Toward the Development of the Physical-Energy Foundations for Processing Bodies with Ionic Bonds

A. A. Vorob'ev

§ 1. Introduction

In developing new materials with specified properties, it is important to have, as guidance, a general scientific physicochemical approach. For this it is necessary to know the dependence of the properties of various types of chemical compounds on their chemical composition. Such a task is worked out on compounds of uncomplicated, known composition and structure, with subsequent transfer of the regularities found to materials of complex composition used in production.

The classical theory of ionic lattices satisfactorily explained and yielded practically applicable formulas for determining the coefficient of all-round compression, polarizability, dielectric permittivity, wave propagation, and the spectrum of natural vibrations of ions, and so forth.

Quantum theory explained the nature of the homeopolar bond and refined the formulas of lattice theory and heat-capacity theory.

Despite the successes in the theoretical description of certain physical properties of ionic crystals, there is still no unified theory explaining, from a single standpoint, the dependence on chemical composition of the thermal, mechanical, optical, and electrical properties of a solid body.

Theoretically, the properties of crystals that are not associated with a violation of their continuity are satisfactorily explained.

In the preceding period, theories of the solid state developed theories of individual properties, as a rule independently of one another—for example, the theory of heat capacity or the theory of electrical breakdown, as well as theories of other properties of crystals—and independently of the chemical composition of the body. Such properties of a solid body or phenomena as electronic and ionic electrical conductivity, losses, breakdown, diffusion, and others were outside the circle of questions pertaining to the theory of the solid state.

Comparison of the properties of crystals and the establishment of links among them, and of the dependence of properties on chemical composition, is a necessary stage in the development of a unified theory of the solid state.

Below, a connection is established among various properties of ionic crystals.

N. S. Kurnakov [1] recommended assessing the dynamics of change in the properties of a substance by observing the change in at least two of any of its parameters. One property may prove insensitive to the action under consideration under the given conditions.

For organic compounds and polymers, simple correlations have been established between the composition and structure of a compound and its physical properties [2, 3, 4].

The energy of interaction between the particles that form a solid body, on which its physical properties depend, is determined by the position of the constituent elements entering into the compound in the system of D. I. Mendeleev.

It is of interest to investigate the dependence of the physical properties of the simplest ionic crystals on the position of the components in the system of D. I. Mendeleev, their atomic weight and number. The bond between particles in crystals is most fully expressed by the magnitude of the energy that must be expended in order to separate the ions to an infinitely large distance from one another. This quantity is called the energy of the crystal lattice.

In our method of investigation, the necessary attention was paid to obtaining pure preparations and measuring properties in a series of compounds when either the cation or the anion, or both, are replaced in the formation of solid solutions.

Having data on the change of the property under study under unchanged external conditions, when replacing the anion or cation by the example of halide compounds of metals of the first and second groups and oxides of metals of the second group, it proved possible to formulate a proposition on the change in the properties of compounds as a function of their chemical composition.

Comparison of results obtained by different authors, in particular on the kinetics of various processes in ionic compounds, was also complicated by the fact that studies were usually carried out at one and the same temperature for compounds of different chemical compositions having different Debye temperatures. In such cases the crystal lattices of different compounds are in relatively dissimilar states of loosening and approach to the melting state. Probably, measurements of the properties of ionic compounds of different composition should be performed at the same Debye temperature or the same relative temperature, \(T_{\text{op}}/T_{\text{pl}}\), where \(T_{\text{op}}\) and \(T_{\text{pl}}\) are, respectively, the temperatures at which the measurement is made and the melting temperature.

Among the scientific trends that have taken shape in the study of the dependence of the properties of ionic compounds on their composition, one may note attempts at scientific generalization and at establishing a connection between the properties of ionic compounds and their chemical composition through lattice energy, free volume, ionic radii, the atomic numbers of the atoms entering into the compound, and so forth.

Comparison of the properties of materials and establishment of the relationships between them is a necessary stage in the development of the theory of materials with specified properties.

In the proposed work such an attempt has been made to establish a relationship between various properties of ionic compounds. As the characteristic associated with the properties of ionic crystals, the lattice energy was chosen.

This thermodynamic characteristic is macroscopic. On the other hand, in Kapustinskii’s formula it is expressed through the ionic radii and charges of the ions and is connected with the microscopic structure and composition of the compound.

In practice, the structure and composition of single crystals always differ from the ideal. It was proposed that the defects of a real crystal and their concentration are regularly related to the magnitude of the lattice energy [5].

Ionic structures have close packing; therefore, the motion of ions in a crystal is possible only by jumps to a neighboring vacant site. The motion of an ion in one direction by jumps is accompanied by the displacement of a vacancy of an ion of the same sign in the opposite direction. An essential

quantity in the activation energy of the processes of displacement of ions from one equilibrium position to another is the activation energy for the displacement of ion vacancies. Table 1 gives the values of the activation energy for the displacement of ion vacancies in the lattice of alkali-halide salts, determined from the temperature dependence of electrical conductivity, and, for bromine and potassium ion vacancies, also from diffusion measurements; Table 2 gives the values for oxides of metals of the second group.

Table 1

Activation energy for the displacement of ion vacancies in the lattice of alkali-halide salts

Compound	Vacancy	Activation energy, eV	Lattice energy per ion pair, eV
NaCl	Na$^{+}$	0.85	7.93
NaCl	Cl$^{-}$	1.68	7.93
NaBr	Na$^{+}$	0.8	7.55
NaBr	Br$^{-}$	1.18	7.55
KCl	K$^{+}$	0.68	7.23
LiF	Li$^{+}$	0.65	10.56
LiCl	Li$^{+}$	0.41	8.6
LiBr	Li$^{+}$	0.39	8.05
LiI	Li$^{+}$	0.38	7.51

The magnitude of the activation energy increases with increasing ion radius; an exception is the potassium ion in sylvite, for whose displacement the activation energy is found to be very small [5].

As the lattice energy decreases, the activation energy required for the displacement of a cation vacancy from one position to a neighboring one decreases.

This conclusion makes it possible to infer a correspondence between the macroscopic description of processes through the lattice energy and the microscopic description through the activation energy of the processes of displacement of ions and their vacancies in a solid.

Table 2

Activation energy in eV, calculated from the temperature variation of electrical conductivity for oxides of metals of the second group, according to A. P. Nakhodnova, and lattice energy

Oxide name	Lattice energy per ion pair, eV	Activation energy in eV at electrical conductivity, high-temperature	Activation energy in eV at electrical conductivity, low-temperature
BeO	47.52	2.143	0.480
MgO	41.36	1.954	0.235
CaO	37.53	1.724	0.356
SrO	34.80	1.502	0.146
BaO	32.86	1.379	0.460
ZnO	42.68	1.597	0.297

§ 2. Dissolution of Alkali-Halide Compounds

When alkali-halide compounds interact with water or with an unsaturated aqueous solution, diffusion or diffusion-kinetic dissolution, the formation of hydrates, dissociation, and the reverse process of crystallization from solution are possible. The probability and intensity of each of these processes depend on the bond energy in the chemical compound being dissolved, the nature of the solvent, and the dissolution conditions.

When a crystal dissolves and the system passes from an ordered state to a disordered one, the energy of the system increases. The ordered

the state of the particles in the crystal corresponds to the minimum of potential energy.

The heat of dissolution \(Q_{\text{diss}}\) is composed of the heat expended on the destruction of the crystal structure and equal to the lattice energy \(U\), and the heat of interaction of the dissolved substance with the solvent \(\Delta H < 0\), which is released during dissolution. Thus, approximately,

\[ Q_{\text{diss}} = \Delta H - U. \tag{1} \]

When dissolution in water and the formation of hydrates occur, the quantity \(\Delta H\) is called the heat of hydration. Depending on the magnitudes of the terms \(\Delta H\) and \(U\), the heat of dissolution \(Q_{\text{diss}}\) may have different signs. Since the quantity \(Q_{\text{diss}}\) is small and amounts to several kcal/mol, it may be approximately assumed that

\[ \Delta H \simeq U. \tag{2} \]

This condition can be used for an approximate determination of the lattice energy of a compound. It follows from (2) that the heat of hydration increases linearly with increasing lattice energy. Figure 1 presents the dependence of the heat of hydration of alkali-halide compounds, taken as the sum of the heats of hydration of the ions, on the lattice energy. As can be seen from Fig. 1, the expected linear dependence (2) is satisfactorily confirmed by experimental data.

Fig. 1. Dependence of the heat of hydration of alkali-halide compounds, taken as the sum of the heats of hydration of ions, on the lattice energy

With increasing lattice energy, the chemical stability of ionic compounds increases [6].

A. E. Fersman believed [7] that the lattice energy also determines the solubility of ionic compounds. Table 3 gives the values of solubility in water, lattice energy, and enthalpy of dissolution, i.e., the thermal effect of dissolution in water, for alkali-halide compounds crystallizing in a system of the NaCl type, according to the data of A. A. Vorob’ev [6].

The monotonic behavior of the solubility and the thermal effect in Table 1 is disrupted in cases where the compounds form hydrates with water:
NaBr·2H₂O, NaJ·2H₂O, KF·2H₂O, LiCl·2H₂O, LiBr·2H₂O, LiJ·3H₂O.

