On the acoustical asymptotics of the phonon spectrum of a finite crystal

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Abstract
The acoustical asymptotics of the spectral distribution of the vibrations of separate atoms in dependence on their positions in the crystal are considered for the cases of a semispace and a plane‐parallel plate of arbitrary thickness. The general solution of the problem is obtained in the case of an arbitrary orientation of the boundary surfaces with respect to the crystallographic axes. In the elastically isotropie continuum approximation this solution is analysed in detail. For the surface atoms the tensor of the Debye effective frequencies is constructed. The method is taken from the dynamics of a crystal with extended defects which in the case under consideration are the stress‐free plane boundaries and is based on an accurately solving of integral equations such as the Dyson ones. The results obtained are discussed in connection with the mean square fluctuations of the positions and momenta of the crystal atoms determining the intensity and the energy position of the Mössbauer no‐phonon line and the Debye‐Waller factor in the cross section of the low‐energy particles. They are also discussed with respect to the surface contribution to the low‐temperature heat capacity. In particular, the statistical weight of the Rayleigh waves in the scale of the squared frequencies is shown to be proportional to √ω2 as well as the density of the three‐dimensional phonon states. Thus the surface vibrations do not lead to deviations of order T2 from the T3‐behaviour of the lattice thermal capacity. Such deviations are due to the surface not in itself as a two‐dimensional manifold but to its closeness, leading to quantization of the volume vibrations and to appearance of new specific modes.

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