The thermal properties of alkali halide crystals IV. Analysis of thermal expansion measurements

Abstract

In part II the moments $\overline{\nu^n}$ and the low-frequency expansions of lattice frequency distributions were obtained by analyzing experimental heat capacities. We now derive from thermal expansion data the volume dependence of the moments and of the low-frequency coefficients, analyzing the Gruneisen function $\gamma$(T, V) = $\beta$V/$\chi_S$C$_P$. The volume dependence of the moments and of $\Theta^C_0$ is conveniently expressed by the function $\gamma(n)$ = -dln$\nu_D$(n)/dln V, where $\nu_D$(n) = ${\frac{1}{3}(n+3)\overline{\nu^n}}^1/n$. Individual values of $\gamma$(n) obtained by the analysis are useful in estimating the volume dependence of various crystal properties, notably C$_\nu$ and the Debye-Waller effect. The analysis is carried out for NaCl and KCl. The volume dependence of the low-frequency expansion can in principle be obtained from low-temperature data, but experimental accuracy in fact allows an estimate only of $\gamma_0 \equiv \gamma$(-3); the results are thus wholly expressed by $\gamma$(n) curves for each salt. These curves are determined to within a few parts per cent for -2 $\leqslant$ n $\leqslant$ 0, but the uncertainty increases to about 10% for $\gamma$(-3). For n > 0 the uncertainty increases so rapidly that the curves in this range must be considered as only provisional; this is partly because present thermodynamic data fix the volume dependence of $\gamma$(T, V) only between very wide limits. The values obtained for $\gamma$(-3) agree moderately well with values estimated from the pressure dependence of elastic constants; the comparison neither confirms nor rules out the possibility of at least a shallow minimum in the $\gamma$(T) curves at low temperatures. The general shape of the $\gamma$(n) curves is in rough agreement with the predictions of the Kellerman rigid ion model, with a maximum in $\gamma$(n) for n ~$\gtrsim$ 2. For the Kellerman model the fall in $\gamma$(n) as $n \rightarrow \infty$ is shown to be due to the longitudinal optical modes, which at the long-wave limit have individual $\gamma$ values lower than $\gamma$(2) or $\gamma$(4). The primary C$_P$ data for sodium chloride of Morrison & Patterson (1956) are tabulated in an appendix, because they were not given in the original paper.