Statistics of electron multiplication in the presence of primary and secondary ionisation processes

A theoretical study is made of the development of a gas discharge in which two ionisation processes are present: primary ionisation due to electron impact in the gas and secondary ionisation due to photo-electric emission at the cathode, with negligible delay. The discharge is allowed to develop continuously in time, in contrast to previous work in which the time was considered to progress in discrete steps. Equations are derived which give the probability of finding any number of electrons in the discharge at a given time. These are solved for the particular problem of finding the probability that the discharge should have become extinguished at any time. It is shown that the discrete-time model gives reasonable agreement with these calculations.