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Radiation-induced aggregatization of immobile defects
Solid State Communications, Vo1.39, pp.351-354. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-1098/81/260351-04$02.00/O
RADIATION-INDUCED AGGREGATIZATION OF IMMOBIIE DEFECTS E. Kotomin and V. Kuzovkov Latvian State University, Rainis 19, biga, USSR ( Received April 10 1981 by E. A. Kaner) A model for accumulation kinetics of Rrenkel defects (vacancy-interstitial) produced under irradiation in solids is presented and developed. Defects are assumed to he immobile (low temperatures or doped crystal) and their accumulation is restricted by annihilation of dissimilar defects. The limiting case of an instant annihilation is studied. The complete description of a spatial defect distribution employing the joint probability functions permits to establish the connection between an initial spatial distribution within Ii'renkel pairs and a steadystate defect distribution and concentration under saturation. The greater initial correlation within pairs, the less defect concentration. The model confirms the formation of radiation-induced aggregates of similar defects.
As is well known, an irradiation of most crystals (whether metals or insulators) produces Frenkel pairs of defects (vacancies-interstitials) l). Dissimilar defects separated by a relative distance less than a clear-cut radius, R,, very rapidly and spontaneously annihilate, A + B ---) 0, restoring a perfect crystalline lattice. At low temperatures (typically below 30 K) defects are immobile and annihilation is the main mechanism here limiting their accumulation and leading to a saturation of defect concentration at high irradiation doses. (The same effect is observed in heavily doped crystals at high temperatures when defects become localized by impurities immediately after their production.) It has been shown2) that a high dose irradiation of alkali halides at low temperatures leads to the emergence of loose aggregates of similar defects. It has also been observed3) that the
share of pair defects, i. e. F2 centres, considerably exceeds the corresponding value for a random (Poisson) distribution. These data are also supported by direct computer simulations of defect accumulation in metals4, as well as in alkali halides5). Several attempts have been made to develop a phenomenological theory of radiation induced (Frenkel) defect accumulation. For instance, the authors6' 7, assumed that defects of both kinds, A and B, have volumes around them prohibited for defects of another kind (B and A, respectively). They took into account the possible overlap of prohibited volumes around similar defects (A-A, B-B) which can stimulate their aggregatization while the overlap of prohibited volumes around dissimilar defects leads to their annihilation. However, the simple model employed is restricted by use of the oneparticle distribution function (DE) 351
352
RADIATION-INDUCEDAGGREGATIZATION OF IMMOBILE DEFECTS
only, i. e. defect concentration, which has led necessarily to a very simplified aSSI.IIDptiOn that a total prohibited volume of two close similar defects equals the volume per isolated defect. This results directly in an overestimate: the defect concentration grows infinitely which means an infinite increase with time of local defect density. On the other hand, there have been attempts of a rigirous approach to the accumulation kinetics based on a hierarchy of equations for the I)F's (e. g.8)). However, in the latter approach a set of rather complicated and approximate equations has been obtained, coupling many-particle DI's of different order but no attempt has been made to describe a defect structure (spatial distribution) development corfining ourselves to joint DF's only, or to investigate the limiting case of an instant annihilation. Such an approach, discussed below,
permits to obtain (in the framework of the superposition approximation for three-particle densities) relatively simple kinetic equations for defect accumulation. (i;detailed derivation of these equations has been given by us quite recently').) Our approach takes into account the discreteness of a crystalline lattice which permits to calculate a saturation concentration at high irradiation doses. Let us now consider the fundamental concepts of the model used in deriving equations and the physical results following from them. (A). Inhere are three possible states of a volume per lattice site, i. e. (i) defectless one, (ii) that containing an interstitial, (iii) a volume with a vacancy. (Our approach permits, in principle, to extend the results to more COmplicated cases, e. g. crowdions.) (B). The spatial distribution of defects is
Vol. 39, No. 2
changed with time due to (i) a production of new defects, (ii) close defect annihilation, (iii) defect motion. The latter process will not be considered here, i. e. defects are immobile (low temperatures) (cf. however9, 10)) . The DF's employed are: (i) the one-particle DE',i. e. the defect concentration, c, gives the probability that an arbitrary site contains a defect, (ii) the tviojoint DF's, X and X. The Y Ii?- +' r 1 yields the probability density of finding two dissimilar defects in unit volumes at ? and f' whereas the X yields the same for the similar defects. (Since we have neglected defect motion, the TIE'sX coincide for both vacancies and interstitick.1 ikr set of ec,uationsdiscussed below,i.:complete (in tzrms of the joint DT's) aridpermits to obtairi not only the exact T;imedevciopment of the deSoct concentration, but aiso to investigate the tine devclopmcnt of tile spatial distribution of defects and, in particulal, to confirm the production of earlier,-expectedradiation-iriduceu ag~re~atts of similar defects. l!helatter prdblcm has not beer correctl.jstated beib1.c:. !i'he obtained kinetic equ;tions read') (the aistunces ale taken iii InlIClue ti v.nich ail the tice const-mts LT's are dimenciun~ess) -2C
