J. We then repeat the process for the other ℓ, J sequences, each such sequence being uncorrelated in positions from the others. With this ladder of resonance parameters in place, PURR randomly samples energy values and computes the cross sections at those energy using the SLBW method. This random sampling is consistent with the narrow resonance approximation in which neutrons arrive without a knowledge of the local resonance structure. These cross sections can be used in formulas like () to obtain statistical estimates for the self-shielded cross sections to be used in multigroup codes. They can also be used to obtain a probability distribution function for the total cross section. This is done by accumulating the “hits” in a set of predefined total cross-section bins. This can be used to determine the probability that the total cross section will present a value in that bin’s range and the average total cross section for that range. At the same time, conditional averages can be accumulated for the elastic, capture, and fission reactions. A conditional average is the average value for the reaction over a bin when the associated total cross also falls in that bin. Once all the cross section samples have been processed for the ladder, a new ladder is constructed, and the entire process is repeated. A typical calculation uses  bins and  ladders. The structure of a probability table for one energy is shown in > Table . In MCNP, a random number is used with the cumulative probability to select a bin, and the resulting cross sections are returned to characterize the collision. Intermediate incident energies are obtained by interpolation. ⊡ Table  Contents of an unresolved resonance range probability table Lowest bin bound Upper bin bound for N bins Partial probability for N bins Cumulative probability for N bins Average total σ for N bins Average elastic σ for N bins Average fission σ for N bins Average capture σ for N bins Neutron Slowing Down and Thermalization  Probability 10 –1 10 –2 10 –3 10 –4 100 101 Total cross section (barns) 102 ⊡ Figure  The probability distribution for the total cross section at  keV (solid) and  keV (dashed) in the unresolved resonance range of U- The probability table can be used to generate a picture of the probability distribution for the total cross section as shown in > Fig. . This example is for U-. It demonstrates how the fluctuations get smaller as energy increases, which means that the self-shielding effect also gets smaller. The probability table can also be used to generate Bondarenko self-shielded cross sections as follows: Pi (E)σ x i (E) σ + σ ti (E) , Pi (E) ∑ i σ + σ ti ∑ σ x (E) = i () where x is t, n, f, or γ. An example of the self-shielded total cross section for the unresolved resonance range of U- is shown in > Fig. . In practice, a σ value of about  is characteristic of an oxide reactor pin, and one of  is appropriate for a very fast system with large amounts of U-. In U-, the threshold for the (n, n  ) inelastic level is . keV, well within the unresolved resonance range. The threshold for (n, n  ) is . keV, which is the upper limit of the unresolved resonance range. In this energy range, inelastic scattering competes with elastic scattering and capture – this is the origin of the “competitive width” mentioned above. In the current ENDF procedures, we are instructed to use the unshielded value of the inelastic cross section as given in the evaluation when constructing the cross section libraries. However, theory predicts that the inelastic cross section should have resonance fluctuations in this energy range just as the other channels do. Adding this self-shielding effect would affect the inelastic slowing down, and the nuclear data community is working to understand the effects of this. One processing system currently attempts to include this inelastic self-shielding effect (Sublet et al. ).   Neutron Slowing Down and Thermalization 15 Total cross section (barns)  14 13 12 11 105 Energy (eV) ⊡ Figure  Self-shielded unresolved resonance range total cross section for U- at σ values of infinity (solid),  (dotted), and  barn (dashed)  Time and Space in Slowing Down . Introduction For most reactor-physics applications, the equilibrium slowing down and thermalization discussed above is sufficient. However, to get a more complete understanding of the process, it is useful to consider time dependence. The MCNP Monte Carlo code can be used to do timedependent simulations; for example, what is the behavior of a pulse of neutrons introduced at time zero and position zero as time increases and the neutron flux spreads out from the origin. This is a problem that was analyzed in the past using eigenvalue–eigenfunction theory or age theory. . Time Dependence of the Energy Spectrum In this section, we introduce a pulse of neutrons at some initial energy at the center of a large sphere of a material and watch the evolution of the energy spectrum averaged over the entire assembly as a function of time. The first example is for graphite. The initial energy is taken to be  MeV to avoid the inelastic levels. The time-dependent flux contours are shown in > Fig. . At early times, there are complex transients resulting from the delta-function source interacting with the discontinuous slowing-down kernel. But as time increases, the flux peaks smooth out into Gaussian shapes that slow down with time with little change in shape. Note that the natural time unit for MCNP is the shake (− s). Neutron Slowing Down and Thermalization  If we do the same thing for heavy water, we get similar results (see > Fig. ). Note that the slowing down is faster for heavy water than it was for carbon, and the pulses shapes are slightly broader. This behavior of neutron pulses in large systems is the basis for the experimental technique called the “lead slowing-down spectrometer. ” Theory says that for large A, the width of the pulse √ (the dispersion) should vary as / A). > Figure  shows the slowing-down behavior in a very 25 *10 –9 E *flux(E ) 20 15 10 5 0 10 –5 10 –4 10 –3 10 –2 10 –1 100 Energy (MeV) ⊡ Figure  Neutron spectra for slowing down in carbon. The curves are for % time bins around times of , , , , , , , , and  shakes (one shake is − s) 6 *10 –9 5 E *flux(E ) 4 3 2 1 0 10 –5 10 –4 10 –3 10 –2 10 –1 100 Energy(eV) ⊡ Figure  Neutron spectra for slowing down in heavy water. The curves are for % time bins around times of , , , , , , , , and  shakes (one shake is − s)   Neutron Slowing Down and Thermalization 100 *10–9 80 E *flux(E )  60 40 20 0 10–6 10–5 10–4 10–3 Energy 10–2 10–1 100 ⊡ Figure  Neutron spectra for slowing down from a point source at  MeV in a large lead sphere. The curves correspond to % time bins around times of , , , , ,, ,, ,, ,, and , shakes (one shake is − s) large sphere of lead. Note that the pulses are narrower than they were for the lighter materials and that the slowing down is much slower. If you put a sample inside the block of lead, you can observe its response to different energies by observing it at different times. Theory says that the time for a given energy should vary like /v. > Figure  shows that the calculation shown in > Fig.  obeys this prediction. A classical experiment is the measurement of “slowing down time.” A pulse is introduced at high energies, and the time required to excite a known resonance in a sample is measured. We can simulate that using MCNP. The definition of “high energy” is ambiguous, but neutrons slow down very quickly near the source energy from inelastic scattering, so the effect of using different starting energies is not too serious. Looking at > Fig. , we can pick off the time associated with an energy of  eV and call that the slowing-down time. > Table  shows results for several materials from ENDF/B-VII. . Time Dependence of the Spatial Distribution To look at the spatial distribution of slowing down, we start a pulse of  MeV neutrons in the center of a large sphere of graphite and watch how the integrated flux spreads out with time. See > Fig. . This was an MCNP calculation using five million source particles. At early times, the distribution is still close to the origin, but as time increases it gradually gets broader. The neutrons act like they are diffusing outward. The outward diffusive current is strong at early times because of the large gradient in the flux. As the central flux is depleted, it gets smaller and flatter, and the outward current gets smaller. As a result, the central flux does not decrease as fast as time increases. Neutron Slowing Down and Thermalization  Time (shakes) 105 104 103 102 10–6 10–5 10–4 Energy (MeV) 10–3 10–2 ⊡ Figure  Time vs. peak energy for slowing down in lead (points). The solid curve is the theoretical /v dependence ⊡ Table  Slowing down time for several materials from ENDF/B-VII Material Slowing-down time (µs) Heavy water . Carbon . Sodium  Lead  At long times, the flux shape will approach the “fundamental mode” shape. For a spherical geometry, the shape would be πr  sin ( ) , r R̂ () where R̂ is the extrapolated endpoint radius just outside of the real radius R. For spheres that are large with respect to the mean free path, R̂ will be close to R. What does “long times” mean? > Figure  shows the final stages of the thermalization of the neutron pulse. The dotted line is the theoretical Maxwellian shape for an infinite medium with no absorption. It is clear that the computed flux in this graphite sphere is approaching the theoretical limit after about , µs. As we saw in > Table , the slowing down