STATISTICAL DISTRIBUTION OF \(F\)-CENTERS AMONG MULTIPLE CENTERS

A. N. KRAVETS

(Presented by the scientific seminar of the Laboratory of Dielectrics and Semiconductors)

At present there is reason to believe that the electron centers \(M\), \(R\), and \(N_2\), formed in alkali-halide crystals under the action of ionizing radiation, consist respectively of two, three, and four \(F\)-centers, and that their concentrations are related by the following relations [1, 2, 3, 5]:

\[ n_M = p_1 n_F^2;\quad n_R = p_2 n_F^3;\quad n_N = p_3 n_F^4, \tag{1} \]

where \(p_1\), \(p_2\), and \(p_3\) characterize, respectively, the efficiencies of the radiation reactions \(F \to M\), \(F \to R\), \(F \to N_2\), and by a number of authors [1, 2, 4] are assumed to be independent of the concentration of \(F\)-centers. However, in [3, 6] it is shown that, when a LiF crystal is irradiated with protons, \(p_1\) increases with increasing dose, which is associated with the occurrence of two irreversible processes that increase the probability of a close arrangement of electron centers: 1) the statistical distribution of electron centers among multiple centers and 2) the statistical distribution of absorbed energy among localization multiplicities. The role of each of these processes depends on the irradiation dose, the intensity and energy of the incident particles, the temperature, and the chemical composition of the specimen under study. Clarifying this role is very important for the interpretation of experimental data.

In the present work an attempt is made to establish the influence of the first irreversible process—the statistical distribution of electron centers among multiple centers—on the kinetics of their accumulation and on the efficiencies of the radiation reactions of coagulation of \(F\)-centers into complex centers. For this purpose, a calculation was carried out by the Monte Carlo method of statistical trials, making it possible to establish the character of the accumulation of electron centers in the case of their statistical distribution, which, as we assume, takes place when the dose is uniformly distributed over the volume of the crystal. It should be noted that a problem posed in an analogous way was recently solved in [4], but only for \(F\)- and \(M\)-centers and for comparatively small portions of the kinetic curves, which did not make it possible to reveal the basic regularities of the kinetics of their accumulation. Therefore it remained necessary to solve the problem posed on a broader scale, which has been done in the present work. The generation of electron centers was carried out by throwing points whose coordinates were taken from a table of random numbers [8], onto the nodes

simple square lattice containing \(N = 10^4\) sites. In a real crystal the coordinate axes of such a lattice correspond to the directions \([01\bar{1}]\) and \([0\bar{1}1]\). Thus, the sites of the lattice under consideration may be regarded as anion sites. It is further assumed that two \(F\)-centers can form an \(M\)-center if they are located at neighboring lattice sites. The sites on which points fell will be called “occupied,” and the remaining ones “empty.” An “occupied” site was taken to be an \(F\)-center if the four sites surrounding it were empty. Two nearest “occupied” sites were taken to be an \(M\)-center if the remaining six sites in their nearest neighborhood were “empty.” Similarly, an \(R\)-center in this case is three neighboring “occupied” sites, and an \(N_2\)-center is four neighboring “occupied” sites, provided that the nearest sites surrounding them are empty. The number of nearest neighboring sites that can surround a given center, called the coordination number, was taken to be equal to \(\chi_1 = 4\), \(\chi_2 = 6\), \(\chi_3 = 8\), \(\chi_4 = 10\) for \(F\)-, \(M\)-, \(R\)- and \(N_2\)-centers, respectively. The results of each throw were entered in a table. Thus it was possible to trace the history of the formation of centers, i.e., the kinetics of their accumulation. If the thrown point fell on an already “occupied” site, the number of sites was considered unchanged. In all, three series of throws were carried out, with a dose (the number of thrown points) of four thousand each. In each series the number of centers obtained was averaged over an interval of one hundred points. The results of all series were also averaged, and, when they were compared, the errors of the “experiment” were determined; in our case these turned out to be smaller than in work [4], since we carried out the averaging over a larger interval of points. In carrying out the present “experiment,” as in work [4], it was assumed that the centers being generated are destroyed only when they coagulate into complex centers. Apparently, these conditions may be fulfilled in a real crystal at sufficiently low temperatures, when reverse reactions and radiation annealing of anion vacancies can be neglected.

