Theory of the Cathode Sheath in a Low-Density Gas Discharge

The space-charge equations of a low-density gas discharge are derived and applied in the present report to the study of potential distributions in the cathode sheath. In order to facilitate mathematical computation, the contribution of secondary electrons and charge exchange to the potential distribution is neglected. The space charge is generated, consequently, by the primary electrons of the hot cathode and the ions resulting from volume ionization of the gas in the sheath. The effect of arbitrary positive-ion currents issuing from the positive column to the sheath is also examined.A self-consistent method of solution of the space-charge equations is employed in the present investigation. The method essentially involves finding a cross section which corresponds to a particular potential distribution. A number of cross sections of physical significance can be obtained by this method. The results of our numerical calculations allow us to draw certain semiquantitative conclusions about the mode of evolution of a cathode sheath and positive column with increasing gas pressure and applied voltage. An important conclusion of our calculations is that a positive column will begin to evolve when the number of ionization mean free paths in the discharge gap is of the same order as the square root of the ratio of electron to positive-ion mass.