Temperature-Induced Changes in the Elastic Properties of Sodium Chloride Single Crystals

A. V. Sharko, A. A. Botaki

(Recommended for publication by the Department of General Physics of TPI)

The study of the elastic properties of crystals is of great importance for the development of the microscopic theory of the properties of solids. Despite the extensive work carried out by both theorists [1, 2] and experimentalists, the regularities in the change of mechanical properties with the chemical composition of a series of alkali-metal halides have as yet been little studied. The data available in the literature from experimental studies of temperature-induced changes in the elastic properties of single crystals of alkali-metal halides are of little use for revealing such dependences, since they were obtained by different authors and under different conditions (the only exception being the works of A. V. Stepanov and S. P. Nikanorov with co-workers [3—6], who measured temperature-induced changes in the elastic properties of most single crystals of alkali-metal halides at temperatures above room temperature).

The authors of the present work set themselves the aim of systematically studying temperature-induced changes in the elastic properties of alkali-metal-halide single crystals at temperatures below room temperature. The present work is devoted to the study of the elastic properties of sodium chloride single crystals by the pulsed ultrasonic method. The velocities of propagation of ultrasonic vibrations were measured with a DUK-6V pulsed ultrasonic flaw detector. Using the method of relative measurements, the following were determined: \(C_{l\infty}\), the velocity of propagation of longitudinal waves in an unbounded medium; \(C_l\), the velocity of propagation of longitudinal waves in a thin rod; and \(C_\perp\), the velocity of propagation of transverse waves. A medium is considered unbounded if its transverse dimensions are much greater than the wavelength of the ultrasonic vibrations, and a specimen is considered thin if its transverse dimensions are smaller than the wavelength of the elastic waves in the specimen.

The essence of the relative-measurement method is as follows. From the scale of the depth-measuring device of the flaw detector, calibrated using a reference specimen with a known ultrasonic propagation velocity \(C_0\), the apparent length \(l'\) of the specimen under test was determined; its true length \(l_0\) was determined by direct measurements. Denoting by \(C\) and \(C_0\), respectively, the velocities of propagation of elastic waves in the investigated and reference specimens, and by \(n\) the number of the multiple reflection of the echo pulse, the desired value of the velocity \(C\) can be determined from the relation

\[ l' = n \frac{C_0}{C} l_0 . \tag{1} \]

Specimens for measuring the elastic constants were made from natural single crystals of sodium chloride and from crystals grown artificially by the Kyropoulos method in the crystallization laboratory of TPI from chemically pure starting material. The specimens had the form of rectangular bars with faces coinciding with the cleavage planes of the single crystal. When recording the temperature dependence of the ultrasound propagation velocity, the specimen was glued to the probe head of the flaw detector with a mixture consisting of 88% ED-5 epoxy resin and 12% polyethylene polyamine and was placed in a cryostat. In preparing the specimens for measurements, special attention was paid to the quality and parallelism of the working faces of the specimen, as well as to their coincidence with the \(\{100\}\) planes. The working faces had a finely ground surface followed by additional polishing. In order to eliminate internal stresses and the effects of mechanical treatment, before the final finishing of the specimen surfaces the latter were annealed at a temperature of \(600^\circ\) for 24 hours, followed by slow cooling. The specimens for measurements of \(C_{l\infty}\) and \(C_{\perp}\) had dimensions \(80\times 40\times 40\) mm, and for measurements of \(C_l\), \(6\times 6\times 25\) mm. The density of sodium chloride single crystals for different temperatures [7] was determined from the formula

\[ \rho(t)=2.1680\left[1-1.12\cdot 10^{-4}t-5\cdot 10^{-8}t^2\right], \]

where \(t\) is the temperature in \({}^\circ\mathrm{C}\).

The change in the true length of the specimen \(l(t)\) as a function of temperature was calculated by the formula [8]

\[ l(t)=l_0\left[1+\int_t^{t_0}\alpha(t)\,dt\right], \]

where \(\alpha(t)\) is the coefficient of linear expansion, which, according to Henglein [9], can be determined from the formula

\[ \alpha(t)=\frac{\rho(t)-\rho_0}{\rho(t)\cdot t}, \tag{3} \]

where \(\rho(t)\) and \(\rho_0\) are the densities at temperatures \(t\) and \(0^\circ\mathrm{C}\), respectively. The Young’s modulus in the \(\langle 100\rangle\) direction and the elastic constant \(C_{11}\) were determined from the measured values of \(C_{l\infty}\) and \(C_l\) using the known formulas

\[ C_{l\infty}=\sqrt{\frac{C_{11}}{\rho}};\qquad C_l=\sqrt{\frac{E_{100}}{\rho}}. \tag{4} \]

To find the third independent constant—the shear modulus \(G\)—pulses of transverse waves were sent into the specimen; for this purpose prismatic probe heads at a frequency of \(2.5\) MHz with an ultrasonic-vibration entry angle of \(50^\circ\) were used, which made it possible to transform longitudinal waves into shear waves. In the general case, when an ultrasonic wave is incident on a plane-parallel plate at a certain angle to the normal, both longitudinal and transverse waves arise in the specimen. At the chosen angle of incidence of the ultrasound, total internal reflection of the longitudinal waves occurs, and only transverse waves propagate in the medium under investigation, directed at a certain angle with respect to the crystallographic axes. Taking this angle into account made it possible to determine the orientation factor \(F(\gamma)\) [10]

\[ F(\gamma)=\gamma_1^2\gamma_2^2+\gamma_1^2\gamma_3^2+\gamma_2^2\gamma_3^2, \tag{5} \]

where \(\gamma\) are the direction cosines of the angles formed by the direction of propagation of the transverse waves with the crystallographic axes.

