PHYSICAL RE VIEW 8 VOLUME 19, NUMBEH, 12 15 JUNE 1979

Stochastic theory of multistate diffusion in perfect and defective systems. II. Case studies

Uzi Landman and Michael F. ShlesingerSchool of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332

(Received 9 November 1978)

The time development of a number of physical systems can be described in terms of the temporal passageof the system through its allowable states. These states may correspond to spatial configurations, energy

levels, competing transition mechanisms, or correlated processes, The evolution of, certain of these systemscan be analyzed by mapping onto a continuous-time random walk on a lattice whose unit cell may containseveral internal states. In addition, we study the influence of periodically placed defects, which modify

transition rates, on transport properties in several condensed-matter systems. A propagator formalism is

reviewed where the matrix propagator (whose dimensions equal the number of internal states in a unit cell) is

renormalized owing to the presence of defects. Knowledge of the propagator allows the evaluation ofobservable quantities such as positional moments, diffusion coefficients, occupation probabilities of states,and line shapes of scattering from diffusing particles. The formalism allows the study of diverse transport

systems in a unified manner. We study the multistate diffusion of particles and clusters (specifically dimers)observed via field-ion microscopy on defective surfaces and derive expressions for observables pertinent to the

analysis of these experiments. The lattice structure dependence of particle diffusion in systems containing

defects which influence their neighbors is demonstrated. Next, the anomalous, non-Arrhenius behavior of the

vacancy diffusion constant, observed via tracer-diffusion measurements, is described in terms of competing

monovacancy mechanisms: nearest-neighbor and next-nearest-neighbor single jumps, and double jumps in

which two atoms jump almost simultaneously, collinearly, as suggested by recent molecular-dynamics studies.

We derive an analytic expression for the vacancy diffusion which provides a reinterpretation of fits toexperimental data. As a further application of our method we investigate non-Markovian transport due torelaxation effects. Finally, we provide an analytic expression for the scattering law, S(k, co), of quasielastic

neutron scattering and demonstrate the non-Lorentzian line shape of the scattered intensity for systems

containing defects and for correlated motion.

I. INTRODUCTION: TRANSPORT AND INTERNALSTATES AND DEFECTS

Studies of transport phenomena in condensed-matter systems provide information about thephysical parameters of the underlying mechanismsgoverning the motion. ' These parameters (ac-tivation energies, transition-frequency factors,structural factors, and correlation functions)reflect both potential and structural character-istics-of the system under study. The degree ofmicroscopic detail concerning the nature and themechanism of transport depends upon the experi-mental resolution and range of data, available andupon the method of analysis. The above is in-timately connected with such characteristic spatialand temporalparameters of the system as cor-relation lengths and relaxation times.

In order to achieve a level of description com-patible with the experimental resolution it is oftenconvenient (and in fact sometimes necessary) todecompose the transport process into kineticelementary steps. Often the stochastic evolutionof the system via transition between these stepscan be mapped onto a random-walk lattice withmultiple states in each unit cell (which we call theinternal states"). The number and type of internal

states reflect the degree of microscopic knowledgeof the process.

The mapping of kinetic steps onto a multiple-internal-state random-walk lattice serves as aunifying element in the analysis of a diverse varietyof transport and reaction phenomena. For exam-ple, atomic and molecular clusters adsorbed oncrystalline surfaces are observed, by means offield-ion microscopy (FIM), to exist in severalgeometric configurations. 4 'The cluster-centroidmotion traverses a random-walk lattice with sev-eral internal states per unit cell representing theallowed configurations. As discussed previ-ously, ""'a proper analysis of FIM diffusion data(positional moments and configuration occupationprobabilities) allows a determination of the in-dividual transition rates between configurations,thus yielding a spectroscopy of the internal statesof the cluster. Analysis which disregards the ex-istence of the above internal states (i.e., analysisbased upon the conventional semilogarithmic plotof the cluster-centroid positional variance o'(t)versus 1/@AT) is inadequate for the determinationof the pertinent diffusion parameters.

The mapping onto a multiple-internal-staterandom-walk lattice is not restricted to actual

19 6220 1979 The American Physical Society

STOCHASTIC THEORY OF MU I, TISTATK. . . . II.

distinct physical spatial configurations, but mayrepresent states of the system in a generalizedsense. For example, in an early study of surfacediffusion by Lennard-Jones' a model has beenintroduced based upon two states of the migratingparticle: one state in which the particle vibratesabout the equilibrium position and another in whichthe particle migrates (the mobile state). Thismodel can be mapped onto a random-walk latticewith two states in each unit cell (see Fig. 5, whichin addition to the two internal states includes adefect site). As a further application of our gen-eral procedure, in Sec. IV we investigate dif-fusion via monovacancy mechanisms. %e show thatthe deviation from the Arrhenius behavior of thediffusion constant at temperatures close to themelting point of the material, ' " can be explained"by including several comPeting monovacancymechanisms which are activated at these elevatedtemperatures. Again, the solution to this problemis achieved via a mapping onto a multistate ran-dom-walk lattice. As a final example we consider(in See. V) a mode of transport in which correlationbetween steps exists in the sense that the depar-ture of a particle (or a propagating excitation)from a site depends (in a time-dependent fashion)upon its last jump [i.e, for a one-dimensionalsystem the particle would leave a site with (dif-ferent} probabilities which depend upon whetherit arrived there from the left or right]. The ran-dom-walk lattice for the correlated motion is ob-tained by mapping the above events onto a lattice(see Fig. 9). In Sec. VI we evaluate the quasi-elastic neutron scattering line shape from dif-fusing species whose motion is correlated.

Most crystalline systems contain imperfectionto various degrees, due to a variety of sources.Defects occur either as an intrinsic property ofcrystals (such as thermally activated point de-fects, interstitials, and vacancies, which arepresent in the equilibrium structure of crystalsat above-zero temperatures"), or may be vol-untarily or involuntarily (contamination) introduced.

Defects and impurities are known to alter par-ticle transport in crystalline systems. " In ad-dition, they influence rates of surface-catalyzedreactions" and enter the theory of heterogeneous,active site-catalyzed, processes. " Thus it is ofimportance to understand the physical mechanismsinvolved in the above systems and to formulatetheoretical methods through which the charac-teristics of the defects and transport processescan be determined via the analysis of experi-mental data (FIM, tracer diffusion, quasielasticneutron scattering, Mdssbauer line shapes, andexcitation diffusion in molecular systems).

In the previous paper" (which will hereafter

be referred to as Paper I}, we have developedthe mathematical formalism of continuous-timerandom walks in multistate systems which containa periodic arrangement of defects. " Defects arecharacterized by modified transition rates fromthe allowable states compared to the ideal (defect-free) host-lattice case. The change is expressedin the distribution function governing transitions,which reflects the potential characteristics of thesystem. %'e show that the particle propagator is"renormalized" owing to the presence of defects.Knowledge of the propagator allows the derivationof expressions for a number of observable quan-tities [moments of particle or vacancy position,equilibrium occupation probabilities of sites, andquasielastic neutron scattering line shapes (Sec.VI)], in a unified fashion.

