Sakuma–Hattori equation

The Sakuma–Hattori equation was first proposed by Fumihiro Sakuma, Akira Ono and Susumu Hattori in 1982. 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 not more than three fitting variables. In 2008, BIPM CCT-WG5 recommended its use for radiation thermometry measurement uncertainty budgets below 960 °C. == 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 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: • C is the scalar coefficient • c_2 = hc/k_\text{B} is the second radiation constant (0.014387752 m⋅K) • \lambda_x is the temperature-dependent effective wavelength (in meters) • T is the absolute temperature (in K) == Planckian form ==

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. ; Sakuma–Hattori equation (Planckian form) : S(T) = \frac{C}{\exp\left(\frac{c_2}{AT + B}\right)-1} ; Inverse equation : \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) Discussion The Planckian form is recommended for use in calculating uncertainty budgets for radiation thermometry It is also recommended for use in calibration of radiation thermometers below the silver point. 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. == See also ==