PHYSICAL AND ELECTRICAL PROPERTIES OF IONIC CRYSTALS

A. A. VOROB'EV and E. K. ZAVADOVSKAYA

Introduction

At the present time, in the physics of dielectrics and in solid-state physics in the broader sense, several separate, mutually unrelated doctrines are considered: the doctrine of dielectric polarization and the dielectric coefficient, ionic conductivity, dielectric losses, electronic conductivity, electric strength and breakdown, mechanical strength and mechanical failure, elastic and thermal properties, and others.

For a long time, both in educational and in scientific literature, the above-mentioned approach to the isolated treatment of these questions was developed. Credit must be given to the teachers who developed the methodology for the separate presentation of these questions. Now only a person not versed in the physics of dielectrics and electrical materials science can ask himself the question of why all these properties of a single whole are considered separately, and where the unity is of all these properties of the object of our study—the dielectric.

The isolated study of individual properties of materials leads some researchers far away from the essence of what is being studied, and leads to attempts to detach these properties from the very object of study and from the interconnection of the properties of the object.

What is common in the methodology and physics when presenting the phenomenon of ionic conductivity, or dielectric losses, or electronic conductivity is the ignoring of the basic structure and composition of the dielectric. What is also common in the methodology of explaining the listed properties is the search for possible secondary processes and the exclusion from consideration of the studied body itself and of the processes in it that are regular under the given conditions.

The study of primary phenomena in a homogeneous dielectric is usually replaced by the study of secondary processes in it or of the properties of a poor-quality material.

A mark of good form in the science of dielectrics was considered to be the ability to explain observed phenomena by the role of impurities, defects, and disturbances of the basic structure.

The basic structure and chemical composition of the material were not taken into account in the study of the properties of dielectrics and the processes in them, with the exception of the phenomenon of dielectric polarization. The ionic conductivity of dielectrics is explained by the liberation of weakly bound impurity ions and their subsequent displacements through defective lattice sites. The high-voltage polarization of a dielectric, which plays a significant role in the theory of ionic conductivity, is represented as a secondary phenomenon and is associated with impurity ions. The explanation of dielectric losses proceeds from the presence in the dielectric of defects and impurities. In the main, losses are explained by vibrations of impurity ions in the potential well of a defective region of the lattice.

The quantum-mechanical theory of electronic conduction of dielectrics and semiconductors in strong fields is a well-developed doctrine that has yielded results of great practical value. It is known what role in this doctrine is assigned to impurities and structural defects, and what enormous practical application semiconductors with impurities have found. It is impossible to point to any practical application of semiconductors with an ideal crystal lattice. Semiconductors with impurities and defects, however, have enormous and ever-increasing applications.

The successes of the theory of semiconductors with structural defects and impurities, and their broad practical introduction, contributed to the fact that the behavior and properties of disturbed structures were studied chiefly. One may perhaps say that some researchers working on semiconductors regarded the role of the basic structure as that of a foundation bearing defects and containing impurities. The basic structure itself, after forming energy bands, was subsequently not taken into account in semiconductor theory. Such consideration of a certain group of phenomena in semiconductors will be correct. Some phenomena in semiconductors are indeed determined by impurities and structural defects. But there are also phenomena and conditions in which they are observed when the structure and composition of the semiconductor or dielectric play a role and must be taken into account.

The phenomena of electronic conduction, luminescence, breakdown, etc., are explained on the basis of the band theory of semiconductors and dielectrics, with account taken of the influence of impurities and structural defects. The theoretical schemes that proved successful for semiconductors are also extended to dielectrics. The known role of impurities and defects of the basic structure in semiconductors has been transferred completely to dielectrics. In particular, in the theory of electronic phenomena in ionic crystals a large role is assigned to electron capture centers.

Primary and secondary electronic currents under photoconductivity in colored ionic crystals of the alkali-halide series and oxides of the second group of the Mendeleev table (Be, Mg, Ca, Zn, Sr, Cd, Ba, and others) are associated with transitions of electrons from color centers ($F$ centers) into the conduction band.

Quantum-mechanical theories interpret the phenomenon of electrical breakdown as a consequence of the catastrophic increase of the electronic conductivity of a dielectric in a strong field. The theory is considered satisfactory if a physical mechanism is found for the increase in the concentration of electrons in the conduction band with increasing electric-field strength.

In the case of thermal breakdown, destruction occurs through ionic current. The role of impurity ions and structural defects in ionic conduction has already been noted.

The mechanical strength of dielectrics, measured experimentally, is hundreds of times smaller than the theoretically calculated value. This discrepancy is also explained by the influence of inhomogeneity and structural defects of the substance. The appeal to disturbed structures had its explanation when, owing to imperfect technology, insulating materials were obtained nonuniform, with reduced properties. At present, materials of a high degree of homogeneity and purity are being obtained. The electrical insulating properties of polymers or artificially grown crystals, etc., approach the theoretically calculated values (polarizability, dielectric constant). Therefore it is time to undertake the study of the electrical properties of dielectrics with an undisturbed structure.