The analysis presented in this study is for sys-tems in which the defects form a (periodic) super-lattice. Examples of physical systems in whichsuch structures occur are transport in orderedalloy and multicomponent systems, ordered over-layer assemblies, "and ordered overlayer adsor-ption-surface systems. " For systems in whichdefects occur randomly, the coherent-potential-approximation (CPA) method may be applied. "'"

The paper is organized in the following manner:A brief summary of pertinent expressions derivedin Paper I is given in Sec. II. In Sec. III we dis-cuss several case studies of multistate diffusionon defective lattices and calculate the effect ofdefects and their concentration on the diffusionconstants and probabilities of occupation of sites.%e also show, in Sec. IIIB, that for systems inwhich defects inQuence transition characteristicsof their nearest-neighbor sites, the diffusion con-stant depends upon the lattice structure. In Sec.IV we propose and analyze competing monovacancydiffusion mechanisms as the origin of the observeddeviations from Arrhenius behavior of the dif-fusion constant as a function of temperature.Time-dependent correlated motion is studied inSec. V. In Sec. VI we derive a general formulafor the scattering law of neutrons scattered quasi-elastically from diffusing species. %e apply thegeneral formula to obtain non-Lorentzian lineshapes for single-particle diffusion in a three-di-mensional (3D) defective crystal and for a time-dependent correlated migration of a particle in1D.

II. SYNOPSIS OF MATHEMATICAL FORMALISM

The mathematical formalism of random walkson periodic lattices with multiple internal stateshas been described elsewhere. "' In Ref. 2

(Sec. V) and Paper I we have discussed diffusion

, 6222 UZI LANDMAN AND MICHAEL F. SHLESINGER 19

R'"(k, u ll„o)[I p(0)(k)y(0)( )]-1eik Io

where the superscript (0) indicates that thesequantities are for a perfect lattice, the matricesare of dimension m & nz, and the propagator hasbeen Fourier transformed over space (1-k) andLaplace transformed over time (r-u) [see Eq.(2.3) and Appendix A in Paper I].

(2.3)

on lattices containing a periodic arrangement ofdefects, treated as internal states. A differentmethod for treating propagation on periodic de-fective lattices, one which facilitates the cal-culations of basic transport quantities (in par-ticular for small defect concentrations), via arenormalized propagator, has also been discussedby us in detail in Paper I. In this section webriefly review the pertinent elements of the theoryof multistate transport on ideal and defectiveperiodic lattices.

IThe basic quantity of the theory, which char-acterizes the temporal and spatial (structural)dependence of transitions performed by the prop-agating species is the waiting-time density func-tion

e,, (I, I; ~) -=P„(1,1)y,;, „(~), (2.1)

where l denotes a lattice-site location and j rep-resents an internal state. 4 (;, , )(7.)dv. is the prob-ability that a transition will occur from (1,j) in thetime interval (r, 7 +d~), given that the particleobtained (1,j) at 7 =0. This function is not anexplicit function of (1,j), but is written in termsof parameters which characterize the state (1',j).These parameters reflect the potential character-istics of the state (1',j). In our studies we havechosen the following form for the waiting-timeprobability density function:

)(0)

g,!"(7) = )).,!"e 'd' ', p (I,)(7) = &), e "J, (2 ~ 2)

where X,'.", X,. are the total transition rates forleaving internal state j, associated with ideal anddefective sites, respectively; P;,.(1, 1') is the prob-ability that, given the occurance of a transitionfrom state (1',j), it will lead to the occupation ofstate (l, i). In general, p, i(1, 1') may also dependon the time T at which the transition occurs (suchan example is discussed in Sec. V).

The semi-Markovian evolution of the transportsystem can be studied in terms of the probabilitypropagator R,, (1, t

l1„0), which is the probability

density of reaching (1, i) exactly at time f, havingstarted at (l„j) at f = 0, independent of the numberof steps taken.

On an ideal lattice, with m internal states perlattice site, we have found in matrix notation that

R(k, u;z) =JR(0)(K,u;z)] '

—AD(K, u) e '

where the components of K are

(2.4)

0 & K„(2 /7io, 0 &K, (2((/R, 0 &K,(2)(/y,

0 =(o.Pyl„l I,) ',3nd the defect matrix D is given by

D(K, u;z) =z Q[p, l „(K)y, i „(u){d]

p(0) (K)y (0)(u)] (2.5)

where the sum extends over all defects in one ofthe equivalent groups of defects which repeatperiodically (denoted by {df). Equation (2.4) forthe probability propagator, in analogy to otherusages of propagator techniques, can be inter-preted as a renormalization of the ideal-latticepropagator R "'(k, u; z) by a "self-energy" cor-rection due to the differences between defectiveand ideal lattice sites. 'The conditional probabilityP of being at state (l, i.) at time f [starting from(l„j)at t =0] is related to the propagator R via

I'„(I f ll„o)T

&)„,.&(v')d ']d&

R;,(1, f —) l1„0)C'(-, ;)(0)dv, (2.6)

where the function C (I, )(7), defined above, accountsfor the event that the system rea, ched (f,i) at anearlier time t —7 and remains there at time t.

Physical quantities of interest, such as spatialand temporal moments of the probability distri-bution and equilibrium occupation probabilities ofstates, can be derived from the above as dis-cussed in detail in Paper I [Eqs. (2.33)-(2.42),and Sec. III and IV of Paper I]. In the followingwe investigate several physical transport systemsusing the above method, and discuss methods ofanalysis of experimental data.

For a lattice with a periodic arrangement ofdefects, forming a defect superlattice of unit cellsize (nl„, Pl„yl, ), where I„, l„and I, are the host-lattice unit-cell length and n, P, and y are integers,we have derived in Paper I the equation for theprobability propagator R [see Eqs. (2.23) and (2.27)of Paper I] valid for all k and u values. While thecomplete expression for R is rather complicated,it may be observed that for calculations of physicalquantities in the long-time (diffusion) limit a re-duced form of the propagator can be used. Theprobability propagator in its reduced form is givenas [see Eq. (2.28) in Paper I],

STOCHASTIC THEORY OF MULTISTATE. . . . II. 6223

III. CASE STUDIES OF MULTISTATE DIFFUSION

In this section we discuss several condensed-matter diffusion systems, and investigate theeffects of defects and internal states on the dif-fusion mechanisms and implications of the abovein the analysis of experimental observations.

A. Single-particle diffusion on a 1D defective lattice

The migration of particles and clusters on solidsurfaces under controlled conditions can be ob-served onanatomic scale, by means of the in-genious methods of field-ion microscopy (FIM).'These measurements provide quantitative dataabout the diffusion process in terms of particle(or cluster-centroid) spatial moments [mean po-sition (in case of a biased experiment"') andvariance] and the equilibrium occupation of sitesin the allowed configurations (in the case of amigrating cluster). From FIM data, one can de-termine such characteristic quantities of thediffusion mechanism as activation energies, fre-quency factors, diffusion constants, informationconcerning pairwise interaction potentials betweenadatoms, and cluster dissociation energies. "Consequently, FIM constitutes a microscopicspectroscopic tool for the investigation of solidsu rfaces.

