DISPLACEMENT OF ATOMS IN SOLIDS UNDER THE ACTION OF RADIATION

G. H. Kinchin and R. S. Pease*)

CONTENTS

I. Formation of Displaced Atoms

1. Energetic considerations . . . . . . . . . . . . . . . . . . . . . . 591
   a) Displacement energy . . . . . . . . . . . . . . . . . . . . . . 591
   b) Threshold values of radiation energy . . . . . . . . . . . . . . . 591
2. Number of displaced atoms . . . . . . . . . . . . . . . . . . . . . 592
   a) Moving atoms. Fundamentals of the theory . . . . . . . . . . . . 593
   b) Moving atoms. Collisions of hard spheres . . . . . . . . . . . . 595
   c) Moving atoms. Rutherford collisions . . . . . . . . . . . . . . . 597
   d) Moving atoms slowing down to a complete stop . . . . . . . . . . 599
   e) Fast neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . 600
   f) Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
   g) Gamma rays . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
3. Distribution of displaced atoms and vacancies . . . . . . . . . . . 604
   a) Models based on consideration of collisions . . . . . . . . . . . 604
   b) “Displacement spikes” and “thermal spikes” . . . . . . . . . . . 605
4. Effects in compounds . . . . . . . . . . . . . . . . . . . . . . . 607
   a) Collisions leading to substitution of atoms . . . . . . . . . . . 607
   b) Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 608

II. Recovery of Defects

1. Recombination of interstitial atoms and vacancies . . . . . . . . . 609
   a) Pairs of interstitial atoms and vacancies located close to one another . . . . . . . . . . . . . . . . . . . . . 609
   b) Random distribution of interstitial atoms and vacancies . . . . . 610
2. Extended processes . . . . . . . . . . . . . . . . . . . . . . . . 611
3. Accumulation of disturbances . . . . . . . . . . . . . . . . . . . 611
4. Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613
5. Radiation annealing . . . . . . . . . . . . . . . . . . . . . . . 614
   References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615

*) §§ 2 and 3 from the review by G. H. Kinchin, R. S. Pease, Reports on Progress in Physics 18, 1 (1955). Translated from English by A. Kh. Breger, under the editorship of Prof. G. S. Zhdanov. The article is closely connected with Glen’s review. See UFN, LX, issue 3, p. 445.

I. FORMATION OF DISPLACED ATOMS

1. Energy considerations

a) Displacement energy.

From heats of sublimation and other thermodynamic data it is known that the binding energy of atoms in solids is close to 5 eV. One should expect that more energy is required to displace an atom from its position in the lattice inside a solid. First, when an atom is displaced in a solid, a larger number of bonds must be broken than when an atom is evaporated from the surface; second, between the position of an atom at a lattice site and a stable position in an interstice there must exist some potential barrier. The results of detailed calculations for copper were published by Huntington [Г-99в]*); he found that the required energy depends on the direction (relative to the crystal axes) in which the atom is displaced, and ranges from 18 to 43 eV.

Experimental data obtained from threshold energies for radiation effects lead to the following values: 31 eV for germanium [Г-120а], 25 eV for copper [Г-64], less than 12 eV for copper atoms in Cu₃Au [Г-616], and greater than (or equal to) 25 eV for iron atoms in an iron–copper alloy [Г-53а]**). Threshold energies for sputtering, on the one hand, are not very reliable for use in solving questions of radiation damage; on the other hand, experimental determination of these quantities is rather difficult. However, these values correctly convey the order of magnitude. Recently Verner¹ found that for 23 metals the displacement energies range from 20 eV for Pb to 180 eV for Hf.

Since the displacement energy varies depending on the recoil direction, and also must be different for different substances and cannot be easily calculated, the simplest procedure is, following Seitz [Г-196а], to adopt for all cases a single value of 25 eV and to vary it only in those cases where this follows directly from experimental data. This method has been used in the works published up to now.

b) Threshold values of the radiation energy.

If, for further discussion, the displacement energy is taken to be 25 eV, then one can easily calculate the minimum energy required for the formation of displaced atoms by various types of

*) In those cases where a reference to the literature in this article coincides with a reference in Glen’s review, we give the number according to the bibliography printed after the translation of Glen’s article (UFN, LX, issue 3, p. 556 ff.) with the letter “Г” preceding it. The remaining literature references in the supplement have the usual numbering in accordance with the list given at the end of the review by Kinchin and Pease. — Transl. note.

**) See Table 3 in the translation of Glen’s review. — Transl. note.

Table I

Threshold radiation energy required to displace an atom
at \(E_d = 25\) eV

radiation. The results of such a calculation are given in Table I. In the case of heavy particles the following expression was used for the maximum energy \(E_{\max}\) transferred by a moving particle of mass \(M_1\) and energy \(E\) to a stationary atom of mass \(M_2\):

\[ E_{\max}=\lambda E, \tag{1} \]

where

\[ \lambda=\frac{4M_1M_2}{(M_1+M_2)^2}. \tag{1a} \]

For electron bombardment relativistic effects are significant; in view of this we used the expression

\[ E_{\max}=\frac{2E(E+2mc^2)}{M_2c^2}, \tag{2} \]

where \(m\) is the electron mass, \(c\) is the speed of light. In the case of \(\gamma\)-rays the threshold energy is determined by the energy of photo- or Compton electrons and therefore is approximately equal to the threshold energy for electrons. Direct interaction of \(\gamma\)-rays with the nucleus is significant in rare cases, with the exception of nuclear recoil in the emission of \(\gamma\)-quanta, as in the Szilard–Chalmers process (see \({}^2\)).

2. Number of displaced atoms

Atoms displaced in direct collisions with the bombarding particles will be called “primary knocked-on atoms,” in contrast to secondary atoms (i.e., atoms displaced as a result of collisions with primary ones). In §§ 2, a)–c) we consider collisions of moving atoms in general and obtain expressions both for the number of displaced atoms formed by primary atoms and for the cross sections for the formation of primary knocked-on atoms by bombarding ions. We apply these expressions in calculating the total number of displaced atoms for ne-

which cases, and in the following paragraphs for cases of bombardment by neutrons, electrons, and \(\gamma\)-rays.

At present there are no accurate experimental data for determining the number of displaced atoms, but some rough estimates make it possible to compare the calculated and observed quantities given in Table II. From these data it is seen that the calculated values are usually higher than the experimental ones, which is probably explained by the presence of recombination. In each case the experimental estimate is based on certain assumptions connecting the observed effects with the number of displaced atoms. These assumptions are briefly given in Table II.

