Diffusion in generalized lattice-gas models

Transcription

1 PHYSICAL REVIEW B VOLUME 58, NUMBER Diusion in generalized lattice-gas models O. M. Braun Institute o Physics, Ukrainian Academy o Sciences, UA-50 Kiev, Ukraine DECEMBER 998-II C. A. Sholl Department o Physics and Electronics Engineering, University o New England, Armidale, NSW 35, Australia Received 0 March 998 A technique has been developed to calculate the eact diusion tensor or the lattice-gas model in the low-concentration limit or any comple elementary cell. The analysis takes into account the detailed structure o the elementary cell and the structure o the diusion barrier. The technique is also applicable to non- Marko diusion and, consequently, can include polaronic and inertia eects. The calculation reduces to simple matri-algebraic operations which may be carried out analytically in many cases. Applications to some one- and two-dimensional models are described, including disordered systems and non-marko diusion, in structures with comple elementary cells. S X I. INTRODUCTION Diusion is a undamental physical process which is o signiicant practical importance in applications such as crystal growth, heterogeneous catalytic reactions, emission electronics, microelectronics, etc. see, or eample, Res.,, and reerences therein. One widely used approach to diusion is based on the lattice-gas model. This model assumes that atoms occupy ied lattice sites and can undergo jumps to other vacant sites with some probability. When the system consists o only a single atom which undergoes a random walk on the lattice a key variable o the model is the conditional joint probability g(l;tl 0 ;t 0 ), which is the probability that an atom that was at the site l 0 at time t 0,isatthe lattice site l at a later time t. The standard approach to the diusional problem assumes that the random walk o an atom corresponds to a Poisson-Marko process, so that the distribution g(l;tl 0 ;t 0 ) obeys the master equation e.g., see Res. 3,4 t gl;tl 0 ;t 0 l;l gl;tl 0 ;t 0 l ;lgl;tl 0 ;t 0 l l;l gl;tl 0 ;t 0, l where l;l is the probability o the transition l l per unit time, and l;l ll l l ;l l;l is the transer matri. The matri l;l should satisy the constraint l l;l 0, which ollows rom the conservation o atoms. The jump rates l;l are usually related to the corresponding energy barriers through the Arrhenius law. Because the dynamics o the lattice-gas model is purely relaational, it is useul to take the Laplace transorm with respect to time, ḡz 0 dt e zt gt, Rez0, so that Eq. reduces to z ll l;l ḡl;l 0 zgl;t 0 l 0 ;t 0, 3 l where the right-hand side o Eq. 3 incorporates the initial condition. For a periodic array o sites, it is convenient also to perorm the spatial Fourier transorm, g (k) l epik(l l 0 ) g(l;l 0 ), where the wave vector k is in the irst Brillouin zone. Assuming the initial condition g(l;t 0 l 0 ;t 0 ) ll0, Eq. 3 becomes z k ḡ k;z. Equation 4 has a trivial solution which ehibits diusional motion, i.e., in the limit t the mean-square atomic displacement is proportional to time. In particular, in the simplest model where the atom can jump only to the nearestneighboring sites with a rate, and all jump directions are equivalent i.e., the model is isotropic, the result is r y Dt or t, where is the dimensionality o the system. The diusion coeicient D in this case is equal to Dqa, where q/ is the geometrical actor, and is the coordination number the number o neighbor sites allowed or a jump. Throughout this paper we will only consider diusion in one and two dimensions eplicitly, oten with surace diusion in mind, but the techniques discussed are equally applicable to three-dimensional systems. The above approach has been studied in detail by Kutner and Sosnowska 5 and Kehr et al.; 6 see also the review articles o Dieterich et al. 7 and Haus and Kehr. 8 It may be etended straightorwardly to describe anisotropic lattices and to include jumps to more distant neighbors. In addition the method has been generalized to describe non-marko processes, 9,8 where the probability o a jump depends on the history o previous jumps. This theory has assumed that the elementary cell o the lattice is primitive. Oten, however, the lattice is characterized by a unit cell with several inequivalent sites. Although there are some particular studies o models with comple /98/58/48700/$5.00 PRB The American Physical Society