Table 3

Solubility of alkali-halide compounds in water [8]

Compound	Lattice energy per ion pair, eV	Solubility at 20°C, kg of substance per 1 kg of water	Enthalpy of dissolution \(\Delta H \cdot 10^{-3}\), kcal/kmol; 1 kmol of substance dissolved in 400 kmol of water
LiF	10.56	0.003	0.9
LiCl	8.50	0.785	—8.5
LiBr	8.05	1.770	—11.5
LiJ	7.51	1.650	—14.9
NaF	9.46	0.042	0.4
NaCl	7.93	0.360	1.2
NaBr	7.55	0.905	0.6
NaJ	7.07	1.780	—1.5
KF	8.37	0.950	—3.9
KCl	7.23	0.340	4.5
KBr	6.96	0.658	5.2
KJ	6.55	1.440	5.2
RbCl	6.95	0.912	4.4
RbBr	6.71	1.520	—

If, during the dissolution of salts, compounds with water are not formed and dissolution proceeds with absorption of heat, i.e., \(\Delta H > 0\), then the solubility of the compounds decreases with increasing lattice energy, as shown in Fig. 2, constructed by A. F. Naumov. For compounds forming crystalline hydrates, the thermal effect upon dissolution increases with increasing lattice energy (Table 1).

According to [9], for aqueous salt solutions, if the function \(\Delta H > 0\), then \(T \cdot \Delta S > 0\), where \(\Delta S\) is the increase in the entropy of the system during the dissolution process.

The heat absorbed during the dissolution of such compounds, and the condition \(T \cdot \Delta S > \Delta H\), automatically regulate dissolution. A large increase in the entropy of the system during dissolution of the crystal causes a decrease in the thermodynamic potential of the system and, consequently, the possibility for the dissolution process to proceed. The increase in entropy \(\Delta S\) during dissolution in water of alkali-halide crystals that do not form hydrates will be the smaller, the greater the lattice energy. The data on \(\Delta S\) were taken from the works of S. A. Shchukarev, L. S. Lilich, and others; the comparison with the lattice energy is ours. In this way A. F. Naumov explains the decrease in the solubility of crystals presented in Fig. 1.

Alkali-halide crystals can dissolve in water and aqueous solutions in a diffusion regime, when the dissolution rate is determined only by diffusion, and in a diffusion-kinetic regime, depending on the structure of the crystal [10, 11, 12].

The different mechanism of dissolution of crystals along different crystallographic directions leads to anisotropy of the dissolution rate. The dissolution rate of a crystal in weak solvents recommended for etching is probably limited to a greater extent by the intensity of the process by which particles leave the crystal for the solution than by the rate at which particles move into the bulk of the solution [13]. For the reasons indicated, the dissolution of crystals in aqueous solutions close to saturated may be regarded as a diffusion–kinetic process.

Fig. 2. Relationship between the solubility of alkali-halide crystals of the NaCl type in water at 20°C and the energy of their crystal lattices

According to A. F. Naumov, under a diffusion–kinetic regime the dissolution rate of a crystal becomes an anisotropic quantity. At room temperature, the greatest dissolution rate of alkali-halide crystals is observed on the (110) face, and the smallest on the (100) face.

Table 4 presents the density of ion distribution on various faces of crystals, according to D. I. Vaisburd.

As the relative undersaturation of the solution decreases,

\[ \sigma = \frac{C_0 - C}{C_0}, \]

where \(C_0\) is the concentration of the equilibrium solution over the crystal, \(C\) is the concentration of the solution, and as \(W\)—the velocity of the solvent jet— increases, with its normal incidence the ratio of the rates \(V(110)\) to \(V(100)\) increases.

The dissolution rate is greater the lower the density of ions on the crystal face and the lower the binding energy between ions.

A. F. Naumov obtained the smallest dissolution rate on the (100) face, which corresponds to the greatest ion density. The bonding between ions on the (100) face is the strongest. The dissolution rate in this case is thus in satisfactory correlation with the energy of ion bonding. In the (110) plane the dissolution rate is the greatest, while the density of ion arrangement is lower than on (100),

the bond between neighboring ions in the (110) plane is less than in (100). Consequently, in this case there is again a satisfactory correlation between the dissolution rate and the bond energy between particles.

Table 4

Density of ion distribution over various crystallographic planes, according to D. I. Weisburg

Chemical composition	Cross-sectional area of an interstice in Ų when cut by plane	Cross-sectional area of an interstice in Ų when cut by plane	Cross-sectional area of an interstice in Ų when cut by plane	Correspondence of the cross-section of the interstice and the halogen atom in the direction	Correspondence of the cross-section of the interstice and the halogen atom in the direction	Correspondence of the cross-section of the interstice and the halogen atom in the direction
1	100	110	111	100	110	111
LiF	0.472	1.625	2.8	0.366	1.26	2.17
LiCl	0.084	3.520	5.61	0.032	1.35	2.17
LiBr	—	4.240	6.7	—	1.04	1.64
LiJ	—	5.560	8.55	—	0.956	1.47
NaF	1.09	1.671	2.6	0.845	1.30	2.02
NaCl	0.634	3.080	5.3	0.244	1.18	2.04
NaBr	0.53	3.800	6.32	0.130	0.932	1.55
NaJ	0.254	5.070	7.25	0.044	0.873	1.42
KF	1.21	0.950	2.22	0.940	0.736	1.72
KCl	1.46	2.490	4.75	0.561	0.959	1.83
KBr	1.45	3.140	5.8	0.355	0.770	1.42
KJ	1.40	4.300	7.55	0.241	0.740	1.3
RbF	1.34	0.817	2.06	1.040	0.633	1.6
RbCl	1.78	2.270	4.60	0.685	0.873	1.77
RbBr	1.85	2.90	5.55	0.453	0.710	1.36
RbJ	1.84	3.940	7.25	0.316	0.679	1.25

The anisotropy of dissolution of alkali-halide crystals increases with increasing surface energy and lattice energy.

According to P. A. Savintsev, the rates of contact melting of the corresponding faces of crystals are in the same ratio to one another as the rates of dissolution of these faces; i.e., the melting rate of crystals in contact along the (110) face is the greatest, and the smallest is that in contact along the (100) face.

In aqueous solutions far from saturation, at a low rate of stirring of the solvent, the dissolution process is diffusion-controlled, and the dissolution rate is an isotropic quantity.

According to A. F. Naumov, the structure-sensitive dissolution regime sets in with intensive stirring and a decrease in the undersaturation of the solution. This transition, for crystals with higher lattice energy, occurs at less intensive stirring of the solution and at greater undersaturation. This conclusion follows from Table 5, obtained by A. F. Naumov.

As can be seen, the ratio of the rates $V(110)/V(100)$ at the same undersaturation $\sigma$ increases with increasing crystal-lattice energy. Consequently, the transition to a diffusion-kinetic mechanism of dissolution for crystals with greater lattice energy occurs at

at lower concentrations of their aqueous solutions used as solvents.

When dissolution is carried out in twice-distilled water, the anisotropy of the dissolution rate \(V(110)/V(100)\) increases with increasing lattice energy and with the intensity of stirring of the solution.

Table 5

Compound	Lattice energy per ion pair, eV	\(v_{(110)}/v_{(100)}\), \(\sigma = 1\)	\(v_{(110)}/v_{(100)}\), \(\sigma = 0.05\)	\(v_{(110)}/v_{(100)}\), \(\sigma = 0.002\)
KI	6.55	1.02	1.18	2.05
KBr	6.94	1.03	1.34	2.63
NaCl	7.93	1.07	1.77	3.20
LiF	10.56	2.80	—	—

For radiation-chemical technology and radiation chemistry, the study of dissolution regimes of irradiated or pre-irradiated materials is of interest. A. F. Naumov found that, after irradiation with the dose indicated below, the solubility of irradiated crystals does not change.

Disruption of the structure of an irradiated crystal leads to an increase in the density of dissolution centers on the (100) face and to an increase in the dissolution rate under certain conditions. For comparison, the (100) and (110) faces were selected, since they contain ions of both signs. The (111) plane consists only of ions of one sign.

The distance between ions on the (110) face is greater by a factor of \(\sqrt{2}\) than on the (100) face, and the bond energy is lower. The density of defects on faces with a lower bond energy will be greater. As a result of the lower bond energy and greater defectiveness, the dissolution rate of the (110) faces is higher than that of the (100) faces [14].

A. F. Naumov measured the linear dissolution rate of NaCl, KCl, KBr, KJ, and LiF crystals after their irradiation with x-rays of energy 8 keV at a dose of \(2 \cdot 10^6\) rad, with protons of energy 4.5 MeV and a dose of up to \(10^{19}\) protons, and with neutrons of energy 4.5 MeV at a dose of \(10^{14}\) neutrons/cm\(^2\).

V. D. Kuznetsov [15] showed that, owing to the different magnitude of the bond energy on the crystal faces, cleavage of the crystal in practice always proceeds along the cleavage plane (100), whose bond energy is lower than the others because of the bond energy weakened by the large distance between ions located in neighboring (100) planes.

Table 6 gives data on the measurement of the linear dissolution rate of the (100) and (110) faces of an NaCl crystal in an aqueous NaCl solution after irradiation with x-rays of energy 8 keV. The relative saturation of the solution was \(\sigma = 3.6 \cdot 10^{-3}\). Dissolution of the crystals was carried out in a jet of undersaturated solution incident on the face of the crystal under study along the normal at a velocity \(\omega = 7\) ml/sec. The linear dissolution rate of the crystals along the (100) and (110) faces as a function of the x-ray irradiation dose at an energy of 8 keV (according to A. F. Naumov) is given in Table 6.

It follows from Table 6 that irradiation of the crystal with soft x-rays, which are absorbed in a thin layer, increases the linear dissolution rate on the (100) face and does not change this quantity on the more defective (110) face.

Similar results were also obtained in the study of KCl, KBr, KI, and LiF crystals. It turned out that, at a given velocity of incidence of the solvent jet on the face being dissolved, for a crystal of each composition there exists a threshold value of the solvent undersaturation, \(\sigma_0\).

