Moment Singularity Analysis of Vibration Spectra

The frequency distribution of a crystal is approximated by combining Van Hove's determination of its analytical nature and Montroll's method of moments. The function G(ω²) is represented by an expression with the correct behavior at the singularities and at the maximum and minimum frequencies. The behavior between singular points is adjusted smoothly by leaving n undetermined parameters. These parameters are then fixed by using the correct first n moments. As a test, this procedure was applied to the two-dimensional square lattice with nearest and next nearest neighbor interactions, solved exactly for a particular case by Montroll. The approximated distribution function had the right form at the end points, contained terms of the appropriate logarithmic form, and a jump function (with known coefficients). It also included Legendre polynomials with unknown coefficients, which were determined by moments. The difference between the exact and approximate distribution functions was a few percent using only the zeroth moment (normalization). Using higher moments produced a gradual increase in accuracy.