On a semi-continuum Green function method for lattice dynamics with application to copper

The present paper describes a phenomenological, semi-continuum Green function method for an analysis of the lattice dynamical data for crystals. A parametric representation of the phonon Green function is constructed from which the frequencies and the polarization vectors of the phonons can be calculated in much the same way as from the dynamical matrix in the Born von Karman model. The long range spatial part of the phonon Green function is approximated by the continuum model and the short range part is defined in terms of certain parameters which can be determined by comparing the calculated and the observed dispersion relations in a way similar to the usual force constant analysis. The present method is both physically and computationally superior to the usual procedure involving the direct construction of the dynamical matrix. Since the Green function is determined directly from the observed phonon data there is no necessity for any a priori restrictive assumptions concerning the nature or range of the interatomic forces prevailing in the crystal and, in fact, by adjusting the parametric fit in the Green function representation rather than in the dynamical matrix, a quite general dynamical matrix is implied in which force constants connect every atom in the lattice. As an illustration, the method is applied to copper (fcc). It is found that the calculated dispersion curves, specific heat and Debye-Waller factor agree quite well with the experimental results with the help of only 7 Green function parameters as compared to a 15 parameter force constant model used in an earlier work.