Phonon Bandwidth and Rate Equations in Avalanche Relaxation

Abstract

The bandwidth of phonons emitted in avalanche spin-phonon relaxation processes is studied when the phonon interruption time is shorter than spin-spin relaxation time. The problem has been solved by the use of second-order Heisenberg equations of motion for phonon-number operators of each lattice mode. It is shown that the usual rate equations, based on the assumption of time-independent bandwidth, are not adequate in these circumstances and that the bandwidth of phonons emitted in the relaxation process decreases during the avalanche. Rough numerical estimates in a typical case suggest that the bandwidth of the emitted phonons, at times at which phonon generation may be considered as practically concluded, may still be larger than the spin-resonance linewidth. Finally, it is emphasized that the results obtained are confirmed by considering the relaxation process in the light of the energy-time uncertainty principle.