Accurate Numerical Method for Calculating Frequency-Distribution Functions in Solids

A new method of calculating absolute phonon frequency-distribution functions, which is an extension of the extrapolation method developed by Gilat and Dolling, is presented for cubic crystals. The method involves dividing the irreducible section of the first Brillouin zone into a cubic mesh and approximating the constant-frequency surfaces inside every small cube by a set of parallel planes. This method proves to be of high precision and resolution in obtaining fine details associated with a given model, and it requires relatively short computing time. Applications have been made to nickel, aluminum, and sodium, for which there exist satisfactory force-constant models. New critical points have been found for Al at $\nu =7.104\mathrm{}\pm 0.006$ THz and for Na at $\nu =2.856\mathrm{}\pm 0.010$ THz. Certain critical points associated with the longitudinal phonon band have been resolved more sharply than in earlier calculations.