Recently published investigations have made it possible to establish a certain connection between the electrical properties of a dielectric, its chemical composition, and its structure.

Let us note that the study of electrical, mechanical, and other properties of dielectrics on the basis of secondary processes and disturbances is fruitless.

promising. It makes it possible to improve a given dielectric, but does not always lead to the establishment of the basic law determining the connection between the properties of dielectrics, their composition and structure; it does not make it possible to improve and design dielectrics with predetermined properties. Such a problem can be solved if the connection between the electrical properties of dielectrics, their composition and structure is known.

In the course of our investigations on the theory of dielectrics, we became acquainted rather late with the works of Acad. A. E. Fersman on geochemistry and discovered a parallelism in some of our developments and conclusions.

It is known that the thermal and mechanical strength of natural groupings of atoms in the gaseous, liquid, and solid states are connected both with one another and with the periodic structure of the atom itself. A consequence of the periodicity of the structure of atoms is the periodicity of the properties of matter: the coefficient of expansion and compressibility, the coupling coefficient, atomic heat capacity, melting and boiling points, heats of melting and boiling, etc.

The strength of the bonds of elements is conditioned by the combination of individual atoms into various systems, molecules, and solid crystalline lattices. Acad. A. E. Fersman gives a number of empirical rules establishing the connection between the melting temperature (thermal destruction) of elements and their compounds with their periodic properties, lattice energy, packing density, and others.

The theory of bonding forces in ionic crystals has been developed in detail. Let us present some consequences of this theory that are necessary for the subsequent description of phenomena in dielectrics.

The course of the processes of formation of chemical compounds and crystallization is thermodynamically directed toward a decrease in the free energy (an increase in entropy) of the system.

The stability and strength of a crystal depend on the magnitude of the forces acting between its particles and on the potential energy of the crystal lattice determined by them. Under the action of external conditions—heating, mechanical action, the action of an external electric field—the interaction energy between particles may be overcome, and the lattice destroyed. This is observed in melting, mechanical, and electrical destruction. The magnitude of the energy required to separate the crystal lattice into individual ions and remove them from one another to an infinitely large distance at absolute-zero temperature, or the energy released in the formation of the lattice from free ions, is called the lattice energy.

It should be thought that the energy of a crystal lattice unambiguously determines its stability and various properties. The magnitude of the lattice energy is an important indicator of the energy level of the system.

The lattice energy is the most important component of the thermal effect in the equation of a chemical reaction. The energy of a crystal lattice in general form can be determined on the basis of the theory of the crystal lattice, taking into account the attractive and repulsive forces acting between the cation and anion, and is given by the formula:

\[ U = A \frac{W_A W_B e^2}{a_0} \cdot \frac{n_1 - 1}{n_1}, \tag{1} \]

where
\(A\) is the Madelung constant, depending on the symmetry of the lattice;
\(W_A\) and \(W_B\) are the valences of the ions entering the lattice;
\(a_0\) is the shortest distance between the centers of oppositely charged ions in the lattice;
\(n_1\) is the exponent in the expression for the repulsive forces, determined from the compressibility of crystals.

The thermochemical lattice energy is determined from the Born–Haber cycle by the formula:

\[ U=(I+c)+\left(\frac{D}{2}-E\right)+Q, \tag{2} \]

where \(Q\) is the heat effect of the reaction, \(U\) is the total lattice energy, \(I\) is the total ionization energy of the metal, \(E\) is the electron affinity of the anion, \(c\) is the heat of sublimation, and \(D\) is the heat of dissociation.

The lattice energy may be calculated in large calories per mole of substance, kcal/mole, or in electron-volts per pair of ions (one molecule), or in other units.

For the energy of a crystalline lattice of the NaCl type, Born obtained the formula:

\[ U=545\sqrt[3]{\rho/M}\ \text{kcal/mole}, \tag{3} \]

where \(\rho\) is the density of the substance and \(M\) is its molecular weight. The expression for the lattice energy of crystals of other types is also obtained in the form (3), but with different coefficients.

In the case of binary compounds of the type \(A_mB_m\), A. F. Kapustinskii, for calculating lattice energies with coordination number 6, gives the following simplified formula:

\[ U=256.1\,\frac{(a+b)\,W_A W_B}{R_A+R_B}, \tag{4} \]

where \(a\) is the number of cations, \(b\) is the number of anions,

\(W_A\) and \(W_B\) are the valences of the anion and cation,

\(a_0=R_A+R_B\) are the radii of the corresponding ions for a lattice structure of the rock-salt type.

The approximate formula (4) does not take into account certain quantities entering into the exact expression, for example, the “repulsion” coefficient, the difference in ionic radii for different types of crystalline structures and coordination numbers, and it does not take into account the polarization correction.

