Relationship of Temperature-Induced Changes in Mechanical Properties to Other Physical Characteristics of Alkali Metal Halide Single Crystals

A. A. Botaki, Yu. Ya. Popova, V. N. Suslova, A. V. Sharko

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

The elastic moduli of a solid are always determined by the resistance that the solid offers to a forced change in the equilibrium interionic distance in the crystal lattice. Consequently, the characteristics of the elastic properties of a solid directly evaluate the forces of interionic interaction of the particles of the crystal lattice. Since many physical properties of solids are determined by the interaction forces of the microparticles of a solid, the connection between these properties and the mechanical characteristics of a solid is quite evident. No less important is the mechanical characteristic of the inelastic properties of a solid—internal friction. This parameter is very sensitive to distortions of the crystal lattice, to processes accompanied by changes in local disturbances of the crystal lattice, to the migration of impurities during deformation, etc.

All that has been said makes it promising to study the elastic moduli and internal friction, and their temperature-induced changes, for the investigation of certain physical processes in solids.

Relationship of the Energy of the Crystal Lattice to the Temperature-Induced Change in Young’s Modulus

Measurements performed on single crystals of KJ, KBr, KCl, NaBr, NaCl, LiF [1] showed that, in addition to a linear dependence between temperature and Young’s modulus \(E\), there is also a linear relation between the energy of the crystal lattice \(U\) and the tangent of the slope angle of the straight lines \(E_{100}=f(t)\). This makes it possible to write the equation of a straight line in the coordinates \(\dfrac{dE}{dt},\, U\)

\[ U=\left(-6.70\cdot 10^{-3}\frac{dE}{dt}+5.20\cdot 10^{3}\right)\ \text{J/mol}. \tag{1} \]

Equation (1) can serve as a calculation formula for determining the energy of the crystal lattice from measured values of the tangent of the slope angle of the straight-line dependences of Young’s modulus on temperature.

Calculation of the crystal-lattice energy of CsJ, CsBr, KJ, KBr, KCl, NaCl, NaF, LiF from the temperature-induced changes in Young’s modulus, carried-

according to the data in [2], gives discrepancies with those measured directly in experiment [3, 4] of less than 6%; moreover, this deviation is random in character.

The fact that the change in the mechanical properties of solid solutions upon heating does not differ substantially from analogous changes in pure single crystals [5] served as the basis for applying formula (1) to the calculation of the lattice energy of the solid solutions KCl—KBr, NaCl—NaBr. The results of the calculation are given in Table 1. It follows from the table that introducing a substitutional impurity into a pure single crystal lowers the lattice energy, with the greatest deviation from the additive value occurring for equimolar compositions.

Table 1

Lattice energy of solid solutions of alkali-metal halides

Study of the decomposition of solid solutions from the temperature changes in Young’s modulus and internal friction

The study of the electrical and optical characteristics of decomposing solid solutions of the NaCl—KCl and NaCl—AgCl systems [6, 7], as well as x-ray studies and hardness measurements carried out on NaCl—AgCl alloys [8], showed that the principal regularities of the decomposition of solid solutions established for metallic alloys are also repeated in solid solutions of alkali-metal halides. Analysis of x-ray diffraction patterns obtained as a result of isothermal decomposition of NaCl—AgCl solid solutions made it possible to establish the presence of two types of diffraction patterns, corresponding to two different states separated from each other in time and differing in structure. The first state may be associated with the early stage of decomposition; the second appears in the course of isothermal annealing and is due to the presence of precipitates of the equilibrium phase.

The difficulties that arise in interpreting the x-ray diffraction patterns do not make it possible to clarify fully the mechanism of the transition from the first stage to the second. Hence there arises the need to bring other methods to bear on the study of the decomposition of solid solutions.