Table II

Relation of the calculated and observed numbers of displaced atoms

) In this review we have retained the term “Frenkel defects,” used by the authors. In Glen’s review this defect is called: a “vacancy—interstitial atom” pair. — Translator’s note.*

a) Moving atoms. Fundamentals of the theory. Calculations for moving atoms were made by Seitz \(\Gamma\)-196a, Ozeroff \(\Gamma\)-166, and Brinkman \(\Gamma\)-276. Although most of Seitz’s results are correct, his calculation is based on the use of the Born approximation, which is inapplicable for considering the collisions that occur in this case. The limits of applicability of various approximations have been examined by Bohr \(^{3}\) and Williams \(^{4}\), and for all collisions of interest to us it is necessary to use the classical

approximation. Ozerov mainly considers the case of displacement of atoms by fission products. Brinkman, who corrected an error in Ozerov’s work, considers the case of identical moving and stationary atoms, but does not explicitly calculate the number of displaced atoms. Our theoretical treatment is based on the works of Knipp and Teller \(^{5}\) and Bohr \(^{3}\).

The basic approximation is that the atoms are displaced as a result of the Coulomb interaction between nuclei and that the effect of the screening electrons consists in cutting off this interaction at a distance close to \(a_0/Z^{1/3}\), where \(a_0\) is the Bohr radius and \(Z\) the atomic number. When the kinetic energy of the moving atom is less than the potential of the Coulomb interaction of two nuclei separated by a distance equal to the radius of the screening electron cloud, the presence of this screen leads to a collision of the hard-sphere type. The upper energy limit \(L_A\) for the applicability of this approximation is determined (see \(^{3}\)) by the equation

\[ L_A = 2 E_R Z_1 Z_2 \frac{(Z_1^{2/3}+Z_2^{2/3})^{1/2}(M_1+M_2)}{M_2}, \tag{3} \]

where \(E_R\) is the Rydberg constant \((13.60\ \text{eV})\); \(Z\) are the atomic numbers; \(M\) are the atomic masses. The subscripts 1 and 2 refer to the moving and stationary atoms. At energies exceeding \(L_A\), the presence of screening electron clouds excludes Rutherford collisions with impact parameters

\[ \frac{a_0}{(Z_1^{2/3}+Z_2^{2/3})^{1/2}} . \]

In this case the minimum energy \(E^*\) that can be transferred in a collision is determined by the equation

\[ E^* = 4 E_R^2 Z_1^2 Z_2^2 (Z_1^{2/3}+Z_2^{2/3})\, \frac{M_1}{M_2 E_d}. \tag{4} \]

As long as \(E^*\) is greater than the displacement energy \(E_d\), all Rutherford collisions lead to displacement of atoms; but when the energy \(E\) of the moving atoms exceeds \(L_B\), where

\[ L_B = 4 E_R^2 Z_1^2 Z_2^2 (Z_1^{2/3}+Z_2^{2/3})\, \frac{M_1}{M_2 E_d}, \tag{5} \]

displacements occur only in some of the Rutherford collisions. It is easy to show that when \(E \gg L_B\), one half of the energy lost in Rutherford collisions is dissipated in lattice vibrations without displacement of atoms.

Another important type of interaction is electronic excitation. Following Seitz \(^{\Gamma-196a}\), we assume that for \(E > L_C\)

DISPLACEMENT OF ATOMS IN SOLIDS UNDER THE ACTION OF RADIATION

the expenditure of energy on this process exceeds by many (\(\sim 10^3\)) times the energy losses to all other types of interactions, and for \(E < L_C\) electronic excitation does not occur at all. The magnitude \(L_C\) is determined by the lowest excitation potential \(I_t\) of the valence electrons and is roughly described (for insulators) by the equation

\[ L_C=\frac{M_1}{m}\,\frac{I_t}{8}, \tag{6a} \]

where \(I_t\) is a quite definite quantity corresponding to the edge of the first main optical absorption band.

In metals, where the valence electrons are free, energy losses for electron excitation occur at all energies. Ozerov showed that at sufficiently low energies this energy expenditure may be neglected in comparison with energy losses in collisions. A satisfactory criterion for the energy above which electronic losses predominate has not yet been found; the expressions obtained by Seitz are unsuitable because, in deriving them, he used the Born approximation. It is evident, however, that since the velocity of the moving atom is considerably less than the velocity of an electron at the maximum of the Fermi distribution, only a small fraction of the electrons can be excited, and we adopt (by rough analogy with insulators):

\[ L_C=\frac{1}{16}\,\frac{M_1}{m}\,\varepsilon_0 \simeq \frac{1}{16}\,\frac{M_1}{m}\,\pi a_0^2 E_R\,(3N_0\sqrt{\pi})^{2/3}, \tag{6b} \]

where \(\varepsilon_0\) is the energy of the free electrons according to Fermi;

\(N_0\) is the number of atoms per unit volume.

In the case of heavy moving atoms the assumption of an abrupt cutoff of ionization losses is very crude, owing to large changes in the effective charge of the moving atoms. Knipp and Teller\(^5\) showed that ionization losses and collision losses have approximately the same magnitude over a rather wide energy interval. Our values of \(L_C\) denote the energy limits below which ionization losses may be neglected.

Table III gives some values of \(L_A\), \(L_B\), and \(L_C\).

b) Moving atoms. Collisions of hard spheres. A characteristic feature of collisions of hard spheres is that the probability of transfer of a given amount of energy \(\Delta E\) does not depend on the magnitude of \(\Delta E\) for \(0 \leq \Delta E \leq \lambda E\) and is equal to zero outside this interval. Let us consider a monatomic solid in which a primary knocked-on atom with energy \(E\) can be produced. In the first collision of this atom its energy is distributed between two atoms, and as a result of the second series of collisions the energy is distributed among four atoms. In a similar way, after the \(p\)-th series of collisions this energy is distri-

Table III

Energy limits for collision processes

| Substance | Moving atom | \multicolumn{3}{c}{Energy limits (eV)} |
|---|---|---:|---:|---:|
| | | $L_A$ | $L_B$ | $L_C$ |
| Diamond | Carbon | $5\cdot 10^3$ | $3\cdot 10^5$ | $1.5\cdot 10^4$ (a)*) |
| Diamond | Deuteron | $4\cdot 10^2$ | $8\cdot 10^2$ | $2\cdot 10^3$ (a) |
| Al | Al | $3\cdot 10^4$ | $9\cdot 10^6$ | $1.7\cdot 10^4$ |
| Al | Deuteron | $1\cdot 10^3$ | $2\cdot 10^3$ | $1.3\cdot 10^3$ |
| KCl | Argon | $6\cdot 10^4$ | $4\cdot 10^7$ | $7\cdot 10^4$ (b) |
| KCl | Deuteron | $1.4\cdot 10^3$ | $4\cdot 10^3$ | $3\cdot 10^3$ (b) |
| Cu | Cu | $2\cdot 10^5$ | $4\cdot 10^8$ | $5\cdot 10^4$ |
| Cu | Deuteron | $3\cdot 10^3$ | $8\cdot 10^3$ | $1.6\cdot 10^3$ |
| Au | Au | $2\cdot 10^6$ | $4\cdot 10^{10}$ | $1.2\cdot 10^5$ |
| Au | Deuteron | $1\cdot 10^4$ | $4\cdot 10^4$ | $1.2\cdot 10^3$ |

