Calculation of Thermodynamic Quantities of 1D Ising Model with Mixed Spin-(s,(2t − 1)/2) by Means of Transfer Matrix
<p>Configurations with a mixed spin on a one-dimensional finite block, where <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mrow> <mn>2</mn> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mo>Φ</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mrow> <mn>2</mn> <mi>N</mi> </mrow> </msub> <mo>∈</mo> <mo>Ψ</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">N</mi> <mo>+</mo> </msup> <mo>.</mo> </mrow> </semantics></math></p> "> Figure 2
<p>The graph of free energy <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>β</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> given in (<a href="#FD25-axioms-12-00880" class="html-disp-formula">25</a>) as a function of <math display="inline"><semantics> <mi>β</mi> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math>.</p> "> Figure 3
<p>(<b>Left</b>) The graph of entropy <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>β</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> given in (<a href="#FD26-axioms-12-00880" class="html-disp-formula">26</a>) for <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> as a function of the temperature <span class="html-italic">T</span> in the absence of a magnetic field. (<b>Right</b>) The graph of entropy <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>β</mi> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </semantics></math> given in (<a href="#FD26-axioms-12-00880" class="html-disp-formula">26</a>) for <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>0.2</mn> </mrow> </semantics></math> as a function of <math display="inline"><semantics> <mi>β</mi> </semantics></math> in the absence of a magnetic field.</p> "> Figure 4
<p>(<b>Left</b>) The graph of magnetization <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <mi>T</mi> <mo>,</mo> <mi>H</mi> <mo>)</mo> </mrow> </semantics></math> given in (<a href="#FD31-axioms-12-00880" class="html-disp-formula">31</a>) for <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> as a function of <span class="html-italic">h</span> and <span class="html-italic">T</span>. (<b>Right</b>) The graph of magnetization <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>(</mo> <mi>T</mi> <mo>,</mo> <mi>H</mi> <mo>)</mo> </mrow> </semantics></math> given in (<a href="#FD31-axioms-12-00880" class="html-disp-formula">31</a>) for <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> as a function of <span class="html-italic">h</span> and <span class="html-italic">T</span>.</p> "> Figure 5
<p>(<b>Left</b>) The graph of susceptibility <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>(</mo> <mi>T</mi> <mo>,</mo> <mi>H</mi> <mo>)</mo> </mrow> </semantics></math> given in (<a href="#FD32-axioms-12-00880" class="html-disp-formula">32</a>) for <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math> as a function of <span class="html-italic">h</span> and <span class="html-italic">T</span>. (<b>Right</b>) The graph of susceptibility <math display="inline"><semantics> <mrow> <mi>χ</mi> <mo>(</mo> <mi>T</mi> <mo>,</mo> <mi>H</mi> <mo>)</mo> </mrow> </semantics></math> given in (<a href="#FD32-axioms-12-00880" class="html-disp-formula">32</a>) for <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> as a function of <span class="html-italic">h</span> and <span class="html-italic">T</span>.</p> "> Figure 6
<p>(<b>Left</b>) The graph of the function <span class="html-italic">f</span> given in (<a href="#FD39-axioms-12-00880" class="html-disp-formula">39</a>) for <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.421</mn> </mrow> </semantics></math>. (<b>Right</b>) The graph of the function <span class="html-italic">f</span> given in (<a href="#FD39-axioms-12-00880" class="html-disp-formula">39</a>) for <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>3.421</mn> </mrow> </semantics></math>.</p> "> Figure 7
<p>The graph of the function <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>(</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> given in (<a href="#FD42-axioms-12-00880" class="html-disp-formula">42</a>) versus <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <msup> <mi>e</mi> <mfrac> <mi>J</mi> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </mfrac> </msup> </mrow> </semantics></math> in the ferromagnetic region (<math display="inline"><semantics> <mrow> <mi>J</mi> <mo>></mo> <mn>0</mn> </mrow> </semantics></math>).</p> ">
Abstract
:1. Introduction
2. Preliminaries
Construction of the Partition Function Associated with the Model
- Bulk free energy
- Entropy
- Internal energy
3. Construction of the Partition Function for the Model via Transfer Matrices
3.1. The Partition Function and the Boltzmann Weight
3.2. The Transfer Matrices
3.3. 1D-MSIM with Mixed Spin-
3.4. Investigation of Thermodynamic Quantities in the Translation-Invariant Case
Case I
3.5. Behavior of the Thermodynamic Quantities of 1D-MSIM with Mixed Spin-(s,(2t − 1)/2) in the Absence of a Magnetic Field
