STORED ENERGY IN IONIC CRYSTALS

V. V. BORISOVSKII, A. A. VOROB'EV, E. M. GOLOVCHANSKII,
E. K. ZAVADOVSKAYA

§ 1 Introduction

Changes in the properties of materials under the action of physical factors are associated with rearrangement of their structure and with the formation or recombination of defects. In many cases, loading of materials is accompanied by the action of chemically active substances (for example, the action of atmospheric oxygen). In the latter case, along with the physical process of change in the structure of the material, the formation of new chemical compounds may be observed. Apparently, when considering processes of change in the properties of dielectrics and semiconductors, two possible processes should be considered.

1. Change in physical properties associated with disturbance of the structure, formation of defects, etc.

2. Changes in properties associated with the formation of new chemical compounds. The rate of chemical reactions increases under the action of physical factors as a result of the supply of additional energy.

The free energy of a body in general form may be written as

\[ F = U - TS - ED/2 - \lambda\sigma/2 - W_{\mathrm{п}}, \tag{1} \]

where
\(U\) — internal energy;
\(T\) — absolute temperature;
\(S\) — entropy;
\(E\) — electric-field strength;
\(D\) — electric displacement;
\(\lambda\) — relative deformation;
\(\sigma\) — mechanical stress;
\(W_{\mathrm{п}}\) — absorbed radiation energy.

For a given complex of actions, in the field of mechanical \(\sigma\) and electric \(E\) stresses, the free energy tends toward a minimum value. An increase in free energy leads to thermodynamic instability of the body. The greater the excess of free energy over the minimum value, the higher its thermodynamic instability.

The greatest thermodynamic instability of a material occurs when its temperature, electrical and mechanical loads, and radiation intensity are changed. If the temperature of a body is sharply decreased, its irradiation is stopped, or the electrical and mechanical loads are removed, then, owing to thermodynamic inertia, stored energy will remain in the system. The magnitude of the stored energy is written in the form

\[ W_{\mathrm{з}} = F_1 - F_2, \tag{2} \]

where \(F_1\) and \(F_2\) are the free energies of the body in thermodynamically equilibrium states 1 and 2, respectively. The amount of stored energy can in some cases serve as a measure of the stability (instability) of a compound and of a body. Thus, if state 1 is taken to be the temperature at which the material is obtained (for example, the growth of a single crystal), and state 2 to be normal conditions, then \(W_3\) serves as a thermodynamic measure of the stability of the dielectric. When the properties of a material change, its entropy and internal energy change. The entropy of a body characterizes the degree of ordering of the material and the presence of structural defects.

The internal energy is determined above all by the forces of interaction between atoms, which determines the values of the frequencies of their natural vibrations. The appearance of structural disturbances is accompanied by a change in the vibration frequency of the atoms surrounding the defective site and, consequently, by a change in internal energy. An important factor in describing changes in the properties of bodies is the entropic component of the free energy, since under external action it is primarily the degree of ordering that changes.

A change in the properties of bodies under external action is called aging.

In most cases insulating materials and products are manufactured at a sufficiently high temperature. After the given product has cooled, its free energy will be higher than in the equilibrium state. The subsequent decrease in the free energy of the body occurs in several ways: 1) as a result of recombination of defects, 2) formation of defect complexes. In the first case the concentration of defects in the material decreases and its electrical and mechanical properties improve. In the second case, the formation of defect complexes creates regions with weakened bonds in the dielectric. At the locations of defect complexes, increased concentrations of electric field intensity and mechanical stresses are created. This will lead to a decrease in the electrical and mechanical strength of the dielectric. Such characteristics of a dielectric as \(\operatorname{tg}\delta\), electrical conductivity, measured at low field strength, may decrease when defect complexes are formed.

Thus, depending on the direction of the processes in the dielectric, either an increase or a decrease in the electrical and mechanical strength of the material may occur. In the practice of electrical insulation, the Arrhenius equation is used to evaluate its thermal aging. As a criterion for changes in properties under thermal action on dielectrics, a decrease in their electrical or mechanical strength is used. It may be assumed that at elevated temperature the formation of defect complexes in the material predominates over their recombination.

Apparently, the association of defects into complexes will proceed more intensively under cyclic temperature action. Under cyclic temperature action, owing to nonuniform heating of individual regions, local increased concentrations of defects may be produced; these defects more readily combine into complexes. Accelerated aging of materials under thermal cycles is well known.

Under prolonged action of an electric field in a dielectric, the following processes may occur: 1) an increase in the concentration of defects, 2) formation of defect complexes, 3) destruction of defect complexes, 4) redistribution of defects in the structure. The electric field causes directed motion of charged defects, which will be displaced toward the electrodes. In this case the defects tend, owing to mutual repulsion, to arrange themselves in the near-electrode layer along equi-

potential line. Such a process leads to a decrease in the concentration of defects in the bulk of the dielectric, which results in an increase in the service life of the insulation. Alternating voltage promotes the formation of defect complexes in the bulk of the dielectric and leads to the appearance of high local field intensities and to a reduction in the electric strength of solid dielectrics. Thus, prolonged exposure of a dielectric to an electric field leads to the appearance of additional defects and increases the entropy of the body. The magnitude of the electric strength depends on the distribution of defects. In the process of electrical aging of dielectrics, their electric strength may either increase or decrease.

Fluxes of high-energy particles cause disruption of the structure of a solid. A high concentration of defects may be accompanied by their coagulation and by the formation of damage. The accumulation of damage is accompanied by macroscopic destruction of the body.

Under prolonged irradiation, the free and stored energies of the body increase. In this case both the internal energy of the system and its entropic component will increase.

After irradiation, owing to relaxation of the material, the free energy of a body placed under normal atmospheric conditions decreases. For this reason, the values of the energy stored under the action of irradiation may serve as a measure of the radiation resistance of an insulating material. It should be noted that determining the stored energy by annealing or bleaching may lead to an inaccurate, underestimated value. This is explained by the fact that during annealing single defects will recombine more rapidly. Complexes of vacancies or interstitial atoms, being thermodynamically more stable, may persist.

Practically under all the actions considered above, the aging process of solid dielectrics proceeds in the direction of decreasing free energy.

After insulation is manufactured, it may be thermodynamically unstable. Its transition to a thermodynamically stable state may occur in two ways:

1) by recombination of defects;
2) by formation of defect complexes.

Both paths are thermodynamically possible and lead to the transition of the body to a more stable state.

In a NaCl crystal, the energy that must be expended to form one defect (Schottky) is \(W_g = 2.02\ \text{eV}\), and the entropy changes by \(5.35 \cdot 10^{-4}\ \text{eV/K}\) [1]; the magnitude of the change will be of the same order for other alkali-halide crystals of the NaCl type [2]. The influence of dislocations on the configurational vibrational entropy is small (not more than \(10^{-4}\ \text{eV/K}\) per atomic plane [3]). Considering the reaction \(2F \to M\), where \(F\) and \(M\) are known color centers, it was found [4] that, owing to this reaction, the entropy changes by \(2.21 \cdot 10^{-3}\ \text{eV/K}\). Thus, it may be concluded that the increase in entropy is determined mainly by radiation-chemical reactions of coagulation of \(F\)-centers into larger aggregates. But even this change in entropy at room temperatures will be significantly smaller than the change due to the increase in internal energy, i.e., \(\Delta U \gg T\Delta S\). Then, if one assumes that the energy of defect formation does not depend on the number of defects, the change in free energy, or the energy stored on defects, is determined by the concentration of defects and their formation energy [5].

