The thermal expansion of alkali halides at low temperatures By G. K. White C.S.I.R.O. Division of Physics, Sydney, Australia (Communicated by K. Mendelssohn, F.R.S.—Received 17 August 1964— Revised 13 January 1965) Linear thermal expansions of eight alkali halides have been determined at liquid oxygen temperatures and at temperatures from 30 °K down to 2 °K. For temperatures T ^ 0/20, where 0 is the Debye temperature, the expansion coefficients are well represented by cl = AT3-}-BT5. Values are reported for the Griineisen parameter y = 3olV/Cx, where C/V is the heat capacity per unit volume and x is the compressibility. For CsBr (b.c.c. structure) y appears to be nearly independent of temperature, with a value of 2*0 but for the other crys tals, which have the rock-salt structure, the parameter y varies with temperature, chiefly between 0/10 and 0/5. At room temperature, y lies between 1-45 and 1*7 but at low temperature this generally decreases to a value y0 which is ca. —0-1 for Rbl, +0*3 for KC1, KBr and KI and 1-0 for NaCl and Nal; LiF does not show this decrease, y0 being 1-7.
Thermal expansion of alkali halides 205 a Debye continuum; 0Q is directly related to the elastic constants and y0 is related to their volume derivatives.
206 G. K. White 2. E xperiments The linear expansions were measured in a differential cell, relative to copper, using the three-terminal capacitance method which has been described before (White 19616; Carr, McCammon & White 1964). Alkali halide crystals present more of an experimental problem than do most metals. They cannot be readily mounted in an absolute cell because the attachment of thermometers and heater imposes undesirable constraints. In the differential celt, the large overall con traction relative to copper in cooling from room temperature reduces the electrical capacitance to be measured and therefore the sensitivity from an optimum figure of parts in 1010 to parts in 109.
Thermal expansion of alkali halides 207 Table 1. Calculation of linear expansion coefficient, a, from measured CHANGE IN CAPACITANCE, A C,RELATIVE TO CAPACITANCE AT 4*2 °K.* 108 A (cm) T cell total T 108 a (°K) 10®A(7 (pF) obs. calib. (KI) (°K) (degK-1) 6-56 32-x 50-3 6-5 56-8 6-83 H i 7-10 50-4 77-5 9-5 87-0 7-30 14-8 7-49 67-e 104-0 12-1 116-! 7-80 18-2 8-11 101-4 156-0 17-2 173-2 8-35 25-5 8-58 137-5 211-5 22-3 233-8 13-24 1270 1954 134 2088 13-93 175 14-61 2015 3098 199 3297 15-36 229 16-11 3078 4732 297 5029 * At 4-2 °K, C = 2-72838 pF and the corresponding sensitivity is 1-5 A per aF (10~6 pF).
208 G. K. White been quite reproducible to the number of figures stated in the table. However, systematic errors arising from uncertainty in the cell calibration (i.e. in the expansivity of copper) and cell geometry may amount to ± 0-5 x 10“7 degK-1, which is less than | % in all these halides except lithium fluoride.
Thermal expansion of alkali halides 209 3-2. Comparison with previous data During the course of this work data extending below 80 °K have been reported by the following: Rubin, Johnston & Altman (1961, 1962) for NaCl and KC1 from 290 °K to ca. 20 °K; Yates & Panter (1962, hereafter referred to as Y. & P.) for LiF, NaCl, KC1, KBr and KI from 270 to ca. 30 °K and by Meincke (1963, also see Meincke & Graham 1963) for NaCl, Nal, KC1 and KBr from 285 to 7 °K. Generally these authors have disagreed by less than 1 % above 100 °K but at lower tempera tures the values of Y. & P. are increasingly smaller. Probable errors in individual values of a are stated by Rubin et al. to be 10-7 degK-1, and by Y. & P. to be 4 x 10-7 degK-1; Meincke states his absolute errors to be greatest between 30 and 90 °K being ± 10-7 degK-1 and falling to ± 5 x 10-8 above 200 °K, and ± 1 x 10-8 near 10 °K.
210 G. K. White (h) Caesium bromide. Previous measurements of Krishnan & Srinivasan (1956) extend only down to 134 °K. At 283 °K we are in agreement and extrapolation of their data to 85 °K gives a value close to that presently observed.
Thermal expansion of alkali halides 211 from a Debye function on the assumption that 6 varies with temperature in a similar fashion to the other alkali halides; this implies a minimum value of 98 °K near 8 °K, 6 rising below this to a value 00 ~ 110 °K (calculated from extrapolated values of elastic constants by Schuele 1963). For caesium bromide the only available figure for CP is that determined at room temperature by Bro listed (1914).
212 G. K. White by extrapolating the room temperature values of Reinitz (1961; see also Schuele t963)- Values for the molar volume V were calculated from density values listed in the International critical tables or in the National Bureau of Standards Circular 539 (U.S. Government Printing Office, Washington).
Thermal expansion of alkali halides 213 Barron et al. (1964) have also pointed out that a small minimum in y may appear at low temperatures. If the long-wavelength transverse modes have very low or negative values of yi} and if these modes are highly dispersive so that their relative contribution to the thermodynamic properties increases as the temperature rises, then this could lead to an initial decrease in y. Such a mechanism has been discussed by Blackman (1959) and by Arenstein, Hatcher & Neuberger (1963). Very deep minima in y, centred near 61 12, have been observed in some diamond- structure solids (e.g. Collins & White loc. cit.) for which the dispersion of the trans verse acoustic waves is large. However, the present observations certainly show no marked minima and indeed they suggest that if a significant minimum in y occurs for the alkali halides, it must occur at rather lower temperatures (say T ^ 6/30) than in the diamond structure solids and much lower than those at which the mini mum in 6 is observed. This is not unlikely but it is desirable to have calculations on a rather more realistic model than that of the rigid-ion or considerably more sensitive experiments before we can be certain of this. The inadequacy of the rigid-ion model to represent the measured dispersion relations has been shown clearly by Cowley et al. (1963) and Karo & Hardy (1963). They have shown that polarisation effects must be taken into account in a model which seeks to explain adequately the various thermodynamic properties.