time to  eV for carbon is about  µs. The thermalization time is seen to be much larger than that. This is a consequence of the chemical binding of the carbon atoms in the graphite and the presence of upscatter. After about , µs,   Neutron Slowing Down and Thermalization 10–4 Flux 10–5 10–6 10–7 10–8 0 20 40 60 Radius (cm) 80 100 ⊡ Figure  Spatial dependence of slowing down in carbon. The curves are for % time bins around times of , , , , , and , shakes (one shake is − s) 10–7 10–8 E *flux(E )  10–9 10–10 10–9 10–8 10–7 Energy (MeV) 10–6 ⊡ Figure  Final thermalization of a pulse in a graphite sphere. The solid curves are for % time bins around times of ,, ,, ,, ,, and , shakes (one shake is − s). The dotted curve is a Maxwellian thermal spectrum Neutron Slowing Down and Thermalization  the spatial flux shape in the graphite sphere should have reached the fundamental mode shape shown in the equation above. . Eigenvalues and Eigenfunctions One theoretical approach to time-dependent slowing down and thermalization makes use of eigenvalues and eigenfunctions. In the diffusion approximation, the transport equation for a homogeneous medium with an isotropic source at energy E  and time t =  can be written as  ∂ϕ(E, r, t) + Σ a (E)ϕ(E, r, t) − D(E)∇ ϕ(E, r, t) v ∂t = S ϕ(E, r, t) + δ(E − E  )δ(t)Q(r) , () where Q(r) is the shape of the source, D(E) is the diffusion coefficient, Σ a is the macroscopic absorption cross section, and the scattering operator is given by S ϕ(E) = ∫ ∞  ′ ′ ′ Σ(E → E)ϕ(E ) dE − Σ s (E)ϕ(E) , () where Σ(E ′ → E) is the differential scattering cross section, and Σ s (E) is the integrated scattering cross section. Following the scheme described in (Williams ). We first separate out the spatial part by writing () ϕ(E, r, t) = ∑ ϕ n (E, t)Fn (r) . n The Fn are given by a Helmholtz equation ∇ Fn (r) + B n Fn (r) =  . () subject to appropriate boundary conditions at the edge of the system. The B n are spatial eigenvalues with associated spatial eigenfunctions Fn . We will give an example below. The ϕ n are solutions of  ∂ϕ n (E, t)  + Σ a (E)ϕ(E, t) − D(E)B n ϕ n (E, t) v ∂t = S ϕ n (E, t) + δ(E − E  )δ(t)Q n , () where Q n = ∫ dr Q(r)Fn r) . () We next assume that the flux can be divided into energy and time factors as follows: −λ n t ϕ n (E, t) = ϕ n (E) e . () The equation for the ϕ n becomes [− λn + D(E)B n ]ϕ n (E) = S ϕ n (E) + δ(E − E )δ(t)Q n . v ()    Neutron Slowing Down and Thermalization The homogeneous part of this provides another eigenvalue problem S ϕ n (E) = [− λn + D(E)B n ]ϕ n (E) . v () The eigenvalues of this equation (square brackets) correlate the spatial eigenvalues B n with the time-decay eigenvalues λ n . In principle, this eigenvalue problem can be attacked by expanding ϕ n and Q n using an appropriate set of orthonormal basis functions and solving for the eigenvalues and expansion coefficients. In practice, this is rather difficult in general. As we saw from > Fig. , the temporal eigenfunction corresponding to λ  would have to be similar to the Maxwellian distribution M(E), and higher eigenfunctions would have to represent the faster decaying Gaussian peaks from the slowing-down process. Similarly, the spatial eigenfunction corresponding to B  would be the fundamental mode shape, and the higher modes would correspond to higher values of λ n and would decay away faster. The eigenvalue–eigenfunction approach is useful in understanding how the temporal and spatial modes behave, but Monte Carlo and multigroup methods are much more useful in practice. As an example of the spatial eigenfunctions, consider the simple slab reactor of thickness T. A good set of basis functions is nπx ⎧ ⎪ , cos ⎪ ⎪ ⎪ T̂ ⎪ ⎪ ψ n (x) = ⎨ ⎪ nπx ⎪ ⎪ ⎪ , sin ⎪ ⎪ ⎩ T̂ n = , , , ... () n = , , , ... , and the corresponding eigenvalues are B n = ( nπ  ) , n = , , , .... T̂ () All these eigenfunctions are zero at the extrapolated half widths of the slab −T̂/ and +T̂/. The ψ  function is the fundamental mode, and it is nonzero over the entire slab. The ψ  function is asymmetric with a zero at the midpoint of the slab. The ψ  function is symmetric with two zeros. If one imagines starting with an initial source that is off center, the flux soon after the initial pulse would also be off center and would be represented by a Fourier series containing sine terms that reflects that. But these higher modes will have larger λ values and decay away faster than the lower modes. The spatial flux will gradually become smoother, and by the time that full thermalization is obtained, only the fundamental mode will remain. . Analytic Age Theory To understand age theory and other “continuous slowing down” methods, it is useful to write the P approximation to the transport equation in terms of the lethargy ∂ϕ  (x, u) + Σ t (x, u)ϕ  (x, u) = ∫ Σ s (x, u ′ → u)ϕ  (x, u ′ ) du ′ + Q  (x, u) ∂x ∂ϕ  (x, u) + Σ t (x, u)ϕ  (x, u) =  ∫ Σ s (x, u ′ → u)ϕ  (x, u ′ ) du ′ + Q  (x, u). ∂x () () Neutron Slowing Down and Thermalization  If A is not too small, in the energy range where elastic scattering dominates, and if the absorption is not too large, the collision density Σ s ϕ  tends to be slowly varying. It is reasonable to make Taylor expansions of the P and P collision densities, keeping two terms for the first and one for the second. Making the following definitions ′ ′ ∫ Σ s (x, u → u) du = Σ s (x, u) , ′ ′ ′ ∫ (u − u )Σ s (x, u → u) du = ξ(u)Σ s (x, u) , and ′ ′ ∫ Σ  (x, u → u) du = μ̄(u)Σ s (x, u) , () () () the P equations become ∂ϕ  ∂ + (Σ t − Σ s ) ϕ  = (ξΣ s ϕ  ) + Q  ∂x ∂u ∂ϕ  + (Σ t − μ̄Σ s )ϕ  = Q  . ∂x () () Taking Q  = , the second equation gives “Fick’s Law,” ψ = − ∂ϕ  ∂ϕ   = −D ,  (Σ t − μ̄Σ s ) ∂x ∂x () where D is the diffusion coefficient. Putting this into the first of the P equations gives the “agediffusion equation” − ∂ϕ  ∂ ∂ (D ) + (Σ t − Σ s )ϕ  = (ξΣ s ϕ  ) + Q  . ∂x ∂x ∂u () The quantity ξ is the average increase in lethargy for a neutron–nucleus collision, and μ̄ is the average cosine for scattering. For isotropic CM scattering in a single material, these two quantities take on the simple values ξ =+ μ̄ = α ln α −α ()  . A () The quantity ξΣ s ϕ  is called the “slowing-down density” and is usually denoted by q(x, u). If Q  =  and there is no absorption (Σ t = Σ s ), the age-diffusion equation is reduced to ∂  q(x, u) ∂q = , ∂x  ∂τ () where τ is called the Fermi age, τ(u) = ∫ u  D du ′ . ξΣ s ()    Neutron Slowing Down and Thermalization This is the “Fermi age equation” in plane geometry. Note that τ has the dimensions of square centimeters, strange for an “age.” It is really something like the square of the mean distance to a collision. An interesting variation on age theory comes about if we write the time-dependent and space-independent version for a pulsed source in the form  ∂ϕ(u, t) ∂ = − [ξΣ s ϕ(u, t)] + δ(u)δ(t) . v ∂t ∂u () This equation can be solved using a Laplace transform, giving ξΣ s (u)ϕ(u, t) = δ (t − ∫ u   v ′ ξΣ s (u ′) du ′ ) , () that is, the pulse retains its original shape and slows down with time. We saw this behavior in > Fig. . Age theory is mostly of historical significance in these days of fast computers. It does not work for hydrogen, and it is weak for other light moderators, like heavy water and beryllium. There are other slightly more complicated continuous slowing-down models, such as the Selengut–Goertzel and Greuling–Goertzel methods, that have found some usefulness in fast-reactor analysis.  Concluding Remarks and Outlook This chapter opened with a discussion on how to generate scattering cross sections in the thermal range with the S(α, β, T) representation. Experience has shown that this approach works well for calculating water–uranium critical systems (Tuli et al. ). However, > Figs.  and > Figure  suggest that further improvements for H in H O are possible, especially for the diffusive representation at low neutron energies. The region from . to . eV is also of concern. Some correspondents have been concerned about the effects of the break between the S(α, β, T) representation at low energies and the epithermal treatment above. The discontinuity at the breakpoint looks nonphysical. Current methods use breakpoints all the way from . to  eV. It is not clear whether the S(α, β, T) representation works for energies as high as  eV. At some point, anharmonic effects should begin to show up. At higher energies, bond breaking and atomic displacements could even occur. The current experimental data are not good enough to resolve these questions. Additional high-accuracy differential and integrated experimental data could drive improvements for the water scattering data. > Figure  showed some improvement in matching experimental data for heavy water coming from the treatment of intermolecular interference, but there are still problems around  meV and in the region from . to . eV that suggest that further physics improvements are needed. There may also be problems with the temperature dependence of the interference effect coming from the static structure factor (> Fig. ). There are only a few good heavy-water critical assemblies available for test calculations, so the confidence we have for water–uranium systems is not available for heavy-water systems. A shortage of reliable critical experiments also exists for other moderator materials, including beryllium and graphite. Neutron Slowing Down and Thermalization  A key problem with the S(α, β, T) approach is that it only works for materials for which the free-scattering cross section is constant. This condition is well satisfied for