veloped on diffusioncontrolled reactions between ... - Science Direct

Chemical Physics 76 (1983) 479-487 North-Holland Publishing Company

SOME

PROBLEMS

E. KOTOMIN

OF RECOMBINATION

1

and V. KUZOVKOV

Latvian State ihbersiry. Received

KINETICS.

4 August

Rainis 19. Riga, USSR

1981; in linal form 1 1:ebmary

1983

New kinetic equations describing diffusioncontrotied accumulation and recombination of defects in solids and liquids and taking into account successively the identity of similar defects and spatial correlations of borlr similar and dissimilar defects are presented. Alternative (the so-called symmetric and asymmetric) approaches to the description of the diffusioncontrolled reactions of initially correlated defects are discussed in detail.

l_ Introduction Several phenomenologica! theories have been developed on diffusioncontrolled reactions between defects in solids and liquids [ 1-lo] _All of them differ from each other in their details although the general idea seems to be that a correct approach to the description of the kinetics of radiation-induced defect accumulation and recombination has to be based on a hierarchy of equations for the many-particle distribution functions (DFs). This is why in the first part of the paper we propose to provide a philosophy to underline a correct approach to handle the phenomena_ To illustrate the key problems of the kinetics involved we start with statistics. As is well known, under certain conditions quantum statistics reduces to the classical one which, though self-consistent, has a limited range of validity_ Its initial formulation led to what is known as Gibbs’ paradox [ 111, i.e. an increase of entropy in the case of identical gases mixing at the same temperature and pressure_ In solving this paradox Gibbs displayed admirable intuition_ He assumed that permutation of similar particles (molecules) does 1101 result in new configurations_ This idea actually corresponds to the quant+rm-mechanical presentation of identical particles T_ The particle identity requirement affects ! It should be noted that the necessity

of taking into account the particle identity has been clearly demonstrared recently for ionic conductivity of imperfect crystals [ 121 and glasses 1131.

0301-0104/83/0000-0000/S

03.00 0 1983 North-Holland

considerably an adequate mathematical formalism and its predictions_ Everyone will agree that a statistical theory violating the principle of particle identity should be regarded as unacceptable_ This also goes for the kinetic theory since. broadly speaking_ kinetics is the time development of statistics_ What has just been said may seem rrivial but it must be admitted that the fundamental role of the identity principle of similar particles is not appreciated entirely in the kinetics of diffusion-controlled reactions_ For instance_ the reaction -4 + B * 0 describes creation and recombination of pairs of Frenkel defects (i.e. of vacancies (A) and interstitial atoms (B) in crystals). Since

Frenkel

defects

are created

under

irradiation

with a considerable spatial correlation within the socalled geminare (genetic. proper) pairs AB. for small defect concentrations the mean distance between different pairs greatly exceeds that within pairs. Obviously the early monomolecular recombination stage is mainly due IO correlative recombination whereas the second bimolecular one seems to result from defects from different pairs recombining as they mix during diffusion. Consequently the kinetics of the monomolecular stage is directly determined by defect spatial distribution within geminate pairs over relative distances and so a special DF for the correlated pairs and a kinetics equarion for it have been used in well-known papers [ 1,2,5,6] (so-called asynmemk approach). However, for a quarter of century it has escaped notice that such a DF is JIO~ observable since similar defects in different

E. Kotomin,

480

IT_KuzovkovfSome

problems of recombination kinetics

geminate pairs are identical_ This becomes more obvious when correlated pairs are partly mixed under high irradiation doses and/or during diffusion. The DFs used in the abovementioned papers actually amount to enumerating defects, thus introducing many-particle probabihty densities to find at a given time t the interstitial k at rk etc. Since all similar defects are identical, there is no need to enumerate them. At best it leads to complicated combinations [4] but as will be shown in section 3 it can also lead to physical paradoxes. We organize this paper as follows. Section 2 is to deal with the development of the formalism consistently taking into account similar defect identity. (Preliminary results were given in ref. [lo] but there we confined ourselves to the joint densities only.) The advantage of our approach consists also in incorporating spatial correlations of sinziiar defects which have been neglected in earlier theories [l-9] *. Based on this, the choice between alternative symmetric and asymmetric approaches is discussed in section 3.

