KINETIC EQUATION FOR THE ACCUMULATION OF $F$-CENTERS IN ALKALI-HALIDE SINGLE CRYSTALS UNDER PROTON IRRADIATION

D. I. VAISBURD, I. Ya. MELIK-GAIKAZIAN

(Presented by the scientific seminar of the EDIP problem laboratory)

On the basis of contemporary ideas about the mechanism of vacancy formation in alkali-halide crystals under the action of radiation [2], and about models of hole and electron color centers [3, 4], it may be considered that the process of radiation damage in alkali-halide crystals is based on the disruption of the ionic chemical bond as a result of the ionization of halogen ions and the displacement of ions and atoms of alkali metals and halogens from lattice sites. If ionization of alkali-metal ions increases the heteropolarity of the lattice, then ionization of halogen ions sharply decreases the heteropolarity—the principal condition for the existence of an ionic chemical bond. When the ionic bond is disrupted by radiation, color centers arise. Hole color centers are holes localized on molecular ions consisting of two or more halogen atoms (ions), one of which may be located in an interstice [3].

The stability of a hole center increases if it is associated with defects of the crystal lattice: vacancies, impurity ions, etc. Features of a metallic bond may be observed in electron color centers. An $F$-center is an electron in an anion vacancy, shared among the six surrounding alkali-metal ions. $M$-, $R$-, and, presumably, $N$-centers are, respectively, $F_2$-, $F_3$-, and $F_4$-coagulates of $F$-centers [4].

The formation, coagulation, and accumulation of hole and electron color centers may be regarded as a process of the “nucleation” and growth of covalent and metallic formations in an ionic crystal. In the synthesis of an alkali-halide crystal from an alkali metal and a halogen, a purely ionic compound is formed and the energy of formation is released. It may be assumed that under the action of radiation the reverse process occurs: disruption of the ionic chemical bond and radiolysis of the alkali-halide crystal into metal and halogen, which “nucleate” in the form of electron and hole color centers and then coarsen into a phase. The indicated process is accompanied by the storage of energy.

In the present work, the kinetics of accumulation of $F$-, $M$-, and $R$-centers in alkali-halide single crystals under irradiation with protons of energy 5 MeV at room temperature was investigated. As quali-

As the radiation source, a cyclotron was used, which makes it possible to obtain high radiation intensities at the location of the specimen and to vary them over wide limits. In our experiments the intensity was varied from \(10^4\) to \(10^6\) erg/cm\(^2\) sec.

The penetration depth of the protons was determined from the thickness of the colored layer and, for crystals of different chemical composition, was (m · \(10^{-6}\)):

The data on the position and half-width of the \(F\)- and \(M\)-absorption bands are summarized in Table 1.

Table 1

Position of the maximum and half-width of the \(F\)- and \(M\)-absorption bands in alkali-halide single crystals irradiated with protons at 300°K

Despite the fact that the form of the kinetic curves for the accumulation of centers changed with changes in the chemical composition of the crystals and in the radiation intensity (Figs. 1, 2), in most cases it proved possible to describe the kinetics of accumulation of \(F\)-centers in the first stage and in the linear part of the second stage by the equation proposed in work (5)

\[ n_F = n^*\left[1-\exp(-b^* t)\right]+a^*t-(a^*/c^*-n')\left[1-\exp(-c^*t)\right], \tag{1} \]

where \(n_F\) is the concentration of \(F\)-centers; \(t\) is the irradiation time;

\[ n^*=\frac{n_0 b}{b+\alpha}, \tag{2} \]

\(n_0\) is the concentration of thermal vacancies present in the crystal before irradiation and dispersed throughout the volume of the crystal; \(b\) is the rate of electron capture by thermal vacancies; \(\alpha\) is the relative rate of radiation bleaching of \(F\)-centers formed from thermal vacancies; \(b^*=b+\alpha\);

\[ a^*=\frac{ac}{c+\beta}, \tag{3} \]

\(a\) is the rate of accumulation of radiation vacancies, i.e., those generated by the radiation; \(c\) is the rate of electron capture by radiation va-

vacancies; \(\beta\) is the relative rate of radiation bleaching of \(F\)-centers formed from radiation vacancies; \(c^* = c + \beta\).

