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{{short description|Formula for the thermal radiation emitted by a perfect black body}}
The '''Sakuma-Hattori equation''' is a mathematical model for predicting the amount of [[thermal radiation]], radiometric flux or radiometric power emitted from a perfect [[blackbody]] or received by a thermal radiation detector.
 
In [[physics]], the '''Sakuma–Hattori equation''' is a mathematical model for predicting the amount of [[thermal radiation]], [[Radiant flux|radiometric flux]] or radiometric power emitted from a perfect [[blackbody]] or received by a thermal radiation detector.
 
== History ==
The Sakuma–Hattori equation was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1982.<ref name=Sakuma1/> In 1996, a study investigated the usefulness of various forms of the Sakuma–Hattori equation. This study showed the Planckian form to provide the best fit for most applications.<ref name=Sakuma2/> This study was done for 10 different forms of the Sakuma–Hattori equation containing not more than three fitting variables. In 2008, BIPM CCT-WG5 recommended its use for radiation thermometry measurement uncertainty budgets below 960&nbsp;°C.<ref name=Fischer/>
 
== General form ==
The Sakuma–Hattori equation gives the [[electromagnetic radiation|electromagnetic signal]] from thermal radiation based on an object's [[temperature]]. The signal can be electromagnetic [[flux]] or signal produced by a detector measuring this radiation. It has been suggested that below the silver point,{{efn | Silver point, the melting point of silver 962°C [(961.961 ± 0.017)°C<ref>
 
