Accumulation of F-centers in Ionic Crystals
Abstract Generated abstract
This paper develops kinetic descriptions for the accumulation of F-centers in irradiated ionic crystals, emphasizing that potential anion sites may transform directly into F-centers as well as through anion vacancies. It formulates several three-state models, including nonchain and unbranched chain mechanisms, and derives a general second-order rate equation whose parameters are combinations of elementary transition constants. The resulting solutions predict saturation concentrations and classify possible time dependences of F-center concentration, including curves with maxima, near single-exponential growth, two-stage accumulation, and induction periods. The analysis suggests that experimentally observed two-stage accumulation, such as in LiF and NaCl, can arise within a single mechanism without requiring two simultaneous formation processes.
Full Text
ACCUMULATION OF $F$-CENTERS IN IONIC CRYSTALS
D. I. Vaisburd, A. G. Vaisburd
(Presented by the seminar of the Department of Experimental Physics)
Under the action of high-energy radiation, electronic and hole centers are created and accumulated in ionic crystals. The most numerous and elementary electronic centers are $F$-centers. When $F$-centers coagulate with one another or aggregate with other defects, complex electronic centers are formed. Therefore, the accumulation of $F$-centers and of the hole centers that complement them largely determines the change in the properties of ionic compounds in a radiation field.
It is known that an $F$-center is an electron captured by the effective field of an anion vacancy. All anion sites of a crystal (or all sites of the anion sublattice) may be regarded as potential $F$-centers, or simply as “potential centers.” In a crystal there exist several kinds of “potential centers”: undisturbed anion sites; anion sites that are part of structural defects (for example, anion vacancies that are part of vacancy aggregates or of “impurity ion plus vacancy” complexes); anion sites located near structural defects (for example, sites occupied by anions situated not far from cation vacancies, dislocations, or the surface). The total concentration of “potential centers” is Loschmidt’s number (the number of molecules in $1\ \mathrm{cm}^3$) and lies in the range $10^{22}$–$10^{23}\ \mathrm{cm}^{-3}$. In a radiation field a “potential center” may be transformed into an $F$-center. If the “potential center” is an undisturbed site and the formation of the $F$-center occurs without participation of a pre-radiation structural defect, then the mechanism of $F$-center creation is called structure-insensitive.
If pre-radiation structural defects participate in the formation of $F$-centers, then the creation mechanism is called structure-sensitive. If, in addition, the structural defect is restored after the formation of the $F$-center and thus a single structural defect can participate successively in the creation of several $F$-centers, then the mechanism is called catalytic. Finally, if, during the creation of an $F$-center, “potential centers” are generated or multiplied, then the mechanism is called chain. It is clear that a structure-insensitive mechanism cannot be chain-like.
Until recently, in phenomenological descriptions of the accumulation of \(F\)-centers it was assumed that the formation of an \(F\)-center takes place as follows: first a “potential center” is transformed into an anion vacancy, and then the latter captures an electron and is transformed into an \(F\)-center [1, 2, 3]. Recently it has become clear that “potential centers” can be transformed directly into \(F\)-centers. For example, according to the low-temperature Vitiol–Hersh mechanism [4, 5], an exciton in a single act decays nonradiatively into an \(F\)- and an \(H\)-center.
In [6] one of the authors proposed a chain high-temperature mechanism for the creation of \(F\)-centers, according to which divacancies (dipole pairs consisting of an anion and a cation vacancy) are transformed directly into \(F\)-centers (see below).
In the present work, in describing the dependence of the concentration of \(F\)-centers on irradiation time, the process of their direct formation is for the first time taken into account in general form.
Kinetic models of the accumulation of \(F\)-centers
All anion sites of the crystal may be in three states:
1) a “potential center” of some kind,
2) an \(F\)-center,
3) an anion vacancy (an isolated anion vacancy).