Ionic lattices have a large value of the lattice energy and, as a consequence, great thermal and mechanical stability. Homeopolar lattices consist of strong isolated molecules weakly bound to one another. A consequence of this is their weak thermal and mechanical stability, despite a higher value of the lattice energy than for ionic lattices.

In the case of a homogeneous crystalline structure, its mechanical strength will be related to the lattice energy. In other cases, for example, in the case of layered structures (mica, graphite, carbonic acid in the solid state), destruction of the lattice will proceed along weakened places along the layers, and there will no longer be correspondence between the full value of the lattice energy and its strength. It should be noted that if some logical connection is indicated between the mechanical strength of a structure and the lattice energy, there are as yet no numerical relations between these quantities.

For crystals, Polanyi gives a formula relating the magnitude of the surface energy \(\sigma\) and the strength \(Z\) in the form

\[ \sigma=\frac{Z^2}{2E}\cdot a_0, \tag{5} \]

where \(a_0\) is the interatomic distance and \(E\) is the modulus of elasticity.

According to Academician A. E. Fersman, the strongest (hard) structures should be obtained in the formation of compounds from tetravalent elements, for example: C—Si; C—Ti; C—Mo. Oxides of the elements of the second group of the Mendeleev table—Be, Mg, Ca, Zn, Sr, Ba—which have high values of lattice energy, heats of formation, and melting temperatures, should also have high electrical and mechanical properties.

Academician A. E. Fersman, who studied the thermal and mechanical stability of crystal lattices, drew interesting conclusions. The binding energy of the elements in a metal depends on the magnitude of the lattice energy \(U\). Complete agreement should be expected only for coordination ionic lattices, where there are no geometrically and physically closed formations apart from the ions or atoms themselves.

The situation is different in lattices with closed bonds or in layered lattices, where the cohesive forces are distributed nonuniformly and where there are large bonding forces within the molecule and a very weak bond between individual molecules. In this case, despite a large total value of the lattice energy, the bond between molecules is weak; as a result, the substance has low hardness and volatility.

Thus, a large value of the lattice energy \(U\) implies high strength only for compounds of the heteropolar coordination type, and conversely: strength, stability, difficulty of volatilization, dissociation, and hardness depend not only on \(U\), but also on the type of lattice and its geometrical features—such is the general conclusion of Academician A. E. Fersman.

Thus, science has already shown that many properties of materials are regularly related to the lattice energy, its type, and geometrical features. It must be assumed that the electrical properties of materials should not be an exception to this rule. These regularities must be established. If the type of lattice and its geometrical features are excluded from consideration, that is, if a lattice of one type is considered, then the properties of the material will be unambiguously related to the energy of the crystal lattice.

We attempted to do this with crystals of alkali-halide salts having a rock-salt-type lattice. We compared their electrical properties with the lattice energy and with the lattice constant, which is related to the lattice energy by A. F. Kapustinskii’s formula.

Modern theory of the crystal lattice and the known experimental results contain the data necessary for comparing the electrical properties of ionic lattices with their energetic and structural characteristics.

Dependence of the Physical Properties of Rock-Salt-Type Crystals on Lattice Energy

The physicochemical properties of chemical compounds change regularly when their composition changes. This is a consequence of the regular change in the properties of the elements themselves.

The melting temperature of the elements in the Mendeleev table changes regularly. It increases with valence and decreases with atomic volume. The melting temperature is related to the structure of the lattice and to the packing density. It has been empirically established that, for elements crystallizing in a face-centered cubic lattice, the melting temperature is approximately proportional to the square root of the atomic weight, while for elements crystallizing in a body-centered cubic lattice, the melting temperature is proportional to the cube root of the atomic weight. Other conditions being equal, the melting or boiling temperature of a chemical compound should decrease with increasing ionic radius and, consequently, the lattice constant of the substance. At the same time, the mechanical—

... mechanical strength and hardness, while the solubility of the compound increases. For the melting temperature of chemical compounds Friedrich and Sittig established the following regularities.

1. There is no sharp difference between the melting temperature of atomic and ionic lattices. This is a consequence of the absence of a sharp difference in the values of the energies of these lattices.

2. The melting temperature changes mainly in direct dependence on the valence of the elements of the compound and, to a lesser degree, in inverse dependence on the atomic volume. The melting temperature of binary compounds is approximately expressed by the formula:

\[ T_{m} \sim \frac{m Z_1 Z_2}{n R^3}, \tag{6} \]

where \(Z_1\) and \(Z_2\) are the valences and \(R\) is the atomic radius.

1. The phenomenon of polarization of atoms in a compound lowers the melting temperature. The melting temperature depends to a considerable degree on the distance between the atoms or ions entering into the compound.

