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# Sakuma–Hattori equation

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 Title: Sakuma–Hattori equation Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Sakuma–Hattori equation

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.

## 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[A], a method using the Sakuma–Hattori equation be used.[1] In its general form it looks like:[2]

$S\left(T\right) = \frac\left\{C\right\}\left\{\exp\left\left(\frac\left\{c_2\right\}\left\{\lambda _x T\right\}\right\right)-1\right\}$

where:

 $C$ Scalar coefficient $c_2$ Second Radiation Constant (0.014387752 m⋅K[3]) $\lambda _x$ Temperature dependent effective wavelength in meters $T$ Temperature in Kelvin

## Planckian form

### Derivation

The Planckian form is realized by the following substitution:

$\lambda _x = A + \frac\left\{B\right\}\left\{T\right\}$

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

 Sakuma–Hattori equation (Planckian form) $S\left(T\right) = \frac\left\{C\right\}\left\{\exp\left\left(\frac\left\{c_2\right\}\left\{AT + B\right\}\right\right)-1\right\}$ Inverse equation [4] $T = \frac\left\{c_2\right\}\left\{A \ln \left\left(\frac\left\{C\right\}\left\{S\right\} + 1\right\right)\right\} - \frac\left\{B\right\}\left\{A\right\}$ First derivative [5] $\frac \left\{dS\right\}\left\{dT\right\} = \left\left[S\left(T\right)\right\right]^2 \frac\left\{A c_2\right\}\left\{C\left\left(AT + B\right\right)^2\right\}\exp\left\left(\frac\left\{c_2\right\}\left\{AT + B\right\}\right\right)$

### Discussion

The Planckian form is recommended for use in calculating uncertainty budgets for radiation thermometry[2] and infrared thermometry.[4] It is also recommended for use in calibration of radiation thermometers below the silver point.[2]

The Planckian form resembles Planck's Law.

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

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:

$S\left(T\right) = \int_\left\{\lambda _1\right\}^\left\{\lambda _2\right\}\frac\left\{c_1\right\}\left\{\lambda^5\left\left[\exp\left\left(\frac\left\{c_2\right\}\left\{\lambda T\right\}\right\right)-1\right\right]\right\} d\lambda$

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.[6]

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

## History

The Sakuma–Hattori was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1987.[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.[6] 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.[2]

## 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.[6]

Name Equation Bandwidth Planckian
Sakuma–Hattori Planck III $S\left(T\right) = \frac\left\{C\right\}\left\{\exp\left\left(\frac\left\{c_2\right\}\left\{AT + B\right\}\right\right)-1\right\}$ narrow yes
Sakuma–Hattori Planck IV $S\left(T\right) = \frac\left\{C\right\}\left\{\exp\left\left(\frac\left\{A\right\}\left\{T^2\right\} + \frac\left\{B\right\}\left\{2T\right\}\right\right)-1\right\}$ narrow yes
Sakuma–Hattori – Wien's II $S\left(T\right) = C \exp\left\left(\frac\left\{-c_2\right\}\left\{AT + B\right\}\right\right)$ narrow no
Sakuma–Hattori Planck II $S\left(T\right) = \frac\left\{C T^A\right\}\left\{\exp\left\left(\frac\left\{B\right\}\left\{T\right\}\right\right)-1\right\}$ broad and narrow yes
Sakuma–Hattori – Wien's I $S\left(T\right) = C T^A \left\{\exp\left\left(\frac\left\{-B\right\}\left\{T\right\}\right\right)\right\}$ broad and narrow no
Sakuma–Hattori Planck I $S\left(T\right) = \frac\left\{C\right\}\left\{\exp\left\left(\frac\left\{c_2\right\}\left\{AT\right\}\right\right)-1\right\}$ monochromatic yes
New $S\left(T\right) = C \left\left(1 + \frac\left\{A\right\}\left\{T\right\}\right\right) - B$ narrow no
Wien's $S\left(T\right) = C \exp\left\left(\frac\left\{-c_2\right\}\left\{A T\right\}\right\right)$ monochromatic no
Effective Wavelength – Wien's $S\left(T\right) = C \exp\left\left(\frac\left\{-A\right\}\left\{T\right\}+\frac\left\{B\right\}\left\{T^2\right\}\right\right)$ narrow no
Exponent $S\left(T\right) = C T^A$ broad no

## Notes

• ^ Silver point, the melting point of silver 962°C [(961.961 ± 0.017)°C[7] ] used as a calibration point in some temperature scales.[8] It is used to calibrate IR thermometers because it is stable and easy to reproduce.

## References

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