the light materials like water, heavy water, and graphite. However, it certainly breaks down for uranium oxide. New methods would be required to extend the thermal-scattering treatment for U- to  eV, spanning the important . eV resonance. Some work has been done on this in recent years, but it has not been reduced to common practice as yet. Finally, from time to time, people have expressed the desire for S(α, β, T) data for additional materials. These evaluations are fairly difficult, and there are few people with experience in the field. Progress is slow. The second section discussed thermalization. There are several computer code systems available that treat this well, as demonstrated by the good results obtained for many thermal critical assemblies during ENDF/B-VII data testing (Tuli et al. ). The Monte Carlo approach works best here because of its good treatments of the complex geometry features of many of the critical experiments. The third section discussed steady-state slowing down. At higher energies and for lighter isotopes, both Monte Carlo and S N methods are capable of calculating slowing down well. The limitations come from the nuclear data. As we saw from > Fig. , the slowing-down spectra are sensitive to the angular distributions. These are not always as good as we would like for current evaluations. During the development of the ENDF/B-VII data for U-, good improvement in matching critical-experiment data for both lattice experiments and fast-reflected criticals was obtained, and part of this improvement came from using the best current nuclear models for the elastic angular distributions. The agreement between calculation and experiment is not that good for fast-reflected critical assemblies with other reflectors (lead, iron, nickel, and copper). This suggests that the angular distributions for scattering could be improved for other materials, resulting in better slowing-down calculations. If there are resonances in the higher energy part of the slowing-down range, a more complete representation of the change in the angular distributions across the resonances, such as the variations shown in > Fig. , could be helpful. This would require more extensive use of the RML resonance representation and enhancements in data-processing and transport codes to make use of these data. Another data issue for slowing down comes from the representation of continuum scattering reactions, such as (n, n′c ) and (n, n). These data are often obtained from nuclear model calculations, which normally produce histogram representations of the scattering. In current evaluations, these √ histogram representations are often fairly coarse and do not do a good job of simulating the E dependence of these distributions at low energies. See > Fig. . This histogram issue also comes up for low-energy delayed-neutron spectra. For the heavier isotopes, the slowing down eventually passes into the unresolved resonance range where reactions are treated statistically. Here, the cross sections are treated to include the temperature, but the downscatter shapes are not. Some correspondents have questioned the reliance on the Single-Level Breit–Wigner representation currently used. A multilevel treatment of scattering might be needed. A more complicated resonance representation might be necessary to handle the secondary structure coming from the double-humped fission barriers. In most current methods, self-shielding effects are omitted for the slowing down from inelastic scattering in the unresolved range. In the resolved resonance range for the heavier isotopes, Doppler broadening of the resonances becomes important. We currently handle that well for the integrated cross sections, but the downscatter shapes are taken from the target-at-rest representation. In addition, we do not normally treat the angular distributions for resonance scattering in detail. The technical    Neutron Slowing Down and Thermalization problems with properly Doppler-broadening scattering distributions and angular distibution are very difficult. The current methods work reasonably well for most reactor problems. Resolved resonance self-shielding is handled in good detail by the continuous-energy Monte Carlo codes, but with less accuracy by codes that depend on multigroup self-shielding factors. The discussion of time-dependent slowing down concentrated on providing some understanding of the effects using Monte Carlo examples. The theoretical methods presented help to provide some of this understanding, but they are of mostly historical interest. They are not really used for modern reactor calculations. In summary, the data and methods used to describe neutron slowing down and thermalization are in fairly good shape for reactor calculations, but there are improvements that we can look forward to in future years. References Alcouffe RE, Baker RS, Dahl JA, Turner SA, Ward RC (Rev. May ) PARTISN: a time-dependent, parallel neutron particle transport code system, Los Alamos National Laboratory report LA-UR-. Askew JR, Fayers FJ, Kemshell PB () A general description of the lattice code WIMS. J Brit Nucl Energy Soc :. Bell GI, Glasstone S () Nuclear reactor theory. Van Nostrand Reinhold Company, New York Bondarenko II (ed) () Group constants for nuclear reactor calculations. Consultants Bureau, New York. Briggs JB et al () International handbook of evaluated criticality safety benchmark experiments. Tech. Rep. NEA/NSC/DOC()/I, Nuclear Energy Agency, Paris Butland AT () LEAP and ADDELT, a users guide to two complementary codes on the ICL- for calculating the scattering law from a phonon frequency function. Atomic Energy Establishment Winfrith report AEEW-M- Cullen DE (November ) TART : a coupled neutron-photon -D, time dependent, combinatorial geometry monte carlo transport code. Lawrence Livermore National Laboratory report UCRL-SM- Cullen DE () PREPRO :  ENDF/B pre-processing codes. IAEA-NDS-, Rev. . Nuclear Data Section, International Atomic Energy Agency, Vienna Herman M, (ed) () Data Formats and Procedures for the Evaluated Nuclear Data File ENDF/B-VI and ENDF/B-VII, Brookhaven National Laboratory report BNL-NCS---Rev (ENDF) Henryson H II, Toppel BJ, Stenberg CG () MC: a code to calculate fast neutron spectra and multigroup cross sections. Argonne National Laboratory report ANL- (ENDF-) Honek H () THERMOS, a thermalization transport theory code for reactor lattice calculations. Brookhaven National Laboratory report  Koppel JU, Houston DH (July ) Reference manual for ENDF thermal neutron scattering data. General Atomic report GA- revised and reissued as ENDF- by the National Nuclear Data Center at the Brookhaven National Laboratory Koppel JU, Triplett JR, Naliboff YD (March ) GASKET: a unified code for thermal neutron scattering. General Atomics report GA- (Rev.) Larson NM (October ) Updated Users’ guide for SAMMY: multilevel R-matrix fits to neutron data using Bayes’ equations. Oak Ridge National Laboratory report ORNL/TM-/R Lathrop, KD (November ) DTF-IV, a FORTRAN program for solving the multigroup transport equation with anisotropic scattering. Los Alamos Scientific Laboratory report LA- MacFarlane RE (September ) ENDF/B-IV and -V Cross sections for thermal power reactor analysis. In: Proceedings international conference of nuclear cross sections for technology, Knoxville, October –, , National Bureau of Standards Publication  MacFarlane RE (July ) TRANSX: a code for interfacing MATXS cross-section libraries to nuclear transport codes. Los Alamos National Laboratory report LA--MS Neutron Slowing Down and Thermalization MacFarlane RE (March ) New thermal neutron scattering files for ENDF/B-VI release . Los Alamos National Laboratory report LA-MS MacFarlane RE, Muir DW () The NJOY nuclear data processing system, version , Los Alamos National Laboratory report LA--M Mattes M, Keinert J () Thermal neutron scattering data for the moderator materials HO, DO and ZrHx in ENDF- format and as ACE library for MCNP(X) codes. International Nuclear Data Committee report INDC(NDS), April . X- Monte Carlo Team () MCNP—A General Monte Carlo N-Particle Transport Code, Version , Los Alamos National Laboratory report LA-UR-- (April )  Skold K () Small energy transfer scattering of cold neutrons from liquid argon. Phys Rev Lett : Sublet J-Ch, Ribon P, Coste-Delcalaux M () CALENDF-: user manual. CEA report CEAR- Tuli JK, Oblizinsky P, Herman M (eds) (December ) Special issue on evaluated nuclear data file ENDF/B-VII.. Nucl Dat Sheets ():–  Williams MMR () The slowing down and thermalization of neutrons. North-Holland Publishing Company/Wiley, Amsterdam/New York Wycoff RWG () Crystal Structures. Wiley, New York/London   Nuclear Data Preparation Dermott E. Cullen Lawrence Livermore National Laboratory, Livermore, CA, USA redcullen@comcast.net  . . .. . . . . . .. .. .. . .. .. .. . Overview . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The ENDF/B Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . ENDF/B Tables and Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Importance of Nuclear Data-Processing Codes. . . . . . . . . . . . .. . . . . . . . . . . . . . . First-Order Approximations: Space, Energy, and Time . . . . . . . . .. . . . . . . . . . . . . . . Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Species of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Evaluated Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Neutron-Interaction Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Neutron-Induced Photon Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Photon Interaction Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Approximate Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Monte Carlo Versus Deterministic Codes . