The Universal Constant of Thermionic Emission

Abstract

Theoretical deduction of the formula for thermionic emission.—The constant $A$ of the thermionic emission formula: $I=A{T}^2{e}^{-\frac{b_0}{T}}$ has been shown by Richardson and Dushman by different lines of argument to be a universal constant, and a value for the constant has been deduced by Dushman. The experimental data of Dushman verify this value of $A$ for pure metals, but if the metal is coated there are very wide divergences. In this paper an attempt is made to give a more rigorous deduction of the emission formula, so that it may be plain under what conditions it may be expected that $A$ be universal. Assuming that the entropy of the electron gas is the same as that of an ordinary monatomic gas in the Sackur Tetrode theory, and also assuming that the entropy of a surface charge is zero at 0° Abs. it is shown that an emission formula of the type above necessarily demands that $A$ have the universal value assigned by Dushman, and the necessary and sufficient condition for this is that the difference between the specific heat of the neutral metal and of a charge on its surface vanish ($C_{\mathrm{pm}}-C_{p\rho }=0\mathrm{}$). This condition may also be formulated in terms of the surface heat and is equivalent to $\sigma +\frac{d{P}_s}{\mathrm{dT}}-\frac{P_s}{T}=0\mathrm{}$, where $P_s$ is the surface heat and $\sigma$ the Thomson heat. If $\sigma$ can be neglected, the emission formula holds with the form given when $P_s$ is proportional to absolute temperature.