An Equilibrium Theory of Electrical Conduction

Abstract

Equilibrium electron theory of electrical conduction.---(1) General equation. Accepting the simple electron theory expression for specific conductivity, the concentration of free electrons is supposed to be determined by the reaction: normal $\mathrm{atom}\ensuremath{\leftrightarrows}\mathrm{positive}\mathrm{ion}+\ensuremath{\nu}$ electrons, which is governed by the ordinary laws of chemical equilibrium, and this gives for the specific electrical resistance $\ensuremath{\rho}=C{(\ensuremath{\nu}N)}^{\ensuremath{-}\frac{1}{(\ensuremath{\nu}+1)}}{T}^{a}{e}^{\frac{b}{T}}$, where $\ensuremath{\nu}$ is the valence in the reaction, $N$ the concentration of atomic nuclei, $a=\frac{(\ensuremath{\nu}+4)}{2(\ensuremath{\nu}+1)}$, $b=\frac{({\ensuremath{\phi}}_{0}\ensuremath{-}{\ensuremath{\psi}}_{0})}{(\ensuremath{\nu}+1)R}$, (${\ensuremath{\phi}}_{0}\ensuremath{-}{\ensuremath{\psi}}_{0}$) being the mean energy required to bring about the hypothetical reaction of 0\ifmmode^\circ\else\textdegree\fi{}K. This formula shows fair quantitative agreement with experimental data for both good and poor conductors; in particular, the constant $a$ is about 1.25 for the alkali metals and less for metals of higher valence except in the case of Fe and Ni. For the metals the requirement is that ${\ensuremath{\phi}}_{0}$ be slightly less than ${\ensuremath{\psi}}_{0}$ while for poor conductors ${\ensuremath{\phi}}_{0}$ must be considerably greater than ${\ensuremath{\psi}}_{0}$. (2) Interpretation of constant b in terms of photoelectric and thermionic work functions. ${\ensuremath{\phi}}_{0}$ and ${\ensuremath{\psi}}_{0}$ are identified with the photoelectric energy function and with the corresponding thermionic function respectively. According to the theory proposed the ordinary expression for the thermionic saturation current becomes: $i=B{T}^{\frac{(4\ensuremath{\nu}+1)}{2(\ensuremath{\nu}+1)}}{e}^{\ensuremath{-}\frac{\ensuremath{\omega}}{\mathrm{RT}}}$, where $\ensuremath{\omega}=\frac{({\ensuremath{\phi}}_{0}+\ensuremath{\nu}{\ensuremath{\psi}}_{0})}{(\ensuremath{\nu}+1)}$ in the present notation. Therefore for metals the photoelectric ${\ensuremath{\phi}}_{0}$ and the thermionic $\ensuremath{\omega}$ as experimentally determined should be practically identical, while for poor conductors the experimental ${\ensuremath{\phi}}_{0}$ should considerably exceed $\ensuremath{\omega}$. These conclusions are both in agreement with the facts. (3) Explanation of photo-conduction. This theory suggests that the mean value of ${\ensuremath{\phi}}_{0}$ is diminished by absorption of radiation of the resonance frequency. For poor conductors this would bring about an increase in conductivity. While for metals at ordinary temperatures the conductivity would not be sensibly affected, at very low temperatures metals should prove photo-sensitive.