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Continuous Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Multigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .                   . . . .. . .. .. .. .. Reconstruction of Energy-Dependent Cross Sections . . . .. . . .. . . .. . . .. . . .. . . Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Representation of Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Tabulated Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Linearized Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Reconstructing the Contribution of Resonances . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Resolved-Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Unresolved-Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Adding Resonance and Background Cross Sections . . . . . . . . . . . .. . . . . . . . . . . . . . . Output Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .            . .. . .. . .. .. .. Doppler Broadening . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . What Causes Doppler Broadening?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . The Doppler-Broadening Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Mathematical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Kernel Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Tabulated Broadened Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . TEMPO and Psi–Chi Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .          Dan Gabriel Cacuci (ed.), Handbook of Nuclear Engineering, DOI ./----_, © Springer Science+Business Media LLC    Nuclear Data Preparation .. . .. .. .. Mathematical Comparisons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low Energies . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High Energies .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       . . .. .. .. . . .. .. .. .. . Self-Shielding . . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Introduction. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Narrow, Intermediate, and Wide Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Narrow Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wide Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermediate Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-section Dependence of Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computation of Multigroup Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tabulated Cross Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linearly Interpolable Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .               . . .. .. .. .. .. .. Transfer Matrix . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Introduction. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncorrelated Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angular Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlated Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution of the Inner Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal-Scattering Law Data: S(α, β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           . . Group Collapse . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Introduction. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noncoincident-Group Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     . . . . .. .. .. . . . .. .. .. .. The Multiband Method . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . Introduction. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiband Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiband Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solution for Band Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Solution for Two Bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalization to N Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Many Bands are Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bramblett–Czirr Plate Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Criticality Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shielding Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .                Nuclear Data Preparation .. .  Fusion Reactor Blanket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .   References . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . .. . . .. . .     Nuclear Data Preparation Abstract: Today, new evaluated data are almost always prepared in the now universally accepted ENDF/B format. Between the originally evaluated data as coded in the ENDF/B format and our particle transport codes, which actually use the evaluated data, are the often overlooked data-processing codes. These data-processing codes translate and manipulate the data from the single universal ENDF/B format to a variety of formats used by our individual particle transport codes, that is, in contrast to our universally accepted evaluated data format, ENDF/B, as yet there is no universally accepted format used by all of our application codes. This chapter covers in detail the work done by our data-processing codes to prepare the evaluated data for use in our applications: this includes reconstructing energy-dependent cross sections from resonance parameters, Doppler broadening to a variety of temperatures encountered in real systems, defining data for use in both continuous energy Monte Carlo codes, as well as multigroup Monte Carlo and deterministic methods codes. In this chapter, both WHAT needs to be done by our data-processing codes and WHY have been defined; also, the overall perspective of a general plan, “The Big Picture,” for the historical and current development of the methods used over the last half century as well as today, has been given. The importance that processing code verification projects have played over the past decades as well as today has been stressed here. It should be remembered that computer codes have always been very complicated, and it is almost impossible to verify the results calculated by any one code without any comparison with one or more other independently developed codes. A classic mistake is to assume that checking the results will impede progress, whereas in fact experience has shown that taking the time to verify results can actually in the long run lead to savings in time and major improvements in the reliability of our codes.  Overview . Introduction The chapter on Nuclear Data Preparation for the CRC Handbook of Nuclear Reactors Calculations (Cullen ) was written by me in ; the present chapter is an update of this earlier work. When this earlier work was reread now, what was surprising is that even after  years only a very little has changed. Most of the methods described here are the same that we used back then. The big differences between the Nuclear Data Preparation then and now are mostly due to the enormous advances that have occurred in computer size and power since then. This now allows us to routinely do things in a few minutes that  years ago took us days or weeks or was simply not feasible at all. Closely related are the advances in nuclear model codes, which have resulted in current nuclear data that are much more detailed, and advances in particle transport codes that are capable of using this much more detailed data. Standing as an interface between the evaluators and their evaluations, on the one hand, and the transport codes, on the other hand, are our nuclear data-processing codes; these dataprocessing codes have also made great strides to take advantage of the enormous increase in computer size and power, and to accommodate changes in modern evaluations and their use in our transport codes. In order to document the historical development of the methods that we use today, references to the methods as they were originally developed have been retained; Nuclear Data Preparation  these are references: Weinberg and Wigner (), Glasstone and Edlund (), Kinsey (), Howerton et al. (, ), Igarasi et al. (), Pope (), Goel and Krieg (), Kolesave and Nikolaev (), Garber and Brewster (), Plechaty et al. (a,b), Alder et al. (), Cashwell et al. (), Lichtenstein et al. (), Plechaty and Kimlinger (), Emmett (), Engle (), Mynatt et al. (), Lathrop and Brinkley (), Hardie and Little (), Fowler et al. (), Cullen (, , a,b, a,b, a–c, a–c), Cashwell and Everett (), Seamon (), Cullen et al. (, , , , , a,b), Garber and Kinsey (), Ozer (), MacFarlane et al. (), Weisbin et al. (, ), Rieffe et al. (), Panini (), Vertes (), Pope et al. (in press), Greene et al. (), Lamb (), Wigner and Wilkins (), Meghreblian and Holmes (), Friedman (), Breit and Wigner (), Gregson and James (), Hutchins et al. (), Green and Pitterle (), Patrick (), Mughabghab et al. (), Gyulassy et al. (), Jeans (), Toppel (), Joanou et al. (), Goldstein and Cohen (), Goldstein (, ), Abramowitz and Stegun (), Doyas et al. (), Lathrop (), Bondarenko et al. (), Shakespeare (), Weisbin and LaBauve (), Zijp et al. (), Konshin (), Ganesan (), Ganesan et al. (), Perez et al. (), Bucholhz (), Westfall (), MacFarlane and Boicourt (), Perkins (), Greene (), Hong and Shultis (), Nikolaev and Phillipov (), Stewart (), Levitt (), Cullen and Pomraning (), Plechaty and Kimlinger (in press), GravesMorris (), Askew et al. (), Bremblett and Czirr (), Czirr and Bramblett (), Lewis and Soran (), Plechaty (), and updated references starting with Cullen () have also been added by me. It is hoped that this allows us to preserve the