Until now, most" FIM diffusion studies havebeen performed on perfect-crystal planes, andthe one-dimensional and two-dimensional motionsof particles and particle clusters of various sizeshave been studied. We have previously suggestedmethods for the analysis of FIM data and havegiven a procedure by which transition rates ofindividual steps (between internal states) of themigration mechanism can be extracted from theexperimental results. " In the following we in-vestigate the motion of a single particle on de-fective periodic surfaces and suggest the possi-bility of obtaining, from FIM measurements onsuch surfaces, information about characteristicparameters of the defects.

We consider first a one-dimensional (1D) motionof a single particle [such as that observed in thechanneled motion of a W atom on the (211) face oftungsten"]. As seen from Eq. (I2.53), the posi-tion variance for particle diffusion on a 1D latticeof spacing l and transition rate a for jumping tonearest-neighbor sites is given by o'(t) =al't. Ifo'(t) is measured for known l for a sequence oftimes, the rate a can be found. When the experi-ment is performed at various temperatures and anactivated Arrhenius form is used for a (i.e.,a = v, e ~'+~ where E, is the activation energyfor a transition and T is the absolute temperature),

E, and the frequency factor can be determined froma plot of ln o'(t; T) vs (& T) '.

When defects are introduced substitutionally"into the lattice in a periodic manner at locationsnl (where n is the period of the defect superlattice)

o'(t) =nl'abt/[b(n —1) +a], (3.1)

where b is the rate of leaving a defect. (Note thatthe effect of the defect has been assumed to belocalized, although more general cases can betreated. ) If the rate a is known from the perfect-lattice case, b can be found from the relation

ao2(f)/[anl't —(n —1)o'(t)] = v e & ~& =b . (3.2)

Consequently, data obtained at different temper-atures permit the determination of the defect pa-rameters v, and E, [from a plot of the logarithmof the left-hand side vs (k~T) ']. Note that theusual practice of plotting logo' vs (k~T) ' will notyield. v, and E, as the intercept and slope of astra, ight line, since only a and b, but not o', arein an activated form.

When the rate a cannot be determined indepen-dently, a relation between the rates a and b, ob-tained via detailed balance for occupation of sites,[see Eqs. (I2.42a) and (I2.42b)] can be used:

P„(defect) = nlimuR, (k =0, u)C„(u), (3.3)

where, in our case, 0=1/n. The above yields

aP„(defect) = lim u 1—nb „o a+u

t' b

b+u

a/na/n+b(l —1/n)

' (3.4)

Thus, if the equilibrium population of the defectsites is recorded, the above relation supplementsthe equation obtained from the measurement of thevariance in position, and a and b can be deter-mined from these two relationships.

B. Single-particle diffusion on defective 2D latticesof different structures

- We examine now the diffusion of single particleson two-dimensional square, hexagonal, face-centered-square and -triangular lattices in thepresence of defects. We show that when a defectinfluences the transition rates of its neighbors,a structure (topology) dependence of the diffusionrate emerges. If the defect does not influence itsneighbors, the diffusion rate on the differentlattices is the same as long as the transition ratesfor leaving normal and defective sites are common

6224 UZI LANDMAN AND MICHAEL F. SHL'ESINGER 19

to them, and the jump lengths and defect concen-trations are the same.

To motivate our analysis we consider first ran-dom walks of a single particle on perfect latticesof different structure (with the same bond lengthand the same rate of leaving sites). Obviously,under these circumstances the probability for themean number of steps n taken in time t is the samefor these lattices. 'The mean-square distancetraveled after n steps is

(3.5)

Dss4/ ~ I 4/8 8

8

,8A

.,qxD..qx,4/ ~ I 4/

~ s / 5D

4/ ~ I 4/

~ s&/ ~ s 4/

n

(R„') = g r', =n.(r,)'=nL'f,=l

(3.6)

where the average is over all possible paths, withequal probability of jumps along connecting bonds,and r,- is the displacement caused by the ith step.Since the jumps are uncorrelated,

ly

8 8 8 8

8 A

ixoc )

independent of the lattice connectivity (structure). 'If equivalent localized defects in equal concen-

trations are placed periodically in equal concen-trations on these different lattices, then the dif-fusion constant for particles on these lattices willremain independent of lattice structure. This isdue to the fact that the probability of occupying adefect site as f -~ (the diffusion limit) is inde-pendent of the lattice structure for. the cases de-scribed above, as can be seen from Eq. (3.3).Thus, in the limit of t-~, the same number ofjumps occur on each lattice, with the defect- andnormal-site equilibrium occupation probabilitiesbeing the same for each lattice. When the defectseffect their nearest neighbors, so the effectivedefect concentrations are no longer equal, astructural effect will appear, as shown in thefollowing examples.

a. Rectangular lattice. Consider a defectiverectangular lattice (Fig. 1) in which a migratingparticle leaves a defect site with rate D, the fournearest neighbors to the defect site with rate B,

FIG. 1. Random-walk lattice for motion of a singleparticle on a rectangular lattice with unit cell (l „,l ~).The total rate to leave a normal site, denoted by e, isA, with jump probabilities q„ to the right or left and q~for up and down transitions. Defect sites, denoted by ,are substitutionally placed, forming a defect superlatticewith unit cell (O.l„,Pl~). The total rate of leaving a de-fect site is D. jn addition, we consider that the nearest-neighbor sites to defects, denoted by 0, are influencedagd have a total rate of leaving of B. Diffusion on sev-eral 2D lattices with the same bond lengths and the samedefect concentrations and characterizations is comparedin the text.

and any other site with rate A. In addition, thedefects are arranged periodically with a unit cellof dimension ((}(l„,Pl, ) where (}. and S are integerslarger than 1, and l„and l, are the perfect-latticeunit-cell dimensions. 'The jump probabilitiesq„and q, are not equal in general, and 2(q„+q,) =1.For a square lattice l„=l„and q„=q, =4. The tran-sition function 4(l, l; 7) [see Eq. (2.1)] is given by

0'(l, l'; v) = [Ae "'(1—(}p ~)+(Be '-Ae "')((}},(»&,, 8 }+(}},(„, 8 „})+(De '-Ae "')(};(» 8„}]

}([q ~1-}',a(x, o} qy~l-}', ~(o, x}l (3 7)

where d represents sites with release rates B orD and [See Eq (14.10)]. 1 1. 4 1 5o'(t) = 2q l' —+ —.—+ ——— (r =x, y) .

