Rate of ionisation and density of electrons in a hollow cathode

The calculation of collision processes in He shows that for electrons with energies of hundreds of electron volts the momentum transfer cross-section is less than the total cross-section for inelastic collisions. Thus, the electron energy distribution relaxes faster than their angular distribution. Neglecting the secondary electrons created in ionisations and assuming that in every inelastic collision an electron loses the same energy, a simple one-dimensional kinetic equation for the energy distribution function of the electrons is deduced. This equation is solved numerically for He when the cathode is the source of electrons with an initial energy of 250 eV. The calculations give the rate of ionisation at different distances from the cathode. The distribution of the rate of ionisation in a rectangular (double) hollow cathode is found. It is shown to be uniform at gas pressures P<1 Torr but becomes nonuniform at higher pressures. The spatial distribution of the electron concentration in a rectangular cathode at 'low' and 'high' pressures is also obtained.