At the suggestion of Professor Doctor A. A. Vorob’ev, we applied to the study of the decomposition of the NaCl—AgCl solid solution the resonance method for measuring the temperature changes in Young’s modulus and internal friction [9]. The investigations were carried out: 1) by observing the decomposition process during storage of homogenized specimens (iso-

thermal annealing), and 2) by observing the homogenization process as the temperature is raised. The measurement results are presented in Figs. 1 and 2. Figure 1 shows the dependence of the internal-friction magnitude on time during isothermal annealing. The curve has maxima and characterizes the decomposition of the solid solution with time. These results

Fig. 1. Dependence of the magnitude of internal friction on the storage time of the solid solution AgCl—NaCl

Fig. 2. Dependence of the magnitude of internal friction and Young’s modulus on the temperature of the decomposed solid solution AgCl—NaCl. The peak of internal friction and the defect in the modulus are due to the process of homogenization of the solid solution

agree with the change in microhardness during isothermal annealing [8].

Figure 2 shows the dependences of the change in Young’s modulus and internal friction upon heating specimens of the decomposed solid solution. The peak of internal friction and the defect in the modulus are evidently associated with atomic rearrangements in the crystal lattice of the solid solution during the homogenization process. On the basis of the data obtained, it may be concluded that the resonance method for measuring temperature-induced changes in Young’s modulus and internal friction is applicable to the study of the decomposition of the NaCl—AgCl solid solution.

Temperature-induced changes in crystal anisotropy

The study of temperature-induced changes in the elastic constants makes it possible to establish the influence of temperature on the anisotropy of the elastic properties of single crystals. A quantitative estimate of crystal anisotropy can be obtained by determining the anisotropy factor

\[ A=\frac{C_{44}}{C_{11}-C_{12}}; \]

(\(C_{ik}\) are elastic constants). The elastic constants of single crystals of the NaCl type are determined mainly by the energy of the Coulomb interaction of ions and by the repulsion energy of the filled shells of neighboring alkali and halide ions [11]. The contributions of these energies to the shear constants \(C=C_{44}\), \(C'=\frac{1}{2}(C_{11}-C_{44})\) are different and depend on the distance between ions. Under thermal deformation, the contributions of these energies depend differently on temperature, which leads to a change in the anisotropy of the crystal [12]. When the anisotropy factor is equal to unity, the crystal becomes isotropic, and the elastic constants cease to depend on the direction in which they are measured. Figure 3 presents the de-

dependence of the anisotropy factor on the temperature of an NaCl single crystal. From the graph one can determine that at a temperature of 680°K the anisotropy factor becomes equal to unity. The data of our low-temperature measurements of the change in the anisotropy factor are in good agreement with analogous measurements at temperatures above room temperature by A. V. Stepanov and S. P. Nikanorov with co-workers [13, 14, 15].

If the forces of interatomic interaction are central and, in addition, each particle is located at a center of symmetry of the lattice, then the Cauchy relation \(C_{12}=C_{44}\) is satisfied. With a change in the temperature of a real crystal, the degree of deviation from the Cauchy relation changes. Figure 4 presents the temperature changes of the factor \(B=C_{44}/C_{12}\). From the graph it is seen that the Cauchy relation is satisfied for NaCl at

Fig. 3. Change in the anisotropy factor
\[ A=\frac{2C_{44}}{C_{11}-C_{12}} \]
with temperature for an NaCl crystal

Fig. 4. Dependence of the factor
\[ B=\frac{C_{12}}{C_{44}} \]
on temperature for an NaCl crystal. At 250°K the Cauchy relation \(B=1\) is satisfied

a temperature of 250°K. Our measurements, as well as the data of [14, 15], show that the magnitude of the factor \(B\) increases as the temperature is lowered.

Elastic constants of crystals and the Debye temperature

Data on the temperature changes of the elastic constants of alkali metal halides can be used to determine the Debye temperature [16] from the formula:

\[ \Theta^3=(9N/4\pi V)(h/k)^3(C_{44}/\rho)^{3/2}9/1[8+\sqrt{3}]\,f(S,t), \tag{2} \]

where \(C_{ik}\), \(V\), \(\rho\) are, respectively, the elastic constants, molar volume, and density of the crystal, determined for 0°K; \(N\), \(h\), \(k\) are twice Avogadro’s number (for alkali metal halides) and the Planck and Boltzmann constants.