*) Values of the first excitation level of valence electrons are taken: (a) from Mott’s book⁶ and (b) from the work of Schneider and O’Bryan⁷.

is divided among $2^p$ atoms, and the number of atoms in the energy interval $dE$ will be

\[ N_p(E)\,dE=2^p\left(\ln\frac{\overline E}{E}\right)^{p-1}\frac{dE}{E(p-1)!}. \tag{7} \]

This result is applicable to free immobile atoms, but only those atoms which receive an amount of energy exceeding the threshold energy $E_d$ are displaced. Further, only an atom possessing energy greater than $2E_d$ can carry out such a collision with an immobile atom after which the energy of both atoms will exceed $E_d$. Thus, we assume that atoms possessing energy from $E_d$ to $2E_d$ are displaced, but cannot themselves lead to a further increase in the total number of displaced atoms. It can be shown that the number of such atoms formed in the $p$-th series of collisions is equal to $E_d N_p(2E_d)$, so that the total number of displaced atoms $N_d$ is determined by the equation

\[ N_d=\sum_{p=1}^{p=\infty} E_d N_p(2E_d)=\frac{\overline E}{2E_d}. \tag{8a} \]

This expression is valid only for $\overline E>2E_d$. It is obvious that for $0\leq \overline E\leq 2E_d$

\[ N_d=1. \tag{8b} \]

Since equation (8a) is linear with respect to $\overline E$, it is applicable

also in the case when \(\bar E\) represents the average energy of a group of primary atoms with energies exceeding \(2E_d\). It is easy to show that this equation is also applicable when the masses of the initial moving and stationary atoms are different—provided that

\[ 2E_d/\bar E \ll \lambda . \]

A simple argument, due to Fry (T. M. Fry), also leads to equation (8). The number of displaced atoms can increase only so long as the energy of the moving atom colliding with a stationary atom exceeds \(2E_d\). In such collisions atoms arise with energies in the interval \(0—2E_d\), with a uniform energy distribution and mean energy \(E_d\). Consequently, the number of displaced atoms is equal to \(\bar E/2E_d\). The physical assumptions made in deriving equation (8a) are such that this equation probably gives an overestimated number of displaced atoms.

Seitz asserts that the equation

\[ N_d=\left(\frac{E'}{E_d}\right)^{1/2}, \tag{9} \]

where \(E'\) is the average energy of all primary displaced atoms, is a good approximation in those cases where \(E'\) is several times greater than \(E_d\). Comparison with equations (8a) and (8b) shows that agreement may occur up to \(\bar E\sim 10E_d\). Ozerov gives an approximate solution based on the Thomas–Fermi model, in which the electrostatic interaction of the shielding electron clouds is taken into account. His results, which cannot be represented in a simple analytical form, lead to values of \(N_d\) approximately \(30\%\) lower than those calculated from equation (8a). This calculation contains an error pointed out by Brinkman, as well as some not entirely justified assumptions; consequently, Ozerov’s results should be regarded only as a rough approximation to the exact equation (8a).

c) Moving atoms. Rutherford collisions. If \(E>L_B\), then, applying the usual theory of unscreened nuclei\(^3\), it is easy to calculate the cross section \(\sigma_p\) for a Rutherford collision in which an energy greater than \(E_d\) is transferred, namely:

\[ \sigma_p=4M_1 Z_1^2 Z_2^2 E_R^2\left(1-\frac{E_d}{\lambda E}\right) \frac{\pi a_0^2}{M_2 E_d E} \tag{10} \]

[for the notation, see § 2, a)]; \(\pi a_0^2\) has the value \(8.8\cdot 10^{-17}\ \text{cm}^2\). The factor \(\left(1-\frac{E_d}{\lambda E}\right)\) is very close to unity for all cases, pred-

constituents of interest, and may therefore be neglected. Thus, the cross section increases as the energy of the moving atom decreases. From equation (10) it follows that the number of primary displaced atoms varies inversely as the square of their energy, and their mean energy is determined by the equation

\[ \overline{E}=E_d \ln \frac{\lambda E}{E_d}. \tag{11} \]

As Seitz pointed out, the logarithmic term depends little on the exact values of \(\lambda\) and \(E\); therefore, for most cases encountered in practice the ratio \(\overline{E}/E_d\) is from 3 to 15. At such energies, to calculate the number of atoms displaced by primary atoms knocked out of their sites, one may use the results obtained for collisions of hard spheres. From equations (8a) and (8b), the mean total number of displaced atoms formed per primary knocked-out atom is determined by the equation

\[ N_d=\frac{1}{2}\left\{1+\ln \frac{\lambda E}{2E_d}\right\}. \tag{12} \]

This equation can be applied directly to such cases as the irradiation of thin foils in a cyclotron; for copper irradiated with deuterons of energy \(10\ \text{MeV}\), we obtain \(\sigma_p=6900\) barns; \(\overline{E}=2700\ \text{eV}\) and \(N_d=5.8\).

If \(L_A<E<L_B\), as in the case of fission products and energetic primary knocked-out atoms, then all Rutherford collisions lead to the formation of displaced atoms; in this case \(\sigma_p\) is approximately constant and equal to

\[ \frac{\pi a_0^2}{Z_1^{1/3}+Z_2^{1/3}}. \]

The mean energy of the primary knocked-out atom and the magnitude \(N_d\) can be estimated if, in equation (11), \(E_d\) is replaced by \(E^*\).

Both Brinkman and Ozerov improve our approximation, according to which there is an abrupt cessation of the nuclear interaction, by considering the transfer of energy between two atoms with a hard charge distribution, treated according to the Thomas–Fermi theory. Brinkman gives an equation for \(\sigma_p\) in the case \(Z_1=Z_2\), valid for all energies, which confirms the condition \(E>L_B\) for equation (10) and shows that, under this condition, one should expect a slow variation of \(\sigma_p\) with \(E\). Ozerov’s approximate results show that nuclear Rutherford collisions are the principal source of energy losses up to values at which an interaction of the hard-sphere type becomes dominant. Both of these investigators used an approximate form of the Thomas–Fermi potential and assumed that the radius of the screening electron cloud is equal to \(2.09a_0/Z_2^{1/3}\);

Knipp and Teller give the value \(0.85a_0/Z_2^{1/3}\) (under the assumption \(Z_2 > Z_1\)), while Bohr gives the quantity

\[ \frac{a_0}{(Z_1^{2/3}+Z_2^{2/3})^{1/2}} . \]

We have adopted the quantity given by Bohr.

d) Moving atoms slowing down to complete stopping. We have already shown that the total number of displaced atoms formed during the slowing down of an atom with energy \(E\) to complete stopping is equal to \(E/2E_d\), if \(E<L_A\). Under the condition \(E<L_C\) this result is also valid for \(L_A<E<L_B\), since the principal energy loss in this region is due to Rutherford collisions, which lead to the formation of primary displaced atoms (see also Г-206a). If \(E>L_B\), then only part of the energy is spent on these collisions, and the above result turns out to be incorrect; for the case \(E\gg L_B\), the total number of displaced atoms is approximately equal to \(E/4E_d\).