historical record, and much of this chapter presents details of methods used in today’s data-processing codes,ll in particular the ENDF/B preprocessing codes – PREPRO (Cullen a–c) and NJOY (MacFarlane and Muir ), as they exist today. (Note that the SIGMA convention of extending cross sections outside of their tabulated range as /V differs from the original convention of extending them as constant. The /V convention is an improvement based upon accumulated years of experience using the SIGMA method; This result indicates that to minimize the error in the contribution from any √ such√as a resonance peak energy, ER , the Doppler √given energy, width should be defined at E  = / ( E R + E), not simply ER; The Joint Evaluated File (JEF), to be distributed by the NEA-Data Bank, Saclay, on behalf of the contributing OECD countries.) This Handbook of Nuclear Energy will present a variety of methods that are currently used to solve nuclear fission-related problems, with emphasis on reactor core problems. The solution of these problems involves describing the transport of neutrons, photons, and charged particles through matter and the interaction of these “particles” with matter. Throughout the following discussion, for simplicity, neutrons, photons, and charged particles will collectively be referred to as particles. In this handbook, there will be a great deal of discussion concerning solution of the linearized Boltzmann equation in order to determine the distribution of particles or flux in space, direction, energy, and time. For time-dependent problems, we may also have to solve a related system of equations to describe the change in composition of the medium, due to burn-up and radioactive decay. Throughout this discussion, it is important to remember that generally the determination of the distribution of particles or flux is a means to an end, rather than an end itself. That is to say, generally, what we are interested in describing is some effect caused by the interaction of the particles with the medium through which they are transported. For example, in reactor core calculations, we are interested in determining the reaction rates for    Nuclear Data Preparation individual reactions and leakage from the core. These may in turn be used to determine a variety of quantities or properties, including . The static and dynamic multiplication factor of the system, which will allow us to determine whether or not the system is statically stable (i.e., is critical) and whether or not the system is dynamically stable (e.g., has a negative Doppler coefficient). . The energy deposition rate, which will allow us to determine the amount of power generated and the temperature distribution throughout the system. This information can be used to determine how much power a reactor can safely generate and temperature-dependent effects within the reactor. . The rate of disappearance or build-up of materials within the reactor, which will allow us to determine the time-dependent composition of the system. This information can be used to establish a fuel management program and will allow us to determine the radiation damage to the reactor, e.g., gas production, atom displacement, etc. Any number of examples of such kind may be given. The important point to understand is that none of these quantities can be defined directly in terms of the distribution of particles, such as the neutron flux; so it is not sufficient to merely solve the linearized Boltzmann equation to define the flux. Again, let us stress that knowing the flux distribution is merely a means to an end. All the effects we are really interested in depend on nuclear data. Even if one could somehow exactly calculate the distribution of particles or flux, without any adequate nuclear data, it would still not be possible to determine the effects that we are really interested in. The second point is that in discussing the nuclear data we must consider not only the data that are used directly to determine the distribution of particles, but also the data that are then subsequently used in conjunction with the determined distribution of particles or flux to define the quantities that we are really interested in, e.g., energy deposition, gas production, and many other quantities as explained later in this chapter. The third point is that throughout this book many methods and approximations will be introduced; and in all cases, these methods and approximations will attempt to conserve some basic properties of the Boltzmann equation. However, since we are interested in effects, as opposed to determining the particle distributions alone, we will attempt to conserve reactions, rather than flux or cross section. In principle, reactions are a physical observable that we can directly relate to what is happening in any system. . The ENDF/B Format As of the mid-s there were many nuclear data libraries; basically, each laboratory had it own data library, in its own format, for its own computer codes. Each library was designed to get the “best” answers using the laboratories’ own computer codes, for the specific applications that each laboratory was interested in. This often required nonphysical “fixes” or fits to force agreement between computer code results and the laboratory’s finished applications. By the mid-s an effort had begun to establish a nuclear data library, containing evaluated data based solely on the “best” available differential measurements and nuclear model calculations. It was hoped that one common library of data could be universally adopted for use throughout the world. Unfortunately, this goal has not been achieved yet. Even today we have a variety of nuclear data libraries, to name a few: ENDF/B-VII in the United States, JEF in the European Community, JENDL in Japan, CENDL in China, and BROND in Russia. Nuclear Data Preparation  Even over  years after the start of the ENDF/B project we still have significant differences between the important evaluations in these various data libraries; these differences are often based on valid, scientifically based judgments that one set of measured data or one nuclear model calculation is better than another; unfortunately, the differences can also be based on national “pride and prejudice”; simply, the human nature that we cannot seem to avoid. Even though the ENDF/B effort has not led to a common set of evaluated nuclear data, it has led to a common universally accepted format for the data; e.g., all the nuclear data libraries mentioned earlier: ENDF/B-VII, JEF, JENDL, CENDL, and BROAD have all adopted the ENDF/B format, which is now in its sixth version and is named ENDF/B- (Kinsey ; Rose and Dunford ). This common format is a great step forward, because it allows us to more easily compare data from the various libraries, and has also greatly reduced the effort on the nuclear data-processing codes. Compared to the earlier situation wherein each laboratory had to develop its own codes to process its own nuclear data, in its own computer-based format, today there are only a few nuclear data-processing code systems that service the needs of the entire international nuclear community. As today the ENDF/B- format is universally used, it simplifies the writing of this chapter; because the focus can be on this single format and the few nuclear data-processing code systems that are used today are being focused. The latest ENDF/B-VII. data (Oblozinsky et al. ), which are now freely available online (Cullen a–c), are also scrutinized. .. ENDF/B Tables and Interpolation Most of the nuclear data contained in the ENDF/B system are in the form of tabulated values with an interpolation law defining how to interpolate between the tabulated values. Since we are interested in the integral results, these interpolation laws are very important in order to uniquely define our nuclear data at all energies; and not just the energies at which the data are tabulated. The interpolation laws allowed in ENDF/B include histogram, linear or log in the x and y dimensions (five basic interpolation laws), and additional laws for special cases such as charged-particle thresholds. These interpolation laws are very useful during the evaluation process; e.g., low-energy capture and fission cross sections tend to vary as /V, which can be exactly defined using log–log interpolation. Similarly, spectra often have analytical forms such as sqrt(E) at low energy and exponential at high energy, which can be defined exactly using nonlinear interpolation laws. For these reasons, we would like to maintain these quite general interpolation laws for use during evaluation and presentation of original evaluations in the ENDF/B format. But these interpolation laws present problems when we want to use the evaluated data in the ENDF/B format. The most obvious problem is that the sum (redundant) cross sections, such as the total, cannot be exactly defined using any of the ENDF/B interpolation laws, except linear–linear. Less obvious is that although the nonlinear interpolation laws can be used to analytically define integrals of the data, these integrals are often numerically unstable in ways that are difficult to detect, and can cause errors in our calculated results. Soon after my work got started at Brookhaven National Laboratory in , over  years ago, it was realized that the interpolation laws defined for use with ENDF/B were a great advantage to allow evaluators to use analytical shapes for cross sections and secondary-particle    Nuclear Data Preparation distributions. For example, at