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.
All Science Journal Classification (ASJC) codes
- Physics and Astronomy(all)