However, usually, when \(E>L_B\), then \(E>L_C\), and losses to electronic excitation must be taken into account. In the case of large energies, \(E\gg L_C\), \(E\gg L_B\), these losses are dominant and are determined by the equation

\[ -\frac{dE}{dx}=2\pi(qZ_1)^2 e^4 Z_2N_0\left(\frac{M_1}{mE}\right)\ln\frac{4Em}{M_1 I}, \tag{13} \]

where \(e\) is the charge of the electron; \(N_0\) is the number of atoms per unit volume; \(I\) is the mean ionization potential of the stationary atoms, and \(qZ_1\) is the effective charge of the moving atom. A detailed discussion of equation (13) and of the associated quantities \(I\) and \(q\) is given by Bethe and Ashkin\(^8\) and by Allison and Warshaw\(^9\).

Combining equations (13), (10), and (12), we find that if an atom loses energy \(\Delta E\), then it displaces \(\Delta N_t\) atoms, where

\[ \Delta N_t=P\frac{\Delta E}{E_d} \tag{14} \]

and

\[ P=mZ_2\frac{1+\ln\frac{\lambda E}{2E_d}}{4M_3q^2\ln\frac{4E_m}{M_1I}} . \tag{14a} \]

For \(q\simeq 1\), \(P\) is usually equal to \(10^{-3}\)—\(10^{-4}\).

For very light moving atoms (for example, for \(\alpha\)-particles arising as a result of radioactive decay), \(q\) and \(I\) are constant over a large range of energy values. In this case \(P\) depends so insignificantly on \(E\) that, without serious errors,

to use the mean value. In addition, the energy interval \(L_B - L_C\) is small, and equation (14) may be applied with a constant mean value \(P\) to all energies exceeding \(L_C\).

Thus, the total number \(N_t\) of atoms displaced by a very light moving atom of energy \(E_0\) (when it is slowed down to complete rest) is

\[ N_t=\frac{P(E_0-L_C)+bL_C}{E_d}, \tag{15} \]

where \(P\) is determined from equation (14a); \(L_C\) is determined by equation (6), and \(b=\dfrac{1}{2}\), if \(L_B>L_C\), or \(b=\dfrac{1}{4}\), if \(L_B\ll L_C\). Essentially the same result was obtained by Seitz. In those cases where the term \(bL_C\) predominates, a considerable uncertainty is introduced in the process of determining \(L_C\), which is almost purely qualitative.

The case of heavy moving atoms with \(E>L_C\) is more complicated, because the ionization losses are determined mainly by changes in the effective charge. In addition, \(L_B\) is usually so large that equation (10) (if it is applicable at all) becomes inapplicable over a large energy interval. This case has been considered in detail by Ozerov. In essence, equation (13) is replaced by a semiempirical expression, \(E_d\) is replaced by the quantity \(E^*\) from equation (11), and a number of averages are made in order to ease the mathematical difficulties that arise. Such a treatment is essential for the case of fragments. In the case of fast primary knock-on atoms, the initial energy rarely exceeds \(L_C\), and it often proves sufficient to assume that all the excess energy is spent on ionization losses, so that the number of displaced atoms is equal to \(L_C/2E_d\).

d) Fast neutrons. For radiation studies one usually uses neutrons produced in fission. Watt \(^{10}\) gives the energy spectrum of fission neutrons of \(U^{235}\) and \(Pu^{239}\); the spectrum has a maximum at \(0.7\) MeV and a mean energy of \(2\) MeV.

Neutron collisions are simple hard-sphere-type collisions, and therefore the recoil energy spectrum \(N(E)dE\) for atoms that have undergone collisions with neutrons is determined by the equation

\[ N(E)dE=\frac{dE}{E_{\max}}\quad (0\leq E\leq E_{\max}), \tag{16} \]

where \(E_{\max}\) is calculated from equation (1).

Only a negligible fraction \(E_d/E_{\max}\) of the atoms that have collided with neutrons are not displaced, so that \(\sigma_p\)—the cross section for the formation of per-

DISPLACEMENT OF ATOMS IN SOLIDS UNDER THE ACTION OF RADIATION

primary atoms—is the total cross section for neutrons. For heavy elements \(E_{\max}\) usually does not exceed the ionization limit \(L_C\).

Applying the results of the preceding paragraphs, one can calculate the total number \(N_d\) of displaced atoms, on the average per fast neutron:

\[ N_d=\frac{E_{\max}}{4E_d}. \tag{17} \]

For lighter elements \(E_{\max}\) usually exceeds \(L_C\), but, with the exception of very light elements, the primary knocked-out atoms expend all excess energy on ionization; in this case

\[ N_d=\frac{\left(2-\frac{L_C}{E_{\max}}\right)L_C}{4E_d}. \tag{18} \]

Figure 1 gives the values of \(N_d\) for neutrons with energy \(2\) MeV at different values of the parameter

\[ \frac{L_C m}{M_2}. \]

This parameter is a function that does not vary very sharply with \(M_2\) and for many substances is close to \(1\) eV. The form of these curves shows that neutrons with energy \(2\) MeV produce the maximum number of displacements in elements of intermediate and small atomic weight. Heavy nuclei receive less recoil energy; lighter nuclei receive a larger amount of energy, but expend a considerable part of this energy on ionization\({}^{11}\).

Fig. 1. Average number of displaced atoms per one primary knocked-out atom under bombardment by neutrons with energy \(2\) MeV, as a function of atomic weight.

Ozerov \(\Gamma\)-166 applied the more complicated expressions derived by him, given in § 2, c), to calculate the number of atoms displaced by fast neutrons; both Ozerov and Seitz give some values for the case of fast neutrons slowing down to complete stopping.

e) Electrons. Electrons displace atoms as a result of Coulomb interaction with nuclei. For this, the electrons must pass through the electronic \(K\)-shell and, consequently, one may use expressions derived on the basis of unscreened-