§ 2. Physical models of energy storage under irradiation and calculation of its magnitude

According to [6], an energy of 4 eV is stored on a single \(F\)-center in an NaCl crystal. Starting from the band model, the energy stored by an \(F\)-center in an NaCl crystal under X-ray irradiation [7] is released upon recombination of an electron with a hole and amounts to 7.1 eV. At the second stage of coloration, where Schottky defects are generated, the energy stored on an \(F\)-center will be composed of the energy stored upon localization of the electron and hole (7.1 eV) and the energy stored on a Schottky pair (2.1 eV [8]), and will already amount to 9.9 eV. According to [7], in contrast to [6], the stored energy is determined mainly not by \(F\)-centers, but by pairs of \(F\)- and \(V\)-centers. Subsequently many researchers adopted this point of view; the concept of electron–hole pairs was introduced into the literature.

Prolonged irradiation of alkali-halide crystals at room temperature leads to the formation of complex color centers \(M\)-, \(R\)-, \(N\)-, but their concentration is small compared with the concentration of \(F\)-centers and does not exceed 20% [9]. Thus, the main contribution to the stored energy in alkali-halide crystals is made by electronic \(F\)- and hole \(V\)-centers.

Irradiation of alkali-halide crystals is accompanied by the appearance of dislocations in them [10]. Studies have shown [11] that the value of the stored energy on dislocations may be neglected.

When ionizing radiation interacts with alkali-halide crystals, some bonds are broken. This process may be represented by the following photochemical reaction:

\[ Me^{+}X^{-}+h\nu \to Me^{0}_{\mathrm{gas}}+X^{0}_{\mathrm{gas}}+\Delta H. \]

Here \(\Delta H\) is the heat of reaction. The reaction given above is endothermic. The irradiated crystal possesses an additional heat content, which is called stored energy. \(Me^{+}\), \(X^{-}\), \(Me^{0}_{\mathrm{gas}}\), and \(X^{0}_{\mathrm{gas}}\) are, respectively: the cation and anion, and the neutral atom of the alkali metal and of the halide. \(Me^{0}_{\mathrm{gas}}\) and \(X^{0}_{\mathrm{gas}}\) may be regarded as \(F\)- and \(V\)-centers. At room temperatures only \(V_{2}\)- and \(V_{3}\)-centers are stable. Hypothetical models of these centers are \(X_{3}^{-}\) and \(X_{3}^{-}\), that is, a \(V_{2}\)-center contains a halide in the atomic state, while a \(V_{3}\)-center includes a halide in the molecular state. Consequently, the stored energy can be determined by knowing the heat of formation of the salt \((\Delta H_{3})\), the heat of sublimation of the alkali metal \((\Delta H_{1})\), and the heat of dissociation of the molecular halide \((\Delta H_{2})\).

The stored energy at the first stage of coloration of the crystal, when holes and electrons are localized on preradiation vacancies, is equal to \(\Delta H_{1}+\Delta H_{2}+\Delta H_{3}\) in the case of formation of a \(V_{2}\)-center, and \(\Delta H_{1}+\Delta H_{2}\) in the case of formation of a \(V_{3}\)-center. Table 1 gives the values of the energy stored by \(F\)- and \(V\)-centers in alkali-halide crystals.

From a comparison of columns 5 and 6 of Table 1 it is seen that the stored energy decreases when a \(V_{3}\)-center is formed. This process leads to a lowering of the free energy of the system, i.e., transfers the system into a more stable state. Consequently, at room temperature, under the action of irradiation, \(V_{3}\)-centers should be observed predominantly in alkali-halide crystals. This was indeed described in [14] in a study of the kinetics of accumulation of \(V_{2}\)- and \(V_{3}\)-centers under proton irradiation of NaCl, KCl, KBr. At small doses of proton irradiation the ratio between \(V_{2}\)- and \(V_{3}\)-centers is approximately the same—

Table 1

Energetic characteristics and stored energy of an electron–hole pair ($F$- and $V$-centers) for a series of alkali-halide crystals [12]

The values of $\Delta H_1$, $\Delta H_2$, $\Delta H_3$ are taken from [13].

is the same. With an increase in the irradiation dose, absorption in the \(V_2\)-band hardly increases, whereas absorption in the \(V_3\)-band increases.

The formation of a colloid in alkali-halide crystals is equivalent to the formation of a metallic phase from alkali-metal vapors. The course of this process is accompanied by a decrease in the stored energy by the amount of the heat of sublimation of the alkali metal \(\Delta H_1\). If, in addition to the coagulation of electron centers, coagulation of hole centers also occurs (in this case the stored energy decreases by the amount of the heat of dissociation of the halide), then the stored energy in the irradiated crystal will decrease by \(20\)–\(30\%\) of its initial value.

In [15, 16] the energy of formation of \(F\)-centers was determined experimentally. In various alkali-halide crystals the authors understood this quantity to be the value obtained by dividing the total absorbed energy by the concentration of \(F\)-centers. In this case one obtains a value of \(28\)–\(150\) eV at the first stage of coloration for various alkali-halide crystals, and at the second stage \(10^3\)–\(10^5\) eV per center, the value obtained depending on the irradiation conditions. However, the calculated value cannot be regarded as the energy of formation of \(F\)-centers, since the total absorbed energy is expended on heating the crystal, luminescence, radiolysis, etc. Only a small fraction of the energy goes into the formation of color centers and other radiation defects and is then accumulated in the crystal. It is assumed that in alkali-halide crystals the energy is stored mainly in electron-hole pairs. The energy of formation of an electron-hole pair should be calculated by dividing the stored energy by the concentration of \(F\)-centers. The value of the stored energy per electron-hole pair calculated in this way does not depend on the type of irradiation, intensity, etc., and therefore can characterize

Table 2

Stored energy per electron-hole pair (\(F\)- and \(V\)-centers) in alkali-halide crystals under different types of irradiation

the energy of formation of an electron–hole pair in the compound under consideration.

Using the stored-energy values determined for different types of irradiation, as well as the concentrations of \(F\)-centers calculated by the Smakula formula [17], various authors [17, 18, 19] have found the amount of energy stored in an electron–hole pair (\(F\)- and \(V\)-center) for a number of alkali-halide crystals (Table 2).

The stored energy has been studied in greatest detail for NaCl crystals. The calculated values of the stored energy per electron–hole pair for NaCl agree well with one another within the measurement error. The exception is the value \(12.4 \pm 0.8\) eV obtained in [19]. In the opinion of the authors themselves, such a high value is probably due to the fact that the concentration of centers more complex than \(F\)- and \(M\)-centers was not taken into account. They believe that the value of the energy stored in an electron–hole pair (\(F\)- and \(V\)-centers) may amount to only \(8.1 \pm 1.1\) eV. In contrast to [17], for KCl, KBr, and KI crystals the results of the authors of the present work do not agree with [18] and coincide, within the measurement error, with theoretical calculations. It is of interest to note that the stored-energy value obtained under gamma-ray irradiation is higher than under proton irradiation. It is possible that this result is connected with the greater localization in the thin layer of stopped protons, nonuniformity in the distribution of the concentration of color centers, and errors in the measurements of the stored-energy magnitude.

§ 3. Kinetics of Stored-Energy Accumulation in Alkali-Halide Crystals under Various Types of Radiation

According to [20], it may be assumed that the magnitude of the stored energy in alkali-halide crystals should depend on the binding energy of the ions of the crystal lattice, the defectiveness of the crystals and their purity, and also on the type and intensity of irradiation. The common method for determining the concentration of defects in the crystal lattice—the optical method—does not permit absorption measurements after irradiation with large doses. Measurement of stored energy can be carried out on specimens over a very wide range of doses. In the present section, the kinetics of stored-energy accumulation in “pure” alkali-halide crystals under various types of radiation (proton, electron, gamma, and reactor) will be considered.

Fig. 1. Kinetics of stored-energy accumulation in alkali-halide crystals (protons 4.5 MeV, intensity \(12.5 \times 10^{10}\) protons/cm\(^2\)·s)

In proton irradiation, studies were carried out on single crystals of NaCl, KCl, KBr, and KI, grown from the melt by the Kyropoulos method, as well as on rock salt. Irradiation of the specimens was performed in air with protons of energy 4.5 MeV. The radiation intensity, unless otherwise specified, was \(12.5 \cdot 10^{10}\) protons/cm\(^2\)·s. The irradiation time ranged from 10 min to 20 h.

