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. ʘ0and the coefficients of the terms inv2,v4andv6in the low-frequency expansion of the spectrum have been derived from the data for the regionT< ʘD/20. The values of ʘ0agree 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 ʘD/6 yield ʘ∞and the first three even moments of the spectrum (μ2,μ4, andμ6) when fitted to the Thirring expansion for ʘD. 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 withμ1and the geometric mean frequencyvg, have been derived from integrals of the form ʃF0(Cv/Ts)dT, with an accuracy comparable to that of the primary experimental heat capacities. Explicit spectra have not been com­puted, but insteadvg, Θ0and theμnhave all been correlated in a graph of the functionvD(n)= {1/3(n+ 3)μn}1/n. Potassium bromide is used as an illustrative example. The sharp curvature of the functionvD(n)for negative values ofnindicates that moments forn< — 1 give critical information about the form of the spectrum. The zero-point energies of the crystals have been calculated from μ1and 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.