Let \(N\) be the concentration of “potential centers,” \(n\) the concentration of \(F\)-centers, and \(n_a\) that of anion vacancies. In the initial state \(N=N_0\). The anion sites may pass from one state to another. It is assumed that the probabilities of transition per unit time are constants (rate constants).
Scheme I. (see Table 1, cell 1.1). If the probability of transition from one state to another is characterized by one constant, then six rate constants are required: \(a\)—for transformation of a “potential center” into an \(F\)-center; \(\alpha\)—of an \(F\)-center into a “potential center”; \(g\)—of a “potential center” into an anion vacancy; \(\gamma\)—of an anion vacancy into a “potential center”; \(b\)—for capture of an electron by an anion vacancy and transformation into an \(F\)-center; \(\beta\)—for transformation of an \(F\)-center into an anion vacancy.
In the case of noncatalytic and nonchain mechanisms,
\[ N=N_0-n-n_a, \tag{1} \]
and within Scheme I the accumulation of \(F\)-centers is described by a system of two first-order kinetic equations
\[ dn_a/dt=gN_0-(b+g+\gamma)n_a+(\beta-g)n, \tag{2} \]
\[ dn/dt=aN_0+(b-a)n_a-(a+\alpha+\beta)n, \tag{3} \]
which is transformed into the second-order equation:
\[ d^2n/dt^2+(a+b+g+\alpha+\beta+\gamma)(dn/dt)+ \]
\[ +\left[(a+g+\gamma)(b+\beta+\alpha)+(a-b)(\gamma-\alpha)\right]n=(ab+a\gamma+bg)N_0. \tag{4} \]
Scheme II. The preceding scheme has certain shortcomings. First, in the solution of equation (4) it is not possible to get rid of radicals, which complicates its analysis (Table 1). Second,
\[ n=n_\infty\left(1-\frac{A}{A-B}e^{-Bt}+\frac{B}{A-B}e^{-At}\right) +\frac{aN_0}{A-B}\left(e^{-Bt}-e^{-At}\right) \]
Table 1
| No. | Scheme | Parameters | \(A\) | \(B\) | \(C\) | \(n_\infty = CN_0/AB\) |
|---|---|---|---|---|---|---|
| 1 | Schematic diagram with centers \(N\), \(n_a\), \(n\); transitions labeled \(a\), \(b\), \(g\), \(\alpha\), \(\beta\), \(\gamma\). | — | \(\displaystyle \frac12(a+g+\gamma+b+\beta+\alpha)+\frac{\sqrt{\frac14(a+g+\gamma+b+\beta+\alpha)^2-(a-b)(\gamma-\alpha)-(a+g+\gamma)(b+\beta+\alpha)}}{-(a+g+\gamma)(b+\beta+\alpha)}\) | \(\displaystyle \frac12(a+g+\gamma+b+\beta+\alpha)-\frac{\sqrt{\frac14(a+g+\gamma+b+\beta+\alpha)^2-(a-b)(\gamma-\alpha)-(a+g+\gamma)(b+\beta+\alpha)}}{-(a+g+\gamma)(b+\beta+\alpha)}\) | \(ab+\alpha\gamma+bg\) | \(\displaystyle \frac{(ab+\alpha\gamma+bg)N_0}{(a+g+\gamma)(b+\beta+\alpha)+(a-b)(\gamma-\alpha)}\) |
