PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Vol. 138　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1965

DISTRIBUTION LAW IN A SOLID OF ABSORBED AND STORED RADIATION ENERGY BY LOCALIZATION MULTIPLICITIES

D. I. VAISBURD

(Presented by the seminar of the Problem Laboratory of Electronics,
Dielectrics, and Semiconductors)

The properties of \(F\)-centers formed from radiation vacancies¹, and a number of phenomena associated with the accumulation of \(F\)-centers and their coagulation into complex electron color centers \((M-, R-, N-)\), fit well into a model according to which \(F\)-centers formed from radiation vacancies are distributed nonuniformly in the volume of a crystal, and their local concentration exceeds the average by one or two orders of magnitude. The concept of high local concentrations of \(F\)-centers formed from radiation vacancies makes it possible to explain:

1) the identical properties of \(F\)-centers from radiation vacancies and from vacancies produced by deformation with respect to photostability and coagulation into \(M\)-centers [1];

2) the coagulation of \(F\)-centers into \(M\)-centers (\(F_2\)-centers) and appreciable concentrations of \(M\)-centers already at \(F\)-center concentrations of \(10^{17} \div 10^{18}\ \mathrm{cm}^{-3}\), when the distance between \(F\)-centers, assuming their uniform distribution throughout the volume, is 20–60 lattice parameters [2];

3) the increase in the photo- and thermal stability of \(F\)-centers with increasing concentration [1, 5];

4) the narrowing of the electron spin resonance line of \(F\)-centers as their concentration increases, which may be associated with the appearance of exchange interaction between the centers [10, 11].

In analyzing the mechanism that could lead to the appearance of high local concentrations of \(F\)-centers, we encounter the following contradiction.

The Seitz mechanism and its modifications [3, 4, 1], according to which the formation of vacancies under the action of radiation occurs at dislocation jogs, can explain high local concentrations of \(F\)-centers, but is not consistent with the overwhelming majority of other facts. Therefore the mechanism of dislocation jogs cannot be the predominant one, much less the only one.

The mechanism of vacancy formation in an ideal lattice as a result of multiple ionization of halogen ions, based on Varley’s idea [5], is continually being improved [6, 7, 8, 9] and at present agrees—

agrees with the majority of experimental facts, but does not explain high local concentrations of \(F\)-centers.

In the present work it is assumed that the possibility of the appearance of high local concentrations of color centers, free radicals, etc., is inherent in the very distribution, in a solid, of absorbed and stored radiation energy.

Because radiation energy is transported by particles and quanta, the increase in the absorbed and stored radiation energy in a solid is not a continuous process, but a quantum, stepwise process. With each absorbed quantum or particle, the absorbed energy increases by the amount of the energy of the quantum or particle.

For most types of radiation (the exceptions are high-energy electrons, \(\gamma\)-quanta with energies greater than \(2.5 \div 10\) MeV, and cosmic particles), the energy of the quantum or particle absorbed by the solid is localized, i.e., the volume in which practically all the energy of the absorbed quantum or particle is distributed is much smaller than the volume of the target.

The energy of heavy charged particles (protons, deuterons, tritons, recoil nuclei, energetic atoms and ions) is localized in the volume immediately adjacent to the axis of the particle track, and is stored in the form of defects (vacancies, displaced atoms, color centers, free radicals).

It is assumed that the concentration of defects decreases according to a Gaussian law with distance from the track axis. The half-width of the stored-energy distribution curve may be taken as the track diameter.

The volume in which practically all the energy of one absorbed particle or quantum is distributed will be called the elementary volume of energy localization. For heavy charged particles the elementary volume of energy localization can be estimated as the volume of a cylinder with base diameter equal to the track diameter and height equal to the range of the particle in the substance.

The elementary volume of energy localization depends on the type of radiation and on the chemical composition of the irradiated substance. If the irradiated solid has absorbed \(n\) quanta or particles, then there exist probabilities of spatial coincidence in the irradiated solid of the elementary volumes of energy localization of several quanta or particles.

