User:Texliebmann/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 electro-magnetic signal from thermal radiation based on an object's temperature. The signal can be electro-magnetic 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. In its general form it looks like :
 * $$S(T) = \frac{C}{\exp\left(\frac{c_2}{\lambda _x T}\right)-1}$$

where:

Derivation
The planckian form is realized by the following substitution:
 * $$\lambda _x = A + \frac{B}{T}$$

Making this substitution renders the following the Sakuma-Hattori Equation in the Planckian Form.

Discussion
The Planckian form is recommended for use in calculating uncertainty budgets for radiation thermometry and IR thermometry. It is also recommended for use in calibration of radiation thermometers below the silver point.

The Planckian form resembles Planck's Law.


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

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.


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

This integral cannot be solved analytically, which makes its use very cumbersome. The solution is to use the Sakuma-Hattori Equation. The planckian form of the Sakuma-Hattori Equation shown above was found to provide the best curve-fit for interpolation of scales for radiation thermometers.

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

History
The Sakuma-Hattori was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1987. 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. This study was done for 10 different forms of the Sakuma-Hattori Equation containing no for than three fitting variables. In 2008, BIPM CCT-WG5 recommended its use for radiation thermometry uncertainty budgets below 960 °C.

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.