Figure 1 shows the kinetics of stored-energy accumulation as a function of the absorbed energy at room—

temperature. From the plots in Fig. 1 it is seen that, according to the character of the change in the stored energy, the crystals studied can be divided into two groups. The first group includes NaCl and KCl grown from the melt by the Kyropoulos method. In these crystals the following features are observed.

1. The change in stored energy with increasing absorbed energy can be divided into three stages.

2. As the absorbed energy increases, the magnitude of the stored energy increases up to a certain maximum value (8.2 J/g in NaCl, 2.5 J/g in KCl).

3. For the second stage there is a decrease in the magnitude of the stored energy, despite an increase in the absorbed energy.

4. Then an increase in the stored energy is again observed. For KCl crystals it ends in saturation of the dependence of the stored-energy magnitude on the growth of absorbed energy, whereas for NaCl crystals the stored energy continues to increase.

The second group includes rock salt, KBr, and KI. In these crystals the rate of energy accumulation decreases monotonically with increasing absorbed energy and, at large values of the absorbed energy, becomes equal to zero.

A study of the influence of proton-irradiation intensity on the kinetics of stored-energy accumulation in NaCl at room temperature showed that, at the intensities studied, the magnitude of the stored energy is determined only by the absorbed energy. At all intensities a maximum appears at the same absorbed energy; a change in intensity by more than a factor of 6 affects neither the position nor the magnitude of the maximum.

With increasing temperature of the samples in the irradiation field (Fig. 2), the stored energy in NaCl crystals decreases. Its maximum shifts toward lower absorbed energy.

Fig. 2. Kinetics of accumulation of stored energy in NaCl crystals at various temperatures (protons 4.5 MeV, intensity \(12.5 \cdot 10^{10}\) protons/\(\text{cm}^2 \cdot \text{s}\))

Fig. 3. Kinetics of accumulation of stored energy in alkali-halide crystals (electrons 1.2 MeV, intensity \(6.3 \cdot 10^{11}\) electrons/\(\text{cm}^2 \cdot \text{s}\))

In [21, 22] the results are presented of studies of the kinetics of stored-energy accumulation in NaCl, KCl, KBr, and rock-salt crystals under irradiation with electrons of energy 1.2 MeV on an electrostatic generator ESG-2.5. At an irradiation intensity of \(0.63 \cdot 10^{12}\) electrons/\(\text{cm}^2 \cdot \text{s}\), the temperature of the samples during irradiation was \(25\text{--}30^\circ\text{C}\).

Figure 3 presents experimental results on the kinetics of accumulation of stored energy in alkali-halide crystals under electron irradiation. Comparing the kinetics of energy accumulation under electron irradiation (Fig. 3) and proton irradiation (Fig. 1), we see that the character of the change in stored energy for one and the same crystal is the same for both types of irradiation. However, the maximum value of the stored energy in the crystals studied is smaller under electron irradiation than under proton irradiation. The maximum in the accumulation kinetics for NaCl and KCl under electron irradiation is shifted into the region of lower absorbed energy.

The first study of stored energy in NaCl, KCl, KBr, and KJ crystals after γ-irradiation \((\mathrm{Co}^{60})\) was carried out by E. K. Zavadovskaya and A. V. Kuz’mina [23] in the dose interval from \(1 \cdot 10^8\) to \(5 \cdot 10^8\) roentgens at an intensity of \(500\ \mathrm{R/s}\). They showed that the magnitude of the accumulated energy in the indicated dose interval changes according to a linear law. A study of the behavior of stored energy in NaCl at intensities of \(600\ \mathrm{R/s}\) and \(6000\ \mathrm{R/s}\) from a \(\mathrm{Co}^{60}\) source [24] likewise showed that the change in stored energy with absorbed energy follows a linear law, but with increasing intensity the rate of energy accumulation decreases. The authors associated this decrease with the fact that at higher intensity strong heating of the specimens occurs in the radiation field, which leads to an intensification of the processes of destruction of defects arising under irradiation.

Fig. 4. Kinetics of accumulation of stored energy in alkali-halide crystals (γ—contour of the IRT-2000 reactor, intensity \(800\ \mathrm{R/s}\))

A study of the kinetics of energy accumulation in NaCl, KCl, KBr, and rock-salt crystals (Fig. 4) at an intensity of \(800\ \mathrm{R/s}\) in the radiation contour of the IRT-2000 nuclear reactor as a function of absorption magnitude in the dose interval from \(1 \cdot 10^8\) to \(1 \cdot 10^9\) roentgens showed that in NaCl crystals and rock salt a linear increase is preserved; in KCl there is a tendency for a maximum to appear. For KBr crystals the dependence observed is the same as under electron and proton irradiation.

The action of γ-rays on LiF crystals is accompanied by the accumulation of stored energy, whose kinetics is analogous to the change in stored energy in NaCl and KCl crystals under electron and proton irradiation (Fig. 5) [25].

Fig. 5. Change in stored energy in LiF crystals under γ-irradiation

N. Hennen [26] studied the influence of reactor irradiation on the magnitude of stored energy in polycrystalline NaCl (maximum dose \(2 \cdot 10^{18}\) fast neutrons per \(\mathrm{cm}^2\), temperature \(90^\circ\mathrm{C}\)). He showed that the dependence of stored energy on

the irradiation dose is complex (Fig. 6). In the kinetics of energy accumulation the author distinguishes three regions characterized by different rates of accumulation of stored energy.

In alkali-halide crystals the main contribution to the magnitude of the stored energy is made by electron-hole centers, most of them being \(F\)- and \(V\)-centers. The kinetics of energy accumulation from radiation in rock salt, KBr, and KJ apparently cannot be compared with the two stages of accumulation of \(F\)-centers in alkali-halide crystals [27].

Indeed, it is known [28] that in alkali-halide crystals at the first stage the concentration of \(F\)-centers does not exceed \(10^{17}\ \text{cm}^{-3}\). If one assumes that approximately \(8\ \text{eV}\) of energy is stored on an electron-hole pair, then the stored energy at these centers will amount to about \(0.05\ \text{J/g}\). This value lies beyond the sensitivity limits of the instruments used. Therefore, measurements of stored energy are as yet carried out in all crystals at the second stage of coloration, when color centers are formed at vacancies generated by radiation.

Fig. 6. Change in stored energy in NaCl crystals under reactor irradiation (temperature \(90^\circ\)C)

At low absorbed energy in NaCl, KCl, and LiF crystals, the radiation energy is stored mainly at \(F\)- and \(V_2\)-centers. The observed decrease in stored energy at the second stage indicates that, in a certain dose region, the concentration of these centers decreases. Indeed, when LiF is irradiated in a reactor by electrons and protons, a maximum is observed on the accumulation curve of \(F\)-centers [28—31]; in NaCl [32] under electron irradiation, and in KCl under neutron irradiation [32, 33]. It is assumed [29] that the decrease in the concentration of \(F\)-centers is due to the increase with dose of the efficiency of the radiation-chemical reaction \(2F \to M\). If it is assumed that only coagulation of \(F\)- and \(V\)-centers into more complex formations occurs, then this process will reduce the stored energy by no more than 20–30% (as was shown above). In reality, a decrease in stored energy is observed, for example in NaCl, of almost 50%.

Consequently, the processes of defect coagulation are accompanied by their annihilation and, in particular, destruction of \(F\)-centers occurs.