at =po
-
a)c -=2pcat 2
=po(zc
-4GYx-q +$%i*
+f(+-y~X)CI
(1)
Y), (2) - 2Yd3)
in k3qs.(I) to (3) pl =&f(r) is. the probability of production of the correlated ('genetic') Frenkel pair of dissimilar defects (per unit time and volume) being separated by a distance r, G = &, g(r) being the annihilat-
Vol. 39, No. 2
MDIATION-INDUCED AGGBEGATIZATION OF IMMOBILE DEFECTS
ion probability of a pair per unit time. In Eqs. (1) to (3) the definition of the spatial convolution (A*B)= A (?') B (f - f') di? has been uss ed. In the derivation Kirkwood's superposition approximation for the three-particle DE's is used. It is assumed in (l)-(3)that c 44 1 always holds (i. e. large annihilation volume = 4/3iiRz >> 3 contains many lattice sPtes). The annihilation is described
V
by ~~'~'til'<~,otherwise
g=O.
(4)
In the limiting case of an instant annihilation ( GO-+ Qo 1 Eqs. (1) to (3) read It'= Zf+POt)
353
because one can easily imagine defect distribution without aggregates with a higher concentration. Thus our results break down the widely-accepted view that a defect aggregatixation leads to a considerable increase of the saturation concentration. The intuitive idea that a nonequilibrium system strives for a spatial structure with a maximal defect concentration seems to be unjustified. Fig. 1 obtained by computer calculations using Eqs. (5) to (7) shows the time development of the joint DE's for similar and dissimilar defects, X and Y. E'orconvenience the relative defect concentrations, c/c, are given instead of time. Two extreme cases are analysed: (i) the random production of the Frenkel defects within pairs - (no - initial correlation - case a), (ii) the
where Y = 0 if r< Ro. For the sake of simplicity the initial production distribution in Eqs. (5) to (7) has been taken as
Such a choice of the f (r) as well as the condition v. >> 1 is not necessary, since our Eqs. in a particular way may be extended to the case of almost complete crystal saturation with de. fects at voe 1 '3) One can deduce from (5) that the concentration development with time is
with the saturation concentration CO = = l/2 (vi' - vii). The relation (9) has been found earlier empirically 11)). At first sight such a low (e. g. saturation concentration is surprising
Fig. 1. Time development of the DF's for the similar, X, and the aissimilar, Y, defects for the cases of a random initial distribution within Frenkel pairs (a) and strong correlation (b). The DF's: ------- X, =t=;=:= Y.
354
RADIATION-INDUCEDAGGREGATIZATION OF IMMOBILE DEFECTS
strong initial correlation within Frenkel pair (case b) with Rg/Ro = 1.2. One can conclude from the curves in(a)t when relative concentration approaches 0.5, the share of close similar defects essentially exceeds that for the Poisson (random) distribution (a horizontal line X = c2), which means the formation of radiation-induced loose aggregates containing similar defects only (cf.4* 5)). Its typical size may be estimated as 2R, which is in agreement with computer simulations. The share of the similar defects exceeds the Poisson value twice at c/c, = 0.5 and 3 and 10 times at c/c, = 0.8 and 0.99, respectively. This is in c_ualitative agreement with the experiment for the F2 centres accumulation in al-: kali halides3). The conclusion can be drawn that a most powerful aggregatization occurs mainly at high irradiation doses. The mean distance between loose aggregates slightly increases with time and amounts to about 1.5 Ro. In the case(b the strong initial correlation the DF of the dissimilar defects, Y, has a very sharp peak at low concentrations within the interval Ros r,< Rg (Rg = 1.2 Ro). Of extreme importance is the formation of a corresponding decay ('negative' correlat-
Vol. 39, No. 2
ion) for the DF of similar defects within this interval. It indicates the coup& of two joint DF's. At the same time the formation of aggregates is suppressed here (along with a much lower saturation concentration, cf.8) , as compared with the case (a\ On approaching saturation the Y peak disappears and so does the X. At saturation thex shows the formation of aggregates quite similar to the case of no initial correlation, but a mean size is about half that in the case of negligible correlation. So what has been said above is the rigorous confirmation of the radiationinduced defect aggregatization (earlier obtained by both computer simulations4' 5, and experimentally2, 3). The important conclusion may be drawn also that a correct approach to kinetic equations has to involve b&the DF's for the similar and the dissimilar defects. As is shown in9$ lo) it is also valid for diffusion-controlled reactions, where decay in the Y arises due to defect migration. Acknowledgments - Many thanks are due to Prof. V. Antonov-Romanovskii, Drs. Yu. Kalnin', F. Pirogov, I. Ele and D. Miller for many stimulating discussions.