...of the Coulomb field. However, we cannot use equation (10), because the electrons usually have relativistic velocities. The basis of the calculation is Mott’s relativistic expression$^{12}$ for the differential scattering cross section. This expression cannot be brought to a completely explicit form. Dageid and Green$^{\Gamma-62}$, using the $\alpha^2$ approximation of Mackinley and Feshbach$^{13}$, obtain

\[ \sigma_p = \frac{4Z_2^2 E^2_R E_{\max}}{m^2 c^4 E_d} \,\frac{1-\beta^2}{\beta^4} \left[ 1+2\pi\alpha\beta \left( \frac{E_d}{E_{\max}} \right)^{1/2} - \frac{E_d}{E_{\max}} \left\{ 1+2\pi\alpha\beta+ (\beta^2+\pi\alpha\beta)\ln\left(\frac{E_{\max}}{E_d}\right) \right\} \right]\pi a_0^2, \tag{19} \]

where $E_{\max}$ is determined by equation (2); $\beta$ is the ratio of the electron velocity to the velocity of light; $\alpha=Z_2/137$. Mackinley and Feshbach consider that their $\alpha^2$ approximation is satisfactory for $Z\leq 27$ and $\beta\approx 1$; equation (19) for calculating $\sigma_p$ is probably adequate even outside the indicated interval. Klontz$^{\Gamma-119\text{a}}$ and Denny$^{\Gamma-536}$ arrive essentially at analogous expressions based on the Mott approximation; there are some discrepancies, which can be corrected with the aid of the data of Bete and Ashkin$^8$. Figure 2 shows the dependence of $\sigma_p$ on electron energy for several typical cases.

Fig. 2. Cross section for the formation of primary atoms knocked out as a result of bombardment by electrons, as a function of energy.

For small values of $\beta$, equation (19) reduces to equation (10). The dependence of $\sigma_p$ for electrons on the energy is mainly due to the term $1-\dfrac{E_d}{\lambda E}$, since $\lambda E\sim E_d$, whereas for ions this term may be neglected. Thus...

DISPLACEMENT OF ATOMS IN SOLIDS UNDER THE ACTION OF RADIATION

for atoms, in contrast to the ions \(\sigma_p\) for electrons, increases with energy in the interval of practical interest.

Moreover, relativistic factors usually begin to play a role before one can notice any decrease similar to the decrease according to equation (10). At high energies \(M_2c^2 \gg E \gg mc^2,\ E_{\max} \gg E_d\), the cross section reaches the value

\[ \sigma_p=\frac{8E_R^2 Z_2^2}{M_3c^2E_d}\pi a_0^2 \tag{20} \]

and thereafter no longer changes. In the first approximation the number of primary displaced atoms varies inversely proportionally to their energy, and the mean energy of these atoms is determined by an equation equivalent to equation (11):

\[ \bar E=\frac{E_dE_{\max}}{E_{\max}-E_d}\ln\frac{E_{\max}}{E_d}. \tag{21} \]

In many cases \(E_{\max}\) is only several times greater than \(E_d\); for example, in copper for electrons with energy \(1\ \text{MeV}\) the value is \(E_{\max}=68\ \text{eV}\). Therefore the total number of displaced atoms only slightly exceeds the number of primary displaced atoms.

Dienes\(^ {536}\) showed that the recoil energy going into bremsstrahlung constitutes only a small correction, and carried out the integration of \(\sigma_p\) over the entire electron path.

g) Gamma rays. Until now little attention has been paid to the question of atomic displacement by \(\gamma\)-rays. However, Dardel\(^ {616}\) showed that the action of \(\gamma\)-rays can lead to measurable effects. The basic data needed for calculating cross sections are given in article \(^{8}\). When substances with \(Z_2 \lesssim 50\) are irradiated with \(\gamma\)-rays of energy about \(2\ \text{MeV}\), the predominant type of interaction is the Compton effect. Atoms are displaced by Compton recoil electrons. When copper is irradiated with \(\gamma\)-rays of energy \(5\ \text{MeV}\),

\[ \sigma_p \simeq 0.1\ \text{barn}. \]

For heavy elements the predominant process is the direct displacement of atoms as a consequence of recoil occurring in the photoelectric effect. The cross section for photoelectric absorption \(\sigma_k\) is approximately described (for the case \(E \gg mc^2\)) by the equation

\[ \sigma_k=Z_2\alpha^4\frac{mc^2}{E}\exp\{-\pi\alpha+2\alpha^2(1-\ln\alpha)\}\ \text{barn}. \tag{22} \]

Here \(\alpha=\dfrac{Z_3}{137}\). For \(\gamma\)-rays with energy \(5\ \text{MeV}\), this equation gives, for very heavy elements, cross sections of the order of \(10^{-24}\ \text{cm}^2\).

Not every photoelectric absorption will lead to displacement of the atom in question, since it is necessary to take into account

taking into account the distribution of the emitted electrons relative to the incident \(\gamma\)-ray; in view of this, the quantity \(\sigma_k\), calculated from equation (22), may be greater than \(\sigma_p\).

Apparently, at these energies no other processes associated with \(\gamma\)-rays play as important a role as the two effects just considered. At still higher energies (\(\gg 10\) MeV; the limit depends on \(Z_2\)), the process of pair production is predominant. Therefore one should expect that in such cases atoms will be displaced by the electrons and positrons thereby produced, and also as a result of the recoil of nuclei participating in pair production.

3. Distribution of Displaced Atoms and Vacancies

a) Models based on consideration of collisions

In all cases, the greater part of the displaced atoms leaves the lattice sites occupied by them with energies exceeding \(E_d\) by no more than a factor of 2–3. At these energies a moving displaced atom will, at each mean interatomic distance it traverses, experience, roughly speaking, one collision, and at each collision will on average lose half its energy. If it is assumed that, when slowed down to an energy of 1 eV, the atom is trapped in some interstice, then the displaced atom can usually undergo no more than six or seven collisions before being trapped and, consequently, will be located at a distance of only a few interatomic spacings from the vacancy formed by its displacement.

The simplest case is bombardment by electrons or irradiation with \(\gamma\)-rays, when the energy of the primary atoms knocked out of their lattice sites is insufficient to displace other atoms. In this case the radiation damage is a statistical distribution of Frenkel defects, in which the distance between an interstitial atom and a vacancy can only in rare cases exceed four or five interatomic spacings. When electrons can transfer to the primary knocked-out atoms an amount of energy greater than \(2E_d\), these atoms will in turn produce displacement of other atoms, and some Frenkel defects will be groups consisting of two or more interstitial atoms and the corresponding number of vacancies, with all these defects located at distances of several interatomic spacings from one another.

In the case of a rapidly moving atom, in the first part of its path, where \(E > L_B\) and equations (10) and (11) are applicable, the damage consists of groups made up of several vacancies and interstitial atoms, situated at comparatively large distances-

DISPLACEMENT OF ATOMS IN SOLIDS UNDER THE ACTION OF RADIATION

...from one another along the ion’s path. As the atom is slowed down, the cross section for the formation of primary knocked-out atoms increases, and new groups of defects are formed here at distances closer to one another than in the first segment of the path; when \(E \ll L_B\), then, roughly speaking, one primary knocked-out atom is formed at every interatomic distance traversed (see [276]), and, consequently, at the end of the ion’s path the concentration of interstitial atoms and vacancies is very large. This feature of radiation damage was found experimentally by Bratten and Pearson [25] and Meier [156]. The same also applies to the pleochroic halos observed in mica; the clearest halos are regions of metamict material [100].