| 2 | — | \(a=0\) | \(\displaystyle \frac12(g+\gamma+b+\beta+\alpha)+\frac{\sqrt{\frac14(g+\gamma+b+\beta+\alpha)^2+b(\gamma-\alpha)-(g+\gamma)(b+\beta+\alpha)}}{-\,b(\gamma-\alpha)-(g+\gamma)(b+\beta+\alpha)}\) | \(\displaystyle \frac12(g+\gamma+b+\beta+\alpha)-\frac{\sqrt{\frac14(g+\gamma+b+\beta+\alpha)^2+b(\gamma-\alpha)-(g+\gamma)(b+\beta+\alpha)}}{-\,b(\gamma-\alpha)-(g+\gamma)(b+\beta+\alpha)}\) | \(bg\) | \(\displaystyle \frac{bgN_0}{(g+\gamma)(b+\beta+\alpha)-b(\gamma-\alpha)}\) |
| 3 | — | \(g=0\) | \(\displaystyle \frac12(a+\gamma+b+\beta+\alpha)+\frac{\sqrt{\frac14(a+\gamma+b+\beta+\alpha)^2-(a-b)(\gamma-\alpha)-(a+\gamma)(b+\beta+\alpha)}}{-(a+\gamma)(b+\beta+\alpha)}\) | \(\displaystyle \frac12(a+\gamma+b+\beta+\alpha)-\frac{\sqrt{\frac14(a+\gamma+b+\beta+\alpha)^2-(a-b)(\gamma-\alpha)-(a+\gamma)(b+\beta+\alpha)}}{-(a+\gamma)(b+\beta+\alpha)}\) | \(a(b+\gamma)\) | \(\displaystyle \frac{a(b+\gamma)N_0}{(a+\gamma)(b+\beta+\alpha)+(a-b)(\gamma-\alpha)}\) |
Continuation of Table 1
| No. | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| 4 | $\gamma=\alpha$ | $a+g+\gamma$ | $b+\beta+\gamma$ | $ab+a\gamma-bg$ | $\dfrac{(ab+a\gamma+bg)N_0}{(a+g+\gamma)(b+\beta+\gamma)}$ |
| 5 | $\gamma=\alpha$ $a=0$ |
$g+\gamma$ | $b+\beta+\gamma$ | $bg$ | $\dfrac{bgN_0}{(g+\gamma)(b+\beta+\gamma)}$ |
| 6 | $\gamma=\alpha$ $g=0$ |
$a+\gamma$ | $b+\beta+\gamma$ | $a(b+\gamma)$ | $\dfrac{a(b+\gamma)N_0}{(a+\gamma)(b+\beta+\gamma)}$ |
| 7 | [[diagram: triangular transition scheme between $N$, $N_0$, and $n$ with arrows labeled $a$, $\beta$, $g$, $\gamma$]] | $a+g+\gamma$ | $b+\beta$ | $ab+a\gamma+bg$ | $\dfrac{(ab+a\gamma+bg)N_0}{(a+g+\gamma)(b+\beta)}$ |
| 8 | $a=0$ | $g+\gamma$ | $b+\beta$ | $bg$ | $\dfrac{bgN_0}{(g+\gamma)(b+\beta)}$ |
| 9 | $g=0$ | $a+\gamma$ | $b+\beta$ | $a(b+\gamma)$ | $\dfrac{a(b+\gamma)N_0}{(a+\gamma)(b+\beta)}$ |
Continuation of Table 1
| No. | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| 10 | [[diagram: boxes \(N_0\), \(n_a\), \(n\) connected by arrows labeled \(a\), \(\beta\), \(\gamma\), \(g\), \(b\)]] | \(\gamma\) | \(b+\beta\) | \(ab+a\gamma+bg\) | \(\dfrac{(ab+a\gamma+bg)N_0}{\gamma(b+\beta)}\) |
| 11 | \(g=0\) | \(\gamma\) | \(b+\beta\) | \(a(b+\gamma)\) | \(\dfrac{a(b+\gamma)N_0}{\gamma(b+\beta)}\) |
in scheme I, the very important process of destruction of \(F\)-centers is taken into account by two constants: \(\alpha\) and \(\beta\). Experimentally, what is most often measured is the total probability of destruction of an \(F\)-center.