\[ S=\frac{C_{11}-C_{44}}{C_{12}+C_{44}};\qquad t=\frac{C_{12}-C_{44}}{C_{44}}. \]

The multiplier \(f(S,t)\) is a certain function of the parameters \(S\) and \(t\). When \(C_{11}-C_{12}=2C_{44}\) and \(C_{12}=C_{44}\), the function \(f(S,t)=1\). The value of the function \(f(S,t)\) is determined from Tables 2 and 3 with the aid of Stirling interpolation formulas.

\[ f_{xy}=f_{00}+\frac{1}{2}x(f_{32}-f_{12})+\frac{1}{2}y(f_{23}-f_{21})+\frac{1}{4}xy(f_{33}+f_{11}- \]

\[ - f_{13} - f_{31}) + \frac{1}{2} x^2 (f_{32} + f_{12} - 2f_{00}) + \frac{1}{2} y^2 (f_{23} + f_{21} - 2f_{00}) + \]

\[ + \frac{1}{4} x^2 y (f_{33} - f_{11} + f_{13} - f_{31} - 2f_{23} + 2f_{21}) + \frac{1}{4} x y^2 (f_{33} - f_{11} + f_{31} - \]

\[ - f_{13} - 2f_{32} + 2f_{12}) + \frac{1}{4} x^2 y^2 (f_{33} + f_{13} + f_{31} + f_{11} - 2f_{21} - \]

\[ - 2f_{23} - 2f_{12} - 2f_{32} + 4f_{00}); \tag{3} \]

\[ x = \frac{S - S_{\mathrm{tab.}}}{\Delta S}; \qquad y = \frac{t - t_{\mathrm{tab.}}}{\Delta t}; \]

Table 2

\[ f(S \ge 1,\ t \ge 0). \]

$S_{\mathrm{tab.}}$ and $t_{\mathrm{tab.}}$ are the quantities closest in magnitude to $S$ and $t$, but smaller in value, that are present in Tables 2 and 3. $\Delta S$ and $\Delta t$, the steps of variation of $S$ and $t$, are found as the difference between two adjacent values of $S$ or $t$ in the table. Then, at the intersection of the row corresponding to the value $S_{\mathrm{tab.}}$ and the column with value $t_{\mathrm{tab.}}$, the value of the interpolation source $f_{00}$ is found. The remaining functions are found according to the scheme of Table 4. Thus, for example, $f_{21}$ and $f_{23}$ are the values of the function $f$ lying in the same row as $f_{00}$ in Tables 2 and 3, respectively to the left and to the right of $f_{00}$; $f_{12}$ and $f_{32}$ are located respectively above and below $f_{00}$.

Table 3

\(f(S \ge 1,\, t < 0)\)

We carried out a calculation of the Debye temperature for sodium chloride single crystals. Extrapolation of the low-temperature dependences of the elastic constants to \(0^\circ\mathrm{K}\) gave the following values [17]:

\[ C_{11}=5.625\cdot10^{11}\ \text{dyn}/\text{cm}^2,\qquad C_{12}=0.973\cdot10^{11}\ \text{dyn}/\text{cm}^2,\qquad C_{44}=1.310\cdot10^{11}\ \text{dyn}/\text{cm}. \]

Substituting all the necessary data into the formula for \(f_{xy}\), we find \(f_{xy}=1.4068\).

The density of the NaCl crystal as a function of temperature was determined from the formula [18]

\[ P(t)=2.1680\,[1-1.12\cdot10^{-4}t-5\cdot10^{-8}t^2], \tag{4} \]

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

For \(0^\circ\mathrm{K}\), \(\rho=2.226^2/\text{cm}^3\). After substituting the corresponding values into the formula for \(\Theta\), we find \(\Theta=321.4^\circ\mathrm{K}\), which is in good agreement with the value of the Debye temperature calculated from thermal data at low temperatures [19].

Table 4

In conclusion, the authors express their gratitude to Professor Doctor A. A. Vorob’ev and Professor Doctor E. K. Zavadovskaya for their constant interest in the work and their assistance.

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