Bombardment by fast neutrons should usually lead to the formation of damage only of the type just indicated, with a very high concentration. When the energy of the knocked-out atom is sufficiently small, the mean free path between collisions in which most of the energy is lost approaches the interatomic distance. Under these conditions, the model based on the consideration of collisions shows that the damage will consist of a region containing a very high concentration of vacancies, surrounded by a region with a high concentration of interstitial atoms.

b) “Displacement spikes” and “thermal spikes.” Brinkman [276] believes that in the case of a very high concentration of damage arising under neutron bombardment of heavy elements, it is meaningless to speak of individual collisions, and he assumes that the volume in which this damage is formed over a short interval of time melts and then solidifies essentially with the same crystallographic orientation as the original material. Brinkman calls such regions “displacement spikes.” The size of a displacement spike is determined in the following way. As the moving atom slows down, the cross section for collisions leading to displacements increases. At the place where one collision occurs at every interatomic distance traversed, all the remaining energy of the moving atom is distributed in the displacement spike with an average energy of \(1\ \text{eV}\) per atom.

The mean diameter of a displacement spike formed by neutrons with energy \(2\ \text{MeV}\) in copper, estimated in this way, is \(\sim 75\ \text{Å}\); such a displacement spike contains \(2 \cdot 10^4\) atoms.

Brinkman believes that the principal defects introduced by such spikes are small dislocation loops and a small number of interstitial atoms and vacancies quenched in during solidification.

Another similar concept, namely the model of a “thermal spike,” was proposed by Seitz [196a]. According to

According to this picture, a moving particle heats the substance around its path through the solid in approximately \(10^{-12}\) sec. As in the case of displacement spikes, the consequences of rapid heating and cooling cannot be predicted unambiguously (see Г-123a). It has been established experimentally that, under neutron bombardment, the ordered alloy \(\mathrm{Cu}_3\mathrm{Au}\) becomes disordered more rapidly than would be expected from the number of displaced atoms, and this agrees with the idea of displacement spikes.

The practical significance of these models depends on the limiting values of volume and time for which macroscopic concepts of heat transfer may still be applied. The frequencies of atomic vibrations are of the order of \(10^{12}\)—\(10^{13}\ \mathrm{sec}^{-1}\). During \(10^{-12}\) sec a lattice disturbance is propagated by elastic waves over a distance of \(10\)—\(50\) Å, which approximately corresponds to the mean free path of phonons in insulators at room temperature \(^{14}\). Consequently, in insulators, for time intervals shorter than \(10^{-12}\) sec or for volumes with linear dimensions considerably smaller than \(50\) Å, the macroscopic laws of heat transfer should be applied with caution.

In metals, where energy transport is carried out chiefly by free electrons, elastic disturbances are for the most part reduced to a perturbation of the free electrons over a time of about \(10^{-13}\) sec \(^{14}\). The mean free path associated with the transport of energy by free electrons is \(\sim 100\) Å or more, and therefore, for the application of the macroscopic laws of heat transfer in metals, a considerably larger volume is apparently required than in insulators. On the basis of these rather crude criteria it is clear, for example, that the disturbances caused by the fission products moving in uranium can apparently be considered on the basis of the concept of thermal spikes. A fragment with an energy of \(100\) MeV has a range of about \(4 \cdot 10^{-4}\) cm \(^{\Gamma-166}\); it is easy to show that the cylindrical volume of radius \(100\) Å surrounding the track of such a fragment is heated, upon absorption of this energy, to \(4000^\circ\) C. The classical expressions for thermal conductivity show that the temperature of this track is reduced by half in approximately \(10^{-11}\) sec. In contrast to this, similar calculations show that the best way of treating the disturbances caused by electrons, \(\gamma\)-rays, and energetic \((\overline{E} > L_B)\) light ions is treatment on the basis of individual collisions. It is very possible that both pictures should be applied simultaneously, since the primary knocked-on atoms possess all energies from \(E_d\) to \(E_{\max}\), and it is almost beyond doubt that the behavior of knocked-on atoms of low energy should be considered on the basis of the concept of collisions. In ordinary irradiations in a reactor, the greater part of the primary knocked-on atoms have comparatively low energies, which is connected with the shape

DISPLACEMENT OF ATOMS IN SOLIDS UNDER THE ACTION OF RADIATION

energy distribution function of fast neutrons*). BermanG-13 and ClemensG-118, in their investigations carried out on quartz, showed that the disturbances caused by fast neutrons can be regarded as a mixture of isolated interstitial atoms and comparatively large volumes with considerable disturbances.

The same difficulty was encountered in attempts at a quantitative explanation of metallization; Bredley16 came to the conclusion that none of the concepts existing up to the present time makes it possible to give an adequate explanation of the various experiments. However, he found that there are no facts speaking in favor of the idea of “hot spots” for ions with energies up to 1800 eV bombarding alkali metals. The most clearly expressed application of the idea of “hot spots,” given by Townes17, is not free from objections, namely that the macroscopic laws of heat transfer cannot be applied to volumes containing only a few atoms for a time of the order of \(10^{-16}\) sec. Keywell18 applied the idea of collisions of the hard-sphere type to the consideration of the sputtering of silver and other metals caused by argon ions with energies up to 5000 eV, and obtained very good agreement with the data of his experiments on the assumption that in each collision of an argon ion (when it is slowed down to 39 eV) one silver atom is “sputtered.”

4. Effects in compounds

a) Collisions leading to replacement of atoms

In some collisions of moving atoms with stationary ones, leading to the displacement of the latter from lattice sites, the residual energy of the bombarding atom proves insufficient for this atom to be able to move away from the vacancy formed in this process. Such collisions are called collisions leading to replacement of atoms (replacement collisions)**), since they lead to a change in the type of atoms occupying certain lattice sites, but not to a change in the total number of atoms. It is obvious that replacement collisions have very little significance in monatomic solids, but prove to be substantial when considering radiation effects in compounds.

*) For reactors with a graphite moderator this function, roughly speaking, has the form

\[ F(E)\,dE=\frac{k\,dE}{E}, \]

where \(F(E)\,dE\) is the flux, \(k\) is a constant (see, for example,15).

