A theory is presented for a simplified, one-dimensional model of a spherical hollow cathode discharge. Analytical expressions are derived for the electric field distribution and cathode fall voltage as functions of radial distance across the cathode dark space. The model assumes that the drift velocity of ions obeys the high-field approximation. An empirically derived discharge maintenance condition is assumed which, in addition to the secondary electron emission due to ions, includes the secondary electron emission effects of photons and metastable atoms from the negative glow region. The electric field and cathode fall voltage are shown to be monotonic functions of radial distance across the cathode dark space, which agrees qualitatively with previous theoretical and experimental work on cylindrical hollow cathodes. The sublinear mathematical form obtained for the electric field and its relationship to other cathode geometries is briefly discussed.
P. D. Mixon, "Theoretical Study of the Electric Field Distribution in the Cathode Fall Region of a Spherical Hollow Cathode Discharge," Appl. Spectrosc. 53, 1638-1641 (1999)