Dynamics of a Semi-Infinite Crystal Lattice in a Quasiharmonic Approximation. I. The Static Equilibrium Configuration of a Semi-Infinite Lattice

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Abstract
This is the first of two papers dealing with the dynamics of semi-infinite crystals. The static equilibrium structure of a semi-infinite crystal at zero temperature is considered in the first paper. The discussion of small-amplitude vibrations of the atoms about their static equilibrium positions is taken up in the second paper. The static equilibrium configuration of a semi-infinite, classical, crystal lattice at zero temperature, having a free boundary, is described in terms of a set of displacements relating the actual positions to those which the lattice particles would assume if the semi-infinite lattice were embedded in an infinite lattice. These displacements are calculated from a set of difference equations which specify the conditions of static equilibrium for the semi-infinite lattice in the absence of external forces. Several unexpected results concerning the structure of the physical boundary region were obtained. These will be shown, in a forthcoming paper, to provide a consistent interpretation of some features of low-energy electron diffraction data, which, until now, could not be interpreted. Of particular interest is the conclusion that the two-dimensional period of the semi-infinite lattice in the planes parallel to the boundary may be larger than that deduced from the bulk lattice structure. The diffraction of low-energy electrons from the surface of single crystals of Ge and Si provides a clear illustration of this effect. It is also shown that the change in the spacing of atomic planes (in the direction normal to the boundary) varies in a nonmonotonic fashion with the distance from the boundary. The effect of appreciable distortions in the surface region is considered in some detail.