Fig. 1. Kinetic curves of the accumulation of \(F\)-centers in alkali-halide single crystals under irradiation with protons of energy 5 MeV.

The concentration of \(F\)-centers reached as a result of the first stage, for a number of the alkali-halide single crystals studied, correlates satisfactorily with the average energy of excitation and ionization (Fig. 1). The observed relation can be explained if one takes into account that: 1) protons lose a large part of their energy to ioni-

Fig. 2. Kinetic curves of the accumulation of \(F\)-centers in KBr single crystals under irradiation by a proton beam of different intensity \((J)\).

zation (Fig. 1).

... and excitation; 2) the spatial distribution of energy losses for ionization and excitation is not the same: ionization energy is distributed dispersively through the crystal, while excitation energy is released mainly at defects—exciton traps; 3) if one assumes that, in the first stage, the ionization process is mainly responsible for the formation of \(F\)-centers, whereas excitons, which release energy at \(F\)-centers, are responsible for the radiation bleaching of \(F\)-centers; 4) with an increase in the average excitation and ionization energy along the series of crystals, a redistribution of the energy losses of charged particles occurs owing to an increase in the fraction of losses for excitation and a decrease in the fraction of losses for ionization. Then the degree to which the direct process of \(F\)-center formation predominates over the reverse process will decrease with increasing average excitation and ionization energy. It is characteristic that, in the kinetics of \(F\)-center accumulation in a KI crystal, the first stage is practically absent, i.e., the formation of \(F\)-centers from thermal vacancies does not predominate over their destruction.

If, to equation (3), written for two radiation intensities \(I_1\) and \(I_2\), one adds the experimentally obtained relation \(a \sim I\) [6] and takes into account that \(c \sim I\) [5, 6], then we obtain:

\[ \frac{a_2^*}{a_2}=\frac{I_{21}^{2}-I_{21}^{\delta_{21}}}{I_{21}(I_{21}-1)}; \tag{4} \]

\[ \frac{a_2/a_2^*}{a_1/a_1^*}=I_{21}^{\delta_{21}-1}; \tag{5} \]

\[ \beta=c_2\,\frac{I_{21}^{\delta_{21}}-I_{21}}{I_{21}^{2}-I_{12}^{\delta_{21}}}, \tag{6} \]

where \(I_{21}=I_2/I_1\); \(\delta_{21}=\lg \dfrac{a_2^*}{a_1^*}\big/ \lg \dfrac{I_2}{I_1}\).

The expressions obtained make it possible, from the ratio of radiation intensities and the rates of accumulation of \(F\)-centers on the rectilinear segment of the second stage, to determine \(a^*/a\) and \(\beta/c\), which characterize the degree to which the formation process predominates over the process of radiation bleaching of \(F\)-centers in the second stage. The values of \(\delta_{21}\), \(a_2^*/a_2\), and \(\beta/c_2\) for \(I_{21}=1.5\) are given in Table 2.

Table 2

Values of \(\delta_{21}\), \(a_2^*/a_2\), \(\beta/c_2\) for crystals of different chemical composition

For all the crystals investigated, the concentration of \(M\)-centers was proportional to the square of the concentration of \(F\)-centers in the region of low \(F\)-center concentrations. An increase in the radiation intensity reduces the efficiency of the transformation \(F \to M\) (Fig. 3).

As is seen from Fig. 1 and Fig. 2, beginning with a certain concentration of \(F\)-centers, the rectilinear segment of the curve passes into a segment...

with a decreasing rate of accumulation of \(F\)-centers. The latter is associated with a change in the rate of accumulation of \(M\)-centers.

