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

Abstract

In part II the moments v 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 Grüneisen function γ ( T , V ) = β V / χ s C p . The volume dependence of the moments and of Θ 0 C is conveniently expressed by the function γ ( n ) = − dln v D ( n ) / d ln ⁡ V , where v D ( n ) = 1 2 ( n + 3 ) v n ¯ 1 / n . Individual values of γ( n ) obtained by the analysis are useful in estimating the volume dependence of various crystal properties, notably C V 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 γ 0 ≡ γ( — 3); the results are thus wholly expressed by γ( n ) curves for each salt. These curves are determined to within a few parts per cent for — 2 ≤ n ≤ 0, but the uncertainty increases to about 10% for γ( — 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 γ( T, V ) only between very wide limits. The values obtained for y( —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 γ( T ) curves at low temperatures. The general shape of the γ( n ) carves is in rough agreement with the predictions of the Kellerman rigid ion model, with a maximum in γ( n ) for n > 2. For the Kellerman model the fall in γ( n ) as n → ∞ is shown to be due to the longitudinal optical modes, which at the long-wave limit have individual γ values lower than γ(2) or γ(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.