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

Mahanty has pointed out that by expanding the imaginary part o f a lattice Green’s function in a sine series, direct numerical treatm ent o f the principal value integral which determines the real part o f a Green’s function o f real argument can be circumvented. Here M ahanty’s result is extended to cover Green’s functions o f purely imaginary arguments. Also, alternative expressions which, in certain cases, sim plify numerical com putations are recorded. Whereas the aforemention­ ed formulas are o f quite general valid ity, an included recurrence relation, which links defect induced changes in mom ents o f the frequency distribution w ith the Fourier coefficients, is re­ stricted to the case o f a mass defect in a diagonally cubic lattice. The prospects o f applying Mahanty’s m ethod to moderately complex polyatom ic crystals are assessed on the basis o f com­ putations pertaining to alkaline earth fluorides. I t is found, through studying simple point de­ fects in CaF2, SrF2, and B aF 2, that by truncating the sine series expansions when associated functionals assume satisfactory values at functions which are constants on the set o f phonon frequencies, one obtains sets o f Fourier coefficients containing sufficient information to cover a variety of defect properties. Also the number o f Fourier coefficients is no larger than to preserve a chief merit o f the method, nam ely the provision o f a convenient way o f condensing and storing information. For the type o f lattices considered M ahanty’s w ay o f evaluating Fourier coefficients is found to be impracticable. However, upon supplem enting the sine series expansion method by an adaptation o f Gilat’s extrapolation procedure for frequency spectra, Brillouin zone integra­ tions involving rapidly varying functions become redundant.