{{Cite journal
The Sakuma-Hattori equation gives the [[electromagnetic radiation|electromagnetic signal]] from thermal radiation based on an object's [[temperature]]. The signal can be electromagnetic [[flux]] or signal produced by a detector measuring this radiation. It has been suggested that below the silver point, a method using the Sakuma-Hattori equation be used.<ref name=Sakuma1>F Sakuma, S Hattori, "Establishing a practical temperature standard by using a narrow-band radiation thermometer with a silicon detector", in ''Temperature: Its Measurement and Control in Science and Industry'', vol. 5, edited by J F Schooley, New York, AIP, 421–427 (1982).</ref> In its general form it looks like:<ref name=Fischer>J. Fischer, P. Saunders, M. Sadli, M. Battuello, C. W. Park, Y. Zundong, H. Yoon, W. Li, E. van der Ham, F. Sakuma, Y. Yamada, M. Ballico, G. Machin, N. Fox, J. Hollandt, M. Matveyev, P. Bloembergen and S. Ugur, "[http://www.bipm.org/wg/CCT/CCT-WG5/Allowed/Miscellaneous/Low_T_Uncertainty_Paper_Version_1.71.pdf Uncertainty budgets for calibration of radiation thermometers below the silver point]" (pdf), CCT-WG5 on Radiation Thermometry, BIPM, Sèvres, France (2008).</ref>
|author=J Tapping and V N Ojha
:<math>S(T) = \frac{C}{\exp\left(\frac{c_2}{\lambda _x T}\right)-1}</math>
| title = Measurement of the Silver Point with a Simple, High-Precision Pyrometer
where:
| journal = Metrologia
{| class="wikitable"
| volume = 26
| <math>C</math>
| issue = 2
| Scalar coefficient
| pages = 133–139
|-
| year = 1989
| <math>c_2</math>
| doi = 10.1088/0026-1394/26/2/008
| Second Radiation Constant (0.014387752 m⋅K<ref>{{cite web |publisher=National Institute of Standards and Technology (NIST) |title=2006 CODATA recommended values |url=http://physics.nist.gov/cuu/index.html |month=Dec |year=2003|accessdate=Apr 27, 2010}}</ref>)
| bibcode = 1989Metro..26..133T
|-
| s2cid = 250764204
| <math>\lambda _x</math>
}}
| Temperature dependent effective wavelength
</ref>] used as a calibration point in some temperature scales.<ref>
|-
{{Cite web
| <math>T</math>
| title = Definition of Silver Point - 962°C, the melting point of silver
| Temperature
| url = http://www.eudict.com/?word=silver+point+melting&lang=engchi
|-
| access-date = 2010-07-26}}
|}
</ref>
It is used to calibrate IR thermometers because it is stable and easy to reproduce.}} a method using the Sakuma–Hattori equation be used.<ref name=Sakuma1>{{cite book |first1=F. |last1=Sakuma |first2=S. |last2=Hattori |chapter=Establishing a practical temperature standard by using a narrow-band radiation thermometer with a silicon detector |title=Temperature: Its Measurement and Control in Science and Industry |volume=5 |editor-first=J. F. |editor-last=Schooley |location=New York |publisher=AIP |pages=421–427 |year=1982 |isbn=0-88318-403-6 }}</ref> In its general form it looks like<ref name=Fischer>{{cite journal |first1=J. |last1=Fischer |first2=P. |last2=Saunders |first3=M. |last3=Sadli |first4=M. |last4=Battuello |first5=C. W. |last5=Park |first6=Y. |last6=Zundong |first7=H. |last7=Yoon |first8=W. |last8=Li |first9=E. |last9=van der Ham |first10=F. |last10=Sakuma |first11=Y. |last11=Yamada |first12=M. |last12=Ballico |first13=G. |last13=Machin |first14=N. |last14=Fox |first15=J. |last15=Hollandt |first16=M. |last16=Matveyev |first17=P. |last17=Bloembergen |first18=S. |last18=Ugur |url=http://www.bipm.org/wg/CCT/CCT-WG5/Allowed/Miscellaneous/Low_T_Uncertainty_Paper_Version_1.71.pdf |title=Uncertainty budgets for calibration of radiation thermometers below the silver point |journal=CCT-WG5 on Radiation Thermometry, BIPM, Sèvres, France |year=2008 |volume=29 |issue=3 |page=1066 |doi=10.1007/s10765-008-0385-1 |bibcode=2008IJT....29.1066S |s2cid=122082731 |display-authors=1 }}</ref>
<math display="block">S(T) = \frac{C}{\exp\left(\frac{c_2}{\lambda_x T}\right) - 1},</math>
where:{{Clarify|reason=what is S(T)?|date=June 2023}}
* <math>C</math> is the scalar coefficient
* <math>c_2</math> is the second radiation constant (0.014387752 m⋅K<ref>{{cite web |publisher=National Institute of Standards and Technology (NIST) |title=2006 CODATA recommended values |url=http://physics.nist.gov/cuu/index.html | date=Dec 2003 |access-date=Apr 27, 2010}}</ref>)
* <math>\lambda_x</math> is the temperature-dependent effective wavelength (in meters)
* <math>T</math> is the absolute temperature (in [[kelvin|K]])
 
== Planckian form ==
Line 26 ⟶ 39:
 
The Planckian form is realized by the following substitution:
:<math display="block">\lambda _x = A + \frac{B}{T}</math>
 
Making this substitution renders the following the Sakuma-HattoriSakuma–Hattori equation in the Planckian form.
 
; Sakuma–Hattori equation (Planckian form)
{| class="wikitable"
: <math>S(T) = \frac{C}{\exp\left(\frac{c_2}{AT + B}\right)-1}</math>
|-
|; Sakuma-HattoriInverse equation<ref (Planckian form)name=MSLNZ/>
|: <math>S(T) = \frac{Cc_2}{A \expln \left(\frac{c_2C}{ATS} + B}1\right)} -1 \frac{B}{A}</math>
; First derivative<ref>''ASTM Standard E2758-10 – Standard Guide for Selection and Use of Wideband, Low Temperature Infrared Thermometers'', ASTM International, West Conshohocken, PA, (2010).
|-
 