Table 6

Irradiation dose \(Q \cdot 10^{-5}\), rad	\(v_{(100)} \cdot 10^3\), m/sec	\(v_{(110)} \cdot 10^8\), m/sec
0	2.01	6.12
0.5	2.54	6.10
1.0	2.91	6.22
1.5	3.23	6.05
2.0	3.35	6.09
2.5	3.48	6.14
3.0	3.49	6.03
3.5	3.37	6.16
4.0	3.40	6.11

Only at a relative undersaturation less than \(\sigma_0\) does the linear dissolution rate of a crystal on the (100) face depend on irradiation (Table 7).

As can be seen from Table 7, the value of \(\sigma_0\) increases with increasing lattice energy. These regularities are determined by the magnitude of the chemical-bond energy. With increasing lattice energy of the compound being dissolved, the dissolution rate decreases.

To maintain a low dissolution rate, at which it still depends on the dose of preliminary irradiation, it is necessary to reduce the degree of undersaturation of the solvent as the chemical-bond energy decreases.

Table 7

Magnitude of the relative undersaturation of the solvent \(\sigma_0\), below which the linear dissolution rate of the solvent \(v_{(100)}\) depends on the dose of preliminary irradiation

Compound	Lattice energy per ion pair, eV	Magnitude of relative undersaturation \(\sigma_0\)
KI	6.55	0.008
KBr	6.94	0.012
KCl	7.23	0.025
LiF	10.56	1.000
NaCl	7.93	0.030

From these results one may draw the general conclusion that the linear dissolution rate will be a structure-sensitive quantity at low dissolution rates and small solution undersaturation.

Figure 3 presents changes in the ratio of the linear dissolution rate of the irradiated (100) face, \(V_0\), to the dissolution rate \(V_n\) of the unirradiated (100) face at different values of the relative undersaturation \(\sigma\) for crystals of different chemical composition after irradiation with X-rays of energy 8 keV at a dose of \(10^5\) rad. Irradiation and dissolution were carried out at room temperature.

The strongest increasing dependence is observed at the smallest value of the relative undersaturation, \(\sigma = 0.003\), and, conse-

therefore, low dissolution rates. The greatest increase in the dissolution rate is observed for irradiated crystals with a higher lattice energy. This result is in good agreement with the assumption that radiation defects causing an increase in the dissolution rate are better preserved in the lattice of compounds with a stronger chemical bond [16].

Fig. 3. Change in the ratio of the linear dissolution rate of the irradiated face (100), \(v_0\), to the rate \(v_{\mathrm{n}}\) of the unirradiated face (100) at different values of the relative undersaturation \(\sigma\) for crystals of different chemical composition after irradiation with X-rays of energy 8 keV at a dose of \(10^5\) rad.

The deviation from the general regularity of an increase in the ratio \(V_0/V_{\mathrm{n}}\) with increasing lattice energy for KJ and KBr crystals at \(\sigma > 0.006\) is explained by the transition, under these conditions, from the diffusion-kinetic regime of crystal dissolution to the diffusion regime. In the diffusion regime of dissolution, the linear dissolution rate becomes a structurally insensitive quantity.

The results of the study of dissolution of ionic crystals of different chemical composition make it possible to draw the general conclusion that, for each ionic compound, dissolution conditions and solvents can be selected under which certain quantities characterizing the dissolution process, for example the dissolution rate, will be sensitive to an external action, for example irradiation.

Thus, under certain conditions, the magnitude of the linear dissolution rate will be a relative characteristic of the radiation resistance of an ionic compound.

Irradiation increases the density of defects in a substance, weakens it, and increases the rate at which substances dissolve.

These effects are manifested more strongly in compounds with a higher chemical-bond energy, since the latter have better radiation memory. Substances with lower bond energy change their properties less after irradiation, owing to the rapid annealing of radiation defects. It may be assumed that, in the process of irradiation, in compounds with lower bond energy the density of radiation defects will be greater and their properties will change relatively more strongly than the properties of compounds with higher lattice energy.

In connection with the foregoing, it is of interest to study the dissolution regime and the characteristics of the process under irradiation of the body being dissolved.

§ 3. Electron-hole properties of ionic crystals and their chemical composition

The electronic properties of crystals and their characteristics are in a regular dependence on the lattice energy. The electronic properties may be considered by representing the crystal as a system of interacting ions, or from the point of view of collectivized electrons. In this case the principal filled band of energy levels and the conduction band are associated with a definite state of the ions that form the crystal.

To determine the electronic levels in isolated atoms, let us take the rest energy of a free electron in vacuum as equal to zero. The lowest free level of the free ion $\mathrm{Na}^{+}$, corresponding to the $3s$ state, is located below the zero level by an amount equal to the ionization energy, $5.12$ eV. The highest filled level for the ion $\mathrm{Cl}^{-}$ is $3p$, located below the vacuum level by an amount of the electron affinity energy of the atom $\mathrm{Cl}^{\circ}$, equal to $4$ eV.

Consequently, the free $3s$ level lies below the occupied $3p$ electronic level by $1.12$ eV. As the free ions $\mathrm{Na}^{+}$ and $\mathrm{Cl}^{-}$ approach one another and the share of the energy of Coulomb interaction in the chemical bond increases, at a certain distance between the ions a change occurs in the mutual arrangement of the electronic levels. With a further decrease in the distance between the ions, the occupied upper $3p$ levels of the $\mathrm{Cl}^{-}$ ion are situated lower than the free $3s$ levels of $\mathrm{Na}^{+}$.

When the distance between the ions reaches a value equal to the lattice period, at which the wave functions of neighboring ions of the same sign overlap, the discrete electronic levels of the anions broaden into bands. The lattice period determines the state between the bands and the width of the bands. Levels corresponding to pairs of like-named ions overlap, forming an energy band. Thus, the scheme of energy levels for electrons in crystals proves to be connected with the lattice period, with its energy structure, on the one hand, and, on the other, with the electronic structure and levels of the atom. For example, the lower filled band of the NaCl crystal is connected with the upper filled $3p$ levels of $\mathrm{Cl}^{-}$, while the free upper energy band of the crystal corresponds to the lower unoccupied $3s$ levels of $\mathrm{Na}^{+}$ ions (Fig. 4).

The interaction between the atoms of a solid body is most strongly reflected in the energy conditions of the outer electrons of atoms, which possess the greatest level energy, determined by the influence of neighboring atoms, the distance between which is equal to $a$.

The energy levels of inner electrons at $r<a$ either do not split at all ($1s$), or split only very slightly ($2s$). From this point of view the crystal represents a large molecule consisting of

sets of atoms having the reduced system of energy levels. The inner electrons of these atoms behave practically as in isolated atoms, while the outer electrons belong to the entire crystal—the molecule. The energy of the electrons may lie within the limits of the shaded regions in Fig. 4, which constitute energy bands.

Fig. 4. Splitting of various atomic levels under the influence of the fields of neighboring atoms, the distance between which is equal to \(1/a\)

The valence electrons in atoms have several allowed levels; therefore, in a crystal, generally speaking, several bands of allowed electron-energy states are formed. The width of the forbidden energy bands is commensurate with the width of the allowed bands. As the energy increases (Fig. 4), the width of the allowed energy bands increases, while that of the forbidden bands decreases.

The energy bands are filled with electrons provided that the Pauli exclusion principle is satisfied and the minimum energy of the system is maintained, and pairs of electrons fill the lower levels of the lower allowed energy band.

In real crystals the number of atoms \(N\) per \(\text{cm}^3\) is \(10^{22}\)—\(10^{23}\). The number of levels in a band is of the same order. The total width of a band amounts to several electronvolts, while the distance between adjacent levels reaches approximately \(10^{-22}\) eV.

As a consequence of the above, experimental regularities are observed that describe the connection between electron-hole transitions in alkali-halide crystals and their properties and the energetics of the ionic lattice in general, the lattice energy, and, consequently, the chemical composition of the compound.

As the energy of the crystal lattice decreases, the width of the forbidden energy gap for electrons in the crystal \(\Delta E\) decreases, and the dielectric constant \(\varepsilon\) of the crystal increases. For alkali-halide crystals with \(F\)-centers, Mott’s formula, proposed by him for impurity semiconductors, is approximately fulfilled:

\[ \varepsilon^{-2} \approx \Delta E . \]

The systematic increase in \(\varepsilon\) with decreasing lattice energy is a consequence of the increase in the polarizability of anions in the transition from fluorides to iodides and of cations in the transition from lithium to cesium, as well as of the weakening of Coulomb bonds in the structure and the increase in the van der Waals components of the chemical bonds in these compounds.

In the transition from lithium fluoride to cesium iodide and with an increase in the ionic component of the chemical bond, the dielectric properties and stability of the compounds deteriorate.

With an increase in lattice energy, the refractive index of light and the absorption coefficient over a broad range of radiation energies decrease. The position of the short-wavelength edge of the intrinsic-absorption band, and of the maxima of intrinsic and additive absorption in the infrared, visible, and ultraviolet parts of the spectrum in crystals activated by impurities, shifts toward the short-wavelength part of the spectrum when, in a given series of compounds with an unchanged cation, one passes to crystals with higher lattice energy.

Some physical processes in crystals are not directly connected with the energy of the crystal lattice. For example, the scattering of photons, phonons, and high-energy particles by atoms constituting the crystal.

These processes are determined by the magnitude of the scattering-center cross section, the magnitude of the ionic radius, the number of electrons, and so forth. With increasing radii of the ions included in a compound, the energy of the crystal lattice decreases and, at the same time, the scattering of particles, phonons, and photons increases. Thus, with decreasing lattice energy, an increase is observed in the absorption of radiation and phonons, and a decrease in the thermal conductivity and sound velocity in crystals.