A discussion of the kinetic equation for the accumulation of \(F\)-centers proposed in [5] shows that the indicated equation must be improved by taking into account the mutual transformations of \(F\)-centers and complex electron color centers.

Fig. 3. Dependence of the absorption coefficient at the maximum of the \(M\)-band on the square of the absorption coefficient at the maximum of the \(F\)-band for LiF single crystals irradiated with a proton beam of different intensity \((J)\).

In the present work, the kinetics of the accumulation of \(F\)-centers and complex electron color centers under proton irradiation was studied jointly on LiF single crystals. In the absorption spectrum of the irradiated LiF single crystals, the \(F\), \(M\), \(R_{323m\mu}\), \(R_{370m\mu}\) bands were observed, as well as a certain band whose maximum, with increasing dose, shifted from \(622\,m\mu\) to \(626\,m\mu\), while its half-width increased from \(0.29\) to \(0.42\,\text{eV}\). We shall conventionally denote this band by \(\Pi_{624}\). Data on the positions of the bands and their half-widths are summarized in Table 3.

Table 3

Position and half-width of absorption bands in LiF single crystals irradiated with protons at room temperature

For all bands, a certain increase in the half-width was observed with an increase in the absorption coefficient at the maximum. This increase was greatest for the \(F\)-band, and it is associated with a shift of the short-wavelength half-maximum.

According to [10], the broadening of the \(F\)-band is due to a certain overlap of the \(F\)-band by neighboring bands, but its main cause is the growth of new bands in the region of the \(F\)-band, associated with complex electron color centers.

In the process of accumulation of \(F\)- and \(M\)-centers, the concentration of \(M\)-centers increased in proportion to the square of the concentration of \(F\)-centers (Fig. 3)

\[ n_M=\alpha_1 n_F^2. \tag{7} \]

The universality of relation (7) with respect to the type of radiation has been verified for x-ray radiation and electrons [7, 8, 9], and in the present work for protons.

According to the model of \(R\)-centers as \(F_3\)-centers [4], one should expect that the concentrations of \(F\)-, \(M\)- and \(R\)-centers are related by the relations

\[ n_R=\alpha_2 n_F^3, \tag{8} \]

\[ n_R=\alpha_3 n_M^{3/2}. \tag{9} \]

Indeed, the experimentally obtained dependence of the logarithm of the absorption coefficient at the maximum of the \(R_{370\,m\mu}\) band on the logarithm of the absorption coefficient at the maximum of the \(M\)-band was expressed by a straight line with slope 1.48, which confirms the model of the \(R_{370\,m\mu}\)-center as an \(F_3\)-center.

According to relations (7), (8), (9), there must exist a definite connection between the form of the kinetic curves for the accumulation of \(F\)-, \(M\)- and \(R\)-centers, which is confirmed experimentally (Fig. 4).

Fig. 4. Kinetic curves for the accumulation of \(F\)-, \(M\)-, and \(R\)-centers in LiF single crystals under proton irradiation.

As can be seen from Fig. 4, accumulation of \(F\)-centers according to the linear law \(n_F\sim t\) corresponds to accumulation of \(M\)-centers according to the law \(n_M\sim t^2\), and of \(R\)-centers \(n_R\sim t^3\); accumulation of \(M\)-centers according to the linear law corres—

there will be accumulation of \(F\)-centers according to the law \(n_F \sim t^{1/2}\), and of \(R\)-centers \(n_R \sim t^{3/2}\); accumulation of \(R\)-centers according to a linear law corresponds to accumulation of \(F\)-centers according to the law \(n_F \sim t^{1/3}\) and of \(M\)-centers \(n_M \sim t^{2/3}\).

The kinetics of accumulation of \(F\)-centers in the second stage, beginning with the linear segment, can be described by a certain function of irradiation time \(f(t)\), whose construction is based on the following.

