DETERMINATION OF THE PROBABILITIES OF ELEMENTARY PROCESSES IN THE FORMATION OF \(F\)-CENTERS IN PURE AND DOPED ALKALI-HALIDE CRYSTALS

L. I. Roshchina, I. Ya. Melik-Gaikazyan, E. P. Kurakina, Yu. V. Volkov

(The paper was presented by the scientific seminar of the Laboratory of Radiation Spectroscopy, Research Institute of Applied Physics)

The kinetics of accumulation of \(F\)-centers at pre-radiation defects depends on the elementary processes of formation and destruction of color centers and can be described by the equation [1, 2, 3]

\[ n_i=\frac{N_1 b_1}{b_1+\beta_1}\left[1-e^{-(b_1+\beta_1)t}\right] = n_1\left(1-e^{-B_1 t}\right), \tag{1} \]

\[ B_1=b_1+\beta_1, \]

where \(n_i\) is the concentration of \(F\)-centers at a certain time \(t\);

\(N_1\) is the concentration of defects in the crystal before irradiation that can be converted into \(F\)-centers (source density);

\(b_1\) is the probability of localization of electrons at pre-radiation vacancies;

\(\beta_1\) is the probability of radiation destruction of \(F\)-centers.

For sufficiently long times and low irradiation intensity, the concentration of \(F\)-centers reaches a limiting value

\[ n_i=\frac{N_1\cdot b_1}{b_1+\beta_1} =\frac{N_1}{1+\beta_1/b_1}. \tag{1a} \]

It is known from the literature data [4—11] that the introduction of metallic impurities into a crystal can substantially increase the limiting concentration of \(F\)-centers accumulated in the first stage, and, at higher impurity contents, decrease it.

The introduction of various impurities into alkali-halide crystals has a substantial effect on the concentration of \(F\)-centers, increasing the initial defectiveness of the crystal \(N_1\). In addition, a change in the conditions for localization of holes and electrons may affect the magnitude of the probability of localization of electrons at anion vacancies \((b_1)\) and the radiation destruction of \(F\)-centers \((\beta_1)\).

The aim of the present work is to determine, from experimental curves of \(F\)-center accumulation, the probabilities of the elementary processes of their formation and destruction for “pure” KCl crystals and KCl crystals with additions of Pb, Sr\(^{++}\).

Pure KCl crystals and crystals with impurities were grown from the melt by the Kyropoulos method from “chemically pure” grade salt. The impurities were added to the melt in the form of the salts PbCl\(_2\), SrCl\(_2\). From single-crystal ingots, samples of strictly identical thickness, 0.5 mm, were cleaved and irradiated with X-rays—

...on an X-ray unit URS-70, a tube with a Mo anode, voltage 42 kV; the current was varied from 4 to 14 mA in order to change the intensity of the incident radiation. The absorption spectra were measured on an SF-4A spectrophotometer.

To find the parameters of equation (1) from the experimental kinetic curve of F-center accumulation by the least-squares method, a calculation program was compiled for the “Minsk-1” electronic computer.

The essence of the calculation method [3] is as follows. It is required to find the values \(n_1\) and \(B_1\) that lead to a minimum of the sum of the squares of the deviations of the experimental points from the theoretical dependence (the residual),

\[ A_0(n_1 \cdot B_1)=\sum [\,n_i-n_1(1-e^{-B_1\cdot t_i})\,]^2=\min . \tag{2} \]

Let us introduce the following notation:

\[ x=\{x_1,\ x_2\}\equiv\{n_1B_1\}, \tag{3} \]

and also

\[ \frac{\partial A_0(x)}{\partial x_i}=A_{i0}(x);\quad \frac{\partial^2 A_0(x)}{\partial x_i \partial y_i}=A_{ij}(x);\quad (i=1,2;\ j=0,\ 1,\ 2). \tag{4} \]

Then the solution of equation (2) is equivalent to the solution of the system

\[ A_{i0}(x)=0,\quad (i=1,2). \tag{5} \]