2 PRB 58 DIFFUSION IN GENERALIZED LATTICE-GAS MODELS 4 87 unit cells, 5,6,8 there is no general solution to this problem. The aim o the present work is to develop a systematic and practical method or evaluating the diusion tensor or the low-concentration limit o the lattice gas model or any comple unit cell. The approach o Bookout and Parris, which uses the mapping o a random-walk problem onto the imaginary-time Schrödinger equation, is a possible approach but it seems to be more complicated than the method described below, and it does not allow a direct generalization to incorporate non-marko eects. A quite direct method o obtaining the diusion tensor or Bravais lattices is to take the continuum limit o Eq. see, or eample, Re. 0. This leads to the diusion equation with the diusion tensor given by D ij l,i l,j, where the summation is over the allowed jumps and l,i is the i Cartesian component o the jump o length l which occurs with requency. However, or comple unit cells the continuum limit o Eq. gives a set o coupled partial dierential equations, rather than a single diusion equation, because o the inequivalence o sites in a cell. It is not thereore possible simply to deduce the diusion tensor in the same way as or the Bravais lattice case. Another useul approach to diusion is to rewrite the long-time equation Dt in the orm 5 D N t N t a 0 corr, 6 where a 0 is the mean distance between the sites, N is the total number o jumps o the atom in the time t, is the average jump rate, and corr is the correlation actor deined as corr /(Na 0 ). For a system with a comple unit cell corr, and the equality corr only holds or a Bravais lattice. The average jump rate is straightorward to calculate or a comple structure, but the correlation actor corr is not. A calculation o the diusion tensor, as described below, combined with a calculation o the average jump rate provides a means o obtaining the correlation actor through Eq. 6. The method developed here to ind the diusional tensor or a comple unit cell is based on the properties o the solution o an equation o the orm o Eq. 4 in the small k limit. The problem is reduced to a set o algebraic equations, which in many cases can be solved analytically, and in any case may simply be solved with a computer. In addition, the approach may easily be etended to study non-marko processes as well as Langmuir-type lattice-gas models. The paper is organized as ollows. The technique is described in Sec. II. Then in Sec. III it is applied to the onedimensional lattice, where we obtain the eact result or any lattice structure, including disorder. In Sec. IV we consider some applications to non-markov processes, including the triangular lattice and the split 0 surace o the body centered cubic crystal. Finally, Sec. V concludes the paper with a short discussion o the results. II. TECHNIQUE A. Non-Bravais lattice The sites in a periodic structure with s sites per unit cell may be labeled by two indices l and, where l labels the elementary cell and labels the dierent sites within the cell,,...,s. In this case Eq. will have the inde l replaced by two indices l and, so that the master equation or the joint probability takes the orm t gl,;tl 0, 0 ;t 0 l,;l, gl,;tl 0, 0 ;t 0 l, l,;l, gl,;tl 0, 0 ;t 0. The limit t o the solution o Eq. 7 must be the equilibrium distribution, limgl,;t N lat, 8 t where is the average thermal population o the site, and the actor N lat (N lat is the total number o unit cells in the system is introduced or convenience. In order to deine the parameters l,;l, in Eq. 7, we associate with each site