2. Basic equations First we consider the recombination kinetics A + B --f 0 for two particles with equal diffusion coefficients DA = D, = D. (We use below “particles” and “defects” as synonyms.) The corresponding kinetic equation reads ]41 dB’/ar = D(V;

+ V;) h’-

o(]rl

-t,I)

IV,

(1)

where W = W(r, , r2; r) is the probability density to find particle I (A) at r1 and particle 2 at r2 at time t. For the reaction A + A + B the kinetic equation remains the same, however the identity of similar particles must be taken into account here, i.e. the solution of eq. (1) must be a symmetrical function of r1 and r7 f. This can be achieved formally by symmetrizing the initial particle distribution [4] but it is tzuf * It is generally assumed [ l-91 that because the similar defects do not interact they are not spatially correlated. It will be shown below (see also ref. [IO]) that such a correlation nevertheless emerges due to recombination of dissimilar defects. * It has been consistently taking into account’in ref. [ 141 using for the reaction A + A + B the secondary quantization formalism.

all. Since the particles cannot be enumerated (marked) [ 151, the interpretation of IU should be changed too. 4 For a system of identical particles IV is the probability density to find any particle A at rl and another one at r7 at time t which differs from the definition of W for the case of dissimilar particles A and B. Due to similar particle identity the problem - what is the probability of finding particle 1 at r, , etc. is meaningless (cf. refs. [ 1,2,5]). In other words, the identity principle restricts both a class of solutions of the kinetic equations (due to the symmetry condition) and a class of observables. In our opinion the many-particle (defect) DFs entering the kinetic equations of the asymmetric approach [ 1,2,5] contain an excess information and a more adequate formalism must operate with the “population numbers” [lo] indicating the mean number of particles within a given volume (see below)_ Let us divide the entire system’s volume Y into a great number N of equivalent cells. Each cell with a volume u can contain no more than a single defect. (There is an analogy with the lattice gas model in the theory of phase transitions [ lS]_) We shall make finally a limiting transition u * O,Nd 00 (NV = v), which corresponds to the traditional continuous approximation valid for relatively small defect concentrations. Let us put the population number vr into accordance with every cell defined by a vector r. The possible cell states depend on the type of reaction. Thus three population numbers u = (Y,f3, y indicating the cell occupied by particles A, B or an unoccupied cell, respectively, must be used for describing the reactions A+B-,O,A+B+O+B,A+A-,B.(Thereaction A + B G C, in its turn, needs the use of a fourth population number describing defects C.) In terms of the described cell model the defect motion and reaction are defined uniquely by a change of the complete set of population numbers, { vJN = description of the vrl 9--a %jv- A complete probability defect system is given by the many-particle DF P( {v,)~) which is nothing but the probability to find the system in an arbitrary physical state {v,}~_ The basic problem is to obtain the kinetic equation for its time development. The recursive relations (2) penkits

US to

obtain

the DFs of lower order. Let us

introduce now (II + n’)-point continuous

limiting

transition

PI*,,*4{rlll ; vl,I*)=

DFs pll,Jl, (assuming u -+ 0)

~-(Jz+‘JwbJJ1~

u$4’J

the

-

(3)

Here P( {~l,},~, (~,~}, ,) gives the probability to find particles A in II cells with the coordinates ri (i = 1, ___Jz) and particles B in JZ’ cells with the coordinates ri v = 1 , ._.n’). The physical meaning of the P,~JIvis self-evident: pII ,Iadr, _.. dr,, is the mean number of configurations &th Jz particles A in unit volumes d~i at fi (i JZ) and JZ’ particles B in unit vohmles dri at ri ;“;--, ... II’) [ 15]_ Thus, P(crJ yields the probability to find a particle A in a cell (i.e. the meLln number of particles A, since a single particle only can be situated in a cell). Therefore p1 o(r;) =P(c+)/u is the density (concentration) of parti‘cles A at r. Similarly P~,~(T~;r?) is thejoirzt density of two particles A and B, etc. It can be shown that eq. (3) describes a set of macroscopic (II + n’)-point densities (not probability densities) which are nothing but averaged products of powers of the macroscopic densities P~.~, p. 1 taken at different spatial points [ 151 *. These DFs’are, in principle, observable unlike the commonly used DFs of the asymmetrical approach ascribing numbers and “birth” dates to all particles involved [ 1,2,5] _(The latter DFs are used sometimes only as an intermediate step to obtain many-point densities.) The many-point densities do not obey recursive relations similar to eq. (2) but are normalized through their asymptotic