A (}.P B D A

1 1 I'4 1 5lim C(u)= —+

~

—+———aP (B D A

Using Eq. (14.13) with the above, we get

(3 3)(3.9)

Notice that when the effect of the defects is loca-lized to its site (i.e., when B =A), and the ratio

19 STOCHASTIC THEORY OF MU LTISTATE. . . . II. 6225

between the rates of leaving a defect (D) and anormal site (A) is denoted by c =D/A, then

o', „(t) 1 I -e=1+-(Tg (t) (Xp E. (3.10)

where the perfect-lattice diffusion is denoted byo', ,„(t)= 2q„AI„'t Fo.r defects inhibiting diffusion(i.e., e. &1),o20(t) &o'(f).

b. Hexagonal lattice. For the hexagonal lattice(see Fig. 2) we use the same designation of ratesas described for the square lattice. The periodicunit cell in this case contains four atoms. Thetransition matrix for the perfect lattice is givenby (see Step 4 in Sec.III of PaperI; wefollowthe labeling in Fig. 2),

~l,{o,o) + ~1,(- x, o)

~7~ (o, o)+ ~l (y o) 7,(0, -1)

0

7,(0, 1)

&l,(o, o)

~7,(o,o) + ~l, (l., o)

~1,(o, o) + ~7,(- z, o) (3.11)

From an inspection of Fig. 2 it can be seen thatthe defect (labeled 3 in the figure) affects threeneighboring sites Tw. o of these (sites 2 and 4)are in the perfect-lattice unit cell (dashed line),while the third one (state 4) is in a neighboring

I

unit cell. Thus the repeating-defect group (dj con-sists of four atoms; the defect (3) and its threeneighbors (site 2, and the two equivalent sites, 4).Consequently, the transformed defect matrix forthis system is given by Eq. (3.12a)

0 x(B,A)(l+e"*'")

x(D,A)

2x(B,A)e '"'~)

1D(k, u) =—

3 0 x(B,A)

x (D, A) (1+e'"'")2x (B,A) (1+e'""'") (3.12a)

where

x(V, IV) = V/(V+u) -IV/(IV+u) . (3.12b)

Forming the propagator matrix A(k, u) =M/b,[Steps 5 and 6 of the procedure (see Sec. 111 inPaper I)], and using Eq. (I4.9), we obtain

II|

tl' 1 1 3 1 4g ~ + +

6 A 4oP B D A

and

tl„' 1 1 3 1 4l18 A 4nP B D A)

(3.13a)

(3.13b)

= X

FIG. 2. Random-walk lattice for motion of a singleparticle on a hexagonal lattice. The bond length is L .Normal (), defect P), and nearest-neighbor-to-a-defect (0), sites have total transition rates for leavingof A, D, and B, respectively. Transitions occur withequal probabilities to nearest neighbors. The defectsites (r', p form a superlattice with unit cell (nl „,Pl ~)where l „and I,~ describe a Cartesian unit cell (dashedlines) with four internal states.

l, =v3I, l, =3L,

and thus

(3.14)

L2t 1 1 3 1 4o„'=o'= —+ —+ ——— . (3.15)2 A 4' B D A

where l„and l„are the Cartesian distances be-tween equivalent points in neighboring unit cells.Denoting the bond length of the hexagon by L, wemay write (see Fig. 2)

6226 UZI LANDMAN AND MICHAEL F. SHLESINGER 19

1 1 4 1 5lim 4 (u) = —+ —+ ———

A 2uP B D A(3.16)

In order to allow a comparison between diffusionon the square and hexagonal lattices we assumeequal bond lengths and equal defect concentra-tions, which requires that 4(oP)„,„„,„„=(aP)~„, .Under these conditions, equal bond lengths andequal defect concentrations (of defects of type D),it is observed that o„',„(f)&cr'„(t) if B &A, whilethe reverse holds for B &A, . Thus, in this ex-ample, if the nearest neighbors to defects slowthe migration of particles, then the hexagonallattice is more "immune" to the presence of de-feats than the square lattice. Note, however, thatwhen A =B, o„',„(t)=o„' (t). It is only when therange of influence of the defects extends to theirneighbors that the effect of structure is exhibitedin the diffusion constant.

c. I"CC lattice. For particle motion on squarefcc lattices of side I we consider nearest-neighborand next-nearest-neighbor jumps which, are as-sumed to occur with a probability ratio q,/q, . Thedefects are characterized by a transition rate Dfor leaving, and its nearest neighbors by a transi-tion rate B. The spacing between equivalent de-fects is characterized by (al, Pl) (n, P integers)in the (x, y) directions, (see Fig. 3). Followingthe procedure demonstrated above for the otherlattices we obtain (see Secs. III and IV in Paper I)

=&i+ n=-. —&ik=o

(3.17)

1 1 4 1 5io„' (t) =I'f(-,' —q,) —+- — —+ ———

/'~A 2oP B D A&

(3.18)

Note that for vanishing defect concentration, i.e.,(nP) -~ and q, = 0, the result for a, square latticeof side / with no defects is retreived. If @,= —,',we obtain the result obtained previously [Eq.(3.9)] for a square lattice of side length I/v 2.Note that of2, (f) is always smaller than o„'(t), ifboth are of the same bond length and there are no

defects, since on the fcc lattice one makes jumpsof length l/v 2. In addition, in order to considerequal defect concentrations we must ascribe2(~P),„=(~P)., = 4(~P)„„

d. TH'angular lattice. As a final example weconsider motion on a defective triangular lattice(Fig 4), w.ith the same defect characterizationas in the previous examples. The defects arearranged in a periodic array, with spacing(nl„, Pl, ) in the x and y directions (with l„,l, asshown in Fig. 4) and I denoting the bond lengthof the lattice. Here we obtain

where 4(q, + q, ) = 1 and 0 & q, ~ —,', 0 ~ q, ~ —,'. Finally,

1 1 6 1lim C (u) =—+ —+ ———

A 2oP B D A(3.19)

FIG. 3. Random-walk lattice for motion of single par-ticles on a face-centered-square lattice. The total transi-tion rates for leaving normal sites, defect sites, andnearest-neighbor sites are A, D, and B, respectively.The lattice spacing between equivalent normal sites is L .The probability of a transition to a nearest-neighbor siteis q&, and to a next-nearest-neighbor site is q 2. Thedefect sites (a) form a superlattice with unit cell(«, PL).

FIG. 4. Random-walk lattice for motion of single par-ticles on a triangular lattice. The bond length is L. Thetotal transition rates for leaving normal, defect, andnearest-neighbor-to-a-defect sites areA, D, and B, re-spectively. Transitions to nearest neighbors occur withequal probabilities. 'The defect sites (a) form a super-lattice with a Cartesian unit cell (~l„,Pl ~) where l„andl „describe a unit cell with two states.

19 STOCHASTIC THEORY OF MULTISTATE. . . . II. 6227

and

8'p(k) 1 s', [2 cosk„L + 2 cos(—,

' k„+ k,)LBk' ), 2 68+ 2 cos(—,'k„k, )—L]~k=, =3L',

(3.20)

which yield

(3.21)

To compare results, at equal defect concentration,with the square lattice it is required that we set(2aP)„= (oP)„. For equal defect concentrationsand bond lengths we see that if B &A then a'„& o„' & o„',„, where the equality holds. if 8 =A, in-dependent of D.

C. Two-state single-particle diffusion

on a defective 2D square lattice

As our next topic we study the diffusion of a

particle on a 2D square lattice which contains aperiodic array of defects. At each site the parti-cle is assumed to be in one of two states: State 1,in which the particle merely vibrates about theminimum of the potential well associated withthat lattice site, and State 2, the mobile state,which the particle achieves by absorbing sufficient-energy to surmount the barrier for migration.Such a model was first suggested by Lennard-Jones' and analyzed using statistical-mechanical(partition-function) methods. The random-walklattice for the above model is shown in Fig. 5.

