Correlation between the crystal lattice energy of alkali halide crystals and the temperature variation of Young's modulus
Abstract Generated abstract
This paper examines the relationship between crystal lattice binding energy and the temperature coefficient of Young’s modulus in alkali halide crystals at temperatures above the Debye temperature. Using measurements on single crystals and comparison with published data, it proposes a linear empirical relation that allows lattice energy to be estimated from the slope of the temperature dependence of Young’s modulus, with discrepancies from directly measured values reported as generally below 6 percent. The same relation is applied to NaCl, NaBr and KCl, KBr solid solutions, where the calculated results suggest that substitutional impurity lowers lattice binding energy, with the largest deviation from additive behavior near equimolar composition.
Full Text
Relationship between the Crystal Lattice Energy of Alkali Halide Crystals and the Temperature Variation of Young’s Modulus
A. A. Botaki
(Presented by the scientific seminar of the Laboratory of Energetics and Electronics of Ionic Structures)
The temperature dependence of Young’s modulus \((E = f(t))\) of alkali halide crystals for temperatures above the Debye temperature up to the melting point, as is now well known [1], is linear.
Our measurements, carried out on single crystals of KJ, KBr, KCl, NaBr, NaCl, and LiF [2], showed that, in addition to the aforementioned linear dependence, there is also a linear relationship between the crystal lattice energy and the tangent of the angle of inclination \(\left(\dfrac{dE}{dt}\right)\) of the straight lines \(E=f(t)\).
This made it possible for us to write the equation of a straight line in the coordinates \(U,\ \dfrac{dE}{dt}\),
\[ U=\left(-6.70\cdot 10^{-3}\frac{dE}{dt}+5.20\cdot 10^{5}\right)\frac{\mathrm{erg}}{\mathrm{mol}}, \tag{1} \]
which can serve as a calculation formula for determining the crystal lattice binding energy from the measured values of the tangent of the angle of inclination of the straight lines describing the dependence of Young’s modulus on temperature.
Table 1 contains the initial data needed to obtain this formula. To confirm the correctness of the conclusions drawn, the same table gives values of \(\dfrac{dE}{dt}\) calculated from the results of measurements of the temperature dependence of Young’s modulus presented in the work of S. P. Nikanorov and A. S. Stepanov [1]. It is evident from the table that calculation by formula (1) of the crystal lattice binding energies on the basis of the data of [1] gives a discrepancy of less than 6% from the values measured directly in experiment [3]; moreover, this deviation is random in character and thereby confirms the linear relationship between \(\dfrac{dE}{dt}\) and \(U\).
In the solid solutions NaCl—NaBr and KCl—KBr, the ions Cl\(^{-}\) and Br\(^{-}\) are randomly distributed among the anion vacancies. Such a system is a substance with an impurity introduced by substitution. The physical properties of these solid solutions do not differ in principle from the physical properties of the alkali halide salts composing them.
The temperature dependence of Young’s modulus for these solid solutions is linear [4].
Assuming that the temperature dependence of Young’s modulus for solid solutions of different compositions satisfies relation (1), one can calculate, from the experimentally measured values of \(\frac{dE}{dt}\), the crystal-lattice binding energy for them. The results of the calculation are given in Table 2.
Table 1
Crystal-lattice energy and temperature rate of change of Young’s modulus of single crystals of alkali-metal halides
| Substance | \(-\dfrac{dE}{dt}; \times 10^7\ \dfrac{N}{m^2\cdot deg}\) | \(-\dfrac{dE}{dt}; \times 10^7\ \dfrac{N}{m^2\cdot deg}\) | \(U,\ \dfrac{kcal}{mol}\) | \(U,\ \dfrac{kcal}{mol}\) | \(\dfrac{\Delta U}{U},\ \%\) |
|---|---|---|---|---|---|
| LiF | 7.67 | 7.30 | 244 | 240 | 1.6 |
| NaF | 5.00 | 217 | 204 | 6.0 | |
| NaCl | 3.60 | 3.74 | 185 | 184 | 0.5 |
| NaBr | 3.23 | 178 | |||
| KCl | 3.09 | 3.44 | 171 | 179 | −4.7 |
| KBr | 2.64 | 2.98 | 161 | 172 | −4.2 |
| KI | 2.19 | 2.62 | 156 | 164 | −5.1 |
| CsBr | 1.22 | 152 | 144 | 5.3 | |
| CsI | 1.15 | 145 | 142 | 2.1 | |
| our measurements | according to [1] | according to [3] | calculation by formula (1), from data of [1] | 1.5% average error |
Table 2
Crystal-lattice energy of solid solutions of alkali-metal halides
| Molecular % of 1st component | NaCl—NaBr: \(-\dfrac{dE}{dt}; \times 10^7\ \dfrac{N}{m^2\cdot deg}\) | NaCl—NaBr: \(U;\ \times 10^4\ \dfrac{J}{mol}\) | KCl—KBr: \(-\dfrac{dE}{dt}; \times 10^7\ \dfrac{N}{m^2\cdot deg}\) | KCl—KBr: \(U;\ \times 10^4\ \dfrac{J}{mol}\) |
|---|---|---|---|---|
| 100 | 3.75 | 76.2 | 3.25 | 73.0 |
| 75 | 3.45 | 74.2 | 3.07 | 71.8 |
| 50 | 3.14 | 72.2 | 2.90 | 70.7 |
| 25 | 2.94 | 70.9 | 2.77 | 69.8 |
| 0 | 3.20 | 72.6 | 2.85 | 70.3 |
It follows from the table that the introduction of a substitutional impurity into a pure single crystal lowers the binding energy of the crystal lattice, with the greatest deviation from the additive value occurring for equimolar compositions.
In conclusion, I express my gratitude to Professor Doctor R. I. Garber for the considerations he expressed during the discussion of the work [5], thanks to which the present work arose.
References
- S. P. Nikanorov, A. V. Stepanov. FTT, 6, 1987, 1964.
- A. A. Botaki. Izv. vuzov MV SSO SSSR—Fizika, 4, 92, 1965.
- J. Gilman. UFN, 80, 455, 1963. S. M. Kashhava and Saxena. Jndiais Journ of Phys., 38, 8, 388, 1964.
- A. A. Botaki. Izv. TPI, 140, 204, 1965.
- A. A. Botaki. Abstracts of reports at the 4th All-Union Conference on relaxation phenomena in solids. Voronezh, 1965.