REFERENCES 1) V. M. Agranovich and V. V. Kirsanov, Sov. Phys. uspekhi 125, 3 (1976). 2) L. W. Hobbs, A. E. Hughes and D. Pooleg, Proc. Roy. Sot. A 332, 167 (1973). 3) B. F. Faraday and W. D. Compton, Phys. Rev. =A, 893 (1965). 4) G. LUck and R. Sizmann, Phys. Stat. Sol. 6, 263 (1964); 9, 507 (1966). 5) I. Tale, D. Miller and E. Kotomin, J. Phys. C E3, 2366 (1975). 6) D. Pooley, Brit. J. Appl. Phys. l7, 855 (1966).
7) A. E. Hughes and D. Pooley, J. Phys. C 4, 1963 (1971). 8) K. Dettmann, G. Laibfried and K. Shroeder, Phys. Stat. Sol. 22, 423 (1967). 9) V. Kuzovkov and E. Kotomin, in: "Physics of Phase Transitions" (Latv. State Univ., Riga, 1980, in Russian), Phys. Stat. Sol. (to be published). 10) V. Kuzovkov and E. Kotomin, J. Phys. C ;1z, 1499 (1980). 11) M. A. Elango, Proc. Inst. Phys. Est. SSR (in Russian) $?, 175 (1974).
0038-1098/81/260351-04$02.00/O
RADIATION-INDUCED AGGREGATIZATION OF IMMOBIIE DEFECTS E. Kotomin and V. Kuzovkov Latvian State University, Rainis 19, biga, USSR ( Received April 10 1981 by E. A. Kaner) A model for accumulation kinetics of Rrenkel defects (vacancy-interstitial) produced under irradiation in solids is presented and developed. Defects are assumed to he immobile (low temperatures or doped crystal) and their accumulation is restricted by annihilation of dissimilar defects. The limiting case of an instant annihilation is studied. The complete description of a spatial defect distribution employing the joint probability functions permits to establish the connection between an initial spatial distribution within Ii'renkel pairs and a steadystate defect distribution and concentration under saturation. The greater initial correlation within pairs, the less defect concentration. The model confirms the formation of radiation-induced aggregates of similar defects.