values,

e.g.pl l(r+m)=p, opo 1 =G,etc. Con’sider now the ;eac&ons A + B + 0 and A + B -+ 0 + B (energy transfer to instant sinks (energy acceptors) B) and contributions into time development of the DFs due to possible processes in a defect system. (i) Particle recombirlation. For the reaction A + B + 0 one gets

where a(r -r’) E a(lr - r’l) is the recombination rate of particles A with B at r and r’. The first term in eq. (4) describes all the possible ways how II particles A can recombine with II’ particles B, whereas the other terms describe the recombination of particles B with A (and vice versa) not belonging to a set of (JZ + ~1’) particles_ When dividing eq. (4) by u@+~~‘), we arrive at

The limitation in eq. (4) rhar vectorsr,I,l and r;,m,l cannot coincide with (r},, . (r’},, 1 becomes unimportant in the continuous approximation containing inregrals instead of cell sums. The abovcmenrioned changes of the DFs due to rrcombinarion correspond esacrly to those in ref. [S] _ In the case of the A + B -+ 0 + B reaction one must omit the last term in eqs. (4) and (5) since particles B do not disappear. (ii) Particle crearion. When considering crrarion of single particles A or B as well as pairs XB by irradiation in unoccupied cells. one gets for rhe reaction A +B+O

=

5 5 r_C(ri-

i=l j=l

rJ P({cq3:,.

{Pr’3,**)

(6)

-Ict,

II

,g

(J@,*+l

- r~)p(CQJ+

1; IPr’IJI’>

.

-(4)

* We used earlier [ 101 the population nunlber fomMsnl but witbout nlaking tile lilniting transition v - 0.

Here p(lr - r']) is the probability to create a particle A at r and B st r’ at time f. {o;}:, indicates that c+~ should be replaced by -yri in the population numbers {c9fJI_In the limiting case u --) 0 we assume ~(r - r’) = u’pf(r -r’). where p is the irradiation intensity. f(r) (if(r) dr = 1) is the initial distribution function of particles in just created geminate pairs.

482

E. Koronrin. V. Kuzovkov/So?ne problems o/recombination kincrics

For the reaction a/‘( (oJ,,s

{P/l’,m

=&

P(rl)fJ(ia,ll,s

A + B l- 0 + B one gets

IPfl,,‘)

9

(7)

wilh p(q) = up. Eqs. (6) a~id (7) amlain rhc DFs where or1c or two population nwnbcrs v = 7. Lcf us

processes, is cut-off by making use of the well-known superposition approximation by Kirkwood for three-

rewrite cq. (2) in the form (Pr’I”‘* “/r) = ~‘({(u,I” 9 {P,*~,J’)

f’({+I”*

For the homogeneous system (V + -) the one-point DFs pl o and pg , (= concentrations of particles A and B) are i;idepend&t of the coordinates, whereas the joint DFs p2.0 = X, , pqa2 = X2, p 1 , = Y (describing similar Xt,2 and dissimdar Y defeci pairs) are dependent on the relative coordinates r = Ir - r’l only. The hierarchy of equations for the time development of DFs obtained by summation of all the above-given contributions of creation, recombination and motion

point DFs

(8)

p&r;

When dividing eq. (7) by u(,,+,,‘) and taking into account eq. (3). we can rewrite its right-hand side in the

z

,r>)

P1.1(r;r;)Pl.l(‘;rS)P().2(:r;,rS) -_-_ PI,&) p0.,(7\) P~.J(;~;)

.