low energies both capture and fission cross sections are /V in shape, which can be exactly defined using log–log interpolation; similarly, at low energy many secondary-neutron distributions tend to vary as sqrt(E), which can also be defined using log–log interpolation. However, at the same time, the flexibility of these interpolation laws has a disadvantage when these evaluations are used in applications. The most obvious disadvantage being that it can cause inconsistency in the cross sections. Subsequently, we will see an example wherein the tabulated total cross section is not the sum of its parts, solely because of the use, or in this case, misuse, of nonlinear interpolation. To allow for both the advantages to evaluators and avoid problems during the use of the evaluations, it was realized that the solution was to provide computer codes to allow the evaluated data in the ENDF/B format to be converted from their original nonlinear-tabulated form to linearly interpolable tabulated form, to within any accuracy needed for use in applications; these codes could be used by evaluators to insure that their evaluations are consistent, and also by evaluated data users to insure that whether or not the original evaluator data are consistent, what they use in their applications are consistent. To meet this need, a series of codes were worked out and they are all included in the ENDF/B Pre-processing codes: PREPRO (Cullen a–c, a–c): () LINEAR, to convert tabulatedll data to linearly interpolable form, () RECENT, to reconstruct linearly interpolable cross sections from resonance parameters, () SIGMA, the sigma method to Doppler-broaden linearly interpolable cross sections was invented by me, () FIXUP, to define the redundant cross sections, such as total, by summation. These codes were required to make linearly interpolable cross sections available to ENDF/B-formatted evaluations. To complete the definition of cross sections, the multiband method was developed by me to handle the unresolved-resonance region. Finally to handle secondary distributions, () LEGEND, to linearize angular distributions, and () SPECTRA, to linearize energy distributions were added. As these codes were made available they were adopted by MINX (Weisbin et al. ), and then inherited by MINX’s successor NJOY (MacFarlane et al. ; MacFarlane and Muir ); thus we now have several code systems that use this linearized data concept. Initially, there was a great deal of resistance to the idea of replacing an “exact” cross section, such as /V at low energy by a linearized “approximation.” But eventually, data users accepted the idea as they realized that it was needed for them to be able to have consistent data for use in their applications. Today, nobody seems to question this approach, and it is widely used in our nuclear data-processing codes and our neutron transport codes. This linearizing is now such an integral part of our codes that much of this chapter is devoted to how it is used to reconstruct cross sections from resonance parameters, to Doppler-broaden cross sections, and to calculate multigroup constants. Development of these codes has taken many years to complete; but today, we have complete systems that are freely available to data users, which include the PREPRO (Cullen a–c) and NJOY (MacFarlane and Muir ). Much of this chapter is designed to document what these codes do, and of at least equal importance, why they do it. As successful as these efforts have been we must accept the fact that it is time to pass the torch to the next generation of nuclear data-processing code designers, wherein we will change from our traditional FORTRAN codes to a new generation of C and other language computer codes. Hopefully, this chapter will help make this transition a smooth one by allowing the new generation to understand not only what we did, but also why. Nuclear Data Preparation .  The Importance of Nuclear Data-Processing Codes It is now common practice for transport code users to define the transport codes and nuclear data they are using, as when we say we used MCNP (Cashwell et al. ; X- Monte Carlo Team ) and the ENDF/B-VII. nuclear data (Oblozinsky et al. ; Cullen a–c). This ignores a very important step, namely the nuclear data-processing code; this is what we call the overlooked, but often limiting factor. For example, in addition to telling users that we used MCNP and ENDF/B-VII., it is important for users to know that we used the NJOY data-processing system (MacFarlane and Muir ), and also know what versions of MCNP and NJOY were used. Only then would we be able to describe how our calculations were done. Our recent studies (Cullen et al. , a–c) demonstrate that our transport codes arel now very accurate, and many of the remaining limitations in the accuracy of our solutions can be traced not to transport, or even the nuclear data, but rather to approximations introduced by our data-processing codes. This suggests that rather than being the overlooked factor, more emphasis should be placed on improving the accuracy and reliability of our nuclear dataprocessing codes. In this chapter, we will attempt to clearly define the approximations that are being introduced by our data-processing codes. . First-Order Approximations: Space, Energy, and Time Even today, with all the available computer size and speed it is still not possible for us to exactly solve the linearized Boltzmann equation. Today’s methods still include approximations needed to allow us to solve our problems within sufficient accuracy and in a timely manner to meet our programmatic needs. The most obvious, first-order approximations, is that we do not attempt to define solutions on a continuous spatial and energy basis. Instead, we try to accurately define “average” values; averaged over spatial zones, over-energy groups, and sometimes over time intervals. In terms of our applications wherein we are interested in physical observables, if we can accurately define averages these are usually good enough to meet our programmatic needs. Beyond these first-order approximations, there are other approximations that are related to the accuracy of our nuclear data, and how accurately our nuclear data can be processed into a form that it can be used by our transport codes; this includes multigroup cross sections and groupto-group transfer matrices, which are also averaged over energy ranges and spatial zones and time intervals. . Basic Equations In this handbook, we will be discussing how to solve the linearized Boltzmann equation, which in its energy- and time-dependent form can be written as (),  →   ∂ → N(r, Ω, E, t) + Ω ∗ ∇ N(r, Ω, E, t) + Σ t (r, E, t)N(r, Ω, E, t) v ∂t ∞  dE ′ ∫ dΩ′ Σ(r, E ′− >E, Ω ′ − >Ω)N(r, Ω ′, E ′ , t) + S(r, Ω, E, t), = π ∫ ′  Ω ()    Nuclear Data Preparation where N(r, Ω, E, t) neutron flux per unit volume, energy, and solid angle at time t. v is the neutron speed (not, velocity). Σ t (r, E, t) total macroscopic cross section at location r and time t for a particle of energy E. Generally, the macroscopic cross sections will be spatially dependent since different materials will be used at different positions (e.g., core vs. shield) and time dependent because of burn-up. Σ(r, E ′ − >E, Ω ′ − >Ω) differential cross section, describing the transfer of particles with initial coordinates E ′ , Ω ′ before the interaction to E, Ω after the interaction. Written in this form, it includes the effect of all possible processes, e.g., scatter, fission, (n, n), etc. S(r, Ω, E, t) flux-independent neutron source. The differential cross section can be written in terms of the contributions from the individual reactions in the form ′ ′ ′ ′ ′ ′ Σ(r, E − >E, Ω − >Ω) = ∑ M k (E )Σ k (r, E )Pk (E − >E, Ω − >Ω), () k where the summation is over reactions k, e.g., k = elastic, fission, etc., and M k (E ′ ) Multiplicity or average number of secondary neutrons, e.g.,  for elastic,  for (n, n), and ν(E ′ ) for fission. Σ k (r, E ′ ) Reaction cross section for process k. Pk (E ′ − >E, Ω ′ − >Ω) Probability distribution for process k, describing the transfer of particles with initial coordinates E ′ , Ω ′ before the interaction to E, Ω after the interaction. This is a normalized distribution which is equal to unity when integrated over all final E, Ω. For the linearized Boltzmann as written previously, everything is assumed to happen instantaneously at time t, at a given spatial location r. If we consider delayed neutrons, the equation is further complicated by an additional integral over all earlier times t ′ ; for simplicity, this complication will not be included in the following equations. If we are to consider the changes in composition due to burn-up and/or radioactive decay, we must consider the coupled set of equations describing the changes in the composition for each constituent material N j (r, t) as a function of position and time. dA j (r, t) = −[R j (r, t) + λ j ]A j (r, t) + ∑ [R j′ (r, t)α( j′ − > j) + β( j′ − > j)]A j′ (r, t) dt j′ and ∞ R j (r,t) = ∫ dEσ j (E)N  (r, E, t), ()  where A j (r, t) Atoms of nuclide j λ j Decay constant of nuclide j σ j (E) Microscopic cross section of nuclide j α( j′ − > j) Probability that an interaction with a nuclide j′ atom will create a nuclide j atom Nuclear Data Preparation  β( j′ − > j) Probability that a decay of a nuclide j′ atom will create a nuclide j atom N  (r, E, t) Scalar neutron flux (integrated over direction Ω) R j (r, t) Reaction rate of nuclide j at (r, t) . Species of Particles Within a fission reactor, each fission event results