On a normal site the rates of leaving states 1and 2 are a»»dA =a»+4a», respectively, andthe corresponding rates for a defect site areb» andB =4b»+4b», respectively [each of theindividual rates is taken in the activated form,i.e., a;& =

a&& exp(- E,g/k&T), b(, = p), exp(- c~g/ksT)1i,j = 1, 2]. By inspection of Fig. 5, we can con-struct the following Fourier and Laplace trans-formed defect matrix

0 &2iB.+u A+u

D(k, u) =aai

b»+u a»+u osk I + cosh IB+u A+u X

(3.22)

a2(t) = o",(t) = 2L't(F/G),

where

(3.23a)

Following the steps of the procedure listed above(see Sec.III in P aper I),we obtain for the positionvariances in the x and y directions

I

[i.e., c„'(t) = 2a2+2t]. Furthermore, when transi-tions from the mobile state are dominant (i.e.,a», b» are greater than other rates) and state 2

is highly populated (small activation energy toachieve the mobile state, a» large, and slow rateof decay, a» small),

F — 22 + (2p) 1 22 22

B A

G — 12 + ( P)1 12 12

A B

(3.23b)c„'(t) = 2a„[1+(c(P) '(a„/b „—1)] 'L't . (3.25)

When the defects act as traps (i.e., a»&b»), itis seen that the diffusion is decreased.

D. Dimer migration on 1D and 2D defective lattices

(3.23c)

Note that the above result reduces to the ideal-lattice case

c„'(t) = [ 2a~2» /( a» +a»)]L't (2' =2;, y) (3.24)

for vanishing defect concentration, i.e., (aP)-~.Also, if the rate of decay from the mobile state(2) to the immobile state (1) vanishes (i.e. , a»= 0),the result for a state-1 particle is recovered

The observation and measurement of the cor-related motion of adatoms on surfaces (clustermigration) by means of field-ion-microscopy tech-niques provide information about the adsorbateparticle-substrate and interparticle interactions. 4

Such information is of fundamental interest to theunderstanding of surface adsorption and relatedphenomena such as thin-film growth, damageannealing, and the mechanisms of certain surfacereactions. The surface systems described abovemay contain defects (and most realistic systems

6228 UZI LANDMAN AND MICHAEL F. SHLESINt ER 19

Qi

b22

(ai, Pmil )

22

I L

(ej+I, Js'mal)

Qi

)

~-(CLi+ i, t)IIl)

G G,

G,' G0 ~O

C=) =G 6 0 0

G G G G G G G0,

G', 0, G a~ ~

G G G G

22

!

4l2„

e bra

'22bl2

(aj,

2I Vi-. l &i Tj+l f 1+2+I l

b el g ig e lb-(2& -Q- - lpi -L I i- -g& -t 11 +$1

a b d 1 b' gll I J

l

(b)

FIG. 5. Random-walk lattice is shown in 2D for single-particle motion where state 1 corresponds to an immobilestate and state 2 is an excited mobile state, as first pro-posed by Lennard-Jones. Defective sites [located at(njL, PmI ) for j and m integer] assumed to form asuperlattice with unit cell (+L, PL ) are denoted bysquares, while normal sites are denoted by circles.The total transition rate out of a state is the sum of allthe transition rates leaving the site, e.g. , the total rateof leaving a normal state 2 isA = 4a~2+a(2. The proba-bility of going to a particular site is the rate of going tothat site divided by the total rate of leaving, e.g. , theprobability to go from a normal site 2 to a particularnearest neighbor 2 is a»/A, and that of relaxing to theimmobile state 1 on a normal site is af2/A.

will undoubtedly contain defects to various de-grees, depending upon operating conditions, sam-ple treatment, etc.) which may exhibit themselvesin various ways. Defects may act as inhibitors orpromoters of surface migration. ' In addition,active sites for reactions may also be regardedas defects. " Finally, multicomponent surfacesmay also be viewed as defective systems. A char-acterization of particle motion on such surfacesrequires the determination of parameters charac-terizing the motion, of both normal and defectivesites.

The migration of adatom clusters on surfaceshas been analyzed by us previously, "yieldingexpressions which allow the determination of in-dividual transition rates for transitions betweencluster configurations involved in the migration

mechanism. Here we derive expressions for dimermotion on certain 1D and 2D defective lattices.

a. Channeled motion on a 1D defective lattice.As has been observed, on certain crystal facesthe motion of adparticles is one-dimensional. "'We consider the 1D motion of an adatom dimeron a substitutional periodic defective lattice (seeFig. 6). The defects are located at yl (y= integer),and the rates associated with the various transi-tions are given in the caption to Fig. 6. The real-space motion of the dimer [Fig. 6(a)] ha, s beenmapped onto a defective random-walk lattice [seeFig. 6(b)] with two states per unit cell (corres-ponding to the two dimer configurations). By in-spection of Fig. 6(b) we construct the propagatormatrix R(k, u;z) [see Eq. (2.4)]:where

a, (x; h) = — (1+e'"'),2 x+u (3.2V)

f, (x„x„h)= ' — ' (1+e'"'), (3.26)X~+A X2+Q

FIG. 6. (a) We illustrate the motion of a dimer ('--- ~ )in 1D wj.th two allowable states: straight (1) and stag-gered (2). The dimer migrates by alternating betweenthese two states. We consider the effect of substitu-tional defects (Q, located at y j/ for integer j) forminga superlattice with unit-cell dimension y/ in the x direc-tion. Nondefective sites are denoted by open circles(0). (b) Centroid position-of the dimer is mapped ontoa random-walk lattice with two states per unit cell. Thetotal transition rates to leave states 1 (straight) and 2(staggered) are 2a and 2b respectively, from normalsites, and g+ e and 2d for defective sites.

y —' —a (A, h) —(1/y) [d(g, E, e, E)

i

+J(e, E,g, E) —2H (A, h)]

—H, (B; h) —(1/2y)I, (D, B; h)~

(3.26)

19 STOCHASTIC THEORY OF MlJLTISTATE. . . . II. 6229

+3 foalZ(x„x„x„x„k)= + eX2+Q X4+8

(3.29)

A. =2a, 8 =2b, E=g+e, and D =2d. Using Eq.(14.9), we find for the variance

1 2 1 1 1 2 1 2 1g'(t) = —I2t —+ —+ ——+ —————2 A 8 y E D A 8

(3.30)

Clearly, when the defect concentration vanishes(i.e., y-~), the result for the ideal 1D lattice isrecovered [see Eq. (3.14) in Ref. 2]. The aboveexpression, coupled with measurement of theprobabilities of occupation of the configurationalstates of the cluster on normal and defect sites,could be used for a complete determination of theparameters characterizing migration in this sys-tem.

b. 2D motion of a dimer on a defective lattice.Consider a surface with substitutionally, periodi-cally placed defects, forming a defect superstruc-ture of unit c'ell [(y„t,y, l), where l is the unit-celldimension of the substrate and y„, y, are integers].The real-space motion of the dimer is shown inFig. 7(a). The dimer centroid performs a randomwalk on a lattice with three internal states perunit cell [see random-walk lattice in Fig. 7(b)].The rate of leaving states 1 and 3 is denoted byA, and that of leaving state 2 by 8. %hen a de-fect is placed in cell (y„j,y,m), the transitionrates into and out of cells (y„j,y,m), (y„j+1, y,m),and (y„j,y,m —1) are modified. The rate of leav-ing states 1 and 3 near a defect site is denoted byC, and the rate of leaving state 2 by D. Conse-quently, the propagator on the ideal lattice isdetermined from the transition matrix