The indicated correlation also makes it possible to extend to these properties the classification of the properties of ionic structures by means of a single characteristic—the energy of the crystal lattice.

The properties of structures with other types of chemical bond can also be classified by means of an energy characteristic. For example, for semiconductors with covalent bonding, a correlation is observed between many properties and the magnitude of the atomization energy or the heat of formation of the compound.

As the basis for classifying the properties of certain polymers, as for crystals, one may take the magnitude of the binding forces between the structural units forming the polymer into which it decomposes upon destruction.

An increase in the bond energy between structural elements in glasses is accompanied by an increase in the temperature and heat of vitrification, in the thermal, mechanical, and chemical stability of the glass, and by a decrease in the loss angle and electrical conductivity, as well as by a shift of the region of anomalous dispersion of dielectric permittivity and losses into the range of higher temperatures.

The radiation resistance of solids changes with a change in the type of chemical bond. In order of resistance to the action of ionizing radiation, chemical bonds are arranged as follows: metallic, ionic, covalent, and the least stable, molecular.

When the halide is replaced in a series of alkali-halide compounds in going from fluorine to iodine, the Coulomb component decreases and the van der Waals component of the lattice energy increases. The accumulated experimental material makes it possible to correctly evaluate changes in the properties of compounds when their chemical composition changes, and to proceed to consideration of the regularities determining the influence of the chemical bond in a solid on its properties, which is one of the central tasks of solid-state chemistry at the present time.

The regularities presented show that, with decreasing ionicity of the bond and increasing share of van der Waals energy, the properties of alkali-halide compounds change. A more detailed study of changes in the physical properties of other ionic compounds in which the Coulomb bond is not predominant is of interest. For heterodesmic ionic structures it is also possible to esta-

reveal useful correlations between the chemical bond and the properties of the solid.

Classification of compounds on the basis of the chemical bond can provide information on the direction of searches for new dielectrics and determination of their properties. Consideration of the properties of dielectrics on the basis of the chemical bond may be less complicated than calculations based on band concepts.

§ 4. Change in the properties of glasses of different chemical composition upon their irradiation

The optical absorption of ionic crystals is associated with the lattice energy. The position of the corresponding maxima in the absorption spectra of pure ionic crystals containing color centers shifts toward shorter wavelengths in passing to crystals with higher lattice energy [17—21]. The long-wavelength edge of the intrinsic fundamental absorption of ionic crystals and the short-wavelength boundary in the infrared region of the spectrum shift toward longer wavelengths in passing in a series to compounds with lower lattice energy [17].

These regularities are observed for single crystals and thin layers of ionic compounds, for example, oxides, sulfides, selenides, and tellurides of barium and strontium, obtained by sputtering [22, 23].

The latter makes it possible to expect that, in glasses colored in a radiation field, the composition and bond energy between particles may also influence optical absorption. The structure of glass is represented by a substance whose atoms and ions are located in interstices. The elementary structural element of glass—a steklon—is a noncanonical polyhedron of glass-forming oxide, in which the lattice cation is surrounded by atoms located from it at different distances and at different angles. Displacement of atoms and ions into interstices, without changing the character of the chemical bonds, weakens some of them and changes the covalency (S. M. Brekhovskikh, [24]). In glasses, a substance with the structure of an ideal crystal is an impurity, while the main substance (the steklon) is represented by vacancies and atoms or ions in interstices.

According to S. M. Brekhovskikh [24], the elementary structural element of multicomponent silicate glasses is the steklon Si—O—M, where M is a metal. Such glasses can be formed by partial replacement of a silicon atom by an atom belonging to one and the same or neighboring subgroups of D. I. Mendeleev’s table (Ge, Sn, Pb, P, As, Sb, Bi, B, Al, Ga, In, Te). Formation of glass by incorporation is possible when cations of small charge and large volume are situated outside the tetrahedra, forming weaker ionic bonds with the steklon.

The structure of real glasses is a homogeneous quasi-solid solution of steklons and regular polyhedra.

A sensitive method for determining the structure of glass is the study of the kinetics of the formation and destruction of color centers, paramagnetic centers, and luminescence centers in irradiated glasses [25].

Coloration of glass upon irradiation occurs as a result of the formation in them of color centers [26, 27]. In irradiated glasses of composition SiO₂—Na₂O MₓOᵧ, absorption bands of the F-, R-, and V-types are present. According to [28], the positions of the maxima of the corresponding absorption bands in irradiated NaCl crystals and glasses coincide.

The nature of the F-centers formed in glasses of simple chemical composition upon their irradiation with X-rays was described by Ekot [28, 29]. In fused quartz, under the action of irradiation, oxygen vacancies can form which, capturing one or two electrons,

form electron color centers. All types of radiation lead to the formation in glass of free electrons and excited atoms. The coloration of glass may also occur because electron traps bind ions into neutral atoms, as a result of which colloidal particles may form. Oxidation and reduction processes are also possible during electron exchange between ions [30].

In the presence in glass of ions capable of changing their valence [31], a change in the coloration of the glass may be associated with the attachment or release of electrons.

It is also assumed [28] that, in the structure of irradiated silicate glass, \(F\)-centers are quasions \(Me^{n+} + e^{-}\), consisting of interstitial cations that have captured free electrons, while \(V\)-centers are holes localized on interstitial oxygen ions. Interstitial cations possess an excess positive charge and can localize free electrons around themselves. Oxygen ions in interstices, owing to an excess of negative charge, localize holes on themselves.

The identity of the structure of color centers in irradiated alkali-halide crystals and glasses is also demonstrated by the course of the process of their destruction. Intense destruction of color centers occurs in the first period after irradiation of the glass, followed by a slower decrease in luminescence intensity thereafter.

Investigation of the mechanism of formation and destruction of color centers and paramagnetic centers made it possible to establish the presence in the structure of glass of vacancies and interstitial atoms. They cause a nonuniform distribution throughout the structure of the magnitude of the chemical potential. Analysis of thermal-bleaching curves shows that the energetic potential, in magnitude, is also distributed nonuniformly throughout the structure [28].

The temperature coefficient of linear expansion of glass is related to its chemical composition. The coefficients of linear expansion depend on the bond energy between the oxides in the glass and on the composition of the compounds formed in them.

Figure 5 presents the dependence of the coefficients of linear expansion of glasses of three binary silicate systems \(\mathrm{Li_2O—SiO_2}\); \(\mathrm{Na_2O—SiO_2}\); \(\mathrm{K_2O—SiO_2}\) in the temperature intervals \(100—200^\circ\mathrm{C}\), \(200—300^\circ\), and \(300—400^\circ\mathrm{C}\), according to [15]. The stronger bonds in the lithium system account for the smaller expansion.

The largest coefficient of expansion is exhibited by the oxides of the alkali metals. Its value is the greater, the greater the atomic weight of the metal and the lower the lattice energy of the oxide.

With increasing atomic number of the metal and decreasing lattice energy of the metal oxide, the coefficient of linear expansion increases.

For alkali-borate glasses colored by the action of \(\alpha\)- and \(\beta\)-radiation, the position of the absorption maxima has been determined [14] at the following wavelengths:

\[ \mathrm{Li_2B_4O_7} — 360\ \mathrm{m\mu} \]

\[ \mathrm{Na_2B_4O_7} — 545\ \text{''} \]

\[ \mathrm{K_2B_4O_7} — 620\ \text{''} \]

\[ \mathrm{Rb_2B_4O_7} — 667\ \text{''} \]

Substitution in borosilicate glasses of one alkali-metal ion by another of larger radius is accompanied by a shift of the maximum of the absorption band toward longer wavelengths.

Ekota [10, 11] studied absorption spectra in the range from 1 to 4.5 eV for glasses of molar composition: $\mathrm{Li_2O\cdot 2SiO_2}$; $\mathrm{Na_2O\cdot 2SiO_2}$; $\mathrm{K_2O\cdot 2SiO_2}$; $\mathrm{Rb_2O\cdot 2SiO_2}$ and mixed glasses $x\mathrm{Na_2O}\cdot(1-x)\,\mathrm{K_2O\cdot 2SiO_2}$, where $x$ varies from 0 to 1, and sodium glasses from $1.4\,\mathrm{Na_2O\cdot 2SiO_2}$ to $0.7\,\mathrm{Na_2O\cdot 2SiO_2}$.

Fig. 5. Dependence of the coefficient of thermal expansion of glasses of three binary silicate systems: K—potassium system, Na—sodium, Li—lithium.
1—room temperature–100°C, 2—100–200°C, 3—200–300°C, and 4—300–400°C.

Figure 7 presents the absorption spectra for alkali-silicate glasses of molar composition after their irradiation at room temperature.

Figure 6 presents the dependence of the temperature coefficient of linear expansion for metal oxides on the atomic number of the metal. Figures 7 and 8 plot the ratio of the absorption coefficient to its value at the absorption maximum. Replacement in silicate glasses of lithium by sodium, potassium, and rubidium, i.e., by ions with a larger ionic radius, is accompanied by a shift of the position of the maximum of the $F$ band into the long-wavelength region, as was described [32, 21] for crystals of alkali-halide salts. In the absorption spectrum of borate glasses

Fig. 6. Temperature dependence of the coefficient of thermal expansion of oxides in glass on the atomic number of the metal in the interval \(20\text{–}420^\circ\mathrm{C}\).

Visible graph labels:

• \(a)\), \(b)\)
• \(\alpha \cdot 10^7\)
• Atomic number
• Elements: Li, Na, K, Ca, Zn, Ba, Mg, Be, Si, B, Al, Fe, Zr

Fig. 7. Absorption spectra in alkali-silicate glasses after their coloration by X-rays (after Enot).

