Sakuma–Hattori equation

The Planckian form is realized by the following substitution:

$${\displaystyle \lambda _{x}=A+{\frac {B}{T}}}$$

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

Sakuma–Hattori equation (Planckian form)

${\displaystyle S(T)={\frac {C}{\exp \left({\frac {c_{2}}{AT+B}}\right)-1}}}$

Inverse equation [5]

${\displaystyle T={\frac {c_{2}}{A\ln \left({\frac {C}{S}}+1\right)}}-{\frac {B}{A}}}$

First derivative [6]

${\displaystyle {\frac {dS}{dT}}=\left[S(T)\right]^{2}{\frac {Ac_{2}}{C\left(AT+B\right)^{2}}}\exp \left({\frac {c_{2}}{AT+B}}\right)}$

The Planckian form is recommended for use in calculating uncertainty budgets for radiation thermometry[3] and infrared thermometry.[5] It is also recommended for use in calibration of radiation thermometers below the silver point.[3]

The Planckian form resembles Planck's law.

$${\displaystyle S(T)={\frac {c_{1}}{\lambda ^{5}\left[\exp \left({\frac {c_{2}}{\lambda T}}\right)-1\right]}}}$$

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:

$${\displaystyle S(T)=\int _{\lambda _{1}}^{\lambda _{2}}{\frac {c_{1}}{\lambda ^{5}\left[\exp \left({\frac {c_{2}}{\lambda T}}\right)-1\right]}}d\lambda }$$

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

$${\displaystyle \int _{0}^{\lambda _{2}}{\frac {c_{1}}{\lambda ^{5}\left[\exp \left({\frac {c_{2}}{\lambda T}}\right)-1\right]}}d\lambda =c_{1}\left({\frac {T}{c_{2}}}\right)^{4}\int _{c_{2}/(\lambda _{2}T)}^{\infty }{\frac {x^{3}}{e^{x}-1}}dx}$$

after substituting ${\displaystyle \lambda ={\tfrac {c_{2}}{xT}},\ d\lambda ={\tfrac {-c_{2}}{x^{2}Tdx}}.}$ Then

$${\displaystyle {\begin{aligned}J(c)&\equiv \int _{c}^{\infty }{\frac {x^{3}}{e^{x}-1}}dx=\int _{c}^{\infty }{\frac {x^{3}e^{-x}}{1-e^{-x}}}dx\\[4pt]&=\int _{c}^{\infty }\sum _{n\geq 1}x^{3}e^{-nx}dx\\[4pt]&=\sum _{n\geq 1}e^{-nc}{\frac {(nc)^{3}+3(nc)^{2}+6nc+6}{n^{4}}}\end{aligned}}}$$

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.