Equilibration distance of ions in the cathode fall

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Abstract
The motion of atomic ions in a rare-gas cathode fall is limited by symmetric charge exchange. Nonequilibrium solutions of the Boltzmann equation for this problem are presented. The distance required for the average ion velocity to approach within 10% of the equilibrium drift velocity is calculated for constant and for linearly increasing fields, with a plane ionization source and with a uniform source. The equilibration distances range from two-thirds to six mean free paths. A rare-gas cathode fall is typically 50–100 mean free paths thick; hence the ion motion may be accurately described by the equilibrium drift velocity throughout most of the cathode fall.