Visible graph labels:

• Wavelength, \(m\mu\)
• Absorption coefficient
• Photon energy, eV
• \(\mathrm{Li_2O \cdot 2SiO_2}\)
• \(\mathrm{Na_2O \cdot 2SiO_2}\)
• \(\mathrm{K_2O \cdot 2SiO_2}\)
• \(\mathrm{Rb_2O \cdot 2SiO_2}\)

the maximum of the absorption band shifts toward shorter wavelengths relative to the position of the absorption maximum in silicate glasses. Substitution of silicon by boron, whose ionic radius is half as large, is probably accompanied by an increase in the interaction energy of the particles.

Fig. 8. Absorption spectra in sodium-silicate glasses \(Na_2O—SiO_2\) after coloration by X-rays (according to Ekote)

In Fig. 8 the absorption spectra in \(Na_2O—SiO_2\) glasses after their irradiation with X-rays are presented. It follows from Fig. 8 that the position of the two long-wavelength absorption maxima shifts regularly with changes in the composition of the glasses. A decrease in the content of \(Na_2O\) in the glass is accompanied by a shift of the position of the absorption maximum into the short-wavelength part of the spectrum.

The ionic radius of sodium is twice as large as the ionic radius of silicon. A decrease in the \(Na_2O\) content probably corresponds to an increase in the interaction in the lattice. It is assumed that the absorption bands for glasses in the visible region are determined by color centers, consisting of an electron localized at an oxygen vacancy situated near an alkali metal. The position of the absorption band in the ultraviolet region is constant and does not depend on the composition of the glasses. It is determined by positive holes localized at the vacancy of an alkali-metal ion located ...

near the oxygen ion. With an increase in the bond energy between the individual structural elements of glass, accompanied by an increase in the heat of glass formation, the dielectric losses decrease [34]. Substitution in glasses of ions of larger radius by ions of smaller radius is accompanied by a decrease in the coefficient of linear expansion and in the heat of glass formation, and by a shift of the positions of the maxima of absorption bands, for glasses containing color centers, toward shorter wavelengths.

For each glass composition there is a dose at which noticeable darkening and a decrease in the transparency of the glass begin. When, in a glass, a cation with a larger ionic radius is replaced by a cation with a smaller ionic radius, darkening of the glass begins at a higher irradiation dose, i.e., the radiation resistance of the glass increases [33].

By changing the chemical composition, one can regulate the physical processes leading to the formation of color centers and increase the resistance of glass to the action of irradiation or other external influences.

Multivalent ions contained in glasses, under the action of irradiation, lose electrons and their valence changes. The energy of the bombarding particles is expended on this. The resistance of glasses to irradiation also increases as a result of strengthening of the bond of the multivalent ion in the structure, especially when the ionic radius of the components is small. Impurities of ions such as Fe$^{3+}$, Cr$^{3+}$, Mn$^{4+}$, As$^{5+}$, Sb$^{5+}$, and others increase the resistance of glass to the action of irradiation [33].

The resistance to the action of $\gamma$-irradiation of protective heavy glasses containing a high percentage of lead oxide is increased by introducing arsenic and antimony into them.

Replacement of compounds containing ions of large radii, for example SiO$_2$ by P$_2$O$_5$ and B$_2$O$_2$, or K$_2$O and Na$_2$O by Li$_2$O and MgO, increases the resistance of glasses to the action of $\gamma$-irradiation.

The resistance of glass to the action of $\gamma$-irradiation is increased by the addition of 0.5–1.5 wt.% sulfur [33]. In such glass, sulfur probably captures electrons liberated under the action of irradiation, and polysulfide compounds are formed.

Coesite—silicon dioxide obtained at high pressures and temperature—has a higher density, equal to 3.01 g/cm$^3$, and a packing more compact than any other natural modification of silicon dioxide, and greater resistance to external influences.

After irradiation of coesite at a temperature of about 100°C with an integral dose of $2 \cdot 10^{20}$ fast neutrons/cm$^2$, no lattice expansion greater than 0.1% or other signs of structural disruption were observed. Irradiation of quartz, tridymite, or cristobalite with the same dose leads to complete loosening of their structure.

The high stability of coesite under neutron irradiation is associated with the high density of coesite, which hinders the implantation of atoms during neutron bombardment.

§ 5. Possible role of exciton excitation in the destruction of compounds with an ionic type of bond$^{1)}$

Under the action of irradiation in solids with ionic bonds, structural defects arise, accompanied by changes in physical

$^{1)}$ The author is grateful to Yu. A. Zakharov for a fruitful discussion of the materials presented and for useful advice, which contributed to the development and clarification of the questions touched upon.

and chemical properties, chemical transformations may also occur, leading to the formation of molecules of a new chemical composition or particles of a new solid phase as a consequence of processes of radiation-chemical decomposition—radiolysis.

Radiation-chemical stability has a satisfactory correlation with lattice energy [35—37]. For example, the radiation-chemical stability of nitrates of metals of the second group increases with an increase in lattice energy, with a decrease in the free volume of the unit cell in the crystal per ion \(NO_{-3}\), with a decrease in the cation radius, and with weakening of its force field.

The radiation-chemical stability of ionic states can be increased or decreased by introducing non-isovalent ions, depending on the mechanism of chemical-bond rupture during radiolysis. When rupture of chemical bonds occurs through the transfer of an electron from the anion to the cation, the introduction of divalent ions—for example, cadmium into silver oxalate—increases the radiation resistance of this compound [38—39].

If decomposition occurs as a result of rupture of the intracationic or anionic component of the lattice—for example, when cadmium is introduced into silver carbonate—then the radiation resistance decreases [40]. The reaction of thermal decomposition of ionic compounds, for example azides into atoms, probably occurs through the transfer of an electron from the anion to the cation. The initial act of this reaction is neutralization of the anion, followed by destruction of the lattice by thermal fluctuations.

The stability of alkali-metal azides during their thermal decomposition increases in going from lithium azide to cesium azide, i.e., in going from a compound with greater energy to a compound with lower lattice energy [41—42].

When azides are split into ions, for example in aqueous solutions, the stability of these compounds increases with increasing lattice energy.

According to Varley, the following hypothesis of the multistage destruction of a compound with ionic chemical bonds is considered possible:

1. Under the action of a quantum of radiation or high-energy particles, the anion loses an electron, which passes to the cation. The ionic chemical bond of the anion in the compound is disrupted.

2. Under the action of thermal fluctuations, the anion, having become an atom, is pushed out of its site in the lattice. The bond between the atoms is disrupted. The body loses mechanical strength under the action of thermal fluctuations. Consequently, an activation process of destruction of the body is assumed, in which atoms that have lost their chemical bond leave the molecule under the influence of thermal fluctuations.

3. The scattering of particles of the body that have lost their bond with one another—and, in the limiting case of decomposition of the body, the scattering of atoms—occurs under the action of the energy of the system released in an exothermic reaction, for example, during detonation of an explosive.

The reaction of thermal decomposition and the rupture of chemical bonds with an ionic type of bond occur through the transfer of an electron from the anion to the cation. The same physical process occurs in photochemical reactions, when absorption of photons at the edge of the band of intrinsic fundamental absorption causes the transfer of an electron from the anion to the cation or the formation of an exciton.

Thus, optical excitation of an electron associated with exciton absorption in ionic compounds is based on the same physical processes of electron transfer from the anion to the cation as in the reaction of thermal decomposition of a substance with an ionic type of bond.

One of the physically possible modes of radiation damage is the neutralization of an anion in the crystal lattice and the subsequent capture of an electron either by an anion vacancy or by some other electron trap.

As a result, the anion loses its chemical bond, and thermal motion pushes it out of the site.

The process of radiation damage to the structure begins with the loss of a chemical bond and ends under the action of thermal vibrations or their fluctuations. The onset of the reactions of thermal decomposition of a compound, thermolysis, radiation destruction, and photolysis may be the same.

The two types of disruption of the chemical bond in the destruction of a substance—radiation and thermal—which are based on the physical processes of anion ionization, proceed at different rates; in the reaction of thermal decomposition, essentially all the molecules of the substance participate. In the case of radiation destruction of a body, at an attainable dose rate a very small fraction of the molecules of the body enters into the reaction. In strong radiation damage, their concentration does not exceed \(10^{19}\) damages per \(cm^3\), i.e., less than one molecule in \(10^5\).

Thus, from the standpoint of physical processes, radiation damage represents a controllable slow explosion. The study of the kinetics of radiation damage is a large-scale copying of the detonation of explosives and offers great possibilities for studying its course.

From this standpoint, radiation physics of the solid state constitutes the physics of slow detonation.

It may be assumed that, at a high flux density of neutrons, fission fragments, and gamma radiation, all anions in a body lying in the path of the radiation flux from the region of a nuclear explosion will undergo neutralization. As a result of neutralization of the anions of the body, or of exciton excitation of its electrons, the ionic chemical bond is lost and the body is destroyed by thermal fluctuations. An explosion of the irradiated body occurs without an exothermic reaction. Thus, in addition to the usually described action of a nuclear explosion on a body, the possibility of its complete disintegration into atoms under the action of a flux of ionizing radiation of high intensity must also be taken into account.

The decomposition reaction will occur owing to the ionizing action of the radiation and the loss of the chemical bond in the molecule.

The coefficient of absorption of radiation by a body increases with an increase in the sum of the atomic numbers of the elements comprising the chemical compound. With such substitution of chemical elements, the bond strength decreases, which will facilitate the destructive action owing to the energy stored by the body in the radiation field. The described phenomenon is possible only under brief exposure to radiation.