This system can be solved by means of Newton’s iterative method [9]

\[ x^{m+1}=x^m-A_x^{-1}(x^m)A(x^m), \tag{6} \]

where

\[ A(x^m)\equiv \begin{pmatrix} A_{10}\\ A_{20} \end{pmatrix}, \]

and the matrix \(A_x^{-1}(x^m)\) is the inverse of the matrix

\[ A_{ij}= \begin{pmatrix} A_{11}A_{12}\\ A_{21}A_{22} \end{pmatrix}. \]

Then

\[ \left\{ \begin{aligned} n_1^{(m+1)}&=n_1^m- \frac{ \begin{vmatrix} A_{10}^m A_{12}^m\\ A_{20}^m A_{22}^m \end{vmatrix} }{ \begin{vmatrix} A_{11}^m A_{12}^m\\ A_{21}^m A_{22}^m \end{vmatrix} },\\[1.2em] B_1^{(m+1)}&=B_1^m- \frac{ \begin{vmatrix} A_{11}^m A_{10}^m\\ A_{21}^m A_{20}^m \end{vmatrix} }{ \begin{vmatrix} A_{11}^m A_{12}^m\\ A_{21}^m A_{22}^m \end{vmatrix} }. \end{aligned} \right. \tag{7} \]

If, according to (2–4), the matrix components are introduced in explicit form, we obtain:

\[ \begin{aligned} A_{10}&=\sum \omega_i\varphi_i,\\ A_{11}&=\sum \varphi_i^2,\\ A_{12}=A_{21}&=\sum (n_1\xi_i-\omega_i)\varphi_i \\ &\quad + (n_1 t_i \varphi_i\omega_i+n_1^2\varphi_i^2), \end{aligned} \tag{a} \]

[[unclear: continuation of formula cut off at bottom of page]]

where, for convenience and brevity of notation, it is adopted that

\[ \left\{ \begin{aligned} \xi_i &= 1-e^{-B_1 t_i},\\ \varphi_i &= t_i e^{-B_1 t_i},\\ \omega_i &= n_i-n_1\xi_i . \end{aligned} \right. \tag{76} \]

The summation is carried out over all experimental points of the given curve.

A block diagram of the organization of the program is shown in Fig. 1.

At first the program block is entered (1) and a check summation is performed (2). If the check sum coincides with the known one, then the initial data are entered (3); if not, the program is entered anew until the known check sum is obtained. After the initial data have been entered, they are also summed (4) and the check sum is verified. If the check sum does not coincide, the initial data are entered again; if the check sum does coincide, the entered decimal numbers are converted into the binary system (5). Control is then transferred to the main program.

Fig. 1. Block diagram of the organization of the program

The operator scheme of the program carrying out the calculation of a series of experimental curves has the following form:

\[ K_1\; Z_2\; K_3\; Z_4\; F_5\; P_6\; Ya_7 \]

\[ \begin{array}{cccccccccccc} & & & \overset{19}{\downarrow} & & \overset{8}{\uparrow} & & & & & & \\ \overset{6}{\downarrow} & & \overset{17}{\downarrow} & & \overset{14}{\downarrow} & & \overset{15}{\uparrow} & \overset{11}{\uparrow} & \overset{13}{\downarrow} & & \overset{10}{\uparrow} & & \overset{4}{\uparrow} \end{array} \]

\[ A_8\; Z_9\; K_{10}\; Z_{11}\; F_{12}\; P_{13}\; A_{14}\; H_{15}\; Z_{16}\; P_{17}\; H_{18}\; K_{19}. \]

Here \(K_1\) is the entry of the initial form of the command for sending \(N_i\) of operator \(Z_9\) into the standard cell;

\(Z_2\) is the sending of zero into the counter of the number of curves;

\(K_3\) is the entry of the initial form of the commands of operator \(Z_4\);

\(Z_4\) is the sending of the parameters \(n_1\) and \(B_1\) into the working cells;

\(F_5\) is the readdressing of the commands of operator \(Z_4\);

\(P_6\) is a check for completion of the cycle over the number of curves being calculated (if the condition is satisfied, control is transferred to operator \(Ya_7\); if not, to \(A_8\));

