Dielectric strength of dielectrics and the crystal lattice constant
Abstract Generated abstract
This paper examines the relation between dielectric strength and crystal lattice constant in alkali halide dielectrics, using experimental data and comparisons with earlier theoretical arguments on ionic radius, ionization potential, and lattice bonding. For simple alkali halide crystals, the dielectric strength is shown to increase as the lattice constant decreases, with an approximately linear dependence for much of the data, while deviations are attributed to increased polarization and weakened electron binding at larger lattice constants. The paper then analyzes solid solutions such as KCl with RbCl and KCl with KBr, finding that their dielectric strength is not uniquely determined by the mean lattice constant calculated from additivity or related models. It argues that lattice distortions and local ion displacements in solid solutions can mask changes in average lattice constant until sufficiently high concentrations of the weaker component produce a rapid decrease in dielectric strength.
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DIELECTRIC STRENGTH OF DIELECTRICS AND THE CRYSTAL LATTICE CONSTANT
E. K. Zavadovskaya
Figure 1 presents the dependence of the dielectric strength of crystals of alkali-halide salts on the magnitude of the lattice constant, according to experimental determinations. The existence of a simple dependence between these quantities is noteworthy.
Fig. 1. Dependence of the dielectric strength of crystals of alkali-halide salts on the magnitude of the lattice constant.
We have theoretically shown that the dielectric strength of a dielectric should increase as the distance between the particles forming it decreases [1]. This is what is observed in the present case.
For the linear part of the dependence shown in Fig. 1, the condition is satisfied:
\[ E_{\text{pr}} = 2.12\,(3.46 - a_0), \]
where
\[ E_{\text{pr}} \text{ is in } \frac{\text{MV}}{\text{cm}} \quad \text{and} \quad a_0 \text{ is in ångströms.} \]
Figure 2 shows the dependence of the dielectric strength on the magnitude of the reciprocal lattice constant \(a_0^{-1}\). Seventy-five percent of the points lie well on a straight line described by the equation \(E_{\text{pr}} = 11.8\,(a_0^{-1} - 0.232)\ \text{MV}/\text{cm}\).
Deviations from the linear dependence occur at large values of the lattice constant, when polarization phenomena increase significantly and the bond of the electron with the ions decreases. We also have
it has been shown that the electric strength of dielectrics decreases with an increase in the radii of the ions forming its crystal lattice [2].
In the theory of the crystal lattice it is assumed that the ions forming the crystal lattice are hard spheres of definite radius, whose spheres touch one another.
In this case the lattice constant is determined as the sum of the ionic radii. This assumption, on purely geometrical grounds, cannot be correct when there is a very large difference in the sizes of the two ions.
The radii of the ions of the elements in Mendeleev’s table change periodically. For a given number of electrons, the radius of an ion rapidly decreases as the nuclear charge increases. This is a consequence of the contraction
Fig. 2. Dependence of electric strength on the magnitude of the reciprocal lattice constant.
of the electron orbits. The energy necessary to free an electron from an atom or ion must be connected with their radius.
Indeed, the ionization potential depends on the radius of the atom, since it is determined by the work necessary to move an electron from the atom to infinity. Approximately, for the magnitude of the ionization potential one may write the expression
\[ W_i \sim \frac{e}{R}, \tag{1} \]
where \(R\) is the radius of the atom.
The investigations of Kapustinsky, Khodakov, and others make it possible, for the comparative characterization of atoms, to use the concept of the ionic potential as a parameter determining the chemical nature of the atom. The potential of the force field of an ion is expressed by the condition
\[ W_э \sim \frac{W}{R}, \tag{2} \]
where \(W\) is the valence or charge,
\(R\) is the radius of the ion.
In each period of the Mendeleev table, the course of variation of the radii of ions of the same valence may be compared with the ionization potential. It turns out that, while the ionization potential increases on going to the next noble gas, the ionic radii decrease somewhat; that is, formula (2) given above is confirmed.
The magnitude of the ionic radius decreases all the more upon removal not of one, but of two or three electrons, that is, with increasing valence.