The phenomenon of spatial coincidence in a solid of several elementary volumes of energy localization (in the more general case, the phenomenon of spatial overlap of elementary volumes of energy localization) will be called the phenomenon of multiple localization of radiation energy.

The number of coincident elementary volumes of energy localization will be called the localization multiplicity \(r\). If \(n\) quanta or particles have been absorbed, then there exist probabilities of \(r\)-fold localization of energy, and \(r\) can take the values \(0, 1, 2, \ldots, n\).

In an irradiated solid, in some part of the volume the radiation energy was not localized; some part of the volume underwent single localization of energy, some part—double, some part—triple, and so on.

Problem: To find the probability distribution of the irradiated volume with respect to the multiplicities of energy localization.

As the conditions of the problem, we shall choose the following.

Type of radiation: parallel, uniform in intensity.

a monochromatic beam of heavy charged particles, in particular protons with energies on the order of ten MeV. As the target, take a solid body in the form of a right cylinder with volume equal to unity and height equal to the depth of penetration of the particles into the given substance. The irradiation is parallel to the axis and perpendicular to the base of the cylinder.

Let us solve the stated problem for the case of heavy charged particles, and then show its solution for the case of quantum radiation.

For the conditions indicated above we have been able to find an exact solution, which we present in the form of a theorem.

Theorem on the distribution in a solid of absorbed stored radiation energy by localization multiplicities

If, upon irradiation of a solid, it turns out that:

1) the elementary volume of energy localization is equal to $\vartheta$;

2) at the same time, no more than one particle is incident on the irradiated surface of the solid, perpendicular to the beam;

3) for each particle, the probability of localization of its energy in some part of the specimen is proportional to the volume of this part,

then the probabilistic decomposition of the irradiated volume of the solid according to multiplicities of energy localization is the expansion of an expression of the form $[(1-\vartheta)+\vartheta]^j$ in powers of $j$, that is

\[ B_{j,r}=\binom{r}{j}(1-\vartheta)^{j-r}\vartheta^r, \tag{1} \]

where $B_{j,r}$ is the probability of $r$-fold localization; $j$ is the absorbed dose, expressed as the number of particles.

Before proceeding to the proof, let us dwell on the foundations of the theorem.

Condition (1) asserts that the theorem is valid only for the case when the radiation energy is localized, which is satisfied for most radiations whose action on solids has been investigated.

Condition (2) is a requirement of finite radiation intensity and is practically always fulfilled.

Condition (3) asserts that a particle can, with equal probability, undergo a collision at any point of the irradiated volume, which is generally true if the beam is parallel, homogeneous in intensity, and the mutual repulsion of particles in the beam may be neglected. According to condition (3), it is convenient to choose the target volume equal to unity. Then the magnitudes of volumes and probabilities coincide.

Proof. We shall carry out the proof by the method of complete mathematical induction.

Let us verify that the indicated decomposition is valid for the cases $j=0,1,2,3,4$.

Having verified that the theorem is valid for $j=0,1,2,3,4$, suppose that the theorem is valid for $j=n$. Then, for $j=n$, the probability of $r$-fold localization is

\[ B_{n,r}=\binom{r}{n}(1-\vartheta)^{n-r}\vartheta^r, \]

${}^{2)}\ \binom{r}{j}$ is the notation for binomial coefficients accepted in the literature [13]:

\[ \binom{r}{j}=\frac{j(j-1)\cdots(j-r+1)}{r!}. \]

the probability of $(r-1)$-fold localization

\[ B_{n,r-1}= \begin{pmatrix} r-1\\ n \end{pmatrix} (1-\vartheta)^{\,n-r+1}\vartheta^{\,r-1}. \]