| Inverse equation <ref name=MSLNZ/>
'''''<u>Updated</u>''''': ASTM E2758-15a(2021), https://www.astm.org/e2758-15ar21.html</ref>
| <math>T = \frac{c_2}{A \ln \left(\frac{C}{S} + 1\right)} - \frac{B}{A}</math>
: <math>\frac {dS}{dT} = \left[S(T)\right]^2 \frac{A c_2}{C\left(AT + B\right)^2}\exp\left(\frac{c_2}{AT + B}\right)</math>
|-
| First derivative <ref>''ASTM Work Item 21204 – Standard Guide for the Selection and Use of Wide-band, Low Temperature Infrared Thermometers'', ASTM International, West Conshohocken, PA, Currently balloted with ASTM E20 Committee on Temperature Measurement.</ref>
| <math>\frac {dS}{dT} = \left[S(T)\right]^2 \frac{A c_2}{C\left(AT + B\right)^2}\exp\left(\frac{c_2}{AT + B}\right)</math>
|}
 
=== Discussion ===
 
The Planckian form is recommended for use in calculating uncertainty budgets for [[radiation thermometry]]<ref name=Fischer/> and [[infrared thermometry]].<ref name="MSLNZ">''[http://msl.irl.cri.nz/sites/all/files/training-manuals/tg22-july-2009v2.pdf MSL Technical Guide 22 – Calibration of Low Temperature Infrared Thermometers]'' (pdf), Measurement Standards Laboratory of New Zealand (2008). '''''<u>Updated</u>''''': Version 3. July 2019, [https://www.measurement.govt.nz/download/28]</ref> It is also recommended for use in calibration of radiation thermometers below the silver point.<ref name=Fischer/>
 
The Planckian form resembles [[Planck's law|Planck's Law]].
 
:<math display="block">S(T) = \frac{c_1}{\lambda^5\left([\exp\left(\frac{c_2}{\lambda T}\right)-1\right)]}</math>
 
However the Sakuma-HattoriSakuma–Hattori equation becomes very useful when considering low-temperature, wide-band radiation thermometry. To use Planck's Lawlaw over a wide spectral band, an [[integral]] like the following would have to be considered:
 
:<math display="block">S(T) = \int_{\lambda _1}^{\lambda _2}\frac{c_1}{\lambda^5\left([\exp\left(\frac{c_2}{\lambda T}\right)-1\right)]} d\lambda</math>
 
This integral yields an [[incomplete polylogarithm]] function, which can make its use very cumbersome. The Sakuma-Hattori equation shown above was found to provide the best curve-fit for interpolation of scales for radiation thermometers among a number of alternatives investigated.<ref name=Sakuma2>Sakuma F, Kobayashi M., "Interpolation equations of scales of radiation thermometers", ''Proceedings of TEMPMEKO 1996'', pp. 305–310 (1996).</ref>
The standard numerical treatment expands the incomplete integral in a geometric series of the exponential
<math display="block">\int_0^{\lambda_2} \frac{c_1}{\lambda^5 \left[\exp\left(\frac{c_2}{\lambda T}\right)-1\right]} d\lambda
= c_1 \left(\frac{T}{c_2}\right)^4\int_{c_2/(\lambda_2 T)}^\infty \frac{x^3}{e^x -1} dx
</math>
after substituting <math>\lambda = \tfrac{c_2}{xT}, \ d\lambda = \tfrac{-c_2}{x^2 T dx}.</math> Then
<math display="block">\begin{align}
J(c)&\equiv \int_c^\infty \frac{x^3}{e^x -1}dx
=\int_c^\infty \frac{x^3 e^{-x}}{1- e^{-x}}dx \\[4pt]
&=\int_c^\infty \sum_{n\ge 1}x^3 e^{-nx} dx
\\[4pt]
&=\sum_{n\ge 1} e^{-nc} \frac{(nc)^3+3(nc)^2+6nc+6}{n^4}
\end{align}
</math>
provides an approximation if the sum is truncated at some order.
 
The Sakuma–Hattori equation shown above was found to provide the best curve-fit for interpolation of scales for radiation thermometers among a number of alternatives investigated.<ref name=Sakuma2>Sakuma F, Kobayashi M., "Interpolation equations of scales of radiation thermometers", ''Proceedings of TEMPMEKO 1996'', pp. 305–310 (1996).</ref>
The inverse Sakuma-Hattori function can be used without iterative calculation. This is an addition advatage over integration of Planck's Law.
 