in the release of approximately – MeV of energy. This energy is distributed approximately as follows (), Kinetic energy of fission production Beta decay energy Gamma decay energy Neutrino energy Fission neutron energy Instantaneous gamma ray energy Total  MeV       MeV In addition to the energy released in fission, energy is also released because of exoergic reactions, such as (n, p). Finally, the energy of the neutron as it slows down is distributed to other species of particles because of photon production from inelastic scatter and nuclear recoil due to scatter. Since energy within the reactor core is distributed between neutrons, photons, and charged particles, in principle, there will be similar equations for each species and these equations will be coupled, since one species of particles can produce particles of a different species, e.g., photon production due to neutron capture, and photonuclear neutrons due to photon interactions. However, as applied to nuclear-fission reactor cores, the only transport calculations that will be considered here will be transport of neutrons and photons. The mean-free path of fission products and charged particles is short enough that they may be considered to come to rest and deposit their energy at the point at which they are “created” or emitted by a nucleus, and their transport need not be considered at all. Although the transport of charged particles be ignored, the production of charged particles will be considered, as it is important to consider the production of hydrogen and helium gas due to proton, deuteron, triton, He , and alpha emission. The production of such gases can be determined directly from a known (previously calculated) distribution of neutrons, since evaluated data libraries now contain hydrogen- and helium-production cross sections (Kinsey ; Rose and Dunford ). Similarly, effects such as heat production (KERMA) and displacement production (DPA) can be calculated. From a known neutron distribution and production cross section, the production of gas of type x can be simply calculated from ∞ dG X (r, t) = ∫ dEΣ X (r, E, t)N  (r, E, t), dt  where dG X (r,t) dt Production rate of gas of type x Σ X (r, E, t) Macroscopic production cross section of gas x N  (r, E, t) Scalar flux (integrated over direction Ω) ()    Nuclear Data Preparation The transport of photons is important in reactor core calculations. Since photons tend to transport longer distances than neutrons, the photons tend to have a smoothing influence on energy deposition, by depositing their energy over a wider spatial region. In principle, neutron production due to photon interaction (photonuclear reactions) should also be considered; there are a number of situations wherein such production is important, e.g., systems containing appreciable quantities of beryllium. However, coupling of the neutron and photon transport equations in this way complicates the solution of the equations and for reactor core calculations, it is generally judged as “not to be worth the effort.” Therefore, the only coupling that we will consider may be photon production due to neutron interactions. With this assumption, the neutron transport equation may be solved independently; the then known neutron distribution may be used to define a neutron-induced photon source, and the photon transport equation may then in turn be solved. Therefore, the cross sections that we must consider include the following five categories: . . . . . Neutron interaction Neutron-induced photon production Photon interaction Gas (charged particle) production due to neutron interaction “Effect” production, such as heat (MERMA) and displacement (DPA) Of these five categories, gas and “effect” production can be calculated using a known (previously calculated) neutron distribution and production cross sections, as in (). Since it is completely analogous to our treatment of multigroup cross sections presented later in this chapter, it will not be explicitly considered further in this chapter. Another type of data that must be considered is thermal-scattering law data; this data is briefly discussed here and more extensively in > Chap.  on slowing down and thermalization. Of the remaining three categories, since the emphasis of this handbook is on nuclear power and fission reactor core calculations, major emphasis will be placed on the treatment of neutron interactions. Much of the treatment of photon interaction and production is very similar to the treatment of neutron interactions and secondary-neutron energy distributions. Therefore, photon interaction and production will only be mentioned when its treatment differs from that required for neutron interactions. . Evaluated Data In order to solve (), obviously we need to know the total cross section and the cross section, and multiplicity and secondary energy–angle distributions for each reaction. In order to solve (), we need to know the cross sections for individual reactions, half-lives, and probability of production of nuclides due to interaction or absorption of other nuclides. Of the data required to solve (), only the cross sections need to be processed prior to use; half-lives and production probabilities are basic nuclear data that may be used directly in applications. The treatment of cross sections for use in () is completely analogous to the treatment of cross selctions for use in (); and as such, () will not be explicitly considered further in this chapter. At the present time, there is no single unified theory of nuclear cross sections that will allow us to predict all nuclear cross sections. Therefore, the presently available cross sections are obtained by combining the results of differential and integral experimental measurements with the results of nuclear model calculations, to define the evaluated cross sections that are Nuclear Data Preparation  used in reactor calculations. The presently available, evaluated nuclear data libraries contain almost all the data that we require, and in the following sections, we will briefly review the representations that are used in these data libraries. Examples of data that may not be explicitly present in the evaluated libraries and may require the use of additional models is the calculation of KERMA and DPA; these additional models will be covered elsewhere in this handbook. .. Neutron-Interaction Data Cross Sections Evaluations represent the total, elastic, capture, and fission cross sections in the form of resonance parameters plus a tabulated-background cross section (Kinsey ; Rose and Dunford ). The most general representation at successively higher energies is . . . . Tabulated cross sections at low energy. Resolved-resonance parameters plus a tabulated background cross section. Unresolved-resonance parameters plus a tabulated background cross section. Tabulated cross sections at higher energies. The cross sections for all other reactions are represented in tabular form, e.g., inelastic, (n, n), etc. Since in order to use cross sections they must be defined at all energies, all tabulated cross sections also have an interpolation law associated with them, in order to uniquely define the cross sections at all energies between the energies at which the cross sections are tabulated. The major difficulty in processing the neutron-interaction data for later use in applications is that modern evaluations contain a great deal of resonance structure, which makes it very difficult to accurately define the combination of resonances and background cross sections. For example, > Fig.  illustrates the ENDF/B-VII. (Cullen a–c), U elastic cross section, at  and  K, which has been reconstructed from resonance and background combinations; and in the case of  K data, Doppler broadened, in order to obtain the energy-dependent cross section in tabulated form (Cullen a–c). Here, just this one elastic cross section requires between , energy points (at  K) and , (at  K). The sheer size of these data tables can cause a problem during not only during data processing, but also during subsequent use in applications. > Figure  shows some interesting details that we should mention. First at the top of the figure, we can see that the resolved-resonance region extends from − eV up to  keV, and above this the unresolved-resonance region extends from  keV up to  keV. Note the resonance structure in the resolved-resonance region that abruptly changes to the smooth cross section in the unresolved-resonance region; this is because for this figure in the unresolvedresonance region we only plot the infinitely dilute cross section (explained later). Note also the effect of temperature, with the  K temperature cross section being much smoother than the  K data, in the resolved-resonance region. > Figure  for the – keV energy range, shows the effect of Doppler broadening in better details. Finally, note the range of the cross section that extends over many orders of magnitude, in very narrow energy ranges; this makes the sampling of this data with accuracy, very difficult. For example, if we randomly sample an energy in the energy range of > Fig. , we can see that in most cases we will completely miss the resonances and select the smooth potential cross section between resonances.   Nuclear Data Preparation MAT 9237 92–U –238 –96 . 