The same regularity is also observed for the change in electrical strength, as we have established. With increasing polarizability the electrical strength of crystals of the alkali-halide series decreases. With an increase in the melting temperature of crystals, their electrical strength simultaneously increases.

1. The role of anions is greater than the role of cations in their morphotropic substitution.

Fig. 1. Dependence between lattice energy and melting temperature \(T_{\mathrm{pl}}\,^\circ\mathrm{C}\) for crystals of alkali-halide salts.

For the electrical strength of crystals of alkali-halide salts this rule also holds, as was indicated by A. Hippel and A. A. Vorob'ev.

1. The melting temperature of a substance increases for more symmetrically constructed molecules. Such series are known, for example, in organic chemistry, where sometimes the even members of a homologous series have a higher melting temperature than the intermediate odd ones.

Thus, in the series of acids—oxalic, succinic, and so forth—an even number of \(\mathrm{CH}_2\) groups corresponds to an elevated melting temperature.

The dependence of the melting temperature of crystals of alkali-halide salts on the lattice energy, presented in Fig. 1, is in agreement with the propositions given above. Between the melting temperature of halo-

of the compound of the metal and its lattice energy, an approximately linear dependence is obtained.

With an increase in the lattice energy of a chemical compound, its thermal stability, characterized by the melting point, increases. When the scale is reduced, the lines in Fig. 1 may merge into one. In this case the dependence of the melting point of alkali-halide compounds on the lattice energy becomes almost linear.

Fig. 2. Dependence between the molecular volume and the lattice energy for crystals of alkali-halide salts.

Theoretically it follows that in a simple dependence on lattice energy there should be the thermal effects accompanying physical and chemical processes. Experimental data show that with a decrease in the lattice energy of alkali-halide compounds there decrease the latent heat of fusion,

Fig. 3. Dependence of the specific gravity of crystals of alkali-halide salts on lattice energy.

the temperature of evaporation and the latent heat of evaporation; and also the heat of sublimation of chemical compounds in the solid state, the heat content, heat capacity, etc. Heat capacity, in turn, is related to other physical quantities, which also determines their relation to the lattice energy.

In Fig. 2 the change in the molecular volume, in Å\(^3\), of alkali-halide compounds is presented according to X-ray analysis data, together with the values of the lattice energy for crystals of alkali-halide compounds of the rock-salt type.

In Fig. 3 the dependence of the specific gravity of crystals of alkali-halide salts on the lattice energy is shown. With an increase in the lattice energy there is a general tendency for the specific gravity of the crystals to increase. When the halide is replaced from fluorine to iodine, the specific gravity of the crystals of the compounds increases.

Fig. 4. Dependence of the compressibility coefficient of crystals of alkali-halide salts on the lattice energy.

Replacement of the metal with the halide unchanged is accompanied by a change in the specific gravity of the crystal. This is seen in Fig. 3, where the change in the density of crystals of alkali-halide salts is shown as the lattice energy changes. With increasing lattice energy, the specific gravity of the crystals tends to decrease.

In Fig. 4 the change in the compressibility coefficient is shown for crystals of alkali-halide salts when their composition is changed. The greatest compressibility is possessed by compounds with lower lattice energy, greater polarizability, and the lowest concentration of ions per unit volume.

Dependence of the polarizational properties of molecules and crystals of alkali-halide compounds on the lattice energy

In Fig. 5 the dependence between the number of ions \(n\) in \(1\ \mathrm{cm}^3\) and the lattice constant is constructed for crystals of alkali-halide salts. The quantity \(n\) is determined from the condition:

\[ n=\frac{N\rho}{2(A_1+A_2)}, \tag{7} \]

where \(N=6.06\cdot10^{23}\), \(\rho\) is the specific gravity, and \(A_1\) and \(A_2\) are the atomic weights of the ions.

With increasing lattice constant, the concentration of ions in the lattice continuously decreases. With increasing lattice constant the lattice energy decreases; therefore the concentration of ions in crystals of alkali-halide compounds will increase with increasing lattice energy.

In Figs. 6 and 7 the change in the concentration of ions in the lattice upon replacement of the cation or anion is shown. These results show separately the role of the cation and the anion and are in agreement with the results presented in Fig. 5. Replacement of the cation and anion causes a change in con-

concentrations of ions in the crystal, the influence of each on the change in concentration being approximately the same.

Fig. 5. Dependence between the number of ions in \(1\ \mathrm{cm}^3\) for crystals of alkali-halide salts and the lattice constant.

Tessman, Kahn, and Shockley, using the refractive index \(n\) for crystals, by means of the expression

\[ \alpha_m=\frac{3V_m}{4\pi}\,\frac{n^2-1}{n^2+2}, \]

where \(V_m\) is the volume of the molecule, determined the values of the electronic polarizability \(\alpha_m\) of alkali and halide ions and molecules.

Fig. 6. Change in the number of ions \(n\) in \(1\ \mathrm{cm}^3\) upon replacement of the anion in crystals of alkali-halide salts.