If, in addition to coagulation of \(F\)-centers, their annihilation takes place, then there must be a decrease not only in \(F\)-centers but also in \(V\)-centers. Therefore the decrease in stored energy is associated with annihilation of electron and hole centers in the region of high doses; this annihilation proceeds by a chain reaction, the beginning of the chain being possibly the reaction of transformation of \(V_2\)- into \(V_3\)-centers or breakdown in the radiation field as a result of the accumulation of space charge in the crystal [21]. The following fact speaks in favor of the first assumption. In KCl, at such absorbed energy, in the short-wavelength part of the spectrum there is observed at room temperature a total absorption due to \(V_2\)- and \(V_3\)-centers (Fig. 7). With increasing absorbed energy, the absorption of the \(V\)-bands increases and shifts somewhat into the region of shorter wavelengths. After an absorbed energy of \(7.1\ \text{kJ/g}\) is reached, further ir-

irradiation of the crystal leads to a decrease in absorption in the region of the \(V\)-band and a shift of the maximum into the region of the \(V_3\)-band.

Thus, with increasing absorbed energy the concentration of \(V_2\)- and \(V_3\)-centers first increases and then decreases, and at an absorbed energy of \(\approx 9\ \mathrm{kJ/g}\) mainly \(V_3\)-centers are observed. The decrease in the concentration of hole centers is connected with the fact that, in the transformation \(V_2 \to V_3\)-centers, about \(1.3\ \mathrm{eV}\) is released. In addition, during coagulation of \(F\)-centers into \(M\)-centers, approximately \(0.4\ \mathrm{eV}\) is released [34]. In each act of the transformations \(V_2 \to V_3\) and \(F \to M\), an energy of \(1.7\ \mathrm{eV}\) is released, which increases the efficiency of ionization of color centers. The ionized \(F\)- and \(V\)-centers, owing to their high concentration, annihilate by recombination. In the recombination of an electron–hole pair about \(8\ \mathrm{eV}\) of energy is released, which is quite sufficient to destroy 3–4 new electron–hole pairs.

Fig. 7. Absorption spectra of hole color centers in KCl crystals (protons, \(4.5\ \mathrm{MeV}\), intensity \(12.5 \cdot 10^{10}\ \mathrm{protons/cm^2 \cdot s}\),
\(1 — 17.0 \cdot 10^3,\ 2 — 14.2 \cdot 10^3,\)
\(3 — 10.6 \cdot 10^3,\ 4 — 8.9 \cdot 10^3,\)
\(5 — 7.1 \cdot 10^3,\ 6 — 5.3 \cdot 10^3,\)
\(7 — 3.5 \cdot 10^3,\ 8 — 1.8 \cdot 10^3\ \mathrm{J/g}\))

With a decrease in the concentration of color centers as they are destroyed, the mean distance between them increases. This leads to the result that the energy released during recombination will no longer be sufficient for annihilation, since a large part of the energy is converted into the kinetic energy of atomic motion and ultimately into heat. The reaction of destruction of color centers ceases, and only \(V_3\)-centers are present in KCl crystals.

Thus, in LiF, NaCl, and KCl crystals there exists a certain critical concentration of \(F\)-centers. At this concentration the energy released as a result of the radiation-chemical reaction \(V_2 \to V_3\) will be sufficient to initiate the destruction of electron–hole pairs by a chain reaction.

A repeated increase in the stored energy at the third stage will occur on electron–hole pairs of \(F\)- and \(V_3\)-centers. With increasing concentration of \(F\)- and \(V_3\)-centers, their radiation annealing becomes effective, on the one hand, and coagulation of electronic centers into more complex \(M\)-, \(R\)-, \(N\)-centers, on the other.

The first process leads to a dynamic equilibrium of \(F\)-centers, while the second will contribute to the stored energy. Thus, at large doses, complex electron centers will make a substantial contribution to the stored energy, but the rate of energy accumulation in this case will decrease monotonically, since

\[ U_1 < U_2 < 2U_1;\quad U_2 < U_3 < 3U_1 \text{ and so on,} \]

where \(U_1, U_2, U_3\) are the energy stored in \(F\)- and \(V_3\)-centers, \(M\)- and \(V_3\)-centers, and \(R\)- and \(V_3\)-centers, respectively. As shown in [9], at large doses complex centers contribute about 20% of the stored energy. With a further increase in dose, dynamic equilibrium sets in and

for complex color centers, the growth of stored-energy accumulation ceases.

The absence of a maximum in the rate of accumulation of stored energy in rock salt and KBr should indicate that in these crystals the reaction \(V_2 \to V_3\) does not proceed, or, if it does proceed, the energy released in it is insufficient for ionization and subsequent annihilation of \(F\)- and \(V\)-centers. This may occur in the absence of \(V_2\)- or \(V_3\)-centers at all irradiation doses, or in the case where the concentration of \(V_2\)-centers is very small. Indeed, it was shown [35] that in KBr single crystals mainly \(V_3\)-centers are formed. Consequently, the accumulation of stored energy in KBr follows a saturation curve due to \(F\)- and \(V_3\)-centers and more complex electron centers, similarly to what occurs in KCl and NaCl crystals during repeated accumulation at the third stage.

In studying the influence of dislocations on the formation of \(V\)-centers [36], it was shown that, at their concentration of \(5 \cdot 10^6\) lines per \(\text{cm}^2\), mainly \(V_3\)-centers are present in NaCl crystals under irradiation. In rock salt the dislocation density was of the same order; that is, in these crystals under irradiation, mainly \(V_3\)-centers are likewise formed. On the one hand, the predominant formation of \(V_3\)-centers will lead to the absence of a maximum in the dependence of the accumulation of stored energy on absorbed energy, as for KBr; on the other hand, when \(V_3\)-centers arise, the rate of accumulation of stored energy decreases in comparison with NaCl grown from the melt. The decrease in the rate of accumulation of stored energy is connected with the fact that the energy stored in one electron-hole pair upon the formation of \(V_3\)-centers is less than upon the formation of \(V_2\)-centers.

In KI crystals, even if a maximum is present in the accumulation kinetics, it is very difficult to observe because of the small magnitude of the stored energy (it only slightly exceeds the sensitivity limit of the apparatus).

In most works in which the influence of irradiation intensity on the kinetics of accumulation of color centers in alkali-halide crystals was studied, the number of \(F\)-centers was compared not at the same absorbed energy, but at the same irradiation time. Naturally, at different intensities the crystal received different doses over the same time, and consequently different numbers of \(F\)-centers were formed in it. Under proton irradiation of NaCl, it was shown in [37] that \(n_F = J^{1.08}\) (\(J\) is the irradiation intensity, \(n_F\) is the concentration of \(F\)-centers), that is, the number of \(F\)-centers is practically independent of the irradiation intensity. This may explain the fact that the magnitude of the stored energy in NaCl at room temperature, in the investigated range of intensities, is determined only by the absorbed energy.

When the temperature of crystals is raised in the radiation field, diffusion processes are accelerated, which causes more intensive coagulation of defects; therefore both the rate of accumulation and the magnitude of the stored energy at the first stage decrease. In addition, with increasing temperature, along with ionization of color centers at the expense of the energy released in the process of the transformation \(V_2 \to V_3\), thermal ionization of color centers occurs. These two processes lead to the fact that annihilation of \(F\)- and \(V\)-centers begins at a lower critical concentration and lower absorbed energy. The maximum in the kinetics of accumulation of stored energy shifts, with increasing temperature, into the region of lower absorbed energy.

The study of stored energy in alkali-halide crystals has shown that there is a regular relationship in the change of the properties of materials under the action of radiation to the energy of their crystal lattice. In crystals with a higher lattice energy, more defects remain after irradiation and, consequently, the change in properties is more significant than in crystals with weak bonding of the ions in the lattice.

§ 4. Kinetics of the Accumulation of Stored Energy in Alkali-Halide Crystals with Impurities

Small impurities of Ca++ and Sr++ (up to 0.01 M%) in KCl crystals increase the rate of accumulation of $F$-centers both in the first and in the second stages [38].

