I. Phys Chem. solids Vol. 41, pp. 1019-1022 Pergamon Press Ltd. 1980. Printed in Great Bntain
THERMAL EXPANSION OF LiF AT HIGH PRESSURES?
REINHARDBOEHLERand GEORGEC. KENNEDY* Institute of Geophysics and Planetary Physics, University of California, Los Angeles,CA 90024,U.S.A. (Received 28 January 1980; accepted 25 April 1980)
Abstract-Accurate data on the pressure-volume temperature relationship of LiF are presented and compared with recent X-ray measurements by Yagi[l]. Our measurement is a differential length change measurement between a 25 mm long single crystal of LiF and a tungsten carbide standard as previously described by Boehler and Kennedy[Z]. A piston-cylinder apparatus with fluid cell arrangement was used up to 32 kbar and WC. In spite of the low compressibility of LiF our volume data are accurate within 0.07%. We find a much stronger decline of the thermal expansivity with pressure than Yagi. The values of (I, . KT = (S/U), remain constant within the uncertainty over our pressure and temperature range.
in the results of theoretical deductions of equations of state from finite strain theory or lattice dynamics is very often due to lack of accuracy of input data such as bulk modulus data and assumptions made on the behavior of the thermal expansivity at high pressures. We have also shown that the accurate determination of the Grtineisen parameter at high pressures is only possible when accurate bulk modulus data are available. The bulk modulus can be precisely determined from ultrasonic sound velocity measurements. The pressure range, however, is limited to a few kilobars because of the failure of the transducer-sample bond at higher pressures. In addition the conversion of adiabatic to isothermal data requires the heat capacity and thermal expansion as a function of temperature. The reliable prediction of high pressure compression data by the extrapolation of low pressure ultrasonics is difficult, because the results depend strongly on the choice of the equation used to fit the measured data. The isothermal bulk modulus can also be determined by the differentiation of volume compression data which are usually available up to about 40 kbars. In most cases, however, these measurements are limited to room temperature and the accuracy is not high enough to reliably predict the bulk modulus at high pressures. X-Ray technique allows volume measurements over a large pressure and temperature range. The accuracy of the data, however, is limited due by various factors: The shift of the diffraction lines due to pressure is relatively small compared to the width of the line. Due to absorption in the sample one has to apply an angle correction which is density dependent. The pressure is determined by the use of an internal pressure standard for which the equation of state may be imprecisely known at very high pressures and high temperatures. This is discussed in a recent study by Boehler and Kennedy. Another source of error in X-ray Inconsistancy
tPublication No. 1966,Institute of Geophysicsand Planetary Physics Universityof California,Los Angeles,CA 90024,U.S.A. *Deceased 18March 1980. PCS Vol 41. No. IO-A
measurements using solid high pressure cells is non hydrostaticity in the sample. This problem has been discussed in detail by Sato et a1. and by Singh and Kennedy[6,71. Boehler and Kennedy present a method for the accurate measurement of compressibility and thermal expansion at hydrostatic pressures up to 30 kbars and temperatures up to 500°C. The method is a differential length change measurement between an incompressible material such as tungsten carbide and a more compressible sample. Our results on sodium chloride show higher accuracy in the compression data at high pressures than extrapolated ultrasonic data. The small errors in our measurement are associated with the original length measurement of sample and standard and the uncertainty in the pressure measurement. In the present study of LiF we continue our research on the pressure-volume-temperature relationship of alkali halides. Our results are compared with those of Yagi[l] from X-ray measurements.
The measurements were carried out on various single crystals of LiF of optical quality (Harshaw). The lengths of the specimens are shown in Table 1. We did not observe any change in length before and after a pressure-temperature cycle and we therefore assume that the physical properties of our samples were not altered by the pressure and temperature treatment. The results of our measurements are shown in Table 1 and Fig. 1. At room temperature our compression data agree with those by Yagi[l] and by Vaidya and Kennedy within the experimental error. The dashed line at 300°C shows Yagi’s measurements. The estimate of the absolute errors in pressure, temperature and volume are discussed previously. Our pressures are accurate by 0.4%, the temperatures by 0.3”C and the uncertainty in volume of LiF is 50.07%. Table 2 shows the bulk moduli and their pressure derivatives from a first order Mumaghan and a first order Birch fit of the
BOEHLER and G. C. KENNEDY
Table 1. Length change measurements of LiF Length at 1 atm &I25'C [mn]
1 .9993 .9988 .9984 .9982
20.65 28.20 LiF UC
3.09 8.60 16.18 23.80 31.3)
.9953 .9947 .9941 .9937 .9935
6.83 12.44 19.96 27.52
.9898 .9892 .9886 .9882
8.98 14.J 21.84 29.46
.9868 .9863 .9858 .9853
13.97 19.46 27.01
.9799 .9793 .9787
18.12 23.64 31.12
.9743 .9737 -9732
Table 2. Bulk
1%X 200 25
of LiF and their pressure derivatives. A first order Murnaghan equation (in brackets) were fitted to the experimental data B,(kbar)
and a first order
respectively. A higher order Murnag’han equation did not improve the fit. When we extrapolate our data to higher
pressures using eqns (1) and (2) we find a considerable disagreement between our data and Yagi’s estimates. Our extrapolation yields a 0.7% higher volume at 90 kbar (Yagi’s pressure range) or at the same volume a pressure of 97 kbar. There could be two reasons for this discrepancy: Long extrapolations can yield serious errors because the results depend on the equation used to present the data. For LiF the result is the same if both the Murnaghan or the Birch equation are used. The use of another equation would not improve the reliability of our estimate. The second reason could be an error in the pressure estimate using NaCl as internal standard in X-ray measurements. At 30 kbars we observe that the
Thermal expansion of LiF at high pressures
Fig. I. Compression of LiF as a function of pressure and temperature.
