The Thermionic Work Function and the Slope and Intercept of Richardson Plots

Abstract

This article is a critical correlation of the slope and intercept of experimental Richardson lines with the quantities appearing in theoretical equations based on thermodynamic and statistical reasoning. The equation for experimental Richardson lines is $logi\ensuremath{-}2logT=logA\ensuremath{-}\frac{b}{2.3T}$; $A$ and $b$ are constants characteristic of the surface, $i$ is the electron emission current in amp./${\mathrm{cm}}^{2}$, $T$ is the temperature in degrees K, $logA$ is the intercept and $\ensuremath{-}\frac{b}{2.3}$ is the slope of experimental lines. Statistical theory based on the Fermi-Dirac distribution of electron velocities in the metal shows that $i$ should be given by $logi\ensuremath{-}2logT=logU(1\ensuremath{-}r)\ensuremath{-}\frac{w}{2.3T}$, where $U$ is a universal constant which has the value 120 amp./${\mathrm{cm}}^{2}$ \ifmmode^\circ\else\textdegree\fi{}${\mathrm{K}}^{2}$, $r$ is the reflection coefficient, and $w$ is the work function. A correlation of the experimental and theoretical equations shows that $b=w\ensuremath{-}\frac{\mathrm{Tdw}}{\mathrm{dT}}$, and $logA=logU(1\ensuremath{-}r)\ensuremath{-}\frac{(\frac{1}{2.3})dw}{\mathrm{dT}}$. Only when $r$ is 0 and the work function is independent of the temperature, is it correct to say that the slope is $\ensuremath{-}\frac{w}{2.3}$ and that the intercept has the universal value $logU$. But even when $w$ is a function of $T$, it follows from a thermodynamic argument that the slope is given by $\ensuremath{-}\frac{h}{2.3}$, where the heat function $h$ is defined by $h=(\frac{{L}_{p}}{R})\ensuremath{-}(\frac{5}{2})T$, ${L}_{p}$ is the heat of vaporization per mol at constant pressure. The heat function is related to the work function by the equation $h=w\ensuremath{-}\frac{\mathrm{Tdw}}{\mathrm{dT}}$.From experimental and theoretical arguments it is deduced that the reflection coefficient is probably negligibly small. Hence we conclude that for most surfaces the work function varies with temperature, since the intercepts of Richardson lines are rarely equal to log 120. This conclusion is to be expected since on Sommerfeld's theory, $w$ depends on the number of free electrons or atoms per ${\mathrm{cm}}^{3}$, which in turn varies with temperature due to thermal expansion.The photoelectric work function should equal the thermionic work function but should not in general be equal to -2.3 times the slope of the Richardson line. The Volta potential between two surfaces having work functions ${w}_{1}$ and ${w}_{2}$ should equal $\frac{({w}_{1}\ensuremath{-}{w}_{2})k}{e}$ rather than $\frac{2.3k}{e}$ times the difference between the slopes of the Richardson lines for the two surfaces. The data from photoelectric and Volta potential measurements support the conclusion that the work function depends on temperature.