When the impurity content is increased to 0.04–0.05 M%, a decrease in the rate of accumulation of $F$-centers in the second stage is observed. When Pb++ impurity is introduced into KCl, a decrease in the rate of accumulation of $F$-centers in the second stage is observed in comparison with a crystal without impurities [24].

The introduction of Ca++ impurity into an NaCl crystal increases the rate of energy accumulation at relatively small irradiation doses (Fig. 8). As the irradiation dose is increased in impurity-containing crystals, the rate of accumulation of stored energy decreases, and after a dose of 12 kJ/g it becomes equal to zero.

Fig. 8. Kinetics of accumulation of stored energy in NaCl crystals with impurities under irradiation with 4.5 MeV protons (1 — “pure” NaCl, 2 — 0.004 M% CaCl$_2$, 3 — 0.04 M% CaCl$_2$, 4 — 0.01 M% PbCl$_2$, 5 — 0.01 M% AgI, 6 — 0.25% AgCl, 7 — 0.5% AgCl, 8 — 0.01 M% CdCl$_2$)

As in “pure” crystals, in impurity-containing crystals a maximum is observed in the dependence of the stored energy on the irradiation dose. With an increase in the amount of impurity, the rate of accumulation of stored energy in crystals with impurity increases. Consequently, these studies were carried out in the region of low impurity concentrations, when the rate of accumulation of $F$-centers still increases with increasing irradiation dose.

At large irradiation doses, the magnitude of the stored energy is smaller in crystals with impurity than in “pure” ones. This regularity is a consequence of the fact that the magnitude of the stored energy is determined mainly by the concentration of electron-hole centers. At large doses of proton irradiation, the concentration of $F$-, $M$-, and $V$-centers in crystals with impurity will be smaller than in the initial ones. This may be due to the fact that impurities located at block boundaries, by trapping an electron, can substantially reduce the concentration of color centers. In addition, the Ca++ impurity, possibly at large doses, exhibits its electron-acceptor properties and traps electrons. This process may be a competing process with the formation of $F$-centers.

When Pb++ and Cd++ impurities are introduced into NaCl, the stored energy decreases over the entire dose range (Fig. 8).

Experimental results of investigations of stored energy in NaCl with a silver impurity (NaCl·Ag) showed that the magnitude of the stored energy depends on the concentration of \(Ag^+\) ions. From Fig. 8 it is seen that increasing the impurity concentration leads to a decrease in the stored energy at the same irradiation dose.

Ch. B. Lushchik [39] showed that silver ions located at lattice sites serve as effective traps for electrons. Free holes can be localized at trapping levels of the host material, forming \(V_2\)- and \(V_3\)-centers, and on \(Ag^+\) ions, forming \(V_k\)-centers.

Significant recombination losses occur in the temperature range where stable \(V_k\)-centers are created. For NaCl·Ag this temperature is 318 K, i.e., the temperature of the crystal under irradiation with protons of energy 4.5 MeV. At this temperature free electrons recombine with holes of \(V_k\)-centers [40].

Theoretical consideration of this problem led to the conclusion that in the NaCl·Ag system a hole is localized not on an \(Ag^+\) ion, but on the nearest chlorine ion [40]. Usually impurity centers in alkali-halide crystals can trap both electrons and holes. Because of this feature in alkali-halide crystals, recombination of electrons and holes on an impurity is possible both through recombination of electrons with localized holes and through recombination of holes with localized electrons.

Recombination of electrons and holes on \(Ag^+\) ions leads to a decrease in stored energy. In addition, \(Ag^+\) ions compete with anion vacancies in the localization of electrons, which leads to a decrease in the concentration of \(F\)-centers and in the stored energy.

It was shown in [41] that in KCl doped with sulfur \(Na_2S\), the concentration of cation vacancies decreases, i.e., the number of hole-localization sites decreases. In this connection the concentration of free holes increases, and the process of defect destruction due to recombination of holes and electrons of \(F\)-centers will be ensured. The consequence of this is the lower sensitivity of KCl·S to radiation coloration compared with “pure” KCl.

An analogous result was obtained in the investigation of stored energy in NaCl and NaCl·S. From Fig. 9 it is seen that with increasing concentration of the \(S^{--}\) impurity the kinetics of the energy-storage process changes; consequently, the concentration of radiation defects changes at the same absorbed energy.

Fig. 9. Kinetics of the accumulation of stored energy in NaCl crystals with a sulfur impurity under proton irradiation
(1 — “pure” NaCl, 2 — NaCl·S (0.098 \(M\%\)), 3 — NaCl·S (0.17 \(M\%\)), 4 — rock salt)

The more impurities there are, the closer to one another lie the curves of the change in stored energy in NaCl·S and rock salt. The lattice structure of rock salt, as a result of long-term annealing under natural conditions of occurrence, proves to be more perfect, less disturbed by impurities, than the lattices of synthetic NaCl crystals.

In rock salt, the impurity is concentrated mainly on mosaic blocks. Since doped crystals contain a calcium impurity as a contamination and a specially introduced sulfur impurity, apparently one impurity neutralizes the other here, and thereby the activity of the impurity in the process of destruction of the generated defects decreases. During crystal growth, formation of the compound CaS is possible; in that case the crystal will be partially purified of CaCl\(_2\). At a certain sulfur concentration (0.17%), the kinetics of energy storage in NaCl · S proves to be analogous to the kinetics of energy storage in rock salt up to an absorbed-energy zone of \(12 \cdot 10^3\) J/g. At higher doses the deviation is possibly due to the fact that the NaCl · S lattice, in addition to Ca\(^{++}\) ions, contains a considerable amount of other impurities, which have a substantial influence on the magnitude of the stored energy.

The investigations carried out on doped crystals make it possible to conclude that the impurity changes the kinetics of accumulation of stored energy at large absorbed energies. By regulating the amount of impurity, one can change the rate of accumulation of stored energy in crystals and obtain materials resistant to radiation exposure.

§ 5. Release of Stored Energy during Thermal Annealing of Radiation Defects

Defects arising in a crystal as a result of irradiation can move at a sufficiently high temperature. Thus, the radiation disorder that has arisen can rearrange itself, change, and, ultimately, disappear completely (annealing of defects takes place). Since the activation energy for the motion of defects is different, these defects are annealed in different temperature intervals. Therefore, using the temperature dependence of a change in some property sensitive to defects of a given type, one can estimate, for each region, the number of defects of that type being annealed. One such property is the stored energy, whose magnitude is determined by the concentration of radiation defects.

By analogy with kinetic processes in gases and solutions, annealing processes in a solid can be described by a differential equation of the form

\[ -\left(\frac{dq}{dt}\right)=Kq^x, \tag{1} \]

where \(q\) is the number of possible kinetic processes;
\(t\) is time;
\(K\) is a constant, a characteristic coefficient;
\(x\) is the reaction order.

If, in a solid, the change in the concentration of certain defects \(n_i\) during annealing is measured, then the number of possible kinetic processes \(q\) will be proportional to \(n_i\), namely, \(q=\sigma n_i\) (\(\sigma\) is the cross section for capture of a defect by an atom, or the annihilation cross section). Then the basic differential equation is written in the form

\[ -\frac{dn_i}{dt}=K n_i^x \sigma^{x-1}. \tag{2} \]

The characteristic kinetic coefficient in the Arrhenius equation depends on temperature according to the exponential law

\[ K=A \cdot e^{-E_a/kT}, \tag{3} \]

where \(A\) is a pre-exponential factor which, for a solid, depends on the jump frequency of defects \(\nu\) and the distribution of sinks \(\gamma\) (of the order of unity [41]), \(A=\nu/\gamma\);

\(E_a\) is the energy or heat of activation;

\(T\) is the absolute temperature;

\(k\) is Boltzmann’s constant.