Case II
3.6. Magnetization and Magnetic Susceptibility
4. Nonexistence Phase Transition in the Absence of the External Magnetic Field
Chaoticity of the Model
5. The Average Magnetization for the Mixed Spin-(1,1/2) Ising Model
The Average Magnetization
6. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Ising, E. Beitrag zur theorie des ferromagnetismus. Z. Physik 1925, 31, 253. [Google Scholar] [CrossRef]
- Blume, M. Theory of the first-order magnetic phase change in UO2. Phys. Rev. 1966, 141, 517. [Google Scholar] [CrossRef]
- Capel, H.W. On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting. Physica 1966, 32, 966. [Google Scholar] [CrossRef]
- Lipowski, A.; Suzuki, M. On the exact solution of twodimensional spin S Ising models A. Phys. A 1993, 193, 141. [Google Scholar] [CrossRef]
- Izmailian, N.S.; Ananikian, N.S. General spin-3/2 Ising model in a honeycomb lattice: Exactly solvable case. Phys. Rev. B 1994, 50, 6829. [Google Scholar] [CrossRef]
- De La Espriella, N.; Arenas, A.J.; Paez Meza, M.S. Critical and compensation points of a mixed spin-2-spin-5/2 Ising ferrimagnetic system with crystal field and nearest and next-nearest neighbors interactions. J. Magn. Magn. Mater. 2016, 417, 434–441. [Google Scholar] [CrossRef]
- De La Espriella, N.; Buendia, G.M.; Madera, J.C. Mixed spin-1 and spin-2 Ising model: Study of the ground states. J. Phys. Commun. 2018, 2, 025006. [Google Scholar] [CrossRef]
- Kaneyoshi, T. Phase transition of the mixed spin system with a random crystal field. Phys. A 1988, 153, 556–566. [Google Scholar] [CrossRef]
- Albayrak, E. The study of mixed spin-1 and spin-1/2: Entropy and isothermal entropy change. Phys. A 2020, 559, 125079. [Google Scholar] [CrossRef]
- De La Espriella, N.; Buendia, G.M. Magnetic behavior of a mixed Ising 3/2 and 5/2 spin model. J. Phys. Condens. Matter 2011, 23, 176003. [Google Scholar] [CrossRef]
- Akın, H.; Mukhamedov, F. Phase transition for the Ising model with mixed spins on a Cayley tree. J. Stat. Mech. 2022, 2022, 053204. [Google Scholar] [CrossRef]
- Akın, H. The classification of disordered phases of mixed spin (2,1/2) Ising model and the chaoticity of the corresponding dynamical system. Chaos Solitons Fractals 2023, 167, 113086. [Google Scholar] [CrossRef]
- Seino, M. The free energy of the random Ising model on the Bethe lattice. Phys. A 1992, 181, 233–242. [Google Scholar] [CrossRef]
- Akın, H. Quantitative behavior of (1,1/2)-MSIM on a Cayley tree. Chin. J. Phys. 2023, 83, 501–514. [Google Scholar] [CrossRef]
- Ostilli, M. Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists. Phys. A 2012, 391, 3417–3423. [Google Scholar] [CrossRef]
- Mézard, M.; Parisi, G. The Bethe lattice spin glass revisited. Eur. Phys. J. B 2001, 20, 217–233. [Google Scholar] [CrossRef]
- Qi, Y.; Liu, J.; Yu, N.-S.; Du, A. Magnetocaloric effect in ferroelectric Ising chain magnet. Solid State Commun. 2016, 233, 1–5. [Google Scholar] [CrossRef]
- Akın, H. Calculation of the free energy of the Ising model on a Cayley tree via the self-similarity method. Axioms 2022, 11, 703. [Google Scholar] [CrossRef]
- Akın, H.; Ulusoy, S. A new approach to studying the thermodynamic properties of the q-state Potts model on a Cayley tree. Chaos Solitons Fractals 2023, 174, 113811. [Google Scholar] [CrossRef]
- Salinas, S.R.A. Phase Transitions and Critical Phenomena: Classical Theories. In Introduction to Statistical Physics. Graduate Texts in Contemporary Physics; Springer: New York, NY, USA, 2001. [Google Scholar]
- Amin, M.E.; Mubark, M.; Amin, Y. On the critical behavior of the spin-s ising model. Rev. Mex. Fis. 2023, 69, 021701. [Google Scholar] [CrossRef]
- Wang, W.; Diaz-Mendez, R.; Capdevila, R. Solving the one-dimensional Ising chain via mathematical induction: An intuitive approach to the transfer matrix. Eur. J. Phys. 2019, 40, 065102. [Google Scholar] [CrossRef]
- Mathematica, Version 8.0; Wolfram Research, Inc.: Champaign, IL, USA, 2010.
- Feigenbaum, M.J. Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 1978, 19, 25–52. [Google Scholar] [CrossRef]
- Feigenbaum, M.J. Universal behavior in nonlinear systems. Phys. D 1983, 7, 16–39. [Google Scholar] [CrossRef]
- Hilborn, R.C. Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers; Oxford University Press on Demand: Oxford, UK, 2000. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |
© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Akın, H. Calculation of Thermodynamic Quantities of 1D Ising Model with Mixed Spin-(s,(2t − 1)/2) by Means of Transfer Matrix. Axioms 2023, 12, 880. https://doi.org/10.3390/axioms12090880
Akın H. Calculation of Thermodynamic Quantities of 1D Ising Model with Mixed Spin-(s,(2t − 1)/2) by Means of Transfer Matrix. Axioms. 2023; 12(9):880. https://doi.org/10.3390/axioms12090880
Chicago/Turabian StyleAkın, Hasan. 2023. "Calculation of Thermodynamic Quantities of 1D Ising Model with Mixed Spin-(s,(2t − 1)/2) by Means of Transfer Matrix" Axioms 12, no. 9: 880. https://doi.org/10.3390/axioms12090880