Temperature-induced transition rates for some hydrogenic states

We consider the transition rate β(n,l,T,Z) for a hydrogenlike state characterized by quantum numbers n and l, and nuclear charge Z, induced by a Planck blackbody distribution at temperature T. We expand the Planck distribution in a power series in |E_n-E_n′|/k_BT, where n′ defines one of the set of states representing a transition and where the energies E_n and E_n′ are hydrogenic. We take into account transitions to both bound and continuum states by using sum rules (some not previously reported) for the completeness of hydrogenlike states. The method is not applicable to states of low l and is not accurate for low T. However, for moderate (and high) values of l and T, we obtain analytic upper and lower bounds on β which provide rather accurate estimates of β. The method is quick and convenient as opposed to the straightforward numerical calculation which requires, for contributions of transitions to the continuum, double integration of hypergeometric functions over the Planck distribution. A semiclassical approach is also discussed. Some illustrative numerical results are provided.