Then equation (3) can be written as

\[ -\frac{dn_i}{dt}=n_i^x(\sigma^{x-1}\nu/\gamma)e^{-E_a/kT}. \tag{4} \]

If the samples are heated at a constant rate \(\alpha=T/t\), then equation (4) can be solved as follows.

Taking \(n_{i0}\) as the defect concentration at the beginning of annealing, and \(n_i\) as that at time \(t\), and making the substitution

\[ f_i=n_i/n_{i0}, \]

\[ y=kT/E_a, \]

\[ \beta=[(n_{i0}\sigma)^{x-1}\cdot \nu E_a/k\alpha\gamma], \]

we find that

\[ \frac{df}{dy}=-\beta f\exp(-1/y) \tag{5} \]

or

\[ \int_{1}^{f_i}\frac{df}{f^x}=-\beta\int_{0}^{y}\exp(-1/y)\,dy. \tag{6} \]

As experiments have shown, in alkali-halide crystals \(x=1\), and therefore the solution is obtained in the form

\[ \ln f_i=-\beta\{ye^{-1/y}+Ei(-1/y)\}. \tag{7} \]

Here \(Ei(-1/y)\) is the exponential integral function. Since usually \(1/y=E_a/kT\gg 1\), with high accuracy one may write [43]

\[ E_i(-1/y)\approx e^{-1/y}(y^2-y) \tag{8} \]

and

\[ f_i=\exp(-\beta y^2 e^{-1/y}). \tag{9} \]

Equation (9) expresses the dependence of the reduced concentration of defects on the reduced temperature. As indicated above, the stored energy is proportional to the defect concentration; therefore (9) can be written as

\[ U/U_0=\exp(-\beta y^2 e^{-1/y}), \tag{10} \]

where \(U\) is the stored energy at the given moment of time;

\(U_0\) is that at the beginning of annealing.

Equation (10) shows the dependence of the change in stored energy on the reduced temperature, which is related to the activation energy of some developing process of transformation of radiation defects. Three main factors of the process should be distinguished.

1. The temperature at which a sharp increase in the rate of release of stored energy is observed. The rate of this process is found by differentiating equation (10):

\[ d(U/U_0)/dy=-\beta\exp(\beta y^2e^{-1/y})e^{-1/y}. \tag{11} \]

The temperature position of the maximum rate of energy release is determined from the solution of the equation

\[ \beta y_{\max}^2 e^{-1/y_{\max}}=1, \]

where \(y_{\max}\) is the value of the reduced temperature at the maximum rate of energy release, which, with an error of 2–3%, is equal to \(y_{\max}=(\ln \beta-2\ln|\ln \beta|)^{-1}\).

The activation energy for defect motion is related to the temperature position of the maximum [42]

\[ E_a=\left[d(U/U_0)/dy\right]_{\max} kT_{\max}^2/0.391\,(U_f/U_0)_{\max}, \]

where \(\left[d(U/U_0)/dy\right]_{\max}\) is the rate of energy release at the maximum; \((U_f/U_0)_{\max}\) is the total stored energy released at the maximum.

1. The second factor influencing the process of energy release is the order of the annealing process of radiation defects.

2. The total magnitude of the decrease in stored energy observed as a result of the destruction of its defects at one or another stage of annealing.

Comparison of all the factors involved in the destruction of defects can give a correct picture of the complex processes occurring during annealing.

Most work on the study of the process of energy release during thermal annealing of radiation defects has been carried out on NaCl crystals. Kobayashi [6] investigated, by the method of differential thermal analysis, the release of stored energy in NaCl crystals irradiated with protons of energy 350 MeV. In his experiments, to prevent heating of the specimens, the crystal holder was cooled with water. At a heating rate of \(2^\circ\mathrm{C}/\mathrm{min}\), Kobayashi obtained a spectrum of stored-energy release that has three annealing stages: the first—from room temperature to \(150^\circ\mathrm{C}\), the second—from 150 to \(250^\circ\mathrm{C}\), with a maximum at \(202^\circ\mathrm{C}\), and the third from \(250^\circ\mathrm{C}\) to \(400^\circ\mathrm{C}\), with a small maximum at \(310^\circ\mathrm{C}\).

The author [6] considered that the first annealing stage up to \(150^\circ\mathrm{C}\) is due to coagulation of \(F\)-centers into more complex centers (\(M\)-, \(N\)-centers) and recombination of Frenkel defects. This conclusion was drawn on the basis of comparing the stored-energy data with thermal bleaching of a NaCl crystal irradiated with a dose of \(8\cdot 10^{14}\) protons/\(\mathrm{cm}^2\). At this irradiation dose the total concentration of \(F\)-centers, according to Kobayashi’s data, was \(4\cdot 10^{18}\ \mathrm{cm}^{-3}\) and changed very little up to \(150^\circ\mathrm{C}\). But if it is assumed that all \(F\)-centers are converted into \(M\)-centers, about \(0.12\ \mathrm{J/g}\) will be released. This value is an order of magnitude smaller than the value actually obtained (\(3\ \mathrm{J/g}\)) and was calculated from the condition that, in the transformation \(2F \to M\), \(0.4\ \mathrm{eV}\) is released [34]. The formation of Frenkel defects in NaCl is unlikely. Moreover, if approximately \(8\ \mathrm{eV}\) is stored in one electron–hole pair, then at an \(F\)-center concentration of \(4\cdot 10^{18}\ \mathrm{cm}^{-3}\) the total value of the stored energy in NaCl will be \(2.7\ \mathrm{J/g}\), that is, three times smaller than the value actually obtained (\(9.1\ \mathrm{J/g}\)). Evidently, comparison of the annealing of two specimens having different initial defect concentrations is not sufficiently convincing. In Kobayashi’s opinion, in the region \(150\text{–}250^\circ\mathrm{C}\) the energy is released mainly as a result of the destruction of \(F\)-centers, and in the third stage as a result of annihilation and dissociation of aggregates.

In [44] the release of energy in NaCl irradiated with \(\gamma\)-rays (\(\mathrm{Co}^{60}\)) was studied; a two-stage character of the annealing of stored energy was obtained. This is associated with two types of \(F\)-centers. In the first stage, energy release occurs upon destruction of \(F\)-centers formed at vacancies generated by the radiation; in the second—at post-radiation vacancies. It is difficult to agree with this, since, on the one hand, the stored energy of \(F\)-centers formed at post-radiation vacancies is very small (\(\approx 0.12\ \mathrm{J/g}\)) and lies beyond the sensitivity limits of the apparatus used; on the other hand, these \(F\)-centers therma-

less stable [45], i.e., their destruction should occur at temperatures lower than the destruction of \(F\)-centers formed at anion vacancies generated by irradiation.

A more detailed study of the release of stored energy in NaCl was carried out in [46, 47]. Annealing of the crystals was performed at a rate of \(0.6^\circ\mathrm{C}/\mathrm{min}\) in a furnace of special design. The stored energy was determined by the dissolution method [48]. A batch of samples was irradiated with the same dose of protons with an energy of \(4.5\ \mathrm{MeV}\) \((9 \cdot 10^{14}\ \text{protons}/\mathrm{cm}^2)\) and electrons with an energy of \(1.2\ \mathrm{MeV}\) \((9 \cdot 10^{15}\ \text{electrons}/\mathrm{cm}^2)\). The temperature of the samples in the radiation field did not exceed \(30^\circ\mathrm{C}\).

Before the start of annealing, some of the irradiated crystals were dissolved in order to measure the initial value of the stored energy \(U_0\); the remaining crystals were placed in the furnace, and after every \(5\)--\(10^\circ\mathrm{C}\) several samples were removed, and the remaining part of the stored energy \((U)\) was determined by the dissolution method. The resulting dependence of the relative change in stored energy on temperature is presented in Fig. 10. By differentiating the dependence \(U/U_0=f(T)\), we find the spectrum of release of stored energy as a function of the annealing temperature.