Thus, a decrease in the ionic radius is accompanied by a strengthening of the bond of the valence electron. Upon the liberation of the valence electron, the ion of small radius that is formed will give a stronger bond in the compound.
The following rule has been verified in practice and theoretically substantiated: “With increasing ionic radius, all other conditions being equal, the solubility of compounds should generally increase and their strength and hardness decrease, while the melting or boiling temperature should decrease” (Goldschmidt).
To this it should also be added that, according to Figs. 1 and 2, with increasing ionic radii, all other conditions being equal, the dielectric strength of the dielectric should decrease.
The energy of liberation of electrons, which normally in an undisturbed lattice should be determined by the ionization potential of the atom, will also decrease with increasing lattice constant.
Hippel [3], studying the dielectric strength of alkali-halide crystals, came to the conclusion that replacement of the anion or cation in a rock-salt-type crystal lattice by another of smaller radius or polarizability leads to an increase in dielectric strength.
The lattice constant varies with temperature according to the condition
\[ a_t = a_0 + \alpha t. \tag{3} \]
The increase in the lattice constant with increasing temperature according to formula (3) proves to be very small. In the region of high temperatures near the melting temperature, the lattice constant grows according to an exponential law.
It is of interest to consider how the magnitude of the dielectric strength of solid solutions changes with a change in the lattice constant. It is usually assumed that the lattice constant of a binary solid solution depends linearly on the concentration of the components, namely:
\[ a_{12} = a_1 C_1 + a_2 C_2, \tag{4} \]
where \(a_{12}\), \(a_1\), and \(a_2\) are the corresponding lattice constants of the solution and of the components; \(C_1\) and \(C_2\) are the concentrations of the components.
According to Hippel’s measurements, the dielectric strength of a solid solution, with increasing RbCl content in KCl, at first increases, reaches a maximum at 40% RbCl, and then begins to decrease. The same course of the dependence of dielectric strength on composition is observed for the solid solutions KCl—KBr and NaCl—AgCl.
Figure 3 presents, as a function of the composition of the solid solution of the KCl—RbCl system, the dielectric strength and the lattice constant.
Figure 4 presents the dependence of dielectric strength and of the lattice constant on the component composition of the KCl—KBr solid solution.
The values of the dielectric strength of the points of the system were taken from Hippel’s experimental data. The value of the lattice constant was calculated by us according to the additivity rule. As is seen, the simple dependence between the magnitude of dielectric strength and the lattice constant observed for crystals of alkali-halide salts is not observed for solid solutions of these salts.
Fig. 3. Dependence of the dielectric strength and the lattice constant on the composition of the components of the solid solution of the system
KCl + RbCl.
The same conclusions were drawn by us in analyzing the systems NaCl + AgCl and NaI in KI. We do not know experimentally determined values of the lattice constant of the solid solutions of the indicated systems. It is possible that
Fig. 4. Dependence of the dielectric strength and the lattice constant on the composition of the components of the solid solution of the system
KCl + KBr.
The observed disagreement between the course of the electric strength of solid solutions and their lattice constant is explained by the incorrect determination of the lattice constant according to the additivity rule.
Durham and Hawkins [4] calculated the value of the lattice constant of solid solutions of the systems KBr + KCl, RbBr + RbCl, RbCl + KCl, RbBr + KBr at a temperature of 25°C. They assumed that the ions are not at a constant distance from the nearest neighbor, but occupy the position corresponding to the minimum of the potential energy relative to one another. The values of the interionic distances obtained by them were compared with the corresponding values calculated by me according to the additivity rule. The discrepancy in the values of the interionic distances calculated by these two methods is small. The value of the interionic distances calculated by this second method also changes linearly with the change in the composition of the solution.
Calculations by other authors, who use more or less complicated initial assumptions, also lead to results that differ little from those obtained by the simple additivity rule.
When calculating the lattice constant of a solid solution according to the additivity rule, an important phenomenon of ion displacement is not taken into account; this is pointed out by Wasastjerna [5], on the basis of X-ray analysis of crystals.
