Radiative Cascade Theory

A steady-state cascade theory has been set up for radiative electron transitions. These are assumed to occur between a continuum and various excited states, as well as between any two excited states, of hydrogen-like atoms. The work contains two features which have not previously been fully taken into account: (1) Both spontaneous and induced transition probabilities have been included exactly. (2) In addition to the radiative transitions, the reverse transitions due to absorption of background radiation have also been included. The following graphical results are given: (a) The steady-state occupation probabilities of the excited states as a function of excess electron density. (b) A "sticking probability" $P_n$ (for an electron from a level $n$ to reach the ground state without leaving the atom) as a function of the principal quantum number $n$. (c) The effect of the cascade on the transition rate into the ground state. The calculation is valid for semiconductors and for the analogous astrophysical problem. Temperature dependences have also been studied. The graphs shown bear out quantitatively the expectation that $P_n$ decreases as either the temperature or the principal quantum number increases.