use of the Decker equation yields pressures that are about 0.5 kbar lower than our estimates. This is also reflected in Yagi’s measurements at room temperature. At 90 kbar the underestimate in pressure using Decker’s equation of state would be about 3 kbar. In Fig. 2 the bulk modulus is plotted vs temperature for 0 and 30 kbars and for constant volume together with Yagi’s one atmosphere data. The values of K,, in the literature agree well with one another whereas dK/dT varies from -0.7 kbarpC to -0.3 kba$‘C[l]. We find values of dK/dT of -0.39 and -0.43 at 0 and 30 kbar, respectively. In Table 3 we show the volume of LiF as a function of pressure and temperature. These values are obtained by integrating eqn (1). From these volume data we calculated the pressure and temperature dependence of the volume coefficient of thermal expansion, (Y,.Essentially (Y,is a linear function of pressure and temperature. The results are shown in Table 4. Our values of (&r/tU+ are considerably higher than those determined by Yagi. A possible explanation for this could be the large effect of pressure on the platinum-rhodium emf of thermocouples[lO]. Yagi did not correct for this effect. The maximum correction for the platinum-rhodium
300 T (“C)
Fig. 2. Isothermal bulk modulus of LiF as a function of pressure and temperature.
thermocouple at 30 kbar and 300°C is +8”C. The deviation of Yagi’s data at 300°Cfrom ours 18°C.The correction of chromel-alumel thermocouples in our experiment is negligible in this pressure and temperature range. The Griineisen parameter as a function of pressure can be calculated from our data using the relationship
. KT “‘P.
A Dulong-Petit value for C, can be taken for temperatures above the Debye temperature. For LF the Debye temperature is high (U&‘C) and y behaves anomalous with pressure when calculated from (3). y first increases and then decreases with pressure.
Table 3. V/V,, of LiF at high pressures and high temperatures. Thermal expansion data by Kirby et al. were used to calculate the V/V, values at one atmosphere
R. BOEHLER and G. C. KENNEDY
Table 4. Pressure and temperature dependence of thermal
or.0 x 10’ (et-’ )
‘p I n x 10’ (V’ )
The pressure dependence of the Griineisen parameter has been directly measured for a number of substances using adiabatic pressure pulse technique. In a
shown that q=-(aIny/JIn
is close to unity for many substances which is identical to the above observations.
@lr x to6 ('C-l kbrr-') ragi
Another interesting thermodynamic quantity in equations of state is a, . KT = (JP/t9T), Yagi shows that a. * KT remains nearly constant for 4 alkali halides over a large pressure and temperature range. This is in very good agreement with our results on LiF even though our value of a, . KT is somewhat lower than Yagi’s. In a recent study Anderson [ 1l] shows that y - p = const from seismic data for the lower mantle of the earth, where y is the Griineisen parameter and p is the density. From this follows that a, . KT is independent of pressure and
Acknowledgements-Financial support from the Department of Energy EY-76-S-03-0034and NSF EAR79-083111is gratefully acknowledged.
1. Yagi T., J. Phys. Chem. Solids 39, 563 (1978). 2. Boehler R. and Kennedy G. C., J. Phys. Chem. Solids 41, 517 (1980). 3. Boehler R. and Ramakrishnan J., 1. Geophys. Res. (to be published). 4. Birch F., J. Geophys. Res. 83, 1257(1978). 5. Sato Y., Yagi T., Ida Y. and Akimoto S., High Temp.-High Pressures 7, 315 (1975). 6. Singh A. K. and Kennedy G. C., J. Appl. Phys. 45, 4686 (1974). 7. Singh A. K. and Kennedy G. C., 1. Appl. Phys. 47, 3337
(1976). 8. Vaidya S. N. and Kennedy G. C., J. Phys. Chem. So/ids 32, 951 (1971). 9. Chernov Y. M. and Stepanov A. V., Sou. Phys. Solid State 3, 2097 (l%l). 10. Getting I. C. and Kennedy G. C., J. Appl. Phys. 41, 4552 (1970). 11. Anderson 0. L., L Goephys. Res. 84, 3537(1979).