70 to 9999. % Elastic cross section Resolved U Max ration Min ration 106 0K 104 Cross section (barns)  102 100 10–2 10–4 106 300 K 104 102 100 10–2 10–4 10–5 10–4 10–3 10–2 10–1 100 1 101 102 103 Incident energy (eV) 104 105 106 107 92–U –238 ⊡ Figure  U elastic cross section, entire energy range Secondary-Neutron Distributions The secondary-neutron distributions are represented in one of several available forms depending on whether the secondary energy and scattering angle are considered to be correlated or uncorrelated. For two body reactions, such as elastic and inelastic scattering, the scattering angle and secondary energy are exactly correlated. In order to describe such reactions, the angular distribution is specified and the corresponding secondary energy can be uniquely calculated by considering conservation of energy and momentum. In this case, the angular distribution may be given in either tabulated or Legendre coefficient form and is given in the center-ofmass system. > Figure  illustrates elastic angular distributions for ENDF/B-VII. (Oblozinsky et al. ; Cullen a–c), which were reconstructed from Legendre coefficients using the LEGEND code (Cullen a–c). For uncorrelated reactions, both angular and energy distributions of secondary neutrons are specified and the final distribution is the product of these two distributions. In this case, the angular distribution must be given in the laboratory system. The most common representation for such reactions in current evaluations is to specify the angular distribution as isotropic. > Figure  illustrates some fission spectra for ENDF/B-VII. (Oblozinsky et al. ; Cullen a–c), which are given as uncorrelated data, with tabulated energy spectra and isotopic angular distributions. From this figure, we can see that the fission spectra between  and  MeV incident neutron energy are very smoothly varying with incident neutron energy. Nuclear Data Preparation MAT 9237  92–U –238 –90.00 to 900.0% Elastic cross section Resolved Max ratio 104 0K Cross section (barns) 102 100 10–2 104 300 K 102 100 10–2 1.0 2 1.2 1.4 1.6 Incident energy (KeV) 1.8 2.0 92–U –238 ⊡ Figure  U elastic cross section, – keV energy range Finally we have correlated energy–angle distributions, which are correlated, but not exactly correlated. In this case, we have distributions that are defined as functions of both secondary energy and direction; defined by specifying either the energy spectra at a given set of directions, or the angular distributions at a given set of secondary energies. Recent evaluations are using correlated energy–angle distributions more and more, indicating the improved ability of our nuclear model codes to accurately calculate such distributions. > Figure  illustrates some (n, n′ ) continuum spectra for ENDF/B-VII. (Oblozinsky et al. ; Cullen a–c) that are given as correlated energy–angle distributions, which were reconstructed as energy spectra for viewing using the SIXPAK code (Cullen a–c). These correlated energy–angle distributions were created using a nuclear model code that outputs results as histograms, rather than as continuous in energy. These histogram steps are typically keV wide and at high energy; these are more than adequate, but we can see that at low energy these histograms result in unrealistic “steps” in the distributions; our transport codes are already dealing with this problem (Cullen et al. a–c). .. Neutron-Induced Photon Production Photon production is complicated because in general it will be composed of both discrete and continuum photons; and in addition, there may also be a time-dependent component wherein   Nuclear Data Preparation Elastic angular distributions MAT 9237 92–U –238 101 8 6 1.000 MeV 1.200 MeV 1.400 MeV 1.600 MeV 1.800 MeV 2.000 MeV 2.500 MeV 3.000 MeV 3.500 MeV 4 2 100 Probability/Cosine  8 6 4 2 10–1 8 6 4 2 10–2 8 6 –1.0 1 –0.5 0.0 Cosine (CM) 0.5 1.0 92–U –238 ⊡ Figure  U elastic angular distributions, –. MeV energy range some photons are not produced instantaneously at the time of the neutron collision. Discrete photons may be represented in a number of different forms (Kinsey ; Rose and Dunford ). In one form, the cross section for a given reaction and the resulting transition probability array, between levels, are given. In a second form, one may specify the multiplicity (average number of photons) and cross section for the production of each discrete photon. Both these representations require knowledge of the neutron cross section that induced the reaction. Alternatively, one may simply specify a photon-production cross section for each discrete photon and the continuum. The distribution of neutron-induced photon production is represented in a form similar to the distribution of secondary neutrons. For discrete photons, energy and angle are exactly correlated and only the angular distribution of photons is specified. For continuum photons, both angular- and energy distribution of photons must be specified. In most current evaluations, the angular distribution of photon production is isotropic, which greatly simplifies the calculation and use of photon-production data (> Fig. ). Nuclear data users should be warned that many current evaluations do not include neutroninduced photon-production data. For example, the ENDF/B-VII. library (Oblozinsky et al. ; Cullen a–c) contains  evaluations (isotopes and elemental) and only about half of these include neutron-induced photon production; the fraction is even lower in older data libraries. Almost none of the current evaluations include capture-gamma ray lines, which are Nuclear Data Preparation Fission energy distribution MAT 9237  92–U –238 Probability/(MeV) 10–2 10–4 10–6 10–8 10–10 10–5 1 0.000 MeV 500.0 keV 1.000 MeV 1.500 MeV 2.000 MeV 2.500 MeV 3.000 MeV 4.000 MeV 5.000 MeV 10–4 10–3 10–2 10–1 Secondary energy (MeV) 100 101 92–U –238 ⊡ Figure  U elastic fission spectra, – MeV energy range unique to each isotope. A second warning is that when the neutron-induced photon production is included, energy conservation can be a problem; this can be so severe that photons carry off more energy than is available in a reaction. .. Photon Interaction Data The ENDF/B-VII. photon-interaction data is based entirely on the Livermore Evaluated Photon Data Library (EPDL) (Cullen et al. ). For those interested in fluorescence and/or electron transport, there are also two other libraries included in ENDF/B-VII.: The Evaluated Atomic Data Library (EADL) (Cullen et al. a,b) and the Evaluated Electron Data Library (EEDL) (Cullen et al. a,b); these libraries are of little interest for reactor core calculations and as such will not be discussed further in this chapter. The EPDL library included elemental data for all Z = –. Photon-interaction cross sections are given in a tabulated form with an interpolation law specified between tabulated points. Compared to neutron cross sections, photon-interaction cross sections are relatively smooth and do not create any special processing problems. For uranium, photon-interaction cross sections are shown in > Fig. .   Nuclear Data Preparation (n,n’ ) Continuum energy distribution MAT 9237 92–U –238 5 8.500 MeV 9.000 MeV 9.500 MeV 10.00 MeV 11.00 MeV 12.00 MeV 13.00 MeV 14.00 MeV 15.00 MeV 16.00 MeV 10–1 5 10–2 Probability/(MeV)  5 10–3 5 10–4 5 10–5 5 5 10 10–3 5 5 10–2 10–1 Secondary energy (MeV) 5 100 101 92–U –238 5 ⊡ Figure  U (n, n′ ) continuum spectra, .– MeV energy range In principle, the angular distribution of secondary photons could be specified in the same form as is used for neutrons. However, in the case of photons, a more natural representation is to use a combination of analytical forms and a correction factor: for coherent scattering, this correction factor is named the form factor; and for incoherent scattering, it is named the scattering function. At high energies, the Klein–Nishina formula is an excellent means of describing incoherent scattering; and at low energies, the coherent distribution assumes the simple form ( + μ  ), where μ is the scattering cosine. At other energies, these analytical forms are modified by the form factor (coherent scatter) and scattering function (incoherent scatter), as shown in > Fig. . Generally, when compared with neutrons, the processing of photon cross sections are rather straightforward and present little or no difficulty. However, processing the angular distributions of secondary photons can be difficult for two reasons. First, although analytical forms and correction factors are a convenient means to represent angular distributions, Legendre coefficients or tabulated angular distributions are more convenient for use in many applications. Second, the angular distributions can be very anisotropic and difficult to adequately represent in transport calculations, e.g., in S n calculations, many Legendre coefficients are required to accurately represent the angular distributions. An alternative method of using these correction factors is defined in (Cullen ). Nuclear Data Preparation Nonelastic photon distribution MAT 9237 10–3  92–U –238 5 10–4 5 Probability/(MeV) 10–5 5 10–6 5 10–7 599.0 keV 1.000 keV 1.500 keV 2.000 keV 3.000 keV 4.000 keV 6.000 keV 8.000 keV 5 10–8 5 10–9 5 10–2 2 4 6 1 8 10–1 2 4 Photon energy (MeV) 6 8 100 2 4 92–U –238 ⊡ Figure  U (n, n′ ) nonelastic photon emission spectra . Approximate Methods Unfortunately, an exact solution to Boltzmann equation () is still beyond our capabilities, even with the most modern and powerful computers that we have today, and we must introduce certain simplifying assumptions before attempting a solution. Once we consider introducing simplifying assumptions, we find that there are a variety of them that can be used, each of which leads to a different method of solving the pertinent equations. For exa