\(Ya_7\) is stopping upon completion of the calculation of the entire series of curves;

$A_8$ — addition of one to the counter of the number of curves and printing of the ordinal number of the curve;

$Z_9$ — entry of zeros into the cells where the sums $A_{ij}$ are accumulated;

$K_{10}$ — entry of the initial form of the commands of operator $Z_{11}$ in the cycle over $x_i$;

$Z_{11}$ — entry of the numbers $n_i t_i$ of the next curve into the standard working cells;

$F_{12}$ — redirection of the commands of operator $Z_{11}$;

$P_{13}$ — checking the condition for the end of the cycle over $t_i$ (if the condition is satisfied, a transition is made to operator $H_{15}$, and if the condition is not satisfied, to operator $A_{14}$);

$A_{14}$ — calculation of $A_{ij}$ by formulas (7a, 7b) and transition to operator $Z_{11}$;

$H_{15}$ — calculation and printing of $x^m$ and the residual;

Fig. 2. Kinetic curves of the accumulation of F-centers in crystals of
KCl $+\,1.5\cdot10^{-3}$ mol% PbCl$_2$ at different intensities of X-rays
$I_1:I_2:I_3:I_4:I_5 = 1:1.5:2:2.5:3$

$Z_{16}$ — entry of the values $x^m$ into the cells where $x^m-$ were located;

$P_{17}$ — checking the condition $\Delta x^m = x^m - x^{m-1} < \varepsilon$ (if the condition is satisfied, a transition is made to operator $H_{18}$; if not, to $K_{10}$);

$H_{18}$ — printing a table of the function with the calculated parameters;

$K_{19}$ — entry of the command for entering $n_i$ into an intermediate cell for restoration; transition to operator $Z_4$.

The initial data for each series of curves are placed in memory as follows: each number in binary-coded decimal is written into a pair of cells in the sequence: exponent and mantissa of the numbers.

The approximate values of the parameters \(n_1\) and \(B_1\) of each curve are arranged, in alternating order, in the first group of cells. In the second group are placed the values \(t_i\), which are assumed to be the same for all curves of a given series. In the third group of cells the parameters \(n_i\) are arranged in the following order: first the set \(n_i\) for the first curve is placed, then the set \(n_i\) for the second curve, etc.

The computation time for one iteration is 4–6 sec; the time for printing the table of values of the computed functions is on the order of 20–30 sec.

Fig. 3. Same for crystals \( \mathrm{KCl} + 3 \cdot 10^{-3}\) mol %
\(\mathrm{SrCl_2}\)

During operation of the program, the following are printed:

1) the numbers of the curves being processed;
2) the sums of the squares of the deviation of the theoretical curve from the experimental one (the residual), computed at each stage of iteration;
3) the intermediate and final values of \(n_1\) and \(B_1\);
4) a table of functions with the computed best (minimum residual value) parameters \(n_1\) and \(B_1\).

To determine the parameters of equation (1), we obtained kinetic curves of the accumulation of \(F\)-centers with variation of the intensi-

of the incident radiation. Figures 2 and 3 present the kinetic curves of accumulation of \(F\)-centers in \(\mathrm{KCl}\cdot\mathrm{Pb}\) and \(\mathrm{KCl}\cdot\mathrm{Sr}\) crystals at different x-ray intensities. For the “pure” KCl crystal, the ratio of the kinetic curves is likewise not given here; only for comparison, Fig. 3 gives the kinetic curve of accumulation of \(F\)-centers for KCl at intensity \(J_1\).

By the method described above, the parameters of the curve of equation (1) were calculated. From the determination of the parameter \(N_1\) it follows that its value does not depend on the intensity; therefore, by plotting the dependence of \(n_1\) on intensity, one can find the value \(N_1\), and then \(b_1\) and \(\beta_1\).