The study was carried out in the temperature range from room temperature to \(250^\circ\mathrm{C}\). Above \(250^\circ\mathrm{C}\), NaCl crystals were completely decolorized. In this case the stored energy became equal to zero. The given spectrum of stored-energy release indicates the four-stage character of the process of annealing radiation defects in NaCl crystals: the first stage—from room temperature to \(90^\circ\mathrm{C}\), with a maximum at \(66^\circ\mathrm{C}\); the second—from 90 to \(130^\circ\mathrm{C}\), with a maximum at \(115^\circ\mathrm{C}\); the third—from 130 to \(200^\circ\mathrm{C}\), with a maximum at \(125^\circ\mathrm{C}\); and, finally, the fourth—from 200 to \(250^\circ\mathrm{C}\), with a maximum at \(215^\circ\mathrm{C}\). The peak of the fourth stage is poorly resolved.

To determine the types of defects responsible for the release of stored energy in each of the stages, the destruction of color centers was studied. As the sample is heated, complex processes of formation and decay of defect complexes take place in it, which leads to the disappearance of some absorption bands and the appearance of others. Figure 10 shows the change in the maxima of the absorption bands as a function of the annealing temperature.

Fig. 10. Release of stored energy (a) and annealing of absorption bands (b, c, d, e, f) in NaCl crystals: 1 — change in the stored energy and in the maximum of the absorption bands with annealing temperature; 2 — rate of change of the stored energy and of the maximum of the absorption bands (— — — rate of formation, —— rate of destruction)

In the temperature range \(25\)--\(90^\circ\mathrm{C}\), destruction of \(M\)- and \(V\)-centers is observed, and the maximum rate of destruction of these defects coincides with the maximum rate of release of stored energy. This fact allows one to conclude that the first stage of release—

The processes of decay and annihilation of \(M\)- and \(V\)-centers are responsible for the stored energy. Since in NaCl it is impossible to separate \(V_2\)- or \(V_3\)-centers at room temperature, in order to clarify the question of with which of the hole centers—the \(V_2\)- or the \(V_3\)-center—the \(M\)-center recombines, a calcium impurity was introduced into NaCl. The stored energy released in the first stage in impurity-containing NaCl crystals will be less than in “pure” ones.

Upon irradiation of NaCl crystals with a calcium impurity, preferential formation of \(V_3\)-centers is observed [35].

In the first stage of energy release, recombination of \(M\)-centers with \(V_2\)- and \(V_3\)-centers takes place; in addition, conversion of \(V_2\)- into \(V_3\)-centers may occur, as was already indicated, with energy being released in this process. It may then be assumed that the decrease in the amount of stored energy released in the first stage in impurity-containing crystals is due to a decrease in the concentration of \(V_2\)-centers.

In NaCl crystals with an impurity, in the first stage the stored energy is released mainly upon recombination of \(M\)- and \(V_3\)-centers. \(V_3\)-centers are thermally more stable than \(V_2\)-centers [49]. Therefore the maxima of the rates of destruction of \(M\)-centers and of energy release are shifted into the region of higher temperatures.

In the second stage of release of stored energy, in the temperature range \(90\text{–}130^\circ\mathrm{C}\), \(M\)- and \(N\)-centers are formed, and a band appears with a maximum at \(578\ \mathrm{nm}\), which consists of \(R_1\)- and \(R_2\)-centers [46]. In this region coagulation of the type \(F \to M \to R \to N\), etc., occurs with release of energy. But calculation shows that this process is accompanied by the release of about \(0.3\ \mathrm{J/g}\), whereas in fact \(0.8\ \mathrm{J/g}\) is released. The additional release of energy is due to the destruction of defects associated with the presence of calcium impurity. Indeed, if the content of \(\mathrm{Ca}^{++}\) impurity in NaCl is increased, then in the second stage more stored energy is released than in “pure” NaCl. This is also confirmed by studies of the release of stored energy in NaCl with calcium impurity, presented in [19, 50]. Consequently, the second stage of energy release is apparently due both to a coagulation reaction of the type \(F \to M \to R \to N\) and to the presence of defects caused by the calcium impurity.

When comparing the results of works [6, 18, 19] with the results of work [47], it may be noted that the authors of works [6, 18, 26] could not divide the first stage into the two stages obtained in [47]. This is probably connected with the higher heating rate of the specimens (for example, in [6] \(2^\circ\mathrm{C}/\mathrm{min}\), in [18] \(2^\circ\mathrm{C}/\mathrm{min}\), in [19] \(5^\circ\mathrm{C}/\mathrm{min}\)) and with insufficient sensitivity of the calorimeter.

In the temperature range \(130\text{–}200^\circ\mathrm{C}\), intense destruction of electron and hole centers is observed, accompanied by the release of the greatest amount of energy, with a maximum at \(186^\circ\mathrm{C}\). The data obtained in [47] agree well with the results of the study presented in works [6, 18, 19].

In the range \(200\text{–}250^\circ\mathrm{C}\), defects responsible for absorption in the region of the \(578\ \mathrm{nm}\) band are destroyed without further transformation; the crystal is completely bleached, and the remaining part of the stored energy is released, with a maximum at \(215^\circ\mathrm{C}\).

The temperature positions of the maxima of stored-energy release in NaCl crystals do not coincide with one another according to the data of different authors. It may be assumed that this is due to the different temperature of the specimens in the radiation field, the different initial density of radiation defects, the prior history of the crystal, the presence and

distribution of the impurity, and also by the different heating rates of the specimens during annealing.

In KCl after irradiation with 4.5 MeV protons, at a heating rate of 0.6°C/min, release of stored energy is observed with maxima at 65°C, 160°C, and 210°C, which is due, respectively, to annihilation of \(M\)- and \(V\)-centers, destruction of electron and hole centers, and destruction of the band at 710 nm.

Comparison of the spectra of stored-energy release in NaCl and KCl confirms the proposition that the thermal stability of defects depends on the lattice energy; that is, the lower the lattice energy, the lower the thermal stability of the defects. Therefore in KCl the peaks are shifted toward lower temperatures.

The study of stored energy in a series of alkaline-earth metal fluorides was carried out on crystals grown by the Stockbarger method. The crystals were irradiated in the IRT-2000 reactor with a mixed flux (fast and thermal neutrons and \(\gamma\)-rays). The thermal-neutron flux at the irradiation site was \(16 \cdot 10^{13}\) neutrons/cm\(^2\)·s; \(\gamma\)-irradiation was carried out using a Co\(^ {60}\) source with an intensity of 450 roentgens/s; an RUP-200 unit served as the source of X-rays (voltage 200 kV, current 20 mA). The stored energy was measured by the method of differential thermal analysis. Figure 11 shows the spectra of stored-energy release in CaF\(_2\), SrF\(_2\), and BaF\(_2\) crystals subjected to \(\gamma\)-irradiation with an absorbed dose of \(6.25 \cdot 10^7\) rad. As is seen from Fig. 11, for alkaline-earth metal fluorides there is likewise a dependence of the stored energy on the lattice energy. At the same absorbed energy, more energy is retained in the material with the greater lattice energy.

An analogous dependence is observed under irradiation in the reactor and under X-ray irradiation.

The maxima of the peaks of stored-energy release in CaF\(_2\), SrF\(_2\), and BaF\(_2\) under various kinds of irradiation occur at different temperatures (Table 3).

Fig. 11. Release of stored energy in alkaline-earth metal fluorides

In going from BaF\(_2\) to CaF\(_2\), that is, as the lattice energy of the compound increases, the maxima of the peaks shift toward higher temperatures for all types of irradiation. This circumstance indicates that defects produced by irradiation are thermally more stable in the material with the higher lattice energy, CaF\(_2\); that is, an analogy with alkali-halide crystals is observed.