within the unit cell an energy. The equilibrium occupation numbers should satisy the Boltzmann distribution Z ep, Z ep, 9 where /k B T, T is the temperature, k B is Boltzmann s constant, and the normalization actor Z corresponds to one particle per elementary cell s. s 7 0 The simplest but not unique way to satisy the condition 8 is to eplore the detailed balance condition e.g., see Re. l,;l, l,;l,, which leads to the ollowing constraint or possible values o the rate parameters: l,;l, l,;l, / l,;l,ep. Otherwise the parameters l,;l, are arbitrary; their values are determined by the activation energies o the transitions. The transer matri becomes l,;l, ll l,;l, l,;l,. l, From the periodicity o the lattice it ollows that l,;l, ll,;0,, and we may perorm the Fourier transorm with respect to the cell inde to obtain ;0 k l epikll 0 l,;l0, 0, 3

3 4 87 O. M. BRAUN AND C. A. SHOLL PRB 58 g ;0 k,t epikll 0 gl,;tl 0, 0 ;0. 4 l Taking the Laplace transorm, with the initial condition that the atom was at the site 0 within the cell l 0 at t0 gl,;0l 0, 0 ;0 ll0 0, we obtain the transormed rate equation z ; kḡ ; 0 k,z 0. B. Eigenvalue problem 5 6 To solve Eq. 6 it is convenient to introduce a new matri (k) with elements ; k / ; k /, 7 so that Eq. 6 takes the orm z ; k / ḡ ; 0 k,z / The matri (k) or each k is the Hermitian square ss matri, ; (k) * ;(k). The eigenvalues and eigenvectors o (k) are the solutions o k ukk uk, 9 where u(k)u (k) is a vector with s components. Labeling the s eigenvalues by the inde, Eq. 9 may be rewritten as ; ku k k u k,,...,s. 0 Because the matri is Hermitian, the eigenvectors u () orm a complete orthonormal basis u k* u k, u k* u k. Using the eigenvectors u (k) and the eigenvalues (k), the solution o Eq. 8 can be epressed as ḡ ;0 k,z u ku 0 k* / / z k 0, 3 which may be veriied by direct substitution o Eq. 3 into Eq. 8. It has been proved 5, that the eigenvalues (k) satisy the ollowing conditions. a All (k)0, and (k) is an even unction o k. b One, and only one, eigenvalue (k) is equal to zero when the vector k is equal to zero assuming that the lattice cannot be separated into independent sublattices, so that the transer matri cannot be transormed to the Jordan orm. The lowest eigenvalue will be denoted (k). c The eigenvector u () (k) is a continuous unction o k around the point k0, and or small values o k the eigenvalue (k) has the epansion kd k D y k k y D yy k y, k 0. 4 As shown below, the coeicients D in Eq. 4 are the corresponding diusion coeicients. Thus, the calculation o diusion coeicients reduces to evaluating the k 0 epansion o the lowest eigenvalue (k). The above theory has ollowed the work o Kutner and Sosnowska 5 and Kehr et al. 6 The ollowing analysis describes a method which allows the coeicients D to be ound in general. C. General solution To solve Eq. 9 or the lowest eigenvalue, let us rewrite it in the orm k vk k vk, 5 where v is an unnormalized eigenvector. Consider diusion along the ais, i.e. put k y 0, and evaluate Eq. 5 in the limit k 0. The Taylor epansions o (k) and v(k) are where k 0 k k, k 0, k y 0, 6 vkv 0 v k v k, k 0, k y 0, 7 0 limk, k 0 lim k, k 0 k 8 lim k 0 k k. Substitution o Eqs. 6, 7, and 4 into Eq. 5 yields 0 k k v 0 v k v k which is satisied i D k v 0 v k v k 0 v 0 0, 0 v v 0 0, 0 v v v 0 D v Multiplying both sides o Eq. 3 on the let by v 0, and taking into account that v according to Eq. 30 and that the matri 0 is Hermitian, we get D v 0 * v v 0 /v 0 *v 0. The normalized vector v 0 is v 0 /, 33 34