In scheme II (Table 1, cell 7,1) \(\beta\) is the rate constant for destruction of an \(F\)-center, while \(\gamma\) is the rate constant, averaged over the sum of \(F\)-centers and anion vacancies, for their return to the source, i.e., their transformation into “potential centers”; \(a, b, g\) have the same meaning as in scheme I. Let us denote the sum of anion vacancies and \(F\)-centers by \(f\),
\[ f \equiv n_a+n. \tag{5} \]
According to scheme II, accumulation of \(F\)-centers is described by a system of first-order kinetic equations:
\[ \begin{cases} df/dt=(a+g)N_0-(a+g+\gamma)f,\\ dn/dt=aN_0+(b-a)f-(b+\beta)n, \end{cases} \tag{6} \]
which is transformed into a single second-order equation:
\[ d^2n/dt^2+(a+b+g+\beta+\gamma)(dn/dt)+(a+g+\gamma)(b+\beta)n= \]
\[ =(ab+a\gamma+bg)N_0. \tag{7} \]
Scheme III (Table 1, cell 10,1). In [6] the following mechanism for the creation of \(F\)-centers was proposed:
\[ e^-+v_cv_a \to F+v_c, \]
\[ h^+ + v_c \to h^+v_c, \]
\[ e^0+h^+v_c \to X^0v_c+Me_{int}^+ + v_cv_a, \]
where \(v_cv_a\) is a divacancy consisting of a cation vacancy \((v_c)\) and an anion vacancy \((v_a)\); \(e^-\) is an electron; \(h^+\) is a hole; \(X^0v_c\) is a halogen atom in a cation vacancy; \(Me_{int}^+\) is an interstitial alkali ion. According to this mechanism, the “potential centers” are the anion sites that are part of divacancies; and the creation of \(F\)-centers represents an unbranched chain reaction. In such a reaction, when an \(F\)-center is created, a divacancy is reproduced. Therefore the concentration of “potential centers” may be considered constant,
\[ N=N_0. \]
The constants \(a, b, g, \beta, \gamma\) have the same meaning as in scheme II. Then the accumulation of \(F\)-centers by the unbranched chain reaction can be described by a system of two first-order equations:
\[ df/dt=(a+g)N_0-\gamma f, \tag{8} \]
\[ dn/dt=aN_0+bf-(b+\beta)n, \tag{9} \]
or by a single second-order equation
\[ d^2n/dt^2+(b+\beta+\gamma)(dn/dt)+\gamma(b+\beta)n=(ab+a\gamma+bg)N_0. \tag{10} \]
General equation and its solutions
Consideration of different schemes of accumulation of \(F\)-centers (see, for example, Eqs. (4, 7, 10)) has allowed us to make the following generalization. If the anion sites of the crystal can be found in only three states: “potential center,” \(F\)-center, anion vacancy, then the dependence of the concentration of \(F\)-centers on irradiation time is described by the equation
\[ d^2n/dt^2+(A+B)(dn/dt)+A\cdot Bn=CN_0, \tag{11} \]
where \(N_0\) is the initial concentration of “potential centers”; \(A, B, C\) are combinations of rate constants. For the schemes I, II, III considered, the expressions for \(A, B, C\) in terms of the rate constants are given in Table 1.
Solving equation (11) with the initial conditions
\[ \text{at } t=0 \quad n=0,\quad dn/dt=aN_0, \tag{12} \]
we obtain, for schemes I, II, III, the general formula for the dependence of the concentration of \(F\)-centers on irradiation time
\[ n=\frac{CN_0}{AB}\left(1-\frac{A}{A-B}e^{-Bt}+\frac{B}{A-B}e^{-At}\right) +\frac{aN_0}{A-B}\left(e^{-Bt}-e^{-At}\right). \tag{13} \]
In the special case, if \(A=B\), then
\[ n=(CN_0/A^2)(1-e^{-At}-Ate^{-At})+aN_0te^{-At}. \tag{14} \]
According to (13), the accumulation of \(F\)-centers ends in saturation of their concentration at the level:
\[ n_\infty=CN_0/AB. \tag{15} \]
Specific expressions for \(n_\infty\) for the different schemes are given in Table 1.