Fig. 7. Change in the number of ions \(n\) in \(1\ \mathrm{cm}^3\) upon replacement of the cation in crystals of alkali-halide salts.

Using the values of \(\alpha_m\), they also determined the polarizability of other halogen-containing compounds as the sum of the electronic polarizabilities of their ions.

The refractive index of light in these studies was measured for monochromatic lines of the sodium spectrum on crystals of considerable size. The value of the refractive coefficient for \(\lambda=\infty\) was determined by extrapolation.

Figure 8 shows the dependence between the electronic polarizability of the molecules \(\alpha_m\) and the magnitude of the lattice energy. The polarizability of a molecule depends on the nature of the cation and anion; it decreases successively when Rb is replaced by K, Na, Li. For anions the electronic polarizability decreases when I is replaced by Br, Cl, and F. Rubidium fluoride, which crystallizes in a lattice of the cesium chloride type, is no exception in this series.

Fig. 8. Dependence between the electronic polarizability of molecules and the lattice energy for crystals of alkali-halide salts

The polarizability of the molecule increases with an increase in the lattice constant. A considerable increase in polarizability is observed when ions of one metal and halide are replaced by another metal and halide with larger radii.

Figures 9 and 10 present the dependence of the refractive index \(n_\infty\), determined at \(\lambda=\infty\) and \(\lambda_D=5893\ \text{Å}\), on the lattice energy. From Figs. 9 and 10 it is seen that the refractive index of light in crystals of alkali-halide compounds increases with the successive replacement of metals from Rb to Li and decreases successively when the halides are replaced from I to F. The same results were obtained in measurements of the refractive index of light for the C and F lines of the sodium spectrum.

Figure 11 shows the dependence between the lattice energy and the refraction, taken as the sum of the refractions of the anion and cation. In agreement with the preceding, a decrease in refraction is observed with increasing lattice energy calculated per pair of ions. This dependence makes it possible to see separately the influence of the change in the refraction of the anion and cation on the magnitude of the lattice energy.

If, in the formula for molecular polarization with infinite wavelength, one substitutes instead \(\varepsilon_\infty \sim n_\nu^2\), then one obtains the formula for the molecular refraction \(MR_\nu\), with refractive index \(n\) at the given frequency \(\nu\):

\[ \frac{n_\nu^2-1}{n_\nu^2+2}\cdot\frac{M}{d} = \frac{4\pi}{3}N_\alpha = MR_\nu, \tag{9} \]

where \(M\) is the molecular weight and \(d\) is the density of the substance.

Fig. 9. Dependence of the refractive index \(n_D\) on lattice energy.

Fig. 10. Dependence of the refractive index \(n_\infty\) on lattice energy.

Fig. 11. Dependence between the sum of the anion refraction \(R_A\) and cation refraction \(R_K\), and the lattice energy, for crystals of alkali-halide salts.

Molecular refraction is an additive property of a substance with a clearly expressed dependence on its molecular structure, for example, on the presence of dipoles. As a result, molecular refraction can be calculated from atomic constants. The deviations that arise in this case are explained by the structural features of the substance.

Owing to the interaction between atoms, intramolecular polarization occurs in a molecule, generally differing according to its chemical composition. Intramolecular polarization increases with the packing density in a compound. It is known that as the field strength increases, charge displacement becomes smaller, which corresponds to the coefficient of elastic binding for polarizability in weak fields. Similarly, the polarization of ions in a compound under the action of the field of an incident electromagnetic wave becomes weaker as a result of intramolecular polarization. Thus, intramolecular polarization reduces molecular refraction. Consequently, in the additive calculation of molecular refraction the influence is not taken into account of: a) intramolecular polarization and b) intermolecular polarization, which, as a rule, is weaker than the former.

Fig. 12. Change in molecular refraction for solid salts of alkali-halide compounds.

In crystals of alkali-halide compounds, deformation of the electronic orbits of ions takes place.

Refraction measures the displacement of electron shells, that is, the electronic polarizability in ions or molecules. The force field of a neighboring ion can change the electronic polarizability. Fajans in 1923 showed that in ionic compounds, owing to the influence of the fields of neighboring ions, the refraction of a molecule is usually not equal to the sum of the refractions of the ions forming this compound, and differs from it by the amount:

\[ \Delta R = R_A + R_K - R_M. \]

The magnitude of the deviation \(\Delta R\) may be positive, negative, or equal to zero. For hydrogen halides \(\Delta R\) varies from 0.6 to 5.5. This indicates that the polarizability of the hydrogen-halide molecule is less than the polarizability of the ions.

Figure 12 presents the change in molecular refraction for solid salts of alkali-halide metals.

The positive field of the cations will strengthen the electron shells of neighboring anions; the latter, by their negative field, on the contrary, will reduce the strength of the bond of the electrons with the nucleus of the cation.

