Quantum Statistical Corrections to Astrophysical Photodisintegration Rates

Tabulated rates for astrophysical photodisintegration reactions make use of Boltzmann statistics for the photons involved as well as the interacting nuclei. Here we derive analytic corrections for the Planck-spectrum quantum statistics of the photon energy distribution. These corrections can be deduced directly from the detailed-balance condition without the assumption of equilibrium as long as the photons are represented by a Planck spectrum. Moreover we show that these corrections affect not only the photodisintegration rates but also modify the conditions of nuclear statistical equilibrium as represented in the Saha equation. We deduce new analytic corrections to the classical Maxwell-Boltzmann statistics which can easily be added to the reverse reaction rates of existing reaction network tabulations. We show that the effects of quantum statistics, though generally quite small, always tend to speed up photodisintegration rates and are largest for nuclei and environments for which Q/kT ~ 1. As an illustration, we examine possible effects of these corrections on the r-process, the rp-process, explosive silicon burning, the $\gamma$-process and big bang nucleosynthesis. We find that in most cases one is quite justified in neglecting these corrections. The correction is largest for reactions near the drip line for an r-process with very high neutron density, or an rp-process at high-temperature.