4 PRB 58 DIFFUSION IN GENERALIZED LATTICE-GAS MODELS since it satisies the normalization condition and Eq. 30: 0 v 0 / / ; 0 l l l,;0, / l l,;0, 0, l where we have used Eqs. 7, 3,, and. The vector v can be ound by solving Eq. 3 0 v v 0, 35 which is a set o s linear equations or the components o v. I v is a solution o Eq. 35 then v cv 0, where c is an arbitrary constant, will also be a solution due to Eq. 30. Consequently, one component o the vector v, or eample (v ), may be chosen arbitrarily. This choice does not eect D because in Eq. 33 v 0 v 0 lim k 0 i ly k, / / ; k l l 0 l,l y,;0,, 0, ;l,l y,, 0, where we have also used Eqs. 7, 3,, and. Thus, the problem reduces to the calculation o the matrices 0,, and deined by Eqs. 8, the solution o the system o (s) linearly independent equations 35, and inally the calculation o D as D v 0 * v v 0 * v 0. D. Diusion coeicient 36 It remains to prove that D in Eq. 36 is indeed the diusion coeicient along the ais. To prove this, recall that according to the initial condition 5, the epression,0 g(l,;tl 0, 0 ;0) 0 is equal to the probability that the atom at the origin in the cell l 0 at t0, where it occupies the sites,...,s with probabilities so that,0 g(l,;0l 0, 0 ;0) ll0, will be ound in any o the sites in the cell l at a later time t. Consequently, the meansquare displacement along the ais is equal to t l ll 0 gl,;tl 0,;0., Using Eq. 4, this may be rewritten as 37 tdt as t. 39 Equating the right-hand sides o Eqs. 38 and 39 and taking the Laplace transorms, we get D z lim k 0 k ḡ ; k;z,, z Substituting the epression 3 or ḡ into the right-hand side o Eq. 40 and taking into account Eq. 4, the normalization 0, and also the equation limu kv 0 /, 4 k 0 we inally obtain DD. To ind the diusion coeicient in an arbitrary direction, we need to use the complete epansions instead o the epansions 6 and 7: k 0 k y k y k y k k y yy k y, k 0, vkv 0 v k v y k y v k v y k k y v yy k y, k 0, where y lim k 0 (k)/k k y, etc. Instead o Eqs. 30 to 3 we now get the equations and 0 v 0 0, 0 v v 0, 0 v y y v 0, v 0 v 0 v D v 0, y v 0 y v v y 0 v y D y v 0, yy v 0 y v y 0 v yy D yy v 0. Multiplying Eqs. 46 by v 0 rom the let, using the act that 0 is Hermitian so that v 0 0 0, and taking into account the normalization, we obtain D v 0 * v 0 v 0 * v, tlim k 0 k g ; k;t., 38 On the other hand, according to the deinition o the diusion coeicient, D yy v 0 * yy v 0 v 0 * y v y, D y v 0 * y v 0 v 0 * y v v 0 * v y. 47

5 4 874 O. M. BRAUN AND C. A. SHOLL PRB 58 FIG.. Calculation o the anisotropic diusion coeicient. Consider the atomic displacement along the direction, which is at an angle to the ais. From Fig. it ollows that r cos() cosy sin, so that the mean-square displacement along the direction is equal to r cos y sin y sin cos lim cos k 0 k sin k y sin cos k D t as t. k y g, k;t, Repeating the procedure described above, we obtain the anisotropic diusion coeicient in the orm DD cos D yy sin D y sin v 0 * cos yy sin y sin v 0 v 0 * v cos y v y sin y v v y sin cos. E. Jumps with a memory 50 We now consider the generalization o the technique to non-marko processes e.g., see Re. 9 and the survey o Haus and Kehr, 8 and reerences therein; namely, to random walks or which the memory is not lost ater each step, but only ater a inite number o steps. There are at least two reasons or such a generalization. a Inertia eect. Ater a jump, the inertia o an atom results in persistent motion in the same direction. Thereore, the probability o a jump in the same direction,, may eceed the probabilities o backward and sideways jumps. b Polaronic eect. Just ater an atomic jump, the surrounding substrate atoms do not immediately adjust to the new position o the atom. As a result, the energy o the atom in the new site will eceed the energy which it had in the old site beore the jump. This may result in the probability o the backward jump, b, eceeding the probabilities o the orward and sideways jumps. To generalize the above technique to include memory eects, we introduce a memory inde m j or each site j. For eample, in the case o the one-dimensional lattice, the memory inde could take two values m j, where the value m j corresponds to the atom arriving at site j rom the let-hand side, and m j corresponds to arrival rom the right-hand side. Thus, in the non-marko case the transition rates depend on three indices l, and m. Contrary to the Marko case, a proper choice o the values l,,m;l,,m cannot now be guaranteed by the detailed balance