The Deviation of Anode Currents in Diodes from the Three-Halves Power Law

Abstract

The calculation of the electron currents in a diode with guard rings (tube No. 2 of Kato) shows that the deviation of the anode current $I_a$ at low anode voltages $V$ (filament temperature $T$ and anode voltage $V$ being varied) from the three-halves power law observed by Kato and assumed by him to be due to the initial velocities of electrons ("effective initial velocities") is caused by the corrections of the three-halves power law for cylinders consisting of (a) the barrier correction, (b) the enlargement correction, (c) the correction due to the initial velocities of electrons, (d) the correction due to the contact potential. Taking into account these corrections discussed by Schottky in 1914 we get the space-charge equation for cylinders (equipotential cathodes at uniform temperature along its surface being supposed) $I_a=14.65\mathrm{}·{10}^{-6\mathrm{}}·l{\left(r_a{\beta }^2\right)}^{-1\mathrm{}}{\{V-V_m+\frac{1}{4}{V}_0{\left(\frac{\ln V^{\prime }}{\lambda V_0}\right)}^2+\kappa \}}^{\frac{3}{2}}\mathrm{amperes}$ where $l$ is length of filament, $r_a$ radius of anode, ${\beta }^2$ the well known constant tabulated by Langmuir and Blodgett, $V_m$ barrier potential, $V_0$ average initial energy of electrons in radial direction, $\kappa$ contact potential, $\lambda =1, 5$. The values of the minimum $V_m$ in the potential between filament and anode when calculated according to the equation for parallel planes: $-V_m=\left(\frac{T}{11600}\right)\ln \left(\frac{I_s}{I_a}\right)$ volts (where $I_s$ saturation emission current) are too small; the real values are obtained by the Schottky relation for cylinders. We can compute ${\beta }^2$ (enlargement correction) from the observed currents by combining the Schottky equation: $I_a=I_s{e}^{-n}{\theta }^{-1\mathrm{}}\phi {\left({\theta }^2\alpha n\right)}^{\frac{1}{2}}$ (where $n=\frac{\mathrm{eV}}{\mathrm{kT}}$ the reduced retarding potential between filament and barrier, $\theta$ the ratio $\frac{r_0}{r_m}$ i.e. radius of filament/radius of barrier cylinder, $\alpha ={(1-{\theta }^2)}^{-1\mathrm{}} \mathrm{and} \phi \mathrm{}\left(x\right)\mathrm{}$ the error function) with the corrected space charge equation. These calculations make it probable that the correction member which takes into account the influence of the initial velocities of electrons on the anode currents: $\frac{1}{4}{V}_0{\left(\ln \right(\frac{V^{\prime }}{\lambda V_0}\left)\right)}^2$, as estimated by Langmuir, is too large. It is suggested to replace this member by adding merely the value $V_0$ to the effective anode voltage.