Under prolonged exposure, the energy stored in radiation damage is dissipated owing to the weak Coulomb bond between the particles.

The source of a flux of ionic radiation capable of producing exciton excitation in each molecule and destroying ionic chemical bonds is an electric discharge of the corresponding parameters.

From the arguments presented it follows that the stability of compounds under thermal decomposition (thermolysis) should be compared with radiation-physical stability and photolysis, and not with radiation-chemical stability (radiolysis). This thesis is in agreement with the observed phenomenon, when radiation-chemical stability

increases in a series of compounds as the lattice energy decreases, while the thermal stability decreases.

Considerable difficulties are involved in explaining the high rate at which the thermal decomposition reaction proceeds on the basis of molecular processes. In connection with the exciton mechanism considered for exciting the thermal decomposition reaction, it is possible to explain the neutralization of anions by means of ultraviolet and soft X-ray radiation from the detonation region of the ionizing substance. Then, when the decomposition reaction begins in the substance itself, radiation from this region causes exciton excitation over a considerable distance along the radius from the radiation center.

In this case, exciton excitation will propagate at the speed of propagation of electromagnetic radiation in the medium. The hypothesis of photon excitation of the thermal decomposition reaction has also been advanced earlier (by B. V. Erofeev, Dashkevich, R. F. Vasil'ev, N. N. Semenov, Zenger, and others).

The hypothesis of photon excitation of the thermolysis reaction and its particular form—exciton excitation—are in agreement with experiments (Deb, Evans, Ioffe, and others) on increasing the detonation velocity of silver azide when the radioactive isotope Ag\(^{110}\) is introduced into the lattice.

In addition to the exciton mechanism, an ionization mechanism of radiation absorption is also possible, which will be accompanied by a photocurrent. The exciton and ionization mechanisms of photon absorption in a substance can explain the high propagation velocity of the decomposition reaction.

Characteristic of detonation are the short duration of the process and the high rate at which the reaction proceeds. If the reaction energy were released over a longer time, the result of the reaction would be other transformations of the substance, for example, its heating or melting. The exciton hypothesis of detonation is consistent with the high reaction rate.

Exciton excitation of the substance is the beginning of the reaction. For this reason, the study of exciton excitation, the laws of exciton propagation, and their lifetime under various conditions is of scientific and practical interest. Knowing the energy of the crystal lattice, one can describe the direction of processes in a series of similar ionic compounds, i.e., predict which compounds will be less thermally stable and which more stable, and the decomposition of which compounds is accompanied by the release of greater energy.

The interaction of an exciton with vacancies can be used for determining the concentration of vacancies and other point defects in a crystal, for purposes of defectoscopy of bodies.

In connection with the above, it is of interest to develop methods for generating excitons and to construct exciton generators.

§ 6. On the Physical Foundations of Electron, Ion, and Radiation Technology for Solids with an Ionic Type of Chemical Bond

The incidence of a beam of particles on the surface of a body causes the following physical phenomena in it:

1. Heating of the surface.
2. Sputtering of the surface; erosion; etching.
3. The pressure of the beam creates work hardening and surface strengthening of the body.
4. Lattice defects and color centers are formed. Of practical interest is the strengthening of the surface after concentrated

processing by a stream of particles that develop very high pressures and energy concentration.

The energy absorbed by crystals during their mechanical or thermal treatment, or during irradiation, is expended on heating and on the formation of various types of disturbances of the crystal lattice, which increases the internal energy of the crystal and the energy of elastic deformation. In particular, after irradiation a crystal possesses additional excess nonequilibrium energy, which is called hidden, stored, or Wigner energy. The magnitude of the stored energy associated with radiation damage serves as a measure of its amount and determines the change in the physical and chemical properties of bodies.

The resistance of crystals to external actions increases with increasing lattice energy \[45\]. To produce identical microscopic lattice disturbances—for example, displacement of an ion from a site—more energy must be expended in crystals with higher lattice energy. On the other hand, in crystals with low lattice energy, microscopic disturbances under given conditions are less stable than in crystals with higher energy, in which radiation defects anneal less readily than in crystals with lower lattice energy.

Crystals with lower lattice energy are formed by compounds with a higher effective atomic number. Under isodose irradiation they absorb more energy than crystals with higher lattice energy, which are formed by compounds with a lower effective atomic number.

As a consequence of the more rapid annealing of defects in crystals with lower lattice energy at room temperatures, the amount of stored energy in them will be less than in crystals with higher lattice energy.

If the energy of formation of a defect is known, then, by measuring the total stored energy, one can determine the number of defects. Knowing the number of defects, one can determine the energy of formation of a single defect. If irradiation is carried out at low temperature, when annealing of defects is slowed, then the stored energy represents the energy of formation of the given number of defects. If, during annealing, formation of aggregates of defects or their annealing has occurred, then the stored energy can no longer be interpreted in such a simple way. Radiation damage to bodies is in some respects analogous to the action of plastic deformation and causes the same change in their properties. Unlike strain hardening, radiation damage of materials: 1) differs in the type of microscopic lattice damage; 2) radiation hardening is less than after plastic deformation; 3) radiation damage arises in the microvolume and is not crystallographically oriented \[46\].

The regularities in the change of the physical properties of crystals under deformation and irradiation provide grounds for the assumption that there are common features in residual disturbances after mechanical loading or irradiation \[47\].

Crystals of LiF, NaCl, KCl, and AgCl were studied after irradiation with neutrons, X-rays, or after deformation. After large doses or deformations, single crystals break up into separate blocks, slightly rotated with respect to one another. The maximum misorientation of blocks in LiF crystals reached 7°. Under large plastic deformation the blocks rotated with respect to the initial structure by several tens of degrees. With increasing dose or deformation, the number of blocks increases and their sizes decrease. B. V. Budylin observed that after a large integrated dose of neutron irradiation the crystals split apart. The minimum irradiation dose at which

at which the division of crystals into blocks becomes noticeable, increases with the strength of the crystals, i.e., with the lattice energy. For quartz this dose reached \(1 \cdot 10^{19}\) neutrons/cm\(^2\), and for lithium fluoride \(8 \cdot 10^{17}\) neutrons/cm\(^2\).

The change in the structure of single crystals after irradiation with neutrons or after plastic deformation can be explained by the appearance of point microdefects and their subsequent combination into aggregates [48, 49].

The energy stored in rock salt after relative plastic deformation in compression at room temperature by 6.64% and at a maximum stress of 1.02 kg/mm\(^2\) was 0.4 J/g. Less than 10% of this energy is stored in point defects [50].

To accumulate an energy of \(\simeq 1\) cal/g, i.e., of the order stored upon irradiation of rock salt, it would be necessary to impart to these crystals a relative plastic deformation at room temperature of \(\sim 66\%\). It should be noted that the crystal is destroyed at a relative deformation of 8–10% in the region of the almost linear dependence of the stored energy on the magnitude of deformation.

At low temperatures, in the process of deformation of metals, almost all absorbed energy is stored in the specimen [51]. The energy stored during plastic deformation increases the free energy and weakens the bonding in the lattice. Therefore the Debye temperatures, calculated from X-ray data, in undeformed rock-salt crystals are 260–290°C, and in deformed crystals 150–170°C. The lattice in the slip band is characterized by a large number of displaced atoms and is strongly weakened [52].

With an increase in the concentration of \(F\)-centers from \(2.5 \cdot 10^{17}\) to \(6.5 \cdot 10^{17}\) cm\(^{-}\) after irradiation with X-rays, a linear increase in the magnitude of the stored energy was observed [53].

Upon annealing NaCl crystals containing stable vacancy complexes, four maxima were observed on the curve of heat release as a function of temperature, corresponding to annihilation of mobile defects at low temperatures and to decomposition of vacancy complexes at high temperatures [54].

The magnitude of the stored energy after irradiation in a flux of protons and \(\gamma\)-rays was determined in NaCl crystals [55, 56, 57]. The measurements described in the literature and cited above were performed by the authors using different methods on specimens prepared under nonidentical conditions, which limited the possibilities for their interpretation.

E. K. Zavadovskaya, A. V. Kuz’mina, E. M. Golovchanskii, N. M. Timoshenko, and V. A. Khopler obtained systematic data on the measurement of stored energy in ionic crystals [55, 58–59].

Under identical conditions the crystals were irradiated with protons of energy 4.3 MeV in a flux of \(1.5 \cdot 10^{14}\) protons/cm\(^2\), with X-rays, and with Co\(^{60}\) \(\gamma\)-rays at an intensity of about 500 r/sec. The stored energy was measured in single crystals of CaF\(_2\), SrF\(_2\), BaF\(_2\), LiF, NaCl, KCl, KBr, KJ, and solid solutions of the KCl—KBr system. The measurement was carried out during annealing, and for alkali-halide crystals, except LiF, both during annealing and during dissolution in water.

Figure 9 gives the rate of heat release as a function of temperature during heating of NaCl crystals irradiated with the indicated dose of Co\(^{60}\) \(\gamma\)-rays. With increasing dose, the energy stored in the crystal increases and the low-temperature first maximum rises.

Analogous dependences are observed for crystals with other chemical compositions and types of crystal lattice. Figure 10 gives the rate of heat release during heating of a CaF\(_2\) crystal after irradiation with \(\gamma\)-rays, dose \(5 \cdot 10^7\) roentgens (upper curve), and dose

Fig. 9. Dependence of the rate of heat release on temperature during heating of NaCl crystals irradiated with γ-rays

Visible labels in the figure: NaCl; \(4.62 \times 10^{8}\) r; \(3.70 \times 10^{8}\) r; \(2.59 \times 10^{8}\) r.
Axes: \(dQ/dT \times 10^{-3}\), \( \times 4.18\ \mathrm{erg/deg}\), per specimen; Temperature, \({}^{\circ}\mathrm{C}\).