condition because this condition is not necessarily satisied or a non- Marko process. In the non-marko case it is necessary to ind the stationary solution g eq (,m) o the master equation l,,m;l,,m g eq,m l,,m g eq,m l,,m l,,m;l,,m, 5 and to test whether this solution satisies the Boltzmann distribution m g eq,mn lat. 5 Another problem in the non-marko case is that the matri (k) is not Hermitian. This situation is analogous to that or the Fokker-Planck equation, where the Smoluchowsky equation corresponds to the Marko case, while a more general Fokker-Planck-Kramers equation, which takes into account inertia eects, corresponds to the non-marko case. 3,4 However, the technique described above was based on two requirements: irst, on the completeness o the base o eigenvectors o the matri 0 which is satisied provided 0 is Hermitian at least in the limit k 0, and second, on the second order epansion 4 which ollows rom the law o conservation o atoms. I both these actors are satisied or a non-marko process, the technique can still be applied to ind the diusion tensor. III. ONE-DIMENSIONAL LATTICE As eamples o applications, we begin with a onedimensional Marko model with nearest neighbor jumps. The sites may be enumerated in increasing order by an inde i, and the atom can jump to the let-hand and right-hand sites with the rates i;i. The elementary cell o the lattice has lattice constant a and consists o s sites enumerated by the inde (,...,s) with site energies. The site inde i may be written as ils, where l enumerates the elementary cells. It is convenient to deine the rates, as ; l,;l, ;s l,;l,s, s; l,s;l, i,3,...,s, 53 and it is assumed that they satisy the detailed balance condition. It is also convenient to assume that the inde is cyclic, so that the value s is equivalent to, and 0 tos. Applying the theory o Sec. II gives

6 PRB 58 DIFFUSION IN GENERALIZED LATTICE-GAS MODELS v 0 * v 0 a ;s s, 54 v ;s w s a ;s s Z ep /, v 0,,...,s, 55 v s s; w a Z ep s /, v 0 * v a ;s s. where / is the residence time averaged over all sites. ; The diusion coeicient 36 is thereore s Da The result 58, 57 has been obtained previously by Kehr et al. 6 see also Res. 3 and 8 or a simpler valley model or which all barriers have the same height. The same result has also been obtained or the general case recently using a dierent technique see Re. 8, and reerences therein but the present method is simpler and more general. The result 57 enables the calculation o the correlation actor corr or this one-dimensional model. Using the deinition o corr, the general epression or the diusion coeicient given above, and calculating the average jump rate as it ollows that corr s s s ; ;, s 59 ; s ; ;, 60 taking into account that a 0 a/s. The epressions 58 and 57 or the one-dimensional model may be rewritten in the orm DD F, 6 where D a 0 0, a 0 is the mean distance between sites, 0 is some average transition rate, and the dimensionless correlation actor F is deined by the epression F e e 0 / ;, 6 where the operation is deined as s s. Taking the limit s, we can apply the epressions 6, 6 to a disordered lattice. The result 6, 6 has been obtained previously by Lyo and Richards 5 using another technique. According to Kramers theory, 6 the transition rate, is mainly determined by the activation energy, which is to be overcome by the atom, and depends on the latter eponentially,, 0 ep(, ) where the preactor 0 will be assumed independent o. The activation barrier is equal to the dierence between the saddle energy *, and the well energy. Writing the activation energy as ; ; * ;, 63 where *, is the saddle energy relative to its mean and is the well energy relative to its mean, the correlation actor 6 may be rewritten as F ep ep * ;. 