1. The kinetic curve of accumulation of \(F\)-centers can be divided into segments

\[ n_F = a_j t^{1/j}, \tag{10} \]

where \(j\) refers to that one of the complex electron centers of type \(F_j\) whose accumulation in the given segment proceeds according to a linear law.

1. The kinetic curve of accumulation of \(F\)-centers is continuous with respect to the concentration of \(F\)-centers and its first derivative with respect to time.

2. The second derivative of the concentration of \(F_j\)-centers, where \(j = 1\) for \(F\), \(j = 2\) for \(M\), \(j = 3\) for \(R\), \(j = 4\) for \(N\)-centers, continuously decreases.

Fig. 5. Construction of the kinetic curve of accumulation of \(F\)-centers taking into account the formation of \(F_j\)-centers.

The construction of the function is illustrated in Fig. 5.

\[ f(t) \begin{cases} a_1 t, & \text{for } t_{01} \leq t \leq t_{12},\\ a_2 \left[t - t^{(2)}\right]^{1/2}, & \text{for } t_{12} \leq t \leq t_{23},\\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots\\ a_j \left[t - t^{(j)}\right]^{1/j}, & \text{for } t_{j-1,j} \leq t \leq t_{j,j+1}. \end{cases} \tag{11} \]

\(a_1,\ a_2,\ a_3,\ldots,\ a_j\) are certain constants characterizing the rate of accumulation of \(F\)-centers in the \(j\)-th segment of the kinetic curve; they are functions of the irradiation conditions and chemical composition;

\(t_{12},\ t_{23},\ldots,\ t_{j,j+1}\) are constants characterizing the moment of transition from one segment of the kinetic curve to another;

\(t^{(2)}, t^{(3)}, \ldots, t^{(j)}\) are the geometric vertices of the parabolas of the corresponding segments.

Proceeding from the initial conditions

\[ n_{01}=0;\quad n_{F\mathrm{II}}(t_{01})=0, \tag{12} \]

the conditions of continuity of \(f(t)\)

\[ a_j\,[t_{j,j+1}-t^{(j)}]^{1/j} = a_{j+1}\,[t_{j,j+1}-t^{(j+1)}]^{1/(j+1)}, \tag{13} \]

and the conditions of monotonicity of \(f(t)\)

\[ \frac{1}{j}a_j\,[t_{j,j+1}-t^{(j)}]^{1/j-1} = \frac{1}{j+1}a_{j+1}\,[t_{j,j+1}-t^{(j+1)}]^{1/(j+1)-1}, \tag{14} \]

we obtain

\[ t^{(j)}=\frac{1}{j}\sum_{n=2}^{j}t_{n-1,n}, \tag{15} \]

\[ t_{j,j+1} = \frac{j^j}{(j+1)^j}\, \frac{a_{j+1}^{j(j+1)}}{a_j^{j(j+1)}} + \frac{1}{j}\sum_{n=2}^{j}t_{n-1,n}. \tag{16} \]

In the general case \(a_1, a_2, a_3,\ldots,a_j\) must be replaced by \(a_1^*, a_2^*, a_3^*,\ldots,a_j^*\), which take into account the radiation bleaching of color centers and reverse transformations of the type \(R\to M\), \(M\to F\).

It follows from the form of the function \(f(t)\) that the concentration of \(F\)-centers must tend toward a certain limit, i.e., become saturated.

In order to verify experimentally the form of the kinetic curve, it is necessary to follow the accumulation of \(F\)-centers up to concentrations above \(10^{19}\ \mathrm{cm}^{-3}\), that is, to measure \(F\)-center concentrations an order of magnitude higher than those measured by the optical method.

We made use of the fact that the penetration depth of protons into alkali-halide single crystals, which determines the thickness of the colored layer, depends on the proton energy, and reduced the proton energy by means of an aluminum filter so that the thickness of the colored layer became \(30\ \mathrm{m}\mu\). This made it possible to measure the absorption coefficient at the maximum of the \(F\)-band up to \(2300\ \mathrm{cm}^{-1}\).