The inverse Sakuma–Hattori function can be used without iterative calculation. This is an additional advantage over integration of Planck's law.
== History ==
 
The Sakuma-Hattori was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1987.<ref name=Sakuma1/> In 1996 a study investigated the usefulness of various forms of the Sakuma-Hattori equation. This study showed the Planckian form to provide the best fit for most applications.<ref name=Sakuma2/> This study was done for 10 different forms of the Sakuma-Hattori equation containing not more than three fitting variables. In 2008, BIPM CCT-WG5 recommended its use for radiation thermometry uncertainty budgets below 960 °C.<ref name=Fischer/>
 
== Other forms ==
Line 73 ⟶ 95:
! Planckian
|-
| Sakuma-HattoriSakuma–Hattori Planck III
| <math>S(T) = \frac{C}{\exp\left(\frac{c_2}{AT + B}\right)-1}</math>
| narrow
| yes
|-
| Sakuma-HattoriSakuma–Hattori Planck IV
| <math>S(T) = \frac{C}{\exp\left(\frac{A}{T^2} + \frac{B}{2T}\right)-1}</math>
| narrow
| yes
|-
| Sakuma-HattoriSakuma–Hattori -– Wien's II
| <math>S(T) = C \exp\left(\frac{-c_2}{AT + B}\right)</math>
| narrow
| no
|-
| Sakuma-HattoriSakuma–Hattori Planck II
| <math>S(T) = \frac{C T^A}{\exp\left(\frac{B}{T}\right)-1}</math>
| broad and narrow
| yes
|-
| Sakuma-HattoriSakuma–Hattori -– Wien's I
| <math>S(T) = C T^A {\exp\left(\frac{-B}{T}\right)}</math>
| broad and narrow
| no
|-
| Sakuma-HattoriSakuma–Hattori Planck I
| <math>S(T) = \frac{C}{\exp\left(\frac{c_2}{AT}\right)-1}</math>
| monochromatic
Line 113 ⟶ 135:
| no
|-
| Effective Wavelength -– Wien's
| <math>S(T) = C \exp\left(\frac{-A}{T}+\frac{B}{T^2}\right)</math>
| narrow
Line 125 ⟶ 147:
 
== See also ==
{{Div col|colwidth=20em}}
* [[Stefan–Boltzmann law]]
* [[Planck's law]]
* [[Rayleigh–Jeans law]]
* [[Wien approximation]]
* [[Wien's displacement law]]
* [[Kirchhoff's law of thermal radiation]]
* [[Infrared thermometer]]
* [[Pyrometer]]
* [[Thin-filament pyrometry]]
* [[Thermography]]
* [[Black body]]
* [[Thermal radiation]]
* [[Radiance]]
* [[Emissivity]]
* [[ASTM Subcommittee E20.02 on Radiation Thermometry]]
{{Div col end}}
 
== Notes ==
*[[Stefan–Boltzmann law]]
{{notelist}}
*[[Rayleigh-Jeans law]]
*[[Wien approximation]]
*[[Wien's displacement law]]
*[[Kirchhoff's law of thermal radiation]]
 
*[[Infrared thermometer]]
*[[Pyrometer]]
*[[Thin filament pyrometry]]
*[[Thermography]]
 
*[[Black body]]
*[[Thermal radiation]]
*[[Radiance]]
*[[Emissivity]]
 
* [[ASTM_Subcommittee_E20.02_on_Radiation_Thermometry]]
 
== References ==
<references/>
 
{{DEFAULTSORT:Sakuma-Hattori equation}}
 
[[Category:Statistical mechanics]]
[[Category:Equations]]
[[Category:1982 in science]]
{{catimprove}}
Retrieved from "https://en.wikipedia.org/wiki/Sakuma–Hattori_equation"