If the entire lattice under consideration is taken as a unit area, then the quantity \(S = \dfrac{\chi_1}{N-n}\), where \(n = n_F + 2n_M + 3n_R + \ldots\) is the total number of “occupied” sites, may be regarded as the coagulation area of an \(F\)-center, i.e., the area such that, if one more \(F\)-center falls into it, an \(M\)-center is formed. Similarly, let us introduce into consideration the coagulation areas \(S_2 = \dfrac{\chi_2}{N-n}\); \(S_3 = \dfrac{\chi_3}{N-n}\); \(S_4 = \dfrac{\chi_4}{N-n}\) for \(M\)-, \(R\)- and \(N_2\)-centers, respectively. Using the results of works [6, 7], for the present case one can write the kinetic equations for the accumulation of electron centers in the following form:

\[ n_F = ne^{-\bar{n}s}; \tag{2} \]

\[ n_M = \frac{n}{\kappa \bar{n}s}\, e^{-\bar{n}s} \left(e^{-\kappa \bar{n}s} + \kappa \bar{n}s - 1\right); \tag{3} \]

\[ n_R = \frac{n(\kappa+1)}{\bar{n}s(\kappa_2-\kappa)}\, e^{-\bar{n}s} \left[ \frac{1}{\kappa}\left(e^{-\kappa \bar{n}s} + \kappa \bar{n}s - 1\right) - \frac{1}{\kappa_2}\left(e^{-\kappa_2 \bar{n}s} + \kappa \bar{n}s - 1\right) \right]; \tag{4} \]

\[ n_{N_2}=n\frac{(\kappa+1)(\kappa_2+1)}{\overline{nS}(\kappa_2-\kappa)}e^{-\overline{nS}}\left[ \frac{1}{\kappa(\kappa_3-\kappa)}\left(e^{-\kappa\overline{nS}}+\kappa\overline{nS}-1\right) - \right. \]

\[ {}-\frac{1}{\kappa_2(\kappa_3-\kappa_2)} \left(e^{-\kappa_2\overline{nS}}+\kappa_2\overline{nS}-1\right) - \]

\[ \left. {}-\frac{\kappa(\kappa_3-\kappa)+\kappa_3(\kappa_3-\kappa_2)} {\kappa\kappa_3(\kappa_3-\kappa)(\kappa_3-\kappa_2)} \left(e^{-\kappa_3\overline{nS}}+\kappa_3\overline{nS}-1\right) \right], \tag{5} \]

where

\[ \kappa=\frac{S_2}{S}-1;\qquad \kappa_2=\frac{S_3}{S}-1;\qquad \kappa_3=\frac{S_4}{S}-1, \]

and

\[ \overline{nS}=\int_0^n S(n)\,dn =-\ln\left(1-\frac{n}{N}\right)^{\chi_1}, \]

then

\[ n_F=n\left(1-\frac{n}{N}\right)^{\chi_1}. \tag{6} \]

The expressions for the efficiencies \(p_1=\dfrac{n_M}{n_F^2}\), \(p_2=\dfrac{n_R}{n_F^3}\), and \(p_3=\dfrac{n_{N_2}}{n_F^4}\) can be written using formulas (2–5).

To verify the validity of these formulas, they were compared with the results of a Monte Carlo “experiment.” In this, the quantity \(n=Dn^*\), where \(D\) is the number of points thrown and \(n^*\) is the number of repeated hits on “occupied” sites, which was also recorded in the “experiment.” Figures 1–3 present the results of the “experiment” and the theoretical curves; comparison shows that, at doses close to the number of lattice sites, equations (2–5) predict values that go beyond the error limits of the “experiment.” In a real crystal this discrepancy would occur at a concentration \(n\) of the order of 0.1 of the concentration of anion sites. The presence of this discrepancy can be explained by the fact that the conditions under which equations (2–5) were derived are not sufficiently strict. They do not take into account that, at large \(n\), the law governing the distribution of electron centers among multiple centers will have a more complicated character, namely, it will constitute an inhomogeneous Markov process [9].

Fig. 1. Concentrations of \(F\)- and \(M\)-centers as functions of dose, obtained by the Monte Carlo statistical-test method. Solid curves are theoretical.

Comparison of the results obtained with experimental data can be carried out by comparing the concentrations of electron centers and anion sites. However, such a comparison is made diffi-

although in a real crystal the process of the statistical distribution of centers does not appear in pure form but, apparently, is accompanied by another irreversible process—the statistical distribution of energy among localization multiplicities [3]. The contribution of this process to the kinetics of accumulation of electron centers can be determined experimentally,

Fig. 2. Concentrations of \(R\)- and \(N_2\)-centers as functions of dose, obtained by the method of statistical Monte Carlo trials. The solid curves are theoretical.

Fig. 3. Efficiencies of the radiation-chemical reactions \(F \to M(p_1)\), \(F \to R(p_2)\), \(F \to N_2(p_3)\), obtained from the data presented in Figs. 1 and 2. The solid curves are theoretical.

by varying the magnitude of the elementary volume of energy localization, that is, one of the parameters: the energy of the protons, their intensity, or the stopping power (chemical composition) of the samples under study, while keeping the remaining parameters fixed.

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