On the Motions of Electrons in Gases

Abstract

Motion of electrons through gases when electric field and pressure are such that collisions are elastic.—For electron speeds below a critical value characteristic of each gas, the collisions with molecules are probably as between elastic spheres, since neither ionization nor radiation is produced. The average fraction of its energy lost in elastic collision with a molecule of mass $M$, by an electron of mass $m$, is $f=2\left(\frac{m}{M}\right)(1-\frac{\Omega }{U})$, where $\Omega$ and $U$ are the average kinetic energies of molecules and electrons. But while, if $U>\mathrm{}\Omega$, the electrons lose energy at collisions, they gain energy from the field between collisions, so the kinetic energy tends to increase or decrease to a terminal energy. Assuming Langevin's equation for mobility, we find $U_t \left(\mathrm{in}\mathrm{volts}\right)=.\mathrm{}66\mathrm{} \frac{\mathrm{El}}{\surd f}$ where $\mathrm{El}$ is the product of field strength and mean free path. In the case of plane parallel electrodes the number of collisions per cm and also the ratio of the speed at any point to the terminal speed, are both found to be independent of the field strength. In the case of a straight filament cathode surrounded by a coaxial cylindrical anode, the electrons may acquire their maximum speed near the cathode and subsequently lose speed while approaching the anode, their speed being greater than the terminal speed for this region. Curves are given for various cases. Substituting the value of terminal speed in Langevin's equation leads to the following expression for electron mobility: $\mu =\frac{0.815l\surd \left(\frac{\frac{1}{2}e}{m}\right)}{{[\frac{1}{2}\Omega +{(\frac{{\Omega }^2}{4}+W^2)}^{\frac{1}{2}}]}^{\frac{1}{2}}}$ where the energy due to the field is $eW=\frac{\mathrm{leE}}{1.506\surd f}$. This expression is somewhat similar to that given by Loeb (corrected for numerical error) but leads to values 0 to 10 per cent higher. When the collisions are inelastic the mobility is shown to be greater.