From Fig. 12 it is seen that \(\Delta R > 0\) for almost all salts. Hence it follows that the action of cations on anions is stronger than the reverse. The strengthening influence of a cation increases as its size decreases. Multivalent ions will exert an even greater deforming action. A change in the polarization of ions in the crystal lattice affects the energy that must be expended to destroy the lattice during melting, sublimation, dissolution, electrolysis, etc.

Assuming that the acting field in the dielectric is equal to the average macroscopic field, one obtains a simple expression relating the dielectric coefficient to its electronic component and to the polarizability of ionic displacement \(a\) in the form:

\[ \varepsilon=\varepsilon_{\infty}+4\pi n\alpha, \tag{10} \]

where \(n\) is the number of ion pairs in \(1\ \mathrm{cm}^{3}\). Assuming that the internal field differs from the average macroscopic field by \(\frac{4}{3}\pi I\), where \(I\) is the electric moment of a unit volume of the dielectric, we obtain that the value of the dielectric coefficient \(\varepsilon\) of ionic crystals is related to the polarizability of the ions by the condition:

\[ P=\frac{\varepsilon-1}{\varepsilon+2}=\frac{4\pi}{3}N\left(\alpha_{1}+\alpha_{2}+\alpha_{i}\right), \tag{11} \]

where \(\alpha_{1}\) and \(\alpha_{2}\) are the electronic polarizabilities of the ions,

\[ \alpha_i \]

is the ionic polarizability, equal to \(\alpha_i=\frac{2e^2}{K}\),

where \(K\) is the coefficient of the elastic bond of the ions, \(N\) is the number of ions in \(1\ \mathrm{cm}^{3}\).

The value of the electronic polarizability \(P_{\mathrm{e}}\) is represented by the formula:

\[ P_{\mathrm{e}}=\frac{\varepsilon_{\infty}-1}{\varepsilon+2}. \]

It has been established that the electronic polarization of ionic crystals depends on the polarizability of the individual ions of which the crystal is built and on the number of ions per unit of its volume. Electronic polarization depends on the ratio of the electronic polarizability of an ion to the cube of its radius.

Dependence of the dielectric coefficient of crystals of alkali-halide salts on the lattice constant and the lattice energy

Figure 13 presents the dependence of the electronic component of the dielectric coefficient of crystals of alkali-halide salts on the lattice energy. With increasing lattice energy and bond strength, the electronic component of the dielectric coefficient decreases.

Fig. 13. Dependence of the electronic component of the dielectric coefficient on the lattice energy for crystals of alkali-halide salts.

With an increase in the lattice constant, and consequently also in the radii of the ions composing it, the value of the dielectric coefficient increases. The increase in the dielectric coefficient is determined by the growth of the polarizability of the halide. Replacement of one metal by another in the series of alkali-halide compounds is likewise accompanied by a change in the dielectric coefficient. For example, when sodium is replaced by potassium, despite a considerable—

but, despite the increase in the polarizability of the ion, the dielectric coefficient of the molecule nevertheless decreases. The decrease in the dielectric coefficient of the compound is a consequence of the increase in the radius of the ion and the decrease in their number per unit volume. The polarizability associated with ionic displacement, \(\alpha_i\), calculated for a pair of ions in a heteropolar lattice, is found to be

\[ \alpha_i=\frac{a_0^{3}}{0.58\,(n_1-1)}, \tag{12} \]

where \(a_0\) is the lattice constant and \(n_1>1\) is the exponent in the equation of the repulsive forces due to the interaction of the electron shells.

Fig. 14. Dependence between the ionic component of the dielectric coefficient and the lattice energy.

According to condition (12), the dielectric coefficient must increase proportionally to the ionic polarizability and, consequently, to \(a_0^{3}\).

In Fig. 14 the dependence of the component of the dielectric coefficient due to displacement of the ions on the magnitude of the lattice energy is shown. With an increase in the lattice constant, and consequently with a decrease in the lattice energy, the ionic component of the dielectric coefficient decreases.

Fig. 15. Dependence of the dielectric coefficient on the lattice energy for crystals of alkali-halide salts.

Despite the increase in the polarizability of ions with increasing distance between them, which should have been accompanied by an increase in the dielectric coefficient (Fig. 14), a decrease in it is observed. This increase in the ionic component of the dielectric coefficient with increasing lattice energy is explained by the increase in the concentration of ions with increasing lattice energy of alkali-halide salt crystals, noted earlier.

Fig. 15 shows the dependence of the dielectric coefficient on the magnitude of the lattice energy. With increasing lattice energy, the dielectric coefficient increases. These graphs reproduce the course of the dependence of the ionic component of the dielectric coefficient, which determines the magnitude of the dielectric coefficient of ionic crystals.