Fig. 10. Dependence of the rate of heat release on temperature during heating of CaF\(_2\) crystals after irradiation with γ-rays and X-rays

Visible labels in the figure: 1; 2.
Axes: \(dQ/dT \cdot 10^{-2}\ \mathrm{dyne}\ \mathrm{deg}^{-1}\), per specimen; Temperature, \({}^{\circ}\mathrm{C}\).

Energy stored in various ionic crystals after different types and exposure doses
E. K. Zavadovskaya, A. V. Kuzminoi, E. M. Golovchanskogo, and I. P. Timoshenko

Table 8
(According to data)

Crystal	\(2.5 \cdot 10^7\) r, X-ray, J/g	\(2.5 \cdot 10^7\) r, X-ray, J/mol	\(5 \cdot 10^7\) r, gamma, J/g	\(5 \cdot 10^7\) r, gamma, J/mol	\(5 \cdot 10^{10}\), gamma, J/g	\(5 \cdot 10^{10}\), gamma, J/mol	\(9 \cdot 10^{14}\) protons/cm\(^2\), proton 4.3 MeV, J/g	\(9 \cdot 10^{14}\) protons/cm\(^2\), proton 4.3 MeV, J/mol	Note	Lattice energy, kJ/mol
CaF\(_2\)	12.8	1630	28.0	2180	—	—	—	—	Annealing method	2540
SrF\(_2\)	4.4	585	5.2	406	—	—	—	—	Annealing method	2400
LiF	45	1165	168	4350	1240	32000	—	—	Annealing method	1090
NaCl	—	—	—	—	7.5	428	—	—	Annealing method	760
KCl	—	—	—	—	2.6	201	—	—	Annealing method	690
NaCl	—	—	—	—	9.22	540	10.0	584	Dissolution method	760
KCl	—	—	—	—	6.05	448	2.2	164	Dissolution method	690
KBr	—	—	—	—	3.54	390	1.6	175	Dissolution method	660
KI	—	—	—	—	2.78	462	0.7	116	Dissolution method	620

$2.5 \cdot 10^6$ roentgens after irradiation with X rays (lower curve). If, in crystals irradiated with X rays, two maxima are observed in the heat release, then after irradiation with $\gamma$ rays both maxima merge into one, shifted into the region of higher temperatures. An analogous dependence is observed for $\mathrm{SrF}_2$ crystals.

In $\mathrm{BaF}_2$ crystals containing many impurities, the shape of the curve changes substantially from one sample to another. The magnitude of the stored energy in $\mathrm{CaF}_2$ and $\mathrm{SrF}_2$ after irradiation with X rays has the same value as in alkali-halide crystals that have received a one-hundred-times larger exposure dose. In the calculation per $\partial x c/2$, the magnitude of the stored energy in the crystal is the greater, the greater the lattice energy.

Table 8 presents the magnitude of the stored energy in ionic crystals of different composition after irradiation with X rays or $\gamma$ rays under identical conditions. With a change in chemical composition, the magnitude of the stored energy increases with increasing lattice energy.

The energy stored in radiation damage increases with increasing lattice energy, and annealing of radiation defects occurs at higher temperatures and the defects have a higher activation energy. The stored energy increases with increasing exposure dose. Under irradiation with X rays or $\gamma$ rays, in crystals with higher lattice energy the stored energy accumulates faster than in crystals with lower lattice energy.

Analogous dependences are also observed under proton irradiation, as is seen from Fig. 11, which shows the dependence of the stored energy on dose for crystals of different composition. The magnitude of the stored energy was determined during dissolution in water from the difference between the heats of dissolution of irradiated $Q_{\text{irr}}$ and unirradiated $Q_{\text{unirr}}$ samples when they were dissolved simultaneously in a calorimeter: $\Delta Q = Q_{\text{irr}} - Q_{\text{unirr}}$.

Table 8 gives the values of stored energies obtained upon dissolution of crystals after different doses of $\gamma$ and proton irradiation at room temperature.

The difference $\Delta Q$ depends on the dose of $\gamma$ irradiation, decreasing as indicated in Fig. 12. The following possible explanation of this dependence may be offered. The difference in heats of dissolution $\Delta Q$ is decomposed into two components: the stored energy, $\Delta Q_{\text{st}}$, and $\Delta Q_{\text{chem}}$, a correction taking into account radiolysis products. The correction $\Delta Q_{\text{chem}}$ has the sign opposite to the sign of the stored energy $\Delta Q_{\text{st}}$. As the dose increases, radiolysis products accumulate and the magnitude of $\Delta Q_{\text{chem}}$ increases. No change in the sign of the stored energy $\Delta Q_{\text{st}}$ was observed upon dissolution of crystals irradiated with protons.

After irradiation, several times more energy is stored in alkali-halide crystals than in copper; this is probably connected with the scattering of energy by free electrons [6]. It is possible that this difference is related to the character of radiation damage under X-ray irradiation in alkali-halide crystals. In them, defects are formed as a result of ionization processes, on which 99% of the absorbed energy is expended. In copper, radiation damage occurs mainly as a result of elastic interactions of particles with the atoms of the substance.

When metals are irradiated, almost all of the absorbed energy is transformed into the kinetic energy of lattice ions and is dissipated in the lattice. The energy stored in a solid during its irradiation amounts to less than 0.01% of the absorbed energy. For example, the total kinetic energy imparted to a metal upon irradiation with fast neutrons at a dose of the order of

$10^{19}$ neutrons/cm$^3$ reaches $4 \cdot 10^6$ J/cm$^3$, while the stored energy reaches only about ten kJ/mole. In ionic crystals with a high lattice energy, a relatively larger portion of the absorbed energy is stored than in metals.

The energy stored under irradiation, reaching 8 kJ/g for NaCl, amounts to about 7% of the lattice energy.

Fig. 11. Dependence of stored energy on dose for crystals of different chemical composition

The energy stored in NaCl after irradiation at room temperature is 200 times greater than the energy stored during plastic deformation. As the temperature is lowered, the magnitude of the stored energy increases. Thus, at identical temperatures, irradiation of crystals is a considerably more powerful means of action than their plastic deformation.

Measurement of the stored energy during annealing in NaCl + Ca crystals irradiated with X-rays showed that the stored energy increases with increasing calcium content. The position of the maximum in the dependence of the heat-release rate on temperature shifts toward lower temperatures as the impurity content increases. This result is consistent with the hypothesis of a lower activation energy for annealing of radiation defects in crystals with impurities [59].

In Fig. 12 and Table 9 the value of the stored energy and the heat of formation are given as a function of composition for single crystals of solid solutions of the KCl–KBr system after irradiation with a dose of \(9 \cdot 10^{14}\) protons/cm\(^2\). As can be seen, the stored energy after irradiation

Fig. 12. Dependence of the difference in heats of dissolution \(\Delta Q\) on the dose of \(\gamma\)-irradiation for NaCl and KJ crystals.

in solid solutions is several times smaller than in the original pure crystals. Consequently, crystals of solid solutions will be more resistant to the action of irradiation than pure crystals.

Table 9

Stored energy and heat of formation in solid solutions of the KCl–KBr system after irradiation with an exposure dose of protons \(9 \cdot 10^{14}\) protons/cm\(^2\)
(according to E. K. Zavadovskaya, E. M. Golovchanskii, A. V. Kuzmina)

Composition of the solid solution in mol % in KCl, KBr	Stored energy, J/g	Stored energy, J/mol	Heat of formation, J/g	Heat of formation, J/mol
KCl	2.2	172	—	—
90	1.5	145	3.0	268
75	1.2	121	8.15	755
50	1.25	104	10.6	910
25	1.18	123	5.85	625
10	1.05	115	2.62	304
KBr	1.6	204	—	—

It is evident from Table 9 that the energy stored after irradiation is less than the energy loss during formation of the solid solution, which is associated

with the defectiveness of the solid-solution crystal. Solid-solution crystals have maximum defectiveness at an equimolar composition, which corresponds to a maximum heat of formation of \(10.6\) J/g and a minimum stored energy of \(1.25\) J/g.

A stored energy of about \(8\) J/g corresponds to a concentration of \(F\)-centers in NaCl equal to \(5 \cdot 10^{18}\ \text{cm}^{-3}\). In KCl—KBr solid solutions, under analogous irradiation conditions, the concentration of \(F\)-centers reaches only \(10^{17}\ \text{cm}^{-3}\). Consequently, the increased defectiveness of solid solutions is obtained not at the expense of \(F\)-centers, i.e., halide-ion vacancies, but at the expense of defects of another type. Proton irradiation cannot significantly increase the defect concentration and the stored energy in solid-solution crystals.

Table 10 gives the dependence of stored energy on the composition of the crystal after irradiation with protons at a dose of \(1 \cdot 10^{14}\) protons/cm\(^2\), quenching upon cooling from \(300^\circ\) to \(0^\circ\text{C}\), and plastic deformation.

Table 10

Comparative magnitude of stored energy under irradiation, quenching, and plastic deformation

Crystal	Stored energy	Stored energy	Stored energy	Stored energy	Stored energy	Stored energy
	Irradiation with 4.3 MeV protons, \(1 \cdot 10^{14}\) protons/cm\(^2\)	Irradiation with 4.3 MeV protons, \(1 \cdot 10^{14}\) protons/cm\(^2\)	Quenching	Quenching	Plastic deformation* 6%	Plastic deformation* 6%
	J/g	J/mol	J/g	J/mol	J/g	J/mol
NaCl	2.8	163	2.92	170	0.0418	2.4
KCl	0.8	59	0.50	37	—	—
KBr	0.5	54.5	0.27	29.5	—	—
KI	0.3	50.0	0.25	41.5	—	—

*) Magnitude of the stored energy at a relative deformation of 6%, according to R. I. Garber.

