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采用多频模型计算多声子跃迁几率时,由于能量守恒的要求,按照严格的理论,跃迁几率作为跃迁能量的函数,是在跃迁能量为最小声子能量整倍数处的一系列δ函数组成的。然而用最陡下降法所得的跃迁几率则是跃迁能量的平滑函数。本文指出这两者的联系和差异,并且通过对多频模型中最陡下降法的进一步探讨,对问题作出统一的理论说明。所获得的理论结果表明,最陡下降法是特别适用于多频模型近似连续声子频谱的一种计算方法。For the practical calculation of multiphonon transition probabilities, we have introduced the concept of multi-frequency models, whereby the phonon frequency spectrum is divided into equal intervals of ω0 and the corresponding phonon energies are approximated by integral multiples of hω0. The present paper investigates the method of steepest descent as applied to such calculations with multi-frequency madels. It is fo-and the that the sum of the contributions from the infinite relevant saddle points gives a series of δ-functions situated at phonon transition energies equal to integral multiples of hω0. This singular structure reflects the discrete structure artificially introduced by the multi-frequency model. Averaging the result with respect to phonon transition energy over hω0 just serves to smooth out the artificially introduced siugular structure; this smoothed result is directly given by the contribution from a single saddle point.
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