The actual equilibrium position of the ions in the lattice of a solid solution does not coincide with the corresponding positions in an ideal lattice; the ions are somewhat displaced relative to the equilibrium position.
When the interaction of the components is taken into account, the lattice constant of a solid solution may vary with concentration according to a more complicated law than the additivity rule used by us.
It has been established that the heat of formation for all compositions of NaBr + NaCl is positive. The lattice constants and molar volumes determined experimentally for all solutions NaCl + NaBr exceed the values calculated according to the additivity rule. Consequently, the crystal lattices of the solutions undergo significant distortions. In this connection, the study of the electric strength of this system is of special interest.
Let us note that the difference between the measured and calculated values of the lattice constant and molar volume, as a function of composition, varies along a curve with a maximum.
The empirical additivity rule is rarely justified exactly. It represents only a first approximation; deviations are observed which have a positive sign, when the lattice constant of the solid solution exceeds the value corresponding to the linear law. Deviations with a negative sign are also observed. Deviations with a negative sign occur more often. However, no indications have been established that determine the sign or magnitude of the deviations.
Pines [6] calculated the distortions of the lattice of a solid solution by means of a model of elastic spheres. He calculated the change in the lattice constant of a solid solution and compared it with the results of calculation according to the additivity rule. Pines’s calculations give the correct sign and order of magnitude of the deviations from the additivity rule, except for certain alloys where anomalies occur. The presence of such anomalies is explained by the hypothesis of the completion of electron shells.
An example of a deviation from the additivity rule with a positive sign is the solid solution NaCl + NaBr.
The simple relation between the value of the lattice constant and electric strength, observed for crystals of alkali-halide salts, is not fulfilled for solid solutions of these same salts. The value of the lattice constant does not uniquely determine the magnitude of the electric strength of a solid-solution system.
Figure 5 presents the dependence of the electric strength of crystals of solid solutions of the systems KCl + KBr and KCl + RbCl on the value of the lattice constant, calculated by the formulas of Durkheim and Hawkins.
Over a large concentration interval, the electric strength does not depend on the mean value of the lattice constant. When the concentration of the electrically less strong component reaches 30%, a rapid decrease in electric strength begins as the lattice constant increases.
The following physical explanation may be given for the observed experimental dependence. The formation of a solid solution is accompanied by deformation of the crystal lattice of the solvent and by a change in the lattice constant. A foreign ion deforms the lattice more strongly at short distances and more weakly at large distances. As a result, in some part of the volume the lattice constant will retain its former value; in some regions an increase is possible, and in others, correspondingly, a decrease of the lattice constant.
Fig. 5. Dependence of the electric strength of crystals of solid solutions of the systems KCl + KBr and KCl + RbCl on the value of their lattice constant.
The electrons participating in the breakdown will move in those directions in which the resistance to their motion is least. The electrons will bypass strengthened microscopic volumes, and a change in the lattice constant in small volumes will not affect the electric strength. Only at large concentrations, when, probably, the lattice is rearranged in such a way that the new parameter becomes predominant in some direction, does its influence on the electric strength become appreciable, and the electric strength begins to decrease with increasing lattice constant. Such a conception agrees with the results of studies of the electric strength of crystals along different crystallographic directions.
I express my gratitude to Professor A. A. Vorob'ev for discussion of the questions raised in the article.
References
- Zavadovskaya E. K. DAN, XXXII, 4, 565, 1952.
- Zavadovskaya E. K. DAN, XXXII, 5, 709, 1952.
- A. v. Hippel, Zs. f. Phys., 88, 358, 1934.
- G. S. Durham and Janet A. Hawkins. The Journal of Chemical Physics, 19, 2, 149, 1951.
- Wasastjerna. Acta Soc. Sci. Fennicae, 3, 8, 1944. Soc. Sci. Fennicae, commentationes Phys.-Math., 15, 3, 1950.
- Pines B. Ya. ZhETF, 11, 6, 725–727, 1941.