64 In this ormulation the requency 0 in D is 0 ep(, ), which is the jump rate or the mean activation energy barrier. I the luctuations o the well energies are Gaussian so that Prob ep, 65 where is the dispersion, and the saddle energies luctuate in the same manner but with the dispersion *, then e d, Prob e ep, and, consequently, Fep * In particular, i we put *k B T, we obtain Fe, so that thermal luctuations in the shape o the potential lower the diusion coeicient by a actor o e in the case o the one-dimensional model. Mak et al. 7 have shown that Eqs. 6 to 6 are also true or a two-dimensional valley model or which all the barriers have the same energy but the well energies are random. IV. NON-MARKOFF JUMPS A. Bravais one-dimensional lattice In order to include a memory o one previous step into the theory, we associate with each site an inde m such that m or an atom arriving at a site rom the let-hand neighboring site, and m or arrival rom the right-hand site. Denoting the transition rate o an atomic jump in the same direction as the previous jump by, and the rate o the backward jump by b, the transition rates take the ollowing orm: l,;l,, l,;l, b, l,;l,, l,;l, b. The Fourier transorm o the transer matri is 68

7 4 876 O. M. BRAUN AND C. A. SHOLL PRB 58 k b e ika b e ika b e ika b e ika, 69 and the matrices 0, and are equal to 0 b b b b, b ia 70 b, FIG.. Non-Marko diusion on the triangular lattice. strength o the memory o the previous step, then D may be represented as the product o D a the diusion coeicient or uncorrelated random walks times the correlation actor Fcosh(h)ep(h). a b b. B. Triangular lattice It can be seen that while the matri (k) is non-hermitian, the matrices 0 and are Hermitian, and the technique o Sec. II may be applied. It is easy to show that v ia v 0, b b 0, and the diusion coeicient is now equal to 9 7 D a b b. 7 I we introduce a parameter h using the relations ep h and b ep(h), so that h determines the A second eample o non-marko diusion is the triangular lattice, or which the memory inde m can take si dierent values, m,...,6, corresponding to jumps rom the si nearest-neighboring sites. Taking into account the symmetry o the lattice, we introduce our transition rates b, b, and as shown in Fig.. Depending on the model under investigation, the ollowing variants may occur. a The orward jump model, where b b,so that the atom has a larger-than-average probability o making a transition in the same direction as the previous transition, while the probability or a transition in any other direction is reduced compared to the average value. b The reduced reversal model, where b b, so that the atom has a less-than-average probability o returning to the site visited at the previous step. c The backward jump model, where b b, so that the atom has a larger-than-average probability o returning to the previous site. Using the notation shown in Fig., ater a long but straightorward calculation we get or the transer matri (k) the epression e e b e k6 b e b e e e 6 e e b e b e b e b e e 6 e e b e b e b e b e e 6 e e b e b e b e b e e 6 e e e b e b e b e e 6 e, 73

8 PRB 58 DIFFUSION IN GENERALIZED LATTICE-GAS MODELS where e ep(ik a), e ep(ik a/ik y a y ) and 6 ( b b ). The vectors v 0 and v are now equal to v 0 6, / ia6 / v, 3 b b / / 74 and the isotropic diusion coeicient is given by the epression DD F, 75 where D 3 a is the diusion coeicient or the Marko model, and F describes the non-marko correlation F b b b b FIG. 3. Non-Marko model or the split 0 bcc surace. model, since polaronic and inertia eects could play a signiicant role, especially or jumps between the adsites within a cell. To include the memory o a previous jump, we associate with each adsite a memory inde m in addition to the inde, enumerating the sites within the cell. Let m or an atom arriving at a given site rom an upper site, m or arrival rom a lower site, and m0 or arrival rom the letor right-hand sites. It is now necessary to introduce ive transition rates,,, b, and b as shown in Fig. 3. It is useul to introduce a single inde m, which varies rom to 6, instead o the two indices and m. Using the symmetry o the model, the stationary solution g eq (,m) o the master equation 5 is /( ) and 5 /( ), so that the