Moment Calculations in Lattice Dynamics. I. fcc Lattice with Nearest-Neighbor Interactions

A new method, using a high-speed computer, is introduced for the evaluation of large numbers of spectral moments for crystals with short-range interactions. As an example, the first 34 even moments are obtained for the fcc lattice with nearest-neighbor central interactions. Accurate algebraic expressions for the thermodynamic properties have been obtained, at all temperatures, by the use of the Padé approximant on the high-temperature moment expansions, together with three terms in the low-temperature expansion. To estimate the order of magnitude of the errors to be expected, the method is applied to a linear chain and the results compared with the exact solution. The fractional error has a maximum of order 2×${10}^{-4\mathrm{}}$, and is less than 1×${10}^{-5\mathrm{}}$ for much of the range. An approximation to the spectrum of the fcc lattice is derived by an orthogonal polynomial expansion.