Using the formula \(C_{\perp}=\sqrt{G/\rho}\), from the measured values of \(C_{\perp}\) we find \(G\) in the direction of propagation of the shear waves.

The calculation of the elastic constants \(C_{ik}\) and the elastic compliances \(S_{ik}\) was carried out using the well-known formulas [11]:

\[ S_{11}=\frac{1}{E_{100}}; \]

\[ C_{11}=\frac{S_{11}+S_{12}}{(S_{11}-S_{12})(S_{11}+2S_{12})}; \]

\[ \frac{1}{G}=S_{44}+4\left[(S_{11}-S_{12})-\frac{1}{2}S_{44}\right]F(\gamma); \]

\[ C_{44}=\frac{1}{S_{44}}; \tag{6} \]

\[ C_{12}=\frac{-S_{12}}{(S_{11}-S_{12})(S_{11}+2S_{12})}. \]

The Poisson coefficient was found from the expression [12]

Table 1

\[ \sigma=\frac{1-2\left(\dfrac{C_{\perp}}{C_{l\infty}}\right)^2}{2-2\left(\dfrac{C_{\perp}}{C_{l\infty}}\right)^2}. \tag{7} \]

The measurement data and the elastic constants and anisotropy factors calculated from them, \((S_{11}-S_{12})-\frac{1}{2}S_{44}\) and \(C_{44}; \frac{C_{11}-C_{12}}{2}\), as functions of temperature are given in Tables 1, 2, and 3. In addition, the tables give the temperature changes in the compressibility \(\chi=3(S_{11}+2S_{12})\), the bulk modulus \(\frac{C_{11}+2C_{12}}{3}\), and the measure of resistance to deformation caused by a shearing stress applied in the \(\{110\}\) plane in the \(\langle 100\rangle\) direction, \(\frac{C_{11}-C_{12}}{2}\); plots of these dependences are shown in Fig. 1. Extrapolation of the experimental values of the temperature dependences of the elastic constants to \(0^\circ\)K gives the following values:

Fig. 1.

\[ E_{100}=5.309\cdot10^{11}\ \frac{\mathrm{dyn}}{\mathrm{cm}^{2}}; \qquad G_{100}=C_{44}=1.310\cdot10^{11}\ \frac{\mathrm{dyn}}{\mathrm{cm}^{2}}; \]

\[ C_{11}=5.625\cdot10^{11}\ \frac{\mathrm{dyn}}{\mathrm{cm}^{2}}; \qquad C_{12}=0.973\cdot10^{11}\ \frac{\mathrm{dyn}}{\mathrm{cm}^{2}}; \]

\[ S_{11}=0.1874\cdot10^{-11}\ \frac{\mathrm{cm}^{2}}{\mathrm{dyn}}; \qquad S_{12}=0.0290\cdot10^{-11}\ \frac{\mathrm{cm}^{2}}{\mathrm{dyn}}, \]

\[ S_{44}=0.7368\cdot10^{-11}\ \frac{\mathrm{cm}^{2}}{\mathrm{dyn}}, \]

which is in good agreement with the values for \(0^\circ\)K obtained by Overton and Swim [8].

The determination of the elastic constants, as well as the recording of their low-temperature dependences, was carried out in parallel on natural and artificial—

Table 2

in artificially grown NaCl single crystals. The measurement results showed comparatively good agreement on both specimens, with a certain lowering of the elastic moduli in the rock-salt single crystal, which is apparently explained by the higher dislocation density in the latter. Repeated measurements gave better stability of the measurement results on the artificially grown crystal than on rock salt; therefore the table gives the results of measurements on the artificially grown single crystal.

Table 3

References

1. Huntington. Solid State Phys., 7, 213, 1958.
2. K. A. Aleksandrov, T. V. Ryzheva. Kristallografiya, 6, 289, 1961.
3. A. V. Stepanov and I. M. Eidus. ZhETF, 29, 669, 1955.
4. S. P. Nikanorov, A. V. Stepanov. FTT, 6, 1987, 1964.
5. S. P. Nikanorov, A. V. Stepanov. ZhETF, 37, 1814, 1959.
6. Yu. M. Chernov, A. V. Stepanov. FTT, 3, 2872, 1961.
7. International Critical Tables of Numerical Data (Mc—Graw—Hill Company N. V.), vol. II, p. 43, 1928.
8. Overton Jr. and Swim R. T. Phys. Rev., 84, 758, 1951.
9. Fr. A. Henglein. Zeit Physik Chemie, 115, 91, 1925.
10. B. Boren. Defects in Crystals. Moscow, IL, p. 27, 1969.
11. K. Zener. Elasticity and Anelasticity of Metals. Collection edited by S. V. Vonsovsky. Moscow, IL, p. 22, 1954.
12. Nondestructive Testing (handbook edited by R. McMaster). Moscow, IL, vol. 2, p. 269, 1965.