Shape of the accumulation curve of \(F\)-centers
From equation (13) we find expressions for the first, second, and third derivatives of the concentration of \(F\)-centers with respect to time. We determine the irradiation times at which the corresponding derivatives vanish. For the first derivative
\[ t_{01}=\frac{1}{A-B}\ln\frac{A-C/a}{B-C/a}, \tag{16} \]
for the second derivative
\[ t_{02}=\frac{1}{A-B}\ln\frac{(A-C/a)A}{(B-C/a)B}, \tag{17} \]
for the third derivative
\[ t_{03}=\frac{1}{A-B}\ln\frac{(A-C/a)\cdot A^2}{(B-C/a)B^2}. \tag{18} \]
In the case \(A=B\), \(t_{01}\), \(t_{02}\), \(t_{03}\) are respectively equal to:
\[ t_{01}=\frac{1}{A-C/a};\quad t_{02}=\frac{1}{A-C/a}+\frac{1}{A};\quad t_{03}=\frac{1}{A-C/a}+\frac{2}{A}. \tag{19} \]
From (16), (17), (18) we find that the first derivative \(dn/dt=0\) under the condition:
\[ C/a<B \quad \text{and} \quad C/a<A, \tag{20} \]
the second derivative \(d^2n/dt^2=0\) under the condition
\[ C/a<B \quad \text{and} \quad C/a<A \]
or
\[ C/a>A+B, \tag{21} \]
the third derivative \(d^3n/dt^3=0\) under the condition
\[ C/a<B \quad \text{and} \quad C/a<A \]
or
\[ C/a > A + B - \frac{B}{1 + B/A}. \tag{22} \]
From (16), (17), (18), (20), (21), (22) it follows that, when \(F\)-centers are created from “potential centers” of one kind, the dependence of their concentration on irradiation time may have four forms (Table 2, Figs. 1, 2, 3, 4).
Indeed, four cases are possible:
- If
\[ C/a < B \quad \text{and} \quad C/a < A \tag{23} \]
all three derivatives may be equal to zero. This case corresponds to the curve with a maximum, presented in Table 2 in Fig. 1.
- If
\[ B < C/a < A + B - \frac{B}{1 + B/A} \]
or
\[ A < C/a < A + B - \frac{B}{1 + B/A}, \tag{24} \]
none of the derivatives can be equal to zero. This case corresponds to the curve presented in Table 2, in Fig. 2, and the accumulation of \(F\)-centers proceeds according to a law close to \((1-\exp)\).
- If
\[ A + B - \frac{B}{1 + B/A} < C/a < A + B \tag{25} \]
only the third derivative can be equal to zero. This corresponds to the curve presented in Table 2, in Fig. 3, and the accumulation of \(F\)-centers proceeds according to a two-stage law typical for LiF and NaCl. Thus, to explain such experimental curves there is no need to assume the simultaneous action of two mechanisms of \(F\)-center creation.
- If
\[ C/a > A + B \]
the second and third derivatives may be equal to zero. This case corresponds to the curve presented in Table 2, Fig. 4, and the accumulation of \(F\)-centers proceeds along a curve with an induction period.
It should be noted that curve 4 will always be observed if the probability of direct formation of an \(F\)-center, bypassing the “anion vacancy” state, is equal to zero (Table 2). Indeed, for \(a = 0\), formula (13) is simplified, and the accumulation of \(F\)-centers proceeds according to the law
\[ n = \frac{C N_0}{A B} \left( 1 - \frac{A}{A - B} e^{-Bt} + \frac{B}{A - B} e^{-At} \right) = \]
\[ = \frac{C N_0}{A B} - \frac{C N_0}{A - B} \left( \frac{e^{-Bt}}{B} - \frac{e^{-At}}{A} \right), \]
which was obtained earlier in [3].
In Table 2, for the various schemes, the specific conditions for realization corresponding to the accumulation curve are recorded. The crossed-out cells correspond to impossible cases. The cells denoted by the numbers 1.2; 1.3; 1.4; 3.2; 3.3; 3.4; 4.4; 7.3; 7.4 are not filled in because of the unwieldiness of the corresponding expressions. The latter are easily obtained from the general conditions.
