Closed expressions for macroscopic parameters in the method of induced corepresentations in lattice dynamics
Abstract
The general problem of calculating the frequencies is split into two. In the first part we establish the form of the Hamiltonian H(F,K) which would ensure the most reduced form of the matrix employed to obtain the secular equation for determining the eigenfrequencies. The second part of the problem consists in developing methods of solving the secular equations that would give the correct number of frequencies even with approximate numerical calculations. In this paper we consider the first part of the problem. Its complexity is due to the fact that, on the one hand, the ICR matrices can be taken in substantially different forms (small b-type irreducible corepresentations can be taken in two forms while the forms of complete ICR's of all types depend essentially on how the basis vectors are introduced) and that, on the other hand, only with a fortuitous selection of the forms of the ICR's can we confine ourselves to considering only the coordinates that pertain to the first rows of the complete ICR's. “Necessary” forms of the complete ICR's are established for all possible vectors k (stars K) and types of ICR. In each case invariant second-order combinations are constructed from the displacement coordinates, transforming according to an ICR. The sums of these combinations with indeterminate coefficients are equated to the corresponding H(F,K). The macroscopic parameters (the coefficients of the invariant combinational) are thus expressed in terms of microscopic parameters (the elements of the dynamical matrix). The symmetry of Φ + KΦ is taken into account completely and in the easiest possible way. The procedure for determining the macroscopic parameters was taken from a paper on magnetic phase transitions published by this author in 1980, and can be applied in the theories of spin waves, excitons, etc.