Let now $j=n+1$. Then the probability of $r$-fold localization for $j=n+1$ is:

\[ B_{n+1,r}=B_{n,r}+B_{n,r-1}\vartheta-B_{n,r}\vartheta, \tag{2} \]

\[ B_{n+1,r}= \frac{n(n-1)\ldots(n-r+1)}{r!} (1-\vartheta)^{\,n-r}\vartheta^r+ \]

\[ +\frac{n(n-1)\ldots(n-r+2)}{(r-1)!} (1-\vartheta)^{\,n-r+1}\vartheta^r- \]

\[ -\frac{n(n-1)\ldots(n-r+1)}{r!} (1-\vartheta)^{\,n-r}\vartheta^{\,r+1}= \]

\[ =\frac{n(n-1)\ldots(n-r+1)}{r!} (1-\vartheta)^{\,n-r+1}\vartheta^r+ \]

\[ +\frac{n(n-1)\ldots(n-r+2)}{(r-1)!} (1-\vartheta)^{\,n-r+1}\vartheta^r= \]

\[ =\frac{(n+1)n(n-1)\ldots(n-r+2)}{r!} (1-\vartheta)^{\,n+1-r}\vartheta^r= \]

\[ = \begin{pmatrix} r\\ n+1 \end{pmatrix} (1-\vartheta)^{\,n+1-r}\vartheta^r, \]

which was required to be proved.

According to condition (3) of the theorem, in the case of quantum radiation it is necessary to take into account the loss in the number of quanta as the radiation passes through the substance.

Then the distribution of the absorbed and stored energy of monochromatic radiation by localization multiplicities depends on the target depth coordinate \(x\) and the attenuation coefficient \(\mu\), and, for a plane separated from the irradiated surface by a distance \(x\), takes the form

\[ [(1-\vartheta)+\vartheta]^{\mu j_0 e^{-\mu x}}, \tag{3} \]

where \(j_0\) is the number of quanta incident on the target.

Expression (2) gives the dependence of the probability of \(r\)-fold localization on the absorbed dose \(j\), expressed as the number of particles. Taking into account that \(\vartheta \ll 1\), \(j \gg 1\) (for example, for the TPI cyclotron \(j \simeq 10^{10}\div 10^{16}\) protons), for the cases \(r \ll j\), expression (2) can be simplified:

\[ B_{j,r}= \binom{r}{j}(1-\vartheta)^{j-r}\vartheta^r = \frac{(j\vartheta)^r}{r!}(1-\vartheta)^{j-r} = \frac{(j\vartheta)^r}{r!}e^{-j\vartheta}. \]

Thus,

\[ B_{j,r}=\frac{(j\vartheta)^r}{r!}e^{-j\vartheta} \quad \text{for } j \gg r. \tag{4} \]

The probabilities of \(r\)-fold localization of energy calculated according to (4) are given in Table 1. In Figs. 1 and 2, on the basis of the data in Table 1, the dependences of the probabilities of \(r\)-fold localization (for \(r=0\div 21\)) on dose (for \(j\vartheta=0\div 10\)) and the distribution of the irradiated volume by multiplicities of energy localization for doses \(j\vartheta=1\div 10\) are plotted.

Fig. 1. Dependence of the probability of \(r\)-fold localization of energy on dose in \((j\vartheta)\).

As is seen from Fig. 1, \(B_{j,r}=f(j\vartheta)\) is represented by a curve with a maximum.

Corollary 1. The maximum of the probability of \(r\)-fold localization occurs for the dose \(j\vartheta = r\). Indeed,

\[ \frac{\partial}{\partial (j\vartheta)} B_{j,r} = \frac{r (j\vartheta)^{r-1}}{r!} e^{-j\vartheta} - \frac{(j\vartheta)^r}{r!} e^{-j\vartheta} = \]

\[ = \frac{(j\vartheta)^{r-1}}{r!} e^{-j\vartheta} (r - j\vartheta) = 0; \tag{5} \]

\[ j\vartheta = r. \tag{6} \]

Fig. 2. Probability distribution of the irradiated volume by multiplicities of energy localization for different doses \((j\vartheta)\).

