# Sakuma–Hattori equation

__: Formula for the thermal radiation emitted by a perfect black body__

**Short description**In physics, 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.

## History

The Sakuma–Hattori equation was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1982.^{[1]} 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.^{[2]} 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.^{[3]}

## General form

The Sakuma–Hattori equation gives the 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,^{[lower-alpha 1]} a method using the Sakuma–Hattori equation be used.^{[1]} In its general form it looks like^{[3]}
[math]\displaystyle{ S(T) = \frac{C}{\exp\left(\frac{c_2}{\lambda_x T}\right) - 1}, }[/math]

- [math]\displaystyle{ C }[/math] is the scalar coefficient
- [math]\displaystyle{ c_2 }[/math] is the second radiation constant (0.014387752 m⋅K
^{[6]}) - [math]\displaystyle{ \lambda_x }[/math] is the temperature-dependent effective wavelength (in meters)
- [math]\displaystyle{ T }[/math] is the absolute temperature (in kelvins)

## Planckian form

### Derivation

The Planckian form is realized by the following substitution: [math]\displaystyle{ \lambda _x = A + \frac{B}{T} }[/math]

Making this substitution renders the following the Sakuma–Hattori equation in the Planckian form.

- Sakuma–Hattori equation (Planckian form)
- [math]\displaystyle{ S(T) = \frac{C}{\exp\left(\frac{c_2}{AT + B}\right)-1} }[/math]
- Inverse equation
^{[7]} - [math]\displaystyle{ T = \frac{c_2}{A \ln \left(\frac{C}{S} + 1\right)} - \frac{B}{A} }[/math]
- First derivative
^{[8]} - [math]\displaystyle{ \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^{[3]} and infrared thermometry.^{[7]} It is also recommended for use in calibration of radiation thermometers below the silver point.^{[3]}

The Planckian form resembles Planck's law.

[math]\displaystyle{ S(T) = \frac{c_1}{\lambda^5\left[\exp\left(\frac{c_2}{\lambda T}\right)-1\right]} }[/math]

However the Sakuma–Hattori equation becomes very useful when considering low-temperature, wide-band radiation thermometry. To use Planck's law over a wide spectral band, an integral like the following would have to be considered:

[math]\displaystyle{ 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 standard numerical treatment expands the incomplete integral in a geometric series of the exponential [math]\displaystyle{ \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]\displaystyle{ \lambda = \tfrac{c_2}{xT}, \ d\lambda = \tfrac{-c_2}{x^2 T dx}. }[/math] Then [math]\displaystyle{ \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.^{[2]}

The inverse Sakuma–Hattori function can be used without iterative calculation. This is an additional advantage over integration of Planck's law.

## Other forms

The 1996 paper investigated 10 different forms. They are listed in the chart below in order of quality of curve-fit to actual radiometric data.^{[2]}

Name | Equation | Bandwidth | Planckian |
---|---|---|---|

Sakuma–Hattori Planck III | [math]\displaystyle{ S(T) = \frac{C}{\exp\left(\frac{c_2}{AT + B}\right)-1} }[/math] | narrow | yes |

Sakuma–Hattori Planck IV | [math]\displaystyle{ S(T) = \frac{C}{\exp\left(\frac{A}{T^2} + \frac{B}{2T}\right)-1} }[/math] | narrow | yes |

Sakuma–Hattori – Wien's II | [math]\displaystyle{ S(T) = C \exp\left(\frac{-c_2}{AT + B}\right) }[/math] | narrow | no |

Sakuma–Hattori Planck II | [math]\displaystyle{ S(T) = \frac{C T^A}{\exp\left(\frac{B}{T}\right)-1} }[/math] | broad and narrow | yes |

Sakuma–Hattori – Wien's I | [math]\displaystyle{ S(T) = C T^A {\exp\left(\frac{-B}{T}\right)} }[/math] | broad and narrow | no |

Sakuma–Hattori Planck I | [math]\displaystyle{ S(T) = \frac{C}{\exp\left(\frac{c_2}{AT}\right)-1} }[/math] | monochromatic | yes |

New | [math]\displaystyle{ S(T) = C \left(1 + \frac{A}{T}\right) - B }[/math] | narrow | no |

Wien's | [math]\displaystyle{ S(T) = C \exp\left(\frac{-c_2}{A T}\right) }[/math] | monochromatic | no |

Effective Wavelength – Wien's | [math]\displaystyle{ S(T) = C \exp\left(\frac{-A}{T}+\frac{B}{T^2}\right) }[/math] | narrow | no |

Exponent | [math]\displaystyle{ S(T) = C T^A }[/math] | broad | no |

## See also

- 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

## Notes

- ↑ Silver point, the melting point of silver 962°C [(961.961 ± 0.017)°C
^{[4]}] used as a calibration point in some temperature scales.^{[5]}It is used to calibrate IR thermometers because it is stable and easy to reproduce.

## References

- ↑
^{1.0}^{1.1}Sakuma, F.; Hattori, S. (1982). "Establishing a practical temperature standard by using a narrow-band radiation thermometer with a silicon detector". in Schooley, J. F..*Temperature: Its Measurement and Control in Science and Industry*.**5**. New York: AIP. pp. 421–427. ISBN 0-88318-403-6. - ↑
^{2.0}^{2.1}^{2.2}Sakuma F, Kobayashi M., "Interpolation equations of scales of radiation thermometers",*Proceedings of TEMPMEKO 1996*, pp. 305–310 (1996). - ↑
^{3.0}^{3.1}^{3.2}^{3.3}Fischer, J.*et al*. (2008). "Uncertainty budgets for calibration of radiation thermometers below the silver point".*CCT-WG5 on Radiation Thermometry, BIPM, Sèvres, France***29**(3): 1066. doi:10.1007/s10765-008-0385-1. Bibcode: 2008IJT....29.1066S. http://www.bipm.org/wg/CCT/CCT-WG5/Allowed/Miscellaneous/Low_T_Uncertainty_Paper_Version_1.71.pdf. - ↑
J Tapping and V N Ojha (1989). "Measurement of the Silver Point with a Simple, High-Precision Pyrometer".
*Metrologia***26**(2): 133–139. doi:10.1088/0026-1394/26/2/008. Bibcode: 1989Metro..26..133T. - ↑ "Definition of Silver Point - 962°C, the melting point of silver". http://www.eudict.com/?word=silver+point+melting&lang=engchi.
- ↑ "2006 CODATA recommended values". National Institute of Standards and Technology (NIST). Dec 2003. http://physics.nist.gov/cuu/index.html.
- ↑
^{7.0}^{7.1}*MSL Technical Guide 22 – Calibration of Low Temperature Infrared Thermometers*(pdf), Measurement Standards Laboratory of New Zealand (2008). - ↑
*ASTM Standard E2758-10 – Standard Guide for Selection and Use of Wideband, Low Temperature Infrared Thermometers*, ASTM International, West Conshohocken, PA, (2010).

Original source: https://en.wikipedia.org/wiki/Sakuma–Hattori equation.
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