On the Computation of Lattice Green's Functions by Sine Series Expansions

Abstract

Abstract Mahanty has pointed out that by expanding the imaginary part of a lattice Green’s function in a sine series, direct numerical treatment of the principal value integral which determines the real part of a Green’s function of real argument can be circumvented. Here Mahanty’s result is extended to cover Green’s functions of purely imaginary arguments. Also, alternative expressions which, in certain cases, simplify numerical computations are recorded. Whereas the aforemention­ed formulas are of quite general validity, an included recurrence relation, which links defect induced changes in moments of the frequency distribution with the Fourier coefficients, is re­stricted to the case of a mass defect in a diagonally cubic lattice. The prospects of applying Mahanty’s method to moderately complex polyatomic crystals are assessed on the basis of com­putations pertaining to alkaline earth fluorides. It is found, through studying simple point de­fects in CaF2, SrF2, and BaF2, that by truncating the sine series expansions when associated functionals assume satisfactory values at functions which are constants on the set of phonon frequencies, one obtains sets of Fourier coefficients containing sufficient information to cover a variety of defect properties. Also the number of Fourier coefficients is no larger than to preserve a chief merit of the method, namely the provision of a convenient way of condensing and storing information. For the type of lattices considered Mahanty’s way of evaluating Fourier coefficients is found to be impracticable. However, upon supplementing the sine series expansion method by an adaptation of Gilat’s extrapolation procedure for frequency spectra, Brillouin zone integra­tions involving rapidly varying functions become redundant