**) In what follows, for brevity, we shall call such collisions “replacement collisions.” — Translator’s note.

Kinchin and Pease proposed \((\Gamma\text{-}117a)\) that replacement collisions occur when the energy of the bombarding atom after the collision is less than \(E_d\), and less than the energy initially transferred to the stationary atom. It is further assumed that replacement collisions can take place only when the energy transferred to the stationary atom exceeds some threshold value \(E_\rho\). An approximate calculation of the quantity \(N_\rho\), the number of replacement collisions in the case of neutron bombardment, based on consideration of collisions of the elastic-sphere type and on the assumption of equality of the masses of atoms in the lattice, leads to the equation

\[ N_\rho = 0.5\,N_d \left[ 1 + \frac{\ln \dfrac{E_d}{E_\rho}}{\ln \dfrac{4}{3}} \right], \tag{23} \]

where \(N_d\) is the number of displaced atoms, determined by equation (17). Since \(E_\rho\) must apparently be considerably smaller than \(E_d\), the number of replacement collisions may exceed the number of atomic displacements. This applies, in particular, to the case of ordered alloys, in which \(E_\rho\) may be only slightly greater than the activation energy of internal diffusion (in cracks, necks, etc.), which, according to Huntington’s estimate \(\Gamma\text{-}996\), is \(0.25\ \mathrm{eV}\) for copper.

If the masses of the atoms in a compound differ substantially from one another, the number of replacements will be considerably smaller than that calculated from equation (23), because, as is easy to see from equation (1), a moving atom of mass \(m\) cannot transfer more than half of its energy to an atom of mass \(M_2\) if \(\lambda < \tfrac{1}{2}\). It is easy to show that, according to the proposed mechanism, replacements can occur when

\[ \frac{1}{5.83} < \frac{M_1}{M_3} < 5.83 . \]

b) Disordering. In any compound, both replacement collisions and partial or complete recombination of Frenkel defects may lead to some atoms occupying lattice sites unusual for them. It is well known that such a disordering process occurs in the ordered alloys \(\mathrm{Cu}_3\mathrm{Au}\) and \(\mathrm{Ni}_3\mathrm{Mn}\). For other substances there is little experimental evidence for the existence of this process. In particular, in the case of boron nitride, even at very high doses, disordering was not detected \(\Gamma\text{-}17\). Even in such a case, irradiation should lead to the formation of a certain number of defects in which the displaced atom is located in an inappropriate vacant site or very close to such a site. Such defects may be significant in minerals, since the latter often contain atoms of many different elements.

II. RECOVERY OF DEFECTS

In most cases, when the temperature of a specimen is raised after irradiation, the physical properties return to their original values. In addition, energy is released. In a small number of cases, after very intense irradiation, annealing at high temperatures may lead to recrystallization of the substance with the formation of a new phase having a structure different from that of the initial phase, as, for example, occurs in the case of thorite \(^{19}\) and certain forms of quartz \(^{\Gamma-224}\). Usually one should expect that some recovery of defects also occurs during irradiation, even if the irradiation is carried out at an arbitrarily low temperature.

Recovery as a result of heating usually proceeds in several stages with different values of the activation energy in different temperature intervals, and the course of the recovery process can usually be described by the equation

\[ \frac{dn}{dt}=-cn^\gamma \exp\left(-\frac{\varepsilon}{kT}\right), \tag{24} \]

where \(n\) is the number of defects taking part in the recovery process; \(\varepsilon\) is the activation energy; \(c\) is a constant; \(\gamma\) is often called the “order of the reaction,” by analogy with gaseous chemical reactions. It should be pointed out, however, that this term can sometimes be misleading, since the rate of some recovery processes is determined by the diffusion of defects in the lattice, which leads to apparently high reaction orders that do not correspond to the actual number of components participating in the process, as is the case in chemical reactions.

1. Recombination of Interstitial Atoms and Vacancies

a) Pairs of interstitial atoms and vacancies located close to one another

The recombination of “interstitial atom—vacancy” pairs was considered by Fletcher and Brown \(^{\Gamma-74}\). These authors assumed that vacancies are the more mobile defects and applied the diffusion equations to the process of random migration of vacancies toward spherical “traps,” represented by interstitial atoms. If \(n(0)\) vacancies begin to move at a distance \(b\) from such a trap of radius \(a\) in an infinite isotropic three-dimensional medium, then the number \(n(t)\) of vacancies remaining at time \(t\) is determined by the equation

\[ n(t)=n(0)\left\{1-\frac{a}{b}+\frac{a}{b}\Phi\left(\frac{b-a}{2(Dt)^{1/2}}\right)\right\}, \tag{25} \]

where \(D\) is the vacancy diffusion coefficient. The fraction \(1-\dfrac{a}{b}\) of vacancies is not captured by the trap, but remains “free” and can diffuse indefinitely in the substance. In the early stages of this process of “vacancy release” the lattice cannot be regarded as a continuum; in view of this Fletcher and Brown used numerical solutions applicable to the diamond lattice.

For substances with a layered structure, in which diffusion of interstitial atoms*) can occur only in two directions, the spherical trap used in solving the three-dimensional problem is replaced by a two-dimensional circular trap. The solution has the form

\[ n(t)=n(0)\frac{2}{\pi}\int_{0}^{\infty}\frac{e^{-y^{2}Dt}}{y} \left\{ \frac{Y_{0}(by)J_{0}(ay)-Y_{0}(ay)J_{0}(by)} {J_{0}^{2}(ay)+Y_{0}^{2}(ay)} \right\}\,dy, \tag{26} \]

where \(J_{0}\) and \(Y_{0}\) are Bessel functions of zero order. Spinney**) made calculations from equation (26); some results of this calculation are shown in Fig. 3. In this case “release” of interstitial atoms does not occur. Equation (24) for \(\gamma=6\) gives a good approximation to equation (26).

Fig. 3. Annealing of pairs of interstitial atoms and vacancies located close to one another.

b) Random distribution of interstitial atoms and vacancies. The process of interaction of equal amounts of randomly distributed interstitial atoms and vacancies is described by the kinetic equation of a second-order reaction, namely:

\[ \frac{dn}{dt}=-cn^{2}e^{-\frac{\varepsilon}{kT}}, \tag{27} \]

where \(\varepsilon\) is the activation energy for migration of interstitial atoms.

If the energy of the electrons bombarding the substance exceeds the threshold energy only slightly, so that isolated Frenkel defects are formed, then one may expect defect recovery to occur in two stages. The first stage is described by equation (25) or (26), and the second by equation (27) with the same activation energy. The phenomenon is complicated owing to

*) In what follows we assume that the interstitial atoms are more mobile than the vacancies.

**) Spinney K. T. (unpublished data).

of the existence of an interaction between interstitial atoms and vacancies, considered in papers [Г-15] and [Г-1656]. The authors of these studies came to the conclusion that lattice stresses near a vacancy reduce the activation energy for migration of an interstitial atom located not far from the vacancy, and that under these conditions the diffusion will be anisotropic. However, under short-term irradiations the activation energy should increase in the process of energy release up to a value corresponding to the activation energy for migration in a lattice without defects, and the second stage of defect annealing should not be described by equation (27). Dienes [Г-56д] drew attention to the fact that comparatively small changes in the activation energy have a considerable influence on the apparent order of the reaction.

