The Effect of Space Charge and Initial Velocities on the Potential Distribution and Thermionic Current between Parallel Plane Electrodes

Effect of space charge and cathode temperature on thermionic current and potential distribution.—I. Case of parallel plane electrodes. (a) Current limited by space charge. The results obtained by E. Q. Adams (unpublished), Epstein, Fry and Laue are discussed and summarized, certain errors are pointed out, and the equations are put in a form adapted to easy numerical calculation. Assuming the normal components of the velocities of the emitted electrons have the Maxwell distribution, the integration of Poisson's equation between proper limits leads to a numerical relation between the new variables $\xi =2(x-x_m){\left[\frac{2{\pi }^3{e}^2{i}^{2}m}{k^3{T}^3}\right]}^{\frac{1}{4}}$ and $\eta =\frac{e(V-V_m)}{\mathrm{kT}}$, where $x_m$ and $V_m$ give the position and voltage of the plane of minimum potential, and $k$ is the Boltzmann gas constant. Denoting values at the cathode by the subscript 1, and inserting values of constants: ${\eta }_1=log \left(\frac{i_0}{i}\right)$, where $i_0$ is the saturation current; $V-V_1=\frac{T(\eta -{\eta }_1)}{11,600}$, $\xi -{\xi }_1=9.180\mathrm{}\times {10}^5{T}^{-\frac{3}{4}}{i}^{\frac{1}{2}}(x-x_1)$. These equations and the tables of $\xi \left(\eta \right)$ for various values of $\eta$ enable, for a given cathode temperature $T$, the potential distribution for a given current $i$, or vice versa, to be computed. An approximate solution for the current is: $i=\left[\frac{\left(\frac{2^{\frac{1}{2}}}{9\pi }\right){\left(\frac{e}{m}\right)}^{\frac{1}{2}}{(V-V_m)}^{\frac{3}{2}}}{{(x-x_m)}^2}\right](1+2.66\mathrm{}{\eta }^{-\frac{1}{2}})$, which reduces to the usual three halves power law equation if we neglect $V_m$ and $x_m$ and the correction factor in $\eta$. (b) Equilibrium condition with anode at great distance, current zero. If the only retarding field is that of the space charge, the density of charge is: $\rho =\frac{\mathrm{kT}{\rho }_1}{{[{\mathrm{}\left(\mathrm{kT}\right)\mathrm{}}^{\frac{1}{2}}+x{\left(2\mathrm{}\pi e{\rho }_1\right)}^{\frac{1}{2}}]}^2}$ where ${\rho }_1=i_0{\left[\frac{2\pi m}{\mathrm{kT}}\right]}^{\frac{1}{2}}$. Except near the cathode this is approximately equal to $\frac{\mathrm{kT}}{2\pi e{x}^2}$; hence $\rho$ is proportional to the absolute temperature of the cathode and inversely proportional to the square of the distance away. The potential gradient at the cathode is $X_1={\left[\frac{8\pi {\rho }_1\mathrm{kT}}{e}\right]}^{\frac{1}{2}}$. Equations are also given for the case where an external retarding field $X_{\infty }$ is applied. II. In the case of concentric cylindrical electrodes, the current is: $i=\frac{\left(\frac{8^{\frac{1}{2}}}{9}\right){\left(\frac{e}{m}\right)}^{\frac{1}{2}}{[V-V_m+\frac{1}{4}{V}_0{\left\{log \right(\frac{V}{\lambda V_0}\left)\right\}}^2]}^{\frac{3}{2}}}{r}$, where $V_0$ is the initial energy of the electrons expressed in volts ($\frac{3kT}{2e}$), $r$ is the radius of the anode, and $\lambda$ is a constant between 1 and 2, not yet experimentally determined. The deviations from the three halves power law are not more than one quarter as much as for parallel planes and amount to only about 3 per cent at 130 volts.