1-e&"(k, u)= -H (A, k„)

—~II, (B, k„)

--,'a (B, k, )

—II, (A, k, ) (3.31)

Furthermore,

D (k, u) = — I (C,A, k„)Yx Yy

1I+(D&B, k„)

Yx Y$I

1 I (D, B,k)Yx Yp

1I+(C,A, k, )

Yx Yp(3.32)

and the waiting-time matrix is

lim4(u) =u~o

A '+2(y„y, ) '(C ' A')-~y( ) ~(D ~ B )

A '+2(y.y, )--'(C' A ')„-- (3.33)

Using the results in Paper I [Eq. (14.8)], we ob-tain

l2t 1 2 1 1o„'(t) =——+

A. ~„y, C

(3.34)

where (r =x, y). Again, for vanishing defect con-centrations (i.e., y„y, -~), the result for motionon a perfect lattice is recovered [see Eqs. (3.24)and (3.25), with d =a, in Ref. 2].

IV. MECHANISMS OF VACANCY DIFFUSION

In this section we apply our formalism to theinvestigation of transport via vacancy mechanismsin crystalline materials. A most useful methodfor the measurement of diffusion in solids is viathe radio tracer technique. ' " In this method athin layer of a radio tracer is deposited on thesurface of the solid and the specific activity ofthe tracer monitored as a function of distancefrom the surface is measured by successive sec-tioning of the solid. For these experimental con-ditions the solution to the diffusion equation is

UZI LA%DNA% AIVO MICHAEL F. SHLKSIÃGER

lgI

Oj

B/4

(b)(Y„J,Y„~} &' g„i+I,&„&) @

0/4g

8/4 g ii D/4=I Ii =IP

D/4 I~ I A/2(y j y m-0 &~ {)']~l, )'m-I}

Qs

)I

8/4 ~-4/2A/2 i' 8/4

8/4 ~ ~-8/4

QI Qi Qi

I'IG. 7. (a) Two-dimensional dimer migration. (a)Three possible spatial configurations of a dimer (largefilled circles connected by a dashed line) on a square 2Dlattice (small filled circles). The position of the dimercentroid is denoted by a cross (&). The allowed equiva-lent mirror-image configurations are not shown. Tran-sitions involving a defect (denoted by a filled square)are shown on the left. (b) Random-walk lattice corres-ponding to the centroid motion indicated in (a). Statesaffected by the defect [whose location is at {y„jl,y~.ml )]are denoted by squares to distinguish from those whichare normal, denoted by circles. The arrows denoteallowed transitions and are labeled by the transitionrates. The total rate for leaving normal states 1 and 3is A. , and that of leaving a normal state 2 is B. Thecorresponding total rates for transitions involving adefect are C =g+ e for leaving states 1 and 3, and D forleaving state 2.

[S/( Df) x/a] ( x2/4at)-

where C(x) is the specific activity of the tracerat distance x from the surface, t is the time ofanneal, S is the tracer concentration per unitarea at x = 0 and f = 0, and D is the sought-afterdiffusion coefficient. Performing such measure-ments for a number of temperatures 7.', we obtaina plot of the diffusion coefficient vs I/T. If thediffusion coefficient is of a simple Arrhenius form(D =D,e /"s ) the above plot will yield a straightline whose slope and intercept are E/ks and D„respectively.

Over the past decade it has been observed thatself-diffusion in a number of metals (both bcc andfcc) exhibits non A~-xhenius curved behavior' "in certain temperature ranges. The two modelswhich have been suggested for explaining curvedArrhenius plots in diffusion measurements are:(i) temperature dependence of enthalpy and entropyof defect formation and migration, "and (ii) acombination of single-vacancy and divacancy mi-gration mechanisms. ""Following the secondmodel, phenomeno)ogical two-exponential fits(or three-exponential when an additional inter-stitial mechanism is included) to the data have

' been attempted, yielding adequate agreement.Recent molecular-dynamics studies" of trans-

port in Na (bcc) and Al (fcc) have shown the oc-currance of Nxee monovacancy mechanisms:single jumps (SJ), where the atom migrates toa nearest-neighbor vacancy; second-nearest-neighbor jumps (SNNJ), where an atom migratesto a second-nearest-neighbor vacancy; and doublejumps (DJ) where two atoms, one nearest-neigh-bor and the other second-nearest neighbor to avacancy, perform colinear jumps within a veryshort time delay, thus leading to a double jumpof the vacancy to a non-nearest-neighboring site.In this numerical study, the weighted contributionfrom these three migration mechanisms (whichvaries with temperature) has been suggested asthe origin of the "anomalous" non-Arrheniuscurved behavior of the diffusion coefficient. Fol-lowing the above results, we model the diffusionas a random walk with three states (states 1, 2,and 3 corresponding to SJ, DJ, and SNNJ, re-spectively) and evaluate an analytical expressionfor the diffusion coefficient. As we proceed toshow, the diffusion coefficient which we derivedoes exhibit non-Arrhenius behavior. Note, thatthe model consists of competing monovacancymechanisms and does not invoke the occuranceof divacancies or other species (this does notrule out, however, the possibility of contributionsfrom divacancies or from an inherent temperaturedependence of the activation energies and fre-quency factors).

Consider a simple cubic (sc) lattice containinga vacancy. The filling of the vacancy may occurvia three possible mechanisms of particle mi-gration: SJ, DJ, and SNNJ, which we designateas the three states of the vacancy (1, 2, and 2).Each of the above processes (states) involves a.different type of atomic motion; thus they arecharacterized by different transition rates a&,jg&,

——p&,.xs &~/ sr] with different preexponentialand activation energies. The probability that thevacancy be found in any of the three possiblestates (i.e., that any of the three atomic migration

19 STOCHASTIC THEORY OF MULTISTATK. . . . II. 6231

processes occur) is determined by the rates a»p21e» ~ and g„=v»e ""~ and thus the

contribution from DJ and SNNJ depends upon thecorresponding activation energies and frequencyfactors and on temperature. Once in states 2 or 3,migration occurs via a DJ or a SNNJ, with transi-tion rates b „or c», respectively. We will as-sume (with no loss of generality) these transitionrates (b», c») to be large, so that the diffusionmechanism is determined by the rates g» and a31for populating the DJ and SNNJ states, and therate of performing SJ's g»= v e ~»" ~. By in-spection of the random-walk lattice (Fig. 8) weconstruct the transition matrix for the model:

2g, j 2b, '—C2c,s

A+u ' B+u ' C+zc

I))Cii Cu- ' =Qi, ~i Oi

02 , ' I' Qz

Q~ O~, Q~

Qi Qi 11 OI Ol ' Qi

02. ', , 'gO3) '-g~ O3

Oi-"

Qi'., Oi

02

O~ O~ O~

e(k, u) = (4.2)= = = = interstate tr a nsitions (on site)

gS1A. +u - SJ; DJ; ——-& SNNJ

where

A =6g„+g»+g31,

B =6b

C = 12C13,

C, = cosk„l + cosk l+ cosk l,C, = cos2k„l + cos2k„l + cos2k, l,C, = cos(k„l + k,l)+ cos(k„l + k, l)