As is well known, an irradiation of most crystals (whether metals or insulators) produces Frenkel pairs of defects (vacancies-interstitials) l). Dissimilar defects separated by a relative distance less than a clear-cut radius, R,, very rapidly and spontaneously annihilate, A + B ---) 0, restoring a perfect crystalline lattice. At low temperatures (typically below 30 K) defects are immobile and annihilation is the main mechanism here limiting their accumulation and leading to a saturation of defect concentration at high irradiation doses. (The same effect is observed in heavily doped crystals at high temperatures when defects become localized by impurities immediately after their production.) It has been shown2) that a high dose irradiation of alkali halides at low temperatures leads to the emergence of loose aggregates of similar defects. It has also been observed3) that the
share of pair defects, i. e. F2 centres, considerably exceeds the corresponding value for a random (Poisson) distribution. These data are also supported by direct computer simulations of defect accumulation in metals4, as well as in alkali halides5). Several attempts have been made to develop a phenomenological theory of radiation induced (Frenkel) defect accumulation. For instance, the authors6' 7, assumed that defects of both kinds, A and B, have volumes around them prohibited for defects of another kind (B and A, respectively). They took into account the possible overlap of prohibited volumes around similar defects (A-A, B-B) which can stimulate their aggregatization while the overlap of prohibited volumes around dissimilar defects leads to their annihilation. However, the simple model employed is restricted by use of the oneparticle distribution function (DE) 351
352
RADIATION-INDUCEDAGGREGATIZATION OF IMMOBILE DEFECTS
only, i. e. defect concentration, which has led necessarily to a very simplified aSSI.IIDptiOn that a total prohibited volume of two close similar defects equals the volume per isolated defect. This results directly in an overestimate: the defect concentration grows infinitely which means an infinite increase with time of local defect density. On the other hand, there have been attempts of a rigirous approach to the accumulation kinetics based on a hierarchy of equations for the I)F's (e. g.8)). However, in the latter approach a set of rather complicated and approximate equations has been obtained, coupling many-particle DI's of different order but no attempt has been made to describe a defect structure (spatial distribution) development corfining ourselves to joint DF's only, or to investigate the limiting case of an instant annihilation. Such an approach, discussed below,
permits to obtain (in the framework of the superposition approximation for three-particle densities) relatively simple kinetic equations for defect accumulation. (i;detailed derivation of these equations has been given by us quite recently').) Our approach takes into account the discreteness of a crystalline lattice which permits to calculate a saturation concentration at high irradiation doses. Let us now consider the fundamental concepts of the model used in deriving equations and the physical results following from them. (A). Inhere are three possible states of a volume per lattice site, i. e. (i) defectless one, (ii) that containing an interstitial, (iii) a volume with a vacancy. (Our approach permits, in principle, to extend the results to more COmplicated cases, e. g. crowdions.) (B). The spatial distribution of defects is
Vol. 39, No. 2
changed with time due to (i) a production of new defects, (ii) close defect annihilation, (iii) defect motion. The latter process will not be considered here, i. e. defects are immobile (low temperatures) (cf. however9, 10)) . The DF's employed are: (i) the one-particle DE',i. e. the defect concentration, c, gives the probability that an arbitrary site contains a defect, (ii) the tviojoint DF's, X and X. The Y Ii?- +' r 1 yields the probability density of finding two dissimilar defects in unit volumes at ? and f' whereas the X yields the same for the similar defects. (Since we have neglected defect motion, the TIE'sX coincide for both vacancies and interstitick.1 ikr set of ec,uationsdiscussed below,i.:complete (in tzrms of the joint DT's) aridpermits to obtairi not only the exact T;imedevciopment of the deSoct concentration, but aiso to investigate the tine devclopmcnt of tile spatial distribution of defects and, in particulal, to confirm the production of earlier,-expectedradiation-iriduceu ag~re~atts of similar defects. l!helatter prdblcm has not beer correctl.jstated beib1.c:. !i'he obtained kinetic equ;tions read') (the aistunces ale taken iii InlIClue ti v.nich ail the tice const-mts LT's are dimenciun~ess) -2C
at =po
-
a)c -=2pcat 2
=po(zc
-4GYx-q +$%i*
+f(+-y~X)CI
(1)
Y), (2) - 2Yd3)
in k3qs.(I) to (3) pl =&f(r) is. the probability of production of the correlated ('genetic') Frenkel pair of dissimilar defects (per unit time and volume) being separated by a distance r, G = &, g(r) being the annihilat-
Vol. 39, No. 2
MDIATION-INDUCED AGGBEGATIZATION OF IMMOBILE DEFECTS
ion probability of a pair per unit time. In Eqs. (1) to (3) the definition of the spatial convolution (A*B)= A (?') B (f - f') di? has been uss ed. In the derivation Kirkwood's superposition approximation for the three-particle DE's is used. It is assumed in (l)-(3)that c 44 1 always holds (i. e. large annihilation volume = 4/3iiRz >> 3 contains many lattice sPtes). The annihilation is described
V
by ~~'~'til'<~,otherwise
g=O.
