Theory of High Frequency Gas Discharges. II. Harmonic Components of the Distribution ${\mathrm{Function}}^1$

The type of distribution function employed in the foregoing paper is not sufficiently general to be applicable to all conditions of frequency and field strength. To examine its limitations, and the limitations of corresponding d.c. treatments, the function is developed as a series in Legendre polynomials of $\frac{v_x}{v}$, $x$ being the field direction, and Fourier functions of $\omega t$, $\omega$ being the field frequency. Attention is limited to steady states and to elastic collisions between electrons and gas molecules. By solving the resulting recurrence equations a number of successive approximations has been obtained, and from each approximation the range of validity of the preceding one is determined. Questions of mathematical convergence are not dealt with, since the physical meaning of the results is usually clear and reasonable. The current through the gas is shown to take on d.c. character when $\omega \lambda \ll \left(\frac{m}{M}\right)v$, $\lambda$ being the mean free path and $v$ the (mean) velocity of the electrons producing the current.