Dependence of the ionic conductivity of alkali-halide salt crystals on the lattice constant and lattice energy

The dependence of electrical conductivity on temperature is described by a two-term formula of the type:

\[ \sigma = A_1 e^{-\frac{U_1}{KT}} + A_2 e^{-\frac{U_2}{KT}}, \]

where \(U_1 > U_2\). The energies \(U_1\) and \(U_2\), necessary for the liberation of ions, are called activation energies.

According to Acad. A. F. Ioffe, the electrical conductivity of crystals is caused by the detachment of ions of the main lattice. At low temperatures, sodium ions in NaCl, which are more weakly bound at lattice sites, are torn away; their liberation requires the smaller energy \(U_2\). At higher temperatures, chlorine ions begin to be torn away; their liberation requires the larger energy \(U_1\).

In the opinion of other scientists, conductivity at low temperatures is caused by impurity ions, whose binding energy is smaller, and at high temperatures by ions of the main structure.

With increasing lattice energy, the high-temperature activation energy \(U_1\) remains almost constant, changing within the limits of 10—15% when the halide or metal is substituted. The low-temperature activation energy, with increasing lattice energy, increases almost twofold. Significant changes are observed both when the halide is substituted and when the metal is substituted.

An especially strong change in the low-temperature activation energy occurs when ions with greater polarizability are substituted (ions \(J\), \(Br\)). Since the magnitude of the activation energy determines the ionic electrical conductivity of crystals, one may conclude that electrical conductivity is lawfully connected with the energy of the crystal lattice.

Dependence of dielectric losses in crystals of alkali-halide salts on their composition and other properties

Thanks to the work of K. A. Vodol’yanov at the Siberian Physico-Technical Institute and of his students, the accuracy of measuring the angle of dielectric losses has been considerably increased, and it has become possible to measure a value ten times smaller. As a result, it has proved possible to measure the angle of dielectric losses in crystals and to establish its connection with other physicochemical characteristics of the crystal.

It follows from Fig. 16 that \(\operatorname{tg}\delta\) is directly dependent on the height of the potential barrier separating the lattice particles. As the activation energy decreases, \(\operatorname{tg}\delta\) increases. The value of the activation energy depends on the lattice energy. With an increase in the lattice energy and a strengthening of the entire structure, \(\operatorname{tg}\delta\) decreases (Fig. 17). Hence the connection of \(\operatorname{tg}\delta\) with the magnitude of the lattice constant, the sublimation energy, the heat of formation of the compound, and the melting temperature is obvious.

Fig. 16. Dependence of \(\operatorname{tg}\delta\) on the activation energy for crystals of alkali-halide salts.

The results of the investigations of K. A. Vodopyanov and his students make it possible to conclude that dielectric losses in alkali-halide crystals are determined by their chemical composition and by the strength of the chemical compound. In perfect crystals, dielectric losses are a characteristic of the basic substance and its structure.

Fig. 17. Dependence of electrical strength and \(\operatorname{tg}\delta\) on the lattice energy.

Dependence of Electrical Strength on the Composition and Thermochemical Characteristics of a Substance

Modern theories of electrical breakdown do not relate the magnitude of the electrical strength of dielectrics to their physicochemical characteristics. In the modern quantum-mechanical model of the energy bands of a solid dielectric, there are no accepted characteristics of the dielectric. The magnitude of the width of the forbidden interval in the energy spectrum, which is used in these theories, is likewise not connected with the thermochemical or physicochemical characteristics of the dielectric. An exception is constituted by experimentally established cases, for example, the connection between the width of the forbidden energy interval, between the \(F\)-band and the conduction band, and the lattice constant and lattice energy.

It has been shown that the electrical strength of crystals of alkali-halide salts changes regularly when their composition is changed, and that the magnitude of the electrical strength depends on other properties of the crystals. With increasing lattice energy, the electrical strength of the dielectric also increases.

The magnitude of the electrical strength is also found to stand in corresponding relations with other physicochemical characteristics.

In Fig. 17 the dependence of electrical strength and of the dielectric loss angle on the lattice energy is shown. A regularity is observed: an increase in the dielectric loss angle in crystals is accompanied simultaneously by a decrease in their electrical strength. A decrease in the energy of bonding of particles in the lattice leads to an increase in dielectric losses and a decrease in the electrical strength of the structure.

In Fig. 18 the dependence of electrical strength and of the dielectric loss angle (according to the data of M. A. Krivov) on temperature is shown for crystals of rock salt.

K. A. Vodolazov measured the dielectric loss angle of mixed crystals of the KI—KBr system as a function of temperature at high frequency. According to these data, a decrease in the lattice energy leads to a stronger temperature dependence of the loss angle. For a composition of approximately 50% KI in KBr, when the solid solutions obtained are unstable and easily decompose, K. A. Vodolazov obtained the greatest increase in the loss angle with increasing temperature. As can be seen, in this case also the weakening of the bond between the lattice ions as the temperature rises is accompanied by an increase in the angle of dielectric losses and a decrease in the breakdown voltage of the crystals.