+ cos(k, l + k,l) + cos(k„l —k, l)

+ cos(k„l —k, l) + cos(k, l —k,l),

(4.3a)

(4.3b)

(4.3c)

(4.4a)

(4.4b)

(4.4c)

FIG. 8. Motion of a monovacancy mapped onto a ran-dom-walk lattice (2D section of a sc lattice is shown)with three states per unit cell. State 1 corresponds toan uncorrelated single atomic jump (SJ) into a vacancy(single arrow). State 2 corresponds to two atoms jump-ing colinearly, and nearly simultaneously, leading to adouble jump (DJ) of the vacancy (double arrows). State3 corresponds to a single nearest-neighbor jump (SNNJ)(dashed arrow). After any type of jump we assume thatthe system relaxes immediately to state 1. If the rate-limiting step is the transition from state 1 to states 1, 2,or 3 (with rates a&&, a 2&, anda 3&, respectively), thenthe resulting expression for the monovacancy diffusionconstant is a weighted sum of three terms, each of whichis an exponential if the transition rates are activated(i.e., a&&=~;&e ~&& ~ ~). The resulting expression de-scribes the non-Arrhenius behavior of vacancy diffusionnear the melting temperature.

where l is the unit-cell dimension.Using the waiting-time matrix 4 (k, u), we con-

struct the propagator matrix 8 =M/6 [see Eq.(2.4) and Step 6 in Sec. III of Paper I]. The de-terminant ~ can be written as

Q = 1 —2gC

2g21b1A+u ' (A+u)(B+u)

2a,p C+ —', a„BC+—3a3p Ca»C+ a,P +BC

If the rates g», g», and a» are the rate-limitingsteps (i.e., if the corresponding processes areslower than other processes in the model), thefollowing expression is obtained:

2g31C.s(A + u) (C+u)

Following the method described in detail in Eq.(I4.9b), we write the variance as

a 2(]) l2t (2 p e E ll T/A+B411 8 E21/2BT-

+ 2 p e-E31/2BT)3 31 (4.6)

8 6a„'(I) = tl' lim

u~o r K=O

where b.o is defined in Eq. (I4.6). The aboveyields a general expression for the variance inposition for vacancy diffusion in a sc lattice

g2(t) —f2'(2p e-Ell/2BT+ p Ee2B1/T2

111 e E31/2B T)3 31 (4.6)

and for a fcc solid

for r =x, y, z. Similarly, for a bcc solid we find

6232 UZI LANDMAN AND MICHAEL F. SHLESINGKR 19

o'(t) = t't(2v, e» B + —,' v»e

1 p e 8 13~~BT)3 31 (4.7)

It is seen from the above that a three-exponentialexpression is obtained for the variance in position.However, we did not invoke the occurance of newspecies (like divacancies or interstitials), de-riving our results on the basis of comPeting mono-vacancy mechanisms only. The relative contribu-tions of the three processes involved depend uponthe relative magnitudes of the characteristic acti-vation energies and frequency factors [(v», E»)for SJ, (v», E») for DJ, and (v», E») for SNNJ],and upon the temperature. In certain temperatureranges, the contributions from two or more termsmay become significant and deviations from alinear Arrhenius behavior will occur. It shouldbe noted that the diffusion coefficients derivedabove are for. the vacancy motion. The particle(radio-tracer) diffusion D~ is related to that of thevacancy D„by"

where [v] and [P] are the vacancy and particleconcentrations and f is the correlation-functionfactor. Thus the results obtained in the abovecan be used in the analysis of radiotracer dif-fusion measurements.

V. TIME-DEPENDENT CORRELATED MOTION

Microscopic models of mechanisms of particletransport in solids are characterized by certaintime constants. These time -constants are relatedto the dynamic response of the solid (vibrationalfrequencies, local-mode frequencies, saddle-point crossing frequencies, and lattice relaxationtimes). The relative magnitudes of the charac-teristic time constants determine the contributionsof the corresponding modes of response of thelattice to the transport mechanism. Consider anactivated migration of a particle between twoequivalent sites (1 and 2) in a solid. Starting atsite 1, the outcome of the migration step is whenthe particle attains site 2. In the process of parti-cle. migration the host lattice may deform from itsequilibrium configuration. The return of the sys-tem to its initial equilibrium configuration isgoverned by a relaxation time T,. In the eventthat the time delay between particle jumps (T,)is larger than the lattice relaxation time (T,), eachparticle transition is uncorrelated from itsprevious one. However, when T1&T„atransitionof the particle will be correlated to its previousposition, and thus the stochastic evolution of theparticle position is non-Markovian. We empha-size that the dynamic correlation effects dis-

cussed in this section should not be confused withthe spatial correlation associated with the motionof a cluster, discussed in Sec. III. Correlationsof the type discussed here may occur in particleor excitation (e.g. , polaron) migration in con-densed-matter systems. The effect of such cor-relations on quasielastic neutron scattering lineshapes is discussed in Sec. VI.

Correlated random stalks zither e time is notinvolved and where only the statistics of the ran-dom-walk path is considered have been studiedpreviously. "" If the system is not allowed toreturn to its previous state on the next transitionthen the problem is that of a random walk with arestricted reversal. A generalization of thiscorrelated random walk is a forbidden return toa previous state after a finite number of suceedingtransitions. These types of problems have beentreated first by Montroll and later by others'as approximations to the restricted randem walkof polymer configurations where no polymer unitsmay overlap.

In the following we illustrate the application ofour formalism to the study of correlated motions.We consider a one-dimensional motion which ismapped onto a random-walk lattice with two in-ternal states per unit cell (see Fig. 9). Theparticle achieves state 1 on site j when the transi-tion is from the j —1 site. When the jth site isoccupied by a transition originating from sitej+ 1, state 2 is obtained. Associated with theabove two states are the transition probabilities:q(t) for return to the site where the previous jumporiginated and p(t) for propagation in the otherdirection. 'These transition probabilities areassumed to be of the form

p(t)=-, +we '

q(t) = B —& e ~~

(5.1)

(5.2)

I

I I

I P ~j

I P I P P I

I I

I I I I I

FIG. 9. Correlated non-Markovian motion of a randomwalker in 10 mapped onto a lattice with two states ineach unit cell (dashed lines); state 1 corresponds to theevent that the lattice site was occupied as a result of atransition from the left and is distinguished from state 2,which is occupied when the transition to this lattice sitewas from the right. The probability of the next jumpbeing in the same direction as the preceding one is de-noted byp, and in the reverse direction it is denoted byq. The generalization to correlations over more thanthe preceding jump involve additional internal states.

19 STOCHASTIC THEORY 0 F MU LTIS TATE. . . . II. 6233

where e is a constant (positive or negative) suchthat P(t) and q(t) are smaller than unity. In thelimiting case that the average time betweenparticle jumps T, is much smaller than thecharacteristic lattice relaxation time 7.'„ thetransition probabilities reduce to

1 1p= @+6) tg=2 —E. (5.3)

(5.4)

If, on the other hand, the reverse holds (i.e. , T,» T, ), p = q = —'„and the motion is uncor related.