The kinetic curves for the accumulation of \(F\)- and \(M\)-centers are presented in Fig. 3. Saturation was observed at approximately \(2.1\cdot10^{19}\ \mathrm{cm}^{-3}\), followed by a decrease in the concentration of \(F\)-centers. The form of the experimental curve is satisfactorily described by function (8) up to the maximum.

The decrease in the concentration of \(F\)-centers with increasing concentration of \(M\)-centers indicates that

1) \(\alpha_1\) in expression (7) is a function of the dose or irradiation time;

2) with increasing irradiation dose, an irreversible process occurs that shifts the equilibrium among the \(F_j\)-centers toward centers with larger \(j\).

As the analysis in [1] showed, such an irreversible process may be the process of displacement of the maximum in the distribution of stored radiation energy over localization multiplicities toward larger multiplicities.

When two or more elementary volumes of energy localization coincide, the probability of close arrangement of \(F\)-centers sharply increases and, consequently, so does their coagulation into complex centers.

If one assumes that the probability of formation of an \(F\)-center is proportional to the probability of single localization of energy [1]

\(J\vartheta \exp(-J\vartheta)\), and the probability of formation of \(M\)-centers is the probability of double localization of energy \(\frac{1}{2}(J\vartheta)^2 \exp(-J\vartheta)\), then the efficiency of

\(F \to M\), expressed by the coefficient \(a_1\) from (7), is proportional to

\[ a_1 \sim \exp(J\vartheta). \tag{17} \]

Expression (17) characterizes the dependence of \(a_1\) on dose. At small doses \(J\vartheta \ll 1\), \(a_1 \approx \mathrm{const}\) (where \(J\) is the number of absorbed protons per unit volume), which was observed both in [7, 8, 9] and in the present work (Fig. 3).

If it is assumed that the probability of formation of \(R\)-centers is proportional to the probability of triple localization of energy, \(\frac{1}{6}(J\vartheta)^3 \exp(-J\vartheta)\), then the efficiency of the reaction \(M \to R\), characterized by \(a_2\) from (8), is proportional to

\[ a_2 \sim \exp(J\vartheta/2). \tag{18} \]

The efficiency of coagulation of \(F\)-centers into complex centers increases with increasing dose, and this irreversible process may lead to a decrease in the concentration of \(F\)-centers, and then of \(M\)- and \(R\)-centers, after the maximum has been passed.

The observed course of the kinetic curve for the accumulation of \(F\)-centers can be described by the following equation:

\[ n_F = n^*[1-\exp(-b^*t)]-(a^*/c^*-n')[1-\exp(-c^*t)] + \]

\[ +\exp(-t/\tau)\left(a_1^*t+\frac{a_2^*t^2}{2}+\frac{a_3^*t^3}{6}+\frac{a_4^*t^4}{24}\right). \tag{19} \]

Fig. 6. Kinetic curves of accumulation of \(F\)- and \(M\)-centers in LiF single crystals under irradiation by a proton beam of different intensity \((J)\).

The characteristic time \(\tau\) is related to the chemical composition through the elementary volume of energy localization \(\vartheta\) [1]:

\[ \tau=\frac{k}{j_0^{\vartheta}}, \tag{20} \]

where \(z\) is the penetration depth of protons (in cm), \(j_0\) is the intensity of the proton beam in

\[ \left(\frac{\text{protons}}{\text{cm}^2\,\text{s}}\right). \]

The constants \(a_2^{*}, a_3^{*}, a_4^{*}\) take into account the efficiency of the transformations \(M, R, N \to F;\ N, R \to M;\ N \to R\) and the reverse transformations \(M, R, N \to F;\ N, R \to M,\ N \to R\). Additional experiments are necessary to determine \(a_2^{*}, a_3^{*}, a_4^{*}\).

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