Table 3

Temperatures of the maxima of stored-energy-release peaks in CaF\(_2\), SrF\(_2\), and BaF\(_2\) crystals irradiated by various types of radiation

Comparing the spectra of the release of stored energy of CaF$_2$ and SrF$_2$ crystals irradiated with x rays and $\gamma$ rays, one may conclude that the peaks in the low-temperature region (up to 300°C) have maxima at the same temperatures; under the action of electromagnetic radiation, analogous defects are formed in this region. In the temperature region above 300°C, in SrF$_2$ irradiated with x rays, the peak at 400°C is absent, whereas in CaF$_2$ there is a peak at 520°C. The peak at 360°C is intense in CaF$_2$ irradiated with $\gamma$ rays, weak in the case of x-ray irradiation, and has a maximum at 310°C. The peaks in the temperature region up to 220°C have approximately the same intensity in crystals of one and the same composition.

The results presented above can be explained as follows. Under irradiation with x rays (the absorbed dose is small), the contribution to the stored energy is made by defects produced by the radiation on structural imperfections already present in the crystal before irradiation. Since these defects are formed with minimal energy expenditure, their destruction also occurs easily. These defects apparently determine the peaks in the stored energy at temperatures up to 220°C.

The structural disturbances produced by x-ray irradiation are insignificant, which accounts for the weak peaks in the energy-release spectrum in CaF$_2$ at 310°C and in SrF$_2$ at 280°C, and for the absence of peaks in the high-temperature region. Irradiation with $\gamma$ rays (larger absorbed doses), however, is accompanied, apparently, by the appearance of new defects; in CaF$_2$ a peak is observed at 520°C and a strong peak at 360°C. In SrF$_2$ two peaks also appear—at temperatures of 400°C and 500°C (Fig. 11).

These conclusions are consistent with the results of works [51, 52].

Comparing the spectra of the release of stored energy in CaF$_2$, SrF$_2$, and BaF$_2$ crystals irradiated with $\gamma$ rays and in a reactor, one may judge the analogy of the defects produced by different types of radiation. Thus, for example, for SrF$_2$ the maxima of the peaks of stored-energy release correspond to the same temperatures (Table 3). The same is true for BaF$_2$ crystals, with the only difference that, upon irradiation in a reactor, a weak peak is observed at 50°C, which is not observed under $\gamma$ irradiation.

The peak at 300°C observed for CaF$_2$ irradiated in a reactor is absent in the crystal irradiated with $\gamma$ rays, but is observed for x rays (Table 3). It should be noted that if, for SrF$_2$ and BaF$_2$, the temperatures of the peak maxima coincide under irradiation with $\gamma$ rays and in a reactor, then in CaF$_2$, under irradiation in a reactor, the peaks in the temperature interval up to 400°C are shifted into the region of higher temperatures than in the case of irradiation with $\gamma$ rays (Table 3). It is not yet possible to explain this shift of the peaks.

Studies of stored energy in combination with other investigations (optical absorption, thermoluminescence) make it possible to put forward certain assumptions concerning the nature of the defects responsible for the stored energy.

The question was investigated as to in which defects the energy is stored in irradiated compounds of alkaline-earth metal fluorides. For this purpose, from the absorption spectrum of a CaF$_2$ crystal irradiated with $\gamma$ rays, the concentration of color centers responsible for absorption in the wavelength region 220–1100 nm was calculated. For the observed absorption bands (225, 335, 400, and 580 nm), the concentration was found by the Smakula formula under the assumption that, for all bands, the oscillator strength is equal to unity. The total concentration of color centers was

is equal to \(3 \cdot 10^{17}\ \mathrm{cm}^{-3}\). The stored energy for this case is equal to \(2 \cdot 10^{19}\ \mathrm{eV}\). If simple centers (of the type of \(F\)-centers in alkali-halide crystals) are considered responsible for the stored energy in \(\mathrm{CaF}_2\), then their concentration will be \(1.5 \cdot 10^{18}\ \mathrm{cm}^{-3}\) (it is assumed that the energy of such a center does not exceed the width of the forbidden band of \(\mathrm{CaF}_2\), i.e., \(13.5\ \mathrm{eV}\)).

As can be seen, the concentrations of color centers determined by the optical method and by an approximate calculation based on the magnitude of the stored energy do not coincide. This makes it possible to put forward the following assumptions.

1. The storage of energy in crystals of alkaline-earth fluorides occurs not only at simple, but also at complex color centers (possibly coagulates).

2. The storage of energy occurs not only at defects determined by the optical method in the wavelength interval \(220\)—\(1100\ \mathrm{nm}\), but also at defects of another type.

When irradiated \(\mathrm{CaF}_2\) crystals are heated, four peaks are found on the thermoluminescence curve in the temperature range from room temperature to \(360^\circ\mathrm{C}\). In this temperature range, the defects responsible for the absorption bands at 225, 335, 400, and \(560\ \mathrm{nm}\) are destroyed.

Since the thermoluminescence of crystals is caused by the destruction of electron-hole centers, it may be assumed that the absorption bands at 225, 335, 400, and \(560\ \mathrm{nm}\) in \(\mathrm{CaF}_2\) crystals are due to electron and hole centers. Accordingly, energy is also stored during irradiation.

Heating \(\mathrm{CaF}_2\) crystals in the temperature range \(360\)—\(550^\circ\mathrm{C}\) is not accompanied by luminescence. This gives grounds to assume that the peaks of release of stored energy at temperatures of 360 and \(520^\circ\mathrm{C}\) are due to other defects, but not of the electron-hole type.

Conclusions

1. In ionic crystals, energy is stored mainly at electron-hole centers. The stored energy at one \(F\)-aggregate center is greater than at one \(F\)-center. At high irradiation doses, about 20% of all stored energy is attributable to complex color centers. In this dose region the rate of energy accumulation decreases.

2. The formation of \(F\)-multiple and \(V_3\)-centers is thermodynamically more favorable, since it is accompanied by a decrease in the free energy of the irradiated crystal.

3. According to the nature of the change in stored energy as a function of absorbed energy, alkali-halide crystals can be divided into two groups. In \(\mathrm{KBr}\), \(\mathrm{KI}\), and rock-salt crystals, the accumulation of stored energy occurs in two stages. In the first stage, the energy is stored mainly at \(F\)- and \(V\)-centers. The processes of coagulation of \(F\)-centers determine the second stage. In \(\mathrm{LiF}\), \(\mathrm{NaCl}\), and \(\mathrm{KCl}\) crystals, the change in stored energy with absorbed energy passes through a maximum. The decrease in the rate of accumulation of stored energy at large values of absorbed energy is due to the annihilation of \(F\)- and \(V\)-centers as a result of energy release through the reaction \(V_2 \to V_3\)-centers.

4. The form of the curve of accumulation of stored energy in \(\mathrm{NaCl}\) depends on the initial dislocation density, the type, and the concentration of the impurity. The rate of accumulation of stored energy is lower in crystals with a higher dislocation density. A calcium impurity in \(\mathrm{NaCl}\) increases the rate of accumulation of stored energy in the first stage and suppresses the second. The introduction into alkali-halide crystals of impurities of lead, cadmium, silver, and sulfur decreases the rate of accumulation of stored energy at

the first and second stages. In doped crystals the maximum value of the stored-energy quantity is smaller than in some “pure” ones.

1. As the temperature of the irradiated samples is raised, the stored-energy quantity decreases.

2. For all types of irradiation, under comparable conditions, the greater the lattice energy of the crystal, the greater the stored energy in ionic crystals. With increasing lattice energy, the thermal stability of defects increases.

3. The release of stored energy in ionic crystals occurs in several stages. In alkali-halide crystals, stored energy is released during the decay of electron-hole centers. In CaF₂ the absorption bands lying at wavelengths of 225, 335, 400, and 560 nm are due to electron-hole centers; the decay of these centers is associated with energy release in the temperature range from room temperature to 360°C.

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