(4)
In the limiting case of an instant annihilation ( GO-+ Qo 1 Eqs. (1) to (3) read It'= Zf+POt)
353
because one can easily imagine defect distribution without aggregates with a higher concentration. Thus our results break down the widely-accepted view that a defect aggregatixation leads to a considerable increase of the saturation concentration. The intuitive idea that a nonequilibrium system strives for a spatial structure with a maximal defect concentration seems to be unjustified. Fig. 1 obtained by computer calculations using Eqs. (5) to (7) shows the time development of the joint DE's for similar and dissimilar defects, X and Y. E'orconvenience the relative defect concentrations, c/c, are given instead of time. Two extreme cases are analysed: (i) the random production of the Frenkel defects within pairs - (no - initial correlation - case a), (ii) the
where Y = 0 if r< Ro. For the sake of simplicity the initial production distribution in Eqs. (5) to (7) has been taken as
Such a choice of the f (r) as well as the condition v. >> 1 is not necessary, since our Eqs. in a particular way may be extended to the case of almost complete crystal saturation with de. fects at voe 1 '3) One can deduce from (5) that the concentration development with time is
with the saturation concentration CO = = l/2 (vi' - vii). The relation (9) has been found earlier empirically 11)). At first sight such a low (e. g. saturation concentration is surprising
Fig. 1. Time development of the DF's for the similar, X, and the aissimilar, Y, defects for the cases of a random initial distribution within Frenkel pairs (a) and strong correlation (b). The DF's: ------- X, =t=;=:= Y.
354
RADIATION-INDUCEDAGGREGATIZATION OF IMMOBILE DEFECTS
strong initial correlation within Frenkel pair (case b) with Rg/Ro = 1.2. One can conclude from the curves in(a)t when relative concentration approaches 0.5, the share of close similar defects essentially exceeds that for the Poisson (random) distribution (a horizontal line X = c2), which means the formation of radiation-induced loose aggregates containing similar defects only (cf.4* 5)). Its typical size may be estimated as 2R, which is in agreement with computer simulations. The share of the similar defects exceeds the Poisson value twice at c/c, = 0.5 and 3 and 10 times at c/c, = 0.8 and 0.99, respectively. This is in c_ualitative agreement with the experiment for the F2 centres accumulation in al-: kali halides3). The conclusion can be drawn that a most powerful aggregatization occurs mainly at high irradiation doses. The mean distance between loose aggregates slightly increases with time and amounts to about 1.5 Ro. In the case(b the strong initial correlation the DF of the dissimilar defects, Y, has a very sharp peak at low concentrations within the interval Ros r,< Rg (Rg = 1.2 Ro). Of extreme importance is the formation of a corresponding decay ('negative' correlat-
Vol. 39, No. 2
ion) for the DF of similar defects within this interval. It indicates the coup& of two joint DF's. At the same time the formation of aggregates is suppressed here (along with a much lower saturation concentration, cf.8) , as compared with the case (a\ On approaching saturation the Y peak disappears and so does the X. At saturation thex shows the formation of aggregates quite similar to the case of no initial correlation, but a mean size is about half that in the case of negligible correlation. So what has been said above is the rigorous confirmation of the radiationinduced defect aggregatization (earlier obtained by both computer simulations4' 5, and experimentally2, 3). The important conclusion may be drawn also that a correct approach to kinetic equations has to involve b&the DF's for the similar and the dissimilar defects. As is shown in9$ lo) it is also valid for diffusion-controlled reactions, where decay in the Y arises due to defect migration. Acknowledgments - Many thanks are due to Prof. V. Antonov-Romanovskii, Drs. Yu. Kalnin', F. Pirogov, I. Ele and D. Miller for many stimulating discussions.
REFERENCES 1) V. M. Agranovich and V. V. Kirsanov, Sov. Phys. uspekhi 125, 3 (1976). 2) L. W. Hobbs, A. E. Hughes and D. Pooleg, Proc. Roy. Sot. A 332, 167 (1973). 3) B. F. Faraday and W. D. Compton, Phys. Rev. =A, 893 (1965). 4) G. LUck and R. Sizmann, Phys. Stat. Sol. 6, 263 (1964); 9, 507 (1966). 5) I. Tale, D. Miller and E. Kotomin, J. Phys. C E3, 2366 (1975). 6) D. Pooley, Brit. J. Appl. Phys. l7, 855 (1966).
7) A. E. Hughes and D. Pooley, J. Phys. C 4, 1963 (1971). 8) K. Dettmann, G. Laibfried and K. Shroeder, Phys. Stat. Sol. 22, 423 (1967). 9) V. Kuzovkov and E. Kotomin, in: "Physics of Phase Transitions" (Latv. State Univ., Riga, 1980, in Russian), Phys. Stat. Sol. (to be published). 10) V. Kuzovkov and E. Kotomin, J. Phys. C ;1z, 1499 (1980). 11) M. A. Elango, Proc. Inst. Phys. Est. SSR (in Russian) $?, 175 (1974).