The energy stored after irradiation by a proton flux of \(1 \cdot 10^{14}\) protons/cm\(^2\) and quenching upon cooling in acetone from a temperature of \(300^\circ\) to \(0^\circ\text{C}\) is close in magnitude and is greater the greater the lattice energy of the crystals in J/g. For KCl, KBr, and KJ crystals, which have a similar lattice energy, the stored energy in J/mol differs little in magnitude.

From the data presented it follows that the energetic state of the system changes most of all when impurities are introduced (heat of formation). The stored energy under irradiation and thermal treatment proves to be a quantity of the same order. The latter circumstance is a consequence of the fact that during thermal treatment—quenching—thermal microdefects of the type of vacancies and interstitial atoms are fixed, as also occurs under irradiation.

The possibility of accumulating free energy in a given structure is determined by the experimental conditions and by the chemical composition of the compound. With an increase in bond energy, the magnitude of the energy capacity of the crystal, the magnitude of the free energy, increases. It follows from this that substances with a high lattice energy, capable of retaining the energy accumulated during mechanical or thermal treatment or after irradiation, can be used as memory elements for external actions that have caused the storage of excess free energy in them.

It follows from the materials presented that an energy comparison does not allow one to conclude that irradiation, in its effect on a substance, can be comparable with heat treatment; it exerts a less powerful influence than impurities. The energy stored during plastic deformation preceding the destruction of crystals is two orders of magnitude lower than the energy stored during quenching or irradiation of crystals.

Radiation-induced changes in the properties of bodies with ionic bonds after irradiation are symbatic with the stored energy. The stored energy after irradiation with smaller doses or during quenching has the same order of magnitude as during the introduction of impurities and the formation of solid solutions.

Thus, chemical processes in solids whose characteristics correlate with the free energy in compounds with an ionic type of bond will, after the action of irradiation, change their course in the same way as after heat treatment.

The hypothesis of a strong influence of the consequences of irradiation on the rate of processes determined by free energy, and of a significant change in the properties of bodies with an ionic type of bond, is not confirmed by the presented results of measurements of free energy and by the energy analysis.

The changes occurring in a body during irradiation, and its weakening, are determined by the absorbed energy, which increases on passing to crystals with lower lattice energy. A compound having a lower bond energy in a radiation field absorbs more energy and is weakened relatively more strongly than compounds with higher lattice energy. The absorbed energy exceeds the stored energy by two or more orders of magnitude. In view of this, radiation technology should be developed by influencing processes during irradiation, and not after irradiation.

LITERATURE

1. N. S. Kurnakov. Collected Selected Works. Vols. 1–2, ONTI, Moscow–Leningrad, 1938.
2. A. E. Chichibabin. Fundamentals of Organic Chemistry. Vol. 1, p. 163, Goskhimizdat, Moscow–Leningrad, 1953.
3. V. M. Tatevskii. Moscow State University Press, Moscow, 1953.
4. Ya. K. Syrkin, M. E. Dyatkina. The Chemical Bond and the Structure of Molecules. Goskhimizdat, Moscow–Leningrad, 1946.
5. A. A. Vorob’ev. Physical Properties of Ionic Crystalline Dielectrics. Book 1. Tomsk State University Press, Tomsk, 1960.
6. A. A. Vorob’ev. Physical Properties of Ionic Crystalline Dielectrics. Book 1, Tomsk State University Press, Tomsk, 1960.
7. A. E. Fersman. Geochemistry, vol. 3, ONTI, Moscow–Leningrad, 1937.
8. Concise Handbook for the Chemist. Goskhimizdat, Moscow, 1955.
9. S. A. Shchukarev, L. S. Litsich, V. I. Timofeev. Thermodynamics and the structure of solutions. Proceedings of the Conference, Academy of Sciences of the USSR, pp. 167–171, Moscow, 1959.
10. A. B. Zdanovskii. Kinetics of Dissolution of Natural Salts under Conditions of Forced Convection. Goskhimizdat, Leningrad, 1956.
11. P. A. Savintsev, A. F. Naumov. Physicochemical analysis. Proceedings of a Conference of the Siberian Branch of the Academy of Sciences of the USSR, pp. 179–184, Novosibirsk, 1963.
12. A. F. Naumov. Interuniversity collection of the West Siberian Council for Coordination and Planning of Scientific-Research Work in the Technical and Natural Sciences, pp. 108–111, issue 2, 1963.
13. R. R. Moran. J. Appl. Phys., 29, 1768, 1958.
14. B. Khoyman. Growth and Shape of Crystals, IL, Moscow, 1961.
15. V. D. Kuznetsov. Solid-State Physics, Krasnoe znamya Publishing House, Tomsk, 1937.
16. A. A. Vorob’ev. Physics of alkali-halide crystals. Proceedings of the Second All-Union Conference, pp. 319, 284, Leningrad State University Press, Riga, 1962.
17. A. A. Vorob’ev. Doklady of the Academy of Sciences of the USSR, 98, 573, 1954.
18. A. A. Vorob’ev. News of Higher Educational Institutions, Physics, 1, 160, 1958.
19. A. A. Vorob’ev. Physical Properties of Ionic Crystals. Tomsk State University Press, Tomsk, 1960.
20. A. A. Vorob’ev. Optics and Spectroscopy (in press).
21. A. A. Vorob’ev, E. K. Zavadovskaya. Electrical Strength of Solid Dielectrics, p. 312, Gostekhizdat, Moscow, 1956.
22. A. A. Vorob’ev. UFN (in press).

1. R. I. Zollweg. Phys. Rev., 111, 113, 1958.
2. S. M. Brekhovskikh. Glass and Ceramics, 12, 4, 1964.
3. S. M. Brekhovskikh. Glass and Ceramics, 10, 1957.
4. R. K. Sheline, D. E. Sharp, W. I. Arner. J. Chem. Phys., 19, 1422, 1951.
5. C. L. Rose. Phys. Rev., 26, 108, 1926.
6. S. M. Brekhovskikh. Glass and Ceramics, 12, 1, 1964.
7. R. Iokota. Phys. Rev., 95, 1145 (1954).
8. R. Iokota. J. Phys. Soc. Japan, 7, 1316 (1952).
9. A. I. Zakharov. UFN, 57, 562 (1955).
10. S. S. Bashanov, L. I. Gorogotskaya. Izv. Vyssh. Uchebn. Zaved., Chemistry and Chemical Technology, 2, 858 (1959).
11. R. W. Pohl. Phys. Z. S., 39, 36, 1938.
12. S. M. Brekhovskikh. A. E., 8, 37, 1960.
13. A. N. Oblivantsev. Dissertation, Tomsk, 1966.
14. Yu. A. Zakharov, V. A. Nevostruev. This collection, p. 60.
15. V. V. Boldyrev, L. B. Bystrykh. Zh. Kh., 31, 218, 1964.
16. Yu. A. Zakharov, G. G. Savelev. Kinetics and Catalysis, 6, 611, 1965.
17. Yu. A. Zakharov, V. V. Boldyrev et al. DAN SSSR, 145, 122, 1962.
18. Yu. A. Zakharov, V. V. Boldyrev et al. Proceedings of the Conference on the Effect of Ionizing Radiation on Materials, Tomsk, 1963.
19. F. Bowden, A. Yoffe. Fast Reactions in Solids, IL, Moscow, 1963.
20. P. Gray, T. C. Waddington. P.R.S., A235, 106, 1956.
21. I. V. Vereshchinskii, A. K. Pikaev. Introduction to Radiation Chemistry, p. 301, Academy of Sciences of the USSR, Moscow, 1963.
22. V. V. Sviridov. Photochemistry and Radiation Chemistry of Solid Inorganic Substances, Part I, Publishing House “Vysshaya Shkola,” Minsk, 1964.
23. A. A. Vorobev. Physical Properties, 1, Tomsk State University Publishing House, Tomsk, 1960.
24. B. V. Budylin, A. A. Vorobev. The Effect of Radiation on Ionic Structures. Gosatomizdat, 1962.
25. E. V. Kolontsova. In: The Effect of Nuclear Radiations on Materials, p. 256, Publishing House of the Academy of Sciences of the USSR, Moscow, 1962.
26. K. Huang. Proc. Roy. Soc. A., 190, 1020, 1957.
27. M. A. Krivoglaz. ZhETF, 34, 204, 1958.
28. R. I. Gerber. ZhETF, 6, 176, 1936.
29. V. I. Khotkevich, E. F. Chaikovskii, V. V. Zatkvary, DAN SSSR, 46, 483, 1956.
30. V. I. Startsev, P. N. Aronova. Crystallography, 4, 85, 1959.
31. E. T. Phelps, E. Piarlstein. Phys. Rev., 128, 1575, 1962.
32. Suita Munoru. Repts. Univ. Electro-Communy Sci. and Technol., 14, 50, 1962.
33. A. V. Kuzmina. Dissertation, Tomsk, TPI, 1964.
34. E. K. Zavadovskaya, A. V. Kuzmina. FTT, 6, 3348, 1964.
35. A. V. Kuzmina, V. A. Khollder. In: The Effect of Radiation on the Properties of Materials, Issue 2, Novosibirsk, 1963.
36. E. K. Zavadovskaya, A. V. Kuzmina, E. M. Golovchanskii, N. M. Timoshenko. Izv. Vyssh. Uchebn. Zaved. “Physics” (in press).
37. I. Trnka. Czechosl. Jokrn, Phys., 8, 749, 1958.