Corollary 2. The probability of \(r\)-fold localization has the maximum rate of change at \(j\vartheta = r \pm \sqrt{r}\):

\[ \frac{\partial^2}{\partial (j\vartheta)^2} B_{j,r} = \frac{(r-1)(j\vartheta)^{r-2}}{(r-1)!} e^{-j\vartheta} - \frac{(j\vartheta)^{r-1}}{(r-1)!} e^{-j\vartheta} - \]

\[ - \frac{r (j\vartheta)^{r-1}}{r!} e^{-j\vartheta} + \frac{(j\vartheta)^r}{r!} e^{-j\vartheta} = \frac{(j\vartheta)^{r-2}}{r!} e^{-j\vartheta} \left[ r^2 - r - 2r j\vartheta + (j\vartheta)^2 \right] = 0; \]

\[ r^2 - r - 2r j\vartheta + (j\vartheta)^2 = 0; \]

\[ j\vartheta = r \pm \sqrt{r}. \tag{7} \]

Corollary 3. If \(B_r\) is the maximum probability of \(r\)-fold localization, and \(B_{r+1}\) is the maximum probability of \((r+1)\)-fold localization, then

\[ \frac{B_r}{B_{r+1}} = \left( \frac{r}{r+1} \right)^r \cdot e. \]

According to Corollary 1,

\[ B_r=\frac{r^r}{r!}e^{-r}=\frac{r^{r-1}}{(r-1)!}\cdot e^{-r}, \tag{8} \]

\[ \frac{B_r}{B_{r+1}}=\left(\frac{r}{r+1}\right)^r\cdot e. \tag{9} \]

As is seen from Fig. 2, the dependence

\[ B_{j,r}=f(r)\quad \text{for } j\vartheta=\text{const} \]

is represented by a curve with a maximum; moreover, as the dose increases, the maximum of the distribution shifts at a constant rate toward higher multiplicities, while the height of the maximum decreases according to law (9).

The regularities obtained make it possible to explain the nonuniform distribution of radiation defects and to predict how the nonuniformity of the distribution changes with increasing dose for crystals of different chemical composition when they are irradiated by different types of radiation. The application of the relations derived in the present work to the analysis of experimental data is illustrated in [12].

I thank Professor A. A. Vorob’ev, Doctor of Physical and Mathematical Sciences, and Docent I. Ya. Melik-Gaikazyan for discussing the results obtained.

References

1. P. V. Mitchell, D. A. Wiegand and R. Smoluchowski. Phys. Rev. 121, No. 2, 484 (1961).
2. B. J. Faraday, H. Rabin and W. D. Compton. Phys. Rev. Letters 7, No. 2, 57 (1961).
3. F. Seitz. Revs Mod. Phys 18, 384 (1946); Phys. Rev. 80, 239 (1950).
4. J. J. Markham. Phys. Rev. 88, 500 (1952).
5. J. H. O. Varley. Nature 174, 886 (1954), J. Nuclear Energy 1, 130 (1954).
6. C. C. Klick. Phys. Rev. 120, 760 (1960).
7. R. E. Howard, S. Vosko and R. Smolucholski. Phys. Rev. 122, 1406 (1961).
8. F. E. Williams. Phys. Rev. 126, 70 (1962).
9. J. Sharma and R. Smoluchowski. Bull. Am. Phys. Soc. 7, 178 (1962).
10. A. M. Portis. Phys. Rev. 91, 1071 (1953).
11. N. W. Lord. Phys. Rev. 105, 756 (1957).
12. D. I. Vaisburd, I. Ya. Melik-Gaikazyan. See the present collection, p. 3.
13. J. Riordan. Introduction to Combinatorial Analysis, IL, 1963.

1. By radiation vacancies we mean vacancies generated in a crystal by radiation. ↩