Moments of semiclassical and classical absorption and emission band shapes of impurities in solids

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Abstract
General expressions for all moments are calculated for both semiclassical and classical approximations to absorption and emission band shapes due to impurities whose vibrational interactions with the surrounding lattice of a solid contain both linear and quadratic terms. The first five moments are presented and compared to the exact quantum-mechanical results. The semiclassical approximation is found to be most accurate for vibrational interactions having either linear or linear plus quadratic terms with strong coupling and/or high temperature. The semiclassical approximation is rarely, if ever, appropriate if only quadratic terms occur. The classical approximation is a valid representation of the semiclassical approximation at high temperatures. These conclusions are shown to be consistent with the correspondence principle.