2. Additional processes

The consideration of the recombination process set out above proves too simple to explain all the observed effects. Marx et al. [Г-15] described a number of other possible processes. During annealing, some interstitial atoms may be captured by dislocations or grain boundaries, which must influence the recombination process of interstitial atoms and vacancies. In addition, after all interstitial atoms have disappeared, the excess of vacancies diffuses toward dislocations and grain boundaries with a higher activation energy. Approximate solutions of diffusion problems associated with the process under consideration were given by Fletcher and Brown [Г-74] and by Brinkman et al. [Г-286]. In addition, groups of interstitial atoms and vacancies may form, dissociating with high activation energies.

Thus, although experimental studies of the kinetics of defect recovery can provide valuable information, they are nevertheless insufficient to identify unambiguously the processes occurring in this case. More complete data on recovery processes can be obtained on the basis of experimental measurement of the activation energy, which has a constant value for comparatively large changes in the measured physical property. These values of the activation energy can be compared with estimates obtained from other experimental data (for example, from experiments on cold working, quenching, or self-diffusion) and on the basis of theoretical calculations.

3. Accumulation of imperfections

Let us now consider the case in which irradiation and annealing occur simultaneously. Generally speaking, under irradiation groups of displaced atoms and vacancies are formed. Under the condition that interstitial

... atoms diffuse over distances small compared with the distances between these groups, each group will anneal independently of other groups. Consequently, if upon irradiation \(dx\) displaced atoms were formed instantaneously, then after a time interval \(t\) at temperature \(T\), only \(dx\,F(T,t)\) will remain, where \(F(T,t)\) is a function independent of \(x\), which can be found experimentally by annealing after irradiation at very low temperatures. Thus, at time \(t\) the total number of atoms displaced in the course of irradiation, in which these atoms are formed at a rate \(\dfrac{dx}{dt'}\), is determined by the equation

\[ n(t)=\int_0^t \left(\frac{dx}{dt'}\right) F(T,t-t')\,dt'. \tag{28} \]

If \(\dfrac{dx}{dt}\) is constant during irradiation, then equation (28) shows that the number of displaced atoms present after irradiation for a definite interval of time is proportional to the flux of the bombarding particles. Moreover, it is evident from this equation that for different fluxes the number of displaced atoms is determined not only by the integral dose.

If the annealing of groups of defects can be expressed empirically by equation (24) for \(\gamma>1\) and for a constant activation energy, then the function \(F(T,t)\) takes the form

\[ \left[1+c't\exp\left(-\frac{\varepsilon}{kT}\right)\right]^{1/(1-\gamma)}, \]

where \(c'\) is a new constant.

For \(\gamma>2\) the number of displaced atoms present after irradiation that is long in comparison with \(\left(\dfrac{1}{c'}\right)\exp\left(\dfrac{\varepsilon}{kT}\right)\) is approximately determined by the equation

\[ n(t)\approx \frac{dx}{dt}\,\frac{\gamma-1}{\gamma-2} \left\{\frac{1}{c'}\exp\left(\frac{\varepsilon}{kT}\right)\right\}^{\frac{1}{\gamma-1}} t^{\frac{\gamma-2}{\gamma-1}}, \tag{29} \]

For large values of \(\gamma\) the disturbance is almost proportional to \(x\), and the effect of the simultaneously occurring recovery process consists only in reducing the apparent rate of formation of displacements in proportion to the factor

\[ \left\{\frac{1}{c'}\exp\left(\frac{\varepsilon}{kT}\right)\right\}^{\frac{1}{\gamma-1}}. \]

Such a temperature dependence of the rate of accumulation of disturbances should not occur at small doses if the interstitial atoms and vacancies were distributed randomly. However, this dependence is often observed (see, for example, \[¹⁵\]).

4. Saturation

In equation (29) there is no term that would indicate the presence of any saturation. This is explained by the fact that we have neglected the interaction between different groups of displaced atoms. In fact, during irradiation these groups will sometimes overlap, and at lower temperatures, when the interstitial

Fig. 4. Saturation of the change in interplanar spacing in boron nitride. Boron nitride has a layered lattice, and the interplanar spacing \(c\) is twice the distance between layers of atoms. The doses are expressed as the number of bombarding particles (\(\alpha\)-particles or lithium nuclei) incident on one atom \[¹⁷\].

atoms are immobile, this will lead to exponential saturation of the number of displaced atoms. At higher temperatures, the approach to saturation will be described as a second-order recovery process.

Rather similar ideas follow from another approximation used by Pease \[¹⁷\] in studying saturation effects in boron nitride (Fig. 4). Here the assumption is used that interstitial atoms and vacancies are stable if the distance between them exceeds some critical value, and that at smaller distances they combine instantaneously. Thus, each interstitial atom is surrounded by \(Q\) lattice sites into which it can immediately move, as

only one of them will prove to be free. Statistical considerations lead to the equation

\[ y=\frac{1}{Q}\ln[2-e^{-pQ}], \tag{30} \]

in which \(y\) is the concentration of interstitial atoms; \(p\) is the average number of collisions of each atom with sites occupied by interstitial atoms.

This equation shows that at low doses \(y=p\), and at a dose corresponding to \(p \simeq \frac{1}{Q}\), saturation is established with a concentration of interstitial atoms \(y=\frac{\ln 2}{Q}\). The influence of the irradiation temperature in this model is introduced by assuming that \(Q\) depends on temperature and that \(p\), consequently, also depends on temperature, because in the process of bombardment the interstitial atoms and vacancies combine into groups. Even at the very lowest temperatures \(Q\) can hardly be less than the number of lattice sites that are nearest neighbors to the interstice occupied by an interstitial atom, and therefore the maximum concentration of interstitial atoms in a crystalline substance can hardly be appreciably greater than 5%.

5. Radiation annealing

Sometimes effects are observed which show that recovery processes are accelerated by the radiation itself. For example, Cooper et al. [Г-46a] observed that the rate of increase of the resistance of copper, silver, and gold at \(12^\circ\mathrm{K}\) during irradiation decreases, whereas after irradiation at this temperature no annealing was observed. One obvious mechanism for explaining this consists in assuming local heating produced by the bombarding radiation. However, apparently (see [Г-123a]), in most cases the duration of the heating is too short to cause appreciable annealing. Koehler and Seitz [Г-123a] believe that excitation of electrons facilitates the displacement of interstitial atoms or vacancies. It is also possible that collisions lead to some displacements of vacancies and interstitial atoms. The threshold energy for such displacements is considerably lower than the 25 eV required for displacements, and consequently the probability of displacement of an interstitial atom or vacancy is considerably greater than the probability of formation of such a defect anew. Thus irradiation can lead to displacements that are large in comparison with displacements activated by thermal vibrations, and consequently can lead to some recovery of defects that would not occur in the absence of radiation. Such a process can noticeably reduce the saturation concentration of displacements—

of displaced atoms, given in the preceding paragraph. Such saturation may be significant for limiting the concentration of displaced atoms in clusters formed under bombardment by fast neutrons (§ 1, 3).

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