vector v 0 is equal to C. The split 0 bcc surace As a inal eample, we consider the 0 ace o the bcc structure which gives a rhomboidal lattice or an adatom, and suppose that owing to local substrate distortion around the adatom, the original single adsorption site is split into two symmetric sites separated by a small energy barrier. Such a distorted honeycomb lattice was introduced and studied by Ala-Nissila et al. 4 to described the H-W(0) adsystem in the ramework o the Marko random process. However, it seems reasonable to include also the memory eects into this v 0. Omitting a lengthy calculation, the matri (k) is 77 k e e b e e e b e b b e e e b b e e e 0 0 3, 78

9 4 878 O. M. BRAUN AND C. A. SHOLL PRB 58 where ( b )/3, ( b )/3, and e ep(ik a ik y a y ). The solution o the key equation 35 leads to the ollowing epression or the vector v : where and w v w w w w w, v y y w 0 y w y w 0 w y, w a A b / b b, w a A / b b, 8 w y a y A/ b, 8 Ai b. 83 As a result, we obtain the diusion coeicients b b D a b b, b D yy a y b, and their ratio is D D yy a a y b b b b b. 86 For Marko jumps this ratio reduces to D D yy. V. CONCLUSION 87 A technique has been developed to calculate the diusion tensor or the lattice-gas model at low particle concentration or any comple elementary cell and which can also include memory eects. The calculation reduces to simple matrialgebraic operations, with the only step that could hinder a complete analytic solution being the solution o the matri equation 35. For many cases analytic epressions or the diusion tensor may be obtained. Some one- and twodimensional models were analyzed as eamples, some o which have been studied previously by other techniques, while others such as the one-dimensional model with an arbitrary elementary cell and the two-dimensional non- Marko models, have not been treated previously. Although we have concentrated on one- and two-dimensional eamples, having in mind applications to surace diusion, the technique described is equally applicable three-dimensional models, so that it will also be useul in applications to bulk diusion problems. 5 9 The technique developed in the present work may be programed or a computer calculation in cases where an analytic solution is not possible, so that the eact diusion tensor or walks o a single atom may be ound in practice or a wide class o lattice-gas models. Moreover, the technique allows the study o diusion o a pair o interacting atoms, or a trio o atoms, etc., so that it may be used or the calculation o the eact diusion tensor or stochastic motion o adsorbed islands. ACKNOWLEDGMENTS O. Braun was partially supported by a grant rom the Australian Government Department o Industry, Science and Technology. A. G. Naumovets and Yu.S. Vedula, Sur. Sci. Rep. 4, R. Gomer, Rep. Prog. Phys. 53, C. W. Gardiner, Handbook o Stochastic Methods Springer- Verlag, Berlin, H. Risken, The Fokker-Planck Equation Springer, Berlin, R. Kutner and I. Sosnowska, Acta Phys. Pol. A 47, ; J. Phys. Chem. Solids 38, K. W. Kehr, D. Richter, and R. H. Swendsen, J. Phys. F 8, W. Dieterich, P. Fulde, and I. Peschel, Adv. Phys. 9, J. W. Haus and K. W. Kehr, Phys. Rep. 50, J. W. Haus and K. W. Kehr, J. Phys. Chem. Solids 40, A.R. Allnatt and A.B. Lidiard, Atomic Transport in Solids Cambridge University Press, Cambridge, 993. B. D. Bookout and P. E. Parris, Phys. Rev. Lett. 7, N.G. van Kampen, in Stochastic Processes in Chemistry and Physics North-Holland, Amsterdam, 98.

10 PRB 58 DIFFUSION IN GENERALIZED LATTICE-GAS MODELS J. W. Haus, K. W. Kehr, and J. W. Lyklema, Phys. Rev. B 5, T. Ala-Nissila, J. Kjoll, S. C. Ying, and R. A. Tahir-Kheli, Phys. Rev. B 44, S. K. Lyo and P. M. Richards, Phys. Rev. B 3, H. A. Kramers, Physica Amsterdam 7, ; P.Hänggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 6, C. H. Mak, H. C. Andersen, and S. M. George, J. Chem. Phys. 88, K. W. Kehr and T. Wichmann, in Disordered Materials, edited by D. K. Chaturvedi and G. E. Murch Transtec. Switzerland, 996, p. 5.