Table 2
| Scheme 1 | Scheme 2 | Scheme 3 | Scheme 4 | Scheme 5 | |
|---|---|---|---|---|---|
| Shape of the F-center accumulation curve | Plot of $\eta$ vs. $t$: curve rises to a maximum and then decreases to a plateau; marked times $t_{01}$, $t_{02}$, $t_{03}$ | Plot of $\eta$ vs. $t$: monotonic rise to saturation; marked time $t_0$ | Plot of $\eta$ vs. $t$: monotonic rise to saturation | Plot of $\eta$ vs. $t$: S-shaped rise to a limiting value $\eta_{\lim}$; marked times $t_{02}$, $t_{03}$ | Plot of $\eta$ vs. $t$: sharp rise followed by saturation at $\eta_{\lim}$; marked times $t_{02}$, $t_{03}$ |
| Condition | $C/a<A$ and $C/a<B$ | $A<C/a<A+B-\dfrac{AB}{A+B}$ or $B<C/a<A+B-\dfrac{AB}{A+B}$ | $A+B-\dfrac{AB}{A+B}<C/a<A+B$ | $C/a>A+B$ | |
| No. | 1 | 2 | 3 | 4 | 5 |
| 1. $\alpha<1$ | 1.2 | 1.3 | 1.4 | $b_g\gg a(+g+\alpha+\beta)$ | |
| 2. $a=0$ | — | — | — | — | always fulfilled |
| 3. $g=0$ | 3.2 | 3.3 | 3.4 | — | |
| 4. $\gamma=\alpha$ | $b<a$ and $b_g<\alpha\beta$ | 4.3 | 4.4 | $b_g\gg a(\alpha+g+\gamma+\beta)$ |
Continuation of Table 2
| No. | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| 5 | $\gamma=a$ $a=0$ |
— | — | — | always fulfilled |
| 6 | $\gamma=a$ $g=0$ |
$b<a$ | $b>a$ | — | — |
| 7 | Scheme II | $b<a$ and $a\gamma+bg<a\beta$ | 7.3 | 7.4 | $bg>a(a+g+\beta)$ |
| 8 | $a=0$ | — | — | — | always fulfilled |
| 9 | $g=0$ | $b<a$ and $\gamma<\beta$ | $a<b<a+\dfrac{(b+\beta)^2}{a+\gamma+b+\beta}$ or $\beta<\gamma<\beta+\dfrac{(a+\gamma)^2}{a+\gamma+b+\beta}$ |
$\gamma b>a(a+\gamma+\beta)+\beta(b+\beta)$ | — |
| 10 | Scheme III | — | $\dfrac{bg}{a}<\beta-\dfrac{\gamma(b+\beta)}{\gamma+b+\beta}$ | $\beta-\dfrac{\gamma(b+\beta)}{\gamma+b+\beta}<\dfrac{bg}{a}<\beta$ | $bg>a\beta$ |
| 11 | $g=0$ | — | $b+\beta>\gamma$ | $b+\beta<\gamma$ | — |
References
- R. V. Mitchell, D. A. Wiegand, R. Smoluchowski. Phys. Rev., 121, 484 (1961).
- F. Fisher. Zs. Physik, 1959.
- D. I. Vaisburd, I. Ya. Melik-Gaikazyan, L. I. Roshchina. Theoretical and Experimental Chemistry (in press), 1968. D. I. Vaisburd. Abstract of dissertation, Tashkent, 1966.
- H. Hersh, Phys. Rev., 148, No. 2, 1966.
- I. K. Vitol. Izv. AN, ser. fiz., 30, No. 4, 564, 1966.
- D. I. Vaisburd, S. M. Minaev, N. L. Terent'ev. Izv. AN, ser. fiz. (in press). Proceedings of the 17th Conference on Luminescence, Irkutsk, 1968.