Spin‐dependent Cyclotron Decay Rates in Strong Magnetic Fields

Abstract

Cyclotron decay and absorptionrates have been well studied in the literature, focusing primarilyon spectral, angular, and polarization dependence. Astrophysicalapplications usually do not require retention of information on theelectron spin state, and these are normally averaged in obtaining therequisite rates. In magnetic fields, higher order quantum processes suchas Compton scattering become resonant at the cyclotron frequencyand its harmonics, with the resonances being formally divergent.Such divergences are usually eliminated by accounting for the finitelifetimes of excited Landau states. This practice requires the use ofspin-dependent cyclotron rates in order to obtain accurate determinations ofprocess rates very near cyclotronic resonances, the phase-space domainmost relevant for certain applications to pulsar models. This paperdevelops previous results in the literature to obtain compact analyticexpressions for cyclotron decay rates/widths in terms of a series ofLegendre functions of the second kind; these expressions can be usedexpediently in astrophysical models. The rates are derived using twopopular eigenstate formalisms, namely, that due to Sokolov & Ternovand that due to Johnson & Lippmann. These constitute two sets ofeigenfunctions of the Dirac equation that diagonalize different operators andaccordingly yield different spin-dependent cyclotron rates. This paperillustrates the attractive Lorentz transformation characteristics ofthe Sokolov & Ternov formulation, which is another reason why it ispreferable when electron spin information must be explicitly retained.