Prog. Theor. Phys. Supplement No.23 (1962) pp. 185-206
Time-Dependent Properties of an Isotopically Disordered Lattice
Department of Physics, Faculty of Science, Hokkaido University, Sapporo
An investigation is made of the dynamical and the statistical dynamical behaviors of a harmonic crystal lattice containing isotopes. From equations of motion for the displacement of atoms from their equilibrium positions integral equations are derived on the one hand, and exact expressions for propagators or velocity correlation functions are obtained on the other hand, through the cosine transform of the spectra of the correlation functions. In the latter case, the results are applied to one- and two-impurity problems. It is shown that a close relation exists between the spectra and the frequency spectrum, particularly in the case of a random distribution of isotopes; the auto-correlation function is evaluated from the frequency spectrum.
The short time behavior is investigated from solutions of the integral equation which are obtained by using the method of iteration, and its relation to perturbation calculations and the moment method for the construction of the frequency spectrum is elucidated. While for the study of the long-time behavior, the effect of singularities of the spectra is examined. For a few-impurity problem, particular attention is paid to two typical singularities, the end singularity at the top of the main continuum of the band and the delta-function like spectrum associated with the localized modes. For a random lattice, the long-time behavior is shown to be determined by the end singularities, the exact determination of which is the most difficult problem of perturbation calculations.
Various formal results are applied to one- and three-dimensional lattices.
DOI : 10.1143/PTPS.23.185
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Citing Article(s) :
Progress of Theoretical Physics Vol. 29 No. 2 (1963) pp. 328-330
Frequency Spectrum of a Lattice with Heavy Isotopes
Progress of Theoretical Physics Vol. 33 No. 3 (1965) pp. 363-379
A High-Frequency Resonant Mode in Lattice Vibrations