The results of calculating the parameters \(B_1\), \(b_1\), and \(\beta_1\) for KCl, \(\mathrm{KCl}\cdot\mathrm{Pb}\), and \(\mathrm{KCl}\cdot\mathrm{Sr}\) crystals are presented in Figs. 4 and 5 as functions of the x-ray intensity. It follows from Figs. 4 and 5 that \(\beta_1\) and \(b_1\) increase with increasing intensity, since the probability of localization of electrons at anion vacancies is proportional to the number of

Fig. 4. Dependence of the parameters \(B_1\), \(b_1\), \(\beta_1\) of the kinetic curves of accumulation of \(F\)-centers in KCl on the intensity of x-rays

electrons in the conduction band [1], which increases with increasing x-ray intensity. At the same time, the parameter \(\beta_1\) at first decreases somewhat with increasing intensity, and then remains practically constant at high intensities. Mitchell and co-workers [1], who used high-intensity x-rays, did not observe a change of \(\beta_1\) with \(J\); however, in work [10] by the same authors there is an indication of an increase in \(\beta_1\) at low intensities.

radiation. The reason for such a complex dependence of the quantity \(\beta_1\) on \(J\) has not yet been clarified.

When \(\mathrm{Pb}^{++}\) ions are introduced into a KCl crystal, a certain decrease is observed in the probability of localization of electrons at an anion vacancy per second. This is due to the fact that the addition of an electron-acceptor impurity gives rise to “atomic” impurity centers that compete in electron capture with anion vacancies and, as a consequence, reduce \(b_1\). The improvement of the conditions for localization of electrons at activator levels is accompanied by a decrease in the probability of recombination of electrons and holes; thereby the concentration of free holes increases. These delocalized holes can destroy F-centers [11], and the parameter \(\beta_1\) increases.

Fig. 5. The same for crystals \(\mathrm{KCl}+1.5\cdot10^{-3}\) mol % \(\mathrm{PbCl}_2\) (a) and \(\mathrm{KCl}+3\cdot10^{-3}\) mol % \(\mathrm{SrCl}_2\) (b).

In the \(\mathrm{KCl}\cdot\mathrm{Sr}\) crystal, the conditions for localization of holes are improved owing to their interaction with \(\mathrm{Sr}^{++}V^+\) complexes, followed by the formation of Hayes–Nichols centers [12, 13]. The preferential localization of holes on \(\mathrm{Me}^{++}V^+\) complexes in \(\mathrm{KCl}\cdot\mathrm{Sr}\) is confirmed by the decrease in these crystals of the effect of radiation-induced lowering of ionic conductivity [14]. This process is accompanied by an increase in the stability of F-centers in the radiation field (the parameter \(\beta_1\) decreases).

REFERENCES

1. P. Mitchell, D. Wiegend, R. Smoluchowski, Phys. Rev. 121, 484, 1961.
2. D. I. Vaisburd, I. Ya. Melik-Gaikazyan, L. I. Roshchina. FTT, (in press).
3. D. I. Vaisburd. Candidate dissertation. Tomsk. 1965.
4. D. Schultman, J. Phys. Shem. 57, 8, 1953.
5. E. E. Schneider, Disc. Far. Soc. 31, 176, 1961.
6. L. M. Shamoskii, DAN SSSR 99, 235, 1954.
7. M. L. Kats, B. Z. Semenov. Physics of Alkali-Halide Crystals. Proceedings of the Second All-Union Conference, 1961, 191, Riga, 1962.
8. Ch. B. Lushchik. Tr. IFA AN ESSR No. 3, 3, 1955.

1. I. S. Berezin, N. P. Zhidkov. Computational Methods, vol. 2, Fizmatgiz, 1962.

2. P. V. Mitchell, D. A. Wiegand, R. Smoluchowski. Phys. Rev. 117, 442, 1950.

3. I. A. Parfianovich. Proceedings of the VII Conference on Luminescence, 1958.

4. W. Hayes, G. M. Nichols. Phys. Rev. 117, 993, 1960.

5. W. Hayes. J. Appl. Phys., Suppl. 33, 329, 1962.

6. I. Ya. Melik-Gaikazyan, E. K. Zavadovskaya, M. I. Ignat’eva. FTT, 6, no. 4, 1964.