The thermal properties of alkali halide crystals II. Analysis of experimental results

Abstract

The results reported in part I, together with similar results for sodium chloride, have been analyzed in terms of the spectrum of harmonic vibrations of the crystalline lattice; since the zero-point energy proves to be small, the analysis should conform fairly closely to that of a static lattice. ʘ₀ and the coefficients of the terms in v², v⁴ and v⁶ in the low-frequency expansion of the spectrum have been derived from the data for the region T < ʘ_D/20. The values of ʘ₀ agree well with ʘ (elastic) calculated from the elastic properties of the crystals. After correction for thermal expansion, the results in the temperature range immediately above ʘ_D/6 yield ʘ_∞ and the first three even moments of the spectrum (μ₂, μ₄, and μ₆) when fitted to the Thirring expansion for ʘ_D. For the three potassium salts, and again for the two sodium salts, the ratio ʘ₀/ʘ_∞ appears to depend almost entirely upon the mass ratio of the ions. Values of this ratio suggest that the type of interatomic force is determined primarily by the alkali ion. Negative moments of the spectrum, together with μ₁ and the geometric mean frequency v_g, have been derived from integrals of the form ʃ^F₀ (C_v/T^s)dT, with an accuracy comparable to that of the primary experimental heat capacities. Explicit spectra have not been com­puted, but instead v_g, Θ₀ and the μ_n have all been correlated in a graph of the function v_D(n) = {1/3(n + 3)μ_n}^1/n. Potassium bromide is used as an illustrative example. The sharp curvature of the function v_D(n) for negative values of n indicates that moments for n < — 1 give critical information about the form of the spectrum. The zero-point energies of the crystals have been calculated from μ₁ and compared with values derived by the approximate method of Domb & Salter (1952). The estimated increase in the volume of the crystal caused by zero-point energy ranges from 0.23% for potassium iodide to 0.37% for sodium chloride. By subtracting the heat capacity given by the Thirring expansion we may estimate the effect of anharmonicity of the vibrations. This seems to be roughly determined by the ratio of the amplitude of atomic vibrations to the interatomic distance.