From Fig. 18 it is seen that up to a temperature of about \(250^\circ\text{C}\), the dielectric losses and breakdown voltage of rock salt depend almost not at all on temperature. It is considered that in this temperature interval the breakdown has an electrical form. At higher temperatures, when the breakdown voltage decreases strongly with increasing temperature, the breakdown has a thermal form.

The regularity observed in Fig. 18 may also have another explanation. In some temperature interval the energy of the bond between ions changes little with increasing temperature. Then, at higher temperatures, in connection with an increase in the lattice constant, the bonding forces between particles begin to decrease. This is accompanied by an increase in dielectric losses and a decrease in electrical strength. Thus, considering breakdown as a consequence of the weakening of the bonding forces between particles in the lattice, we do not obtain a fundamental difference between the thermal and electrical forms of breakdown, as has been customary until now.

The difference between the thermal and the electrical form of breakdown probably lies in the manner in which energy is transferred from the electric field to the dielectric and in the development within it of a breakdown channel.

In the electrical form of breakdown, the energy of the field is transferred to the crystal lattice by means of an electronic current, whereas in the thermal form of breakdown an ionic current plays an essential role in the transfer of energy to the lattice.

Conclusion

The physical and electrical characteristics of ionic crystals are naturally connected with their chemical composition and structure. The connection between the various characteristics of crystals of alkali-halide salts is revealed when they are compared with the lattice energy and the lattice constant.

What has been said above about the determining role of the composition and chemical structure of a dielectric in its electrical properties does not exclude, but rather presupposes, the influence on the characteristics of crystals of defects in them and of disturbances of structure. It may be considered that under certain experimental conditions some properties of a crystal will be determined by structural disturbances or by impurities. This circumstance opens broad possibilities for controlling the properties and characteristics of materials. For this it is necessary to establish the laws governing the course of physical processes in the basic structure and when it is disturbed by impurities and defects.

In the arguments presented it has been shown that the thermal, mechanical, and electrical strength of dielectrics increase with an increase in the lattice energy of crystals of the rock-salt type.

The crystalline lattice of the rock-salt type is the most stable and the most widespread in geochemistry. This type of lattice makes it possible to obtain a very compact and symmetrical packing of ions.

In general, the energy of homopolar lattices is somewhat greater than the energy of heteropolar ones, since polarization increases the bond and is an additional term in the expression for the lattice energy. Despite the increase in lattice energy, in molecular lattices a decrease in mechanical and thermal strength is observed as a consequence of the weakening of the bond between individual molecules.

In a crystalline lattice, the substitution of some ions by others of similar volumes is possible. It is also possible to replace some ions by ions of another valence, provided that the valence and radius of the ions are simultaneously changed and their ratio is kept constant. In the case of isomorphous substitution, the process will proceed in the direction of decreasing the energy of the system, increasing the entropy, and increasing the lattice energy. It is known that in the case of alloys of elements the lattice energy usually increases in comparison with the energy of pure metals. The replacement of atoms of vertical groups, standing higher in Mendeleev’s table, by lower-lying ions with smaller ionization potentials is accompanied by a decrease in the bonding energy of the system. Therefore, when replacing Ca, Sr, or Ba, we must each time expect a lower stability of the new system.

As the phenomenon in a dielectric becomes more complicated, it proves to be connected with an ever greater number of other quantities. If the polarizability of an individual ion is determined by the valence and radius of the ion, then the polarizability of a pair of ions is already determined also by the distance between them. The polarizability of a unit volume will be connected, in addition to the indicated quantities, also with the concentration of ions and the energy of their interaction with one another, and so on.

A quantity such as \(\operatorname{tg}\delta\), or the electrical strength of crystals, already proves to be connected with a large number of other characteristics of the crystal. Some of these connections are determinative of the phenomenon, others only influence it to some extent. It is of practical inte—

… determine the specific role of each factor in any given phenomenon. The connection with the lattice energy is the most decisive.

Fig. 18. Dependence of electrical strength and dielectric losses on temperature for rock-salt crystals (according to M. A. Krivov).

The general and particular scientific propositions presented here are also of practical interest. For solid dielectrics, scientific criteria are obtained that make it possible to predict their electrical properties. Our concept leads to the conclusion that a homogeneous ionic dielectric with a high value of the lattice energy will have low electrical conductivity, low losses, and high mechanical, thermal, and electrical strength. The concurrence of all these properties, necessary for a dielectric with high insulating qualities, upon fulfillment of a single condition—namely, a large value of the lattice energy—is a very important circumstance for practice.

The head of the Department of Physical Chemistry of TPI, Associate Professor A. S. Naumova, reviewed the manuscript and made useful comments. Engineer N. K. Novikova helped in preparing the graphs. We take this opportunity to express our gratitude to them.

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