Open AccessMETHOD OF STEEPEST DESCENTS AND MULTI-FREQUENCY MODELS IN MULTIPHONON TRANSITIONS
Author(s) -
Kun Huang,
Dongyuan Liu
Publication year - 1985
Publication title -
wuli xuebao
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.199
H-Index - 47
ISSN - 1000-3290
DOI - 10.7498/aps.34.709
Subject(s) - saddle point , method of steepest descent , phonon , saddle , physics , energy (signal processing) , statistical physics , gradient descent , mathematical analysis , mathematics , quantum mechanics , computer science , geometry , mathematical optimization , machine learning , artificial neural network
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.