(12)

form p,,.,,~ - 0(v), whcrc O(u) tends to zero when u + 0. In other words, the population numbers v = 7 in eqs.(6) %,‘,

and (7) could be omitted.

Eq. (6) now reads

*Iat

(9)

where {r}!, indicates that the vector ri is omitted in this vector set. Eq. (7) becomes

ap,,,#r

=LJ ‘& 4, _tJEr)~‘.

VI,‘,)

.

(10)

Eq. (IO) corresponds also to that in ref. [S]. (iii) Particle motion. In the case of incoherent (hopping) diffusion of particles the change of the DFs is

aP,,,,,m =DA

5 vfp,,,I+D

i=l

5

Bj=l

VI+ “,,, a , (11) 1

where D A, D, are diffusion coefficients. Eq. (11) can also be obtained when describing diffusive motion as a random walk in a lattice in the limiting case u + 0.

3. Two alternative appronches to the theory of diffusion-controlled reactions At present there are two main approaches to the theory of the diffusion-controlled defect rccombination A t B + 0, the so.called asy~w?tetric and sJ’mmenic ones, first presented by Waite [ 1.21 and AntonovRomanovskii [3J (see also refs. [5.6] and refs. [4,8,9], respectively f). In both of them the spatial correlation of only dissimilar defects (A-B) is taken into account, whereas similar defects (A-A, B-B) are assumed to be randomly distributed in a medium. Thus these theories employ single densities (= macroscopic defect concentrations CA and cn) and joint densities for dissimilar (I’) and similar (A’, ,X2) defects. For the latter densities X, =X,=,‘(cA =‘+ = c) is assumed due to their random distribution. In order to decouple the infinite hierarchy of equations for many-point densities the Kirkwood superposition approximation (I 2) is employed in both theories. The main difference between these theories is that in the symmetric one by Antonov-Romanovskii a singZe joint density is employed, whereas in the asymmetric one by Waite fwo joint densities of dissimilar defects are used. These two DFs Y, and Y,, are used to distinguish between the initially correlated geminate l

Some ideas of this section were discussed earlier by us in refs. [ 10.16j.

pairs AB and pairs of uncorrelated defects from different randomly distributed pairs. A relation between tlmc tlleories was cstablislled recently [G]. It 113sbeen shown that they arc equivalent only if the initial correlation within pairs is negligible. In contrast to the concept of correlated distribution in wllich a nxasure of correlation is uniquely described by a deviation of 111~DF frown its asymptotic valtrc c* (Y > c2 means a positive correlation. but Y < Cz a ncgativc one), the concept of iuitially correlated (geminate) or uncorrelated pairs secn~ to be ratller vague since tluxe is IIO criterion of distinguisllhlg between them. The coexistence of two alternative theories for tlle sanle reaction is clearly unsatisfactory. Both of tliem are used for tile interpretation of experimental data, sometimes simultaneously [ 171. The purpose of tllis section is to demonstrate the advantage of an improved symmetric approach (employing tluz A + B + 0 reaction as an example). 3. I. hproved

symmetric approach

The set of kinetic equations for describing diffusion-controlled defect recombination (annealing) incorporating spatial correlations of both similar (A-A, B-B) and dissimilar (A-B) defects due to instantaneous recombination wit11the crear-cut radius R, reads (see eqs. (4) and (11) as we!! as ref. [lo]) acjat = -4nR;DY’(R,)

= -Q

,

(13)

ax,jat = m,v~x, --

cpx,r+Ra rRac3

s

Y(r’)r’dr’

,

v= 1,3 _

[r-R,1

(14)

ayjat = DvzY --

41’ rR,2

r+R, s r-R,

x(r’)r

dr’ ,

r>R,

,

(1%

Here cA = cg = c, Y(r strongly correlated defects within geminate pairs. The mean distance R, within these pairs is of tfx order of R, whereas between different pairs it is R > R,. It is clear that for t < Rt/D monon~olecular ~nneslinp takes place and if R, + R, all defects recombine. i.e. disappear within tlleir own pairs. Ir ~neans thar for I + R2jD the term describing the three-parricle correlations should tend to zero at r < R. This condition is fulfilled in eq. (15) since XV G 0 a~ r