The transition matrix for the above model isgiven by

(5.10)

o'(t) - (p/T', )I2t'+O(qt'), (5.11)

which is not of diffusion character [i.e. , o'(t) (x: t].However, for time t » T,/2q, the first term inEq. (5.10), which is of diffusion character, domi-nates.

Note that in the limit of q -0 (i.e. , immediatereturn to the previously occupied site is negligible)the variance in position is given by

-x (u)e'" ) (1 —x, (u)e'"f

where 6 is the determinant of the matrix 1 —4(k, u),and

1 1/T, 1/T,2 1/T, +u 1/T, +1/T, +u (5.6 }

Using E(I. (2.10a) of Ref. 2, we get

xu '[1 —p~(u)] lp~, ,)(5.7)

where P~, is the initial probability that the particlewill occupy state j'(p, =p, = ~ in the present case),thus accounting for the two alternative ways in

which the particle starts its walk at t=0. Expres-sing R(k, u) =M/& and using E(I. (2.24) of Ref. 2,we obtain the general result

The inverse of the transformed propagator, Lt(k, u)= [1—Q(k, u )] '; is given by

[E(k )]1 ~ +( )

(u)e El(

VI. QUASIELASTIC NEUTRON SCATTERINGFROM DIFFUSING SPECIES

Neutron scattering is a powerful method forobtaining information about the structure and dyna-mics of liquids and solids. 34 The theoreticallysharp elastic zero-energy transfer line (h&u

=O,kkw0} is broadened by diffusive motion,yielding the quasielastic neutron scattering lineshape S(k, &u), where ke and h k are the energyand momentum transfers. This line shape con-tains contributions from coherent scatteringfrom many different atoms and incoherent scat-tering from individual atoms. For certain sys-tems (e.g., hydrogen in metals such as niobium35)the large incoherent scattering cross sectiondominates the coherent one. We will be inter-ested in the incoherent quasielastic neutron scat-tering (QNS) because it can yield information abouttransition rates, diffusion constants, and struc-tural positions of migrating interstitials in solids.A similar analysis also applies to the diffusion-ally broadened Mossbauer resonant absorptionline. "

The incoherent scattering law was derived byVan Hove3' in the form

l' 1+ 2&+ T,/T, 1 —e s-' E,T~ 1-2s+ T,/T, E E

5.8) S(k, e) =— e" ' "G(r, t)dr dt,2r (6 ~ 1)

where

E~ = 1/T, + (1+2E)/T, . (5 9)

In the above expression the first term dominatesin the long-time (diffusion) limit.

For an uncorrelated walk (e = 0 or T, «T, ), the

expression for the variance reduces exactly too,(t) = I2t/T„which is the known result for stmple-particle diffusion on a 1D lattice. In the oppositelimit, for a system where particle jumps occurbefore the lattice relaxes (T, » T,), we obtain

where G is a quantum correlation function. Inthe high-temperature limit (keT &k~) G(r, t) be-comes' '" the classical correlation function fora particle to be at position r at time t if it was atr =0 initially. This approximation holds fortimes of t «10 '3 sec. In our analysis the clas-sical approximation will be employed.

For simple diffusion of a particle in a 3D sclattice where the particle makes transitions witha rate a and with equal probability to its neigh-bors a Lorentzian line shape is obtained34 forS(k, (u):

6234 UZI LANDMAN AND MICHAEL F. SHLESINGER 19

S(&, ~) = (1/7T) p(k)/[p'(k) + (d'], (6.2) S(k, &u) =—Re g d rm, n

xP (r, t~ r =O, t =0)P„ (6.4a)

p(k) = —,'a(3 —cosk, l, —cosk„l„—cosk, l,) . =—Re QP„„(k,i&a)P„%y tl

In the limit of small momentum transfer k,Eq. (6.2) becomes S(k, ~) = (1/v)I'(I'+ e') ',where I'= Dk, with the diffusion coefficient D fora sc lattice equal to -'gl .

We can determine the rate ~ by

limn'S(k, + =0)k =a.k 0

(6.3)

Thus a plot of S(k, a& = 0) vs (vk') ' for small &

yields the transition rate p. In practice, thediffusion constant D is extracted from data for thewidth of the scattering cross section S(k,+) asa, function of k .

The most studied system with QNS is the diffu-sion of hydrogen in metals. 3 ' It has been ob-served that the data is non-Lorentzian and us-ually cannot be explained by the predictions ofsimple jump models. Thus more sophisticateddiffusion mechanisms and models are needed. 40'4'

We give here a, general formula for S(k,~) whichincludes the effects of internal states, clusters,correlations, and periodic defects. We empha-size that only for the simple example given above[Eq. (6.2)] does a single Lorentzian line shapeoccur. In other systems, the dynamics of thesystem (i.e., correlations, cluster motion, andinternal states) and the heterogeneous characterof the medium (defects) would manifest them-selves as deviations from a single Lorentzianline shape.

From Eq. (6.1), in the classical limit, we ob-tain

=—Re gR „(k,ie)~

."" —

)p„,ttlp tl

(6.4b)

, (6.5)

A. +N y+gg ~

where X = 1/T2 and 6—is given as

(6.6)

The term in large parentheses in Eq. (6.4b), foran exponential waiting-time distribution g(t)=A,e' ", is given by

@ „(i(u) =(A+i(u) ' (6.7)

for nz = 1,2. The initial-state occupation prob-abilities are taken to be equal, i.e., p, =p, = —,'.Substituting into Eq. (6.4b), we obtain

where we have averaged over initial positions nand summed over final ones m.

Let us consider first the case of single-parti-cle motion in 1D with temporal correlation, dis-cussed in the previous section. We limit ourdiscussion to the case of T, » T2, i.e., the transi-tion probabilities are p= —,'+ E and q =-, —E

[see Eq. (5.3)]. For this case the matrix R(k,u)given in Eq. (5.5) reduces to (with k in units ofinverse lattice spacing)

'l

X(I —2g cosk)(X [1+2E —(1+2E) cosk] -~ }+2Xtu [1 —(2+ e) cosk]p'[I+ 2q —(1+2g) cosk] -&u'} +(2Xur[1 —(~ + c) cosk]}' (6.8)

In the event of no temporal correlation &=0, theabove result reduces to a Lorentzian [see Eq.(6.2)], and

1 X(1 —cosk)v [A(1 —cosk)]'+ ~' (6.9)

Haus and Kehr42 have studied temporal correlatedmotion, deriving coupled rate equations for

P „(k,&u).

For a system with a periodic arrangement ofdefects, an expression for S(k, a&) given in Eq.(6.4a) may be written explicitly. Using the ex-pression on the right-hand side oi' Eq. (I2.37)[with n = 0 and without the indicated inverse La-place transformation for P, &(k, tu)], we find forsmall (d

STOCHASTIC THEORY OF MULTISTATE. . . . II. 6285

(6.10)

C(k) = 3(cosk, l, + cosk, f, + cosk, l,) .Note that for g=d we recover the Lorentzianline shape. The elastic line (a =0) yields

S(k, 0) = n' [d+ A(a —d)]/ad[1 —C(k)] . (6.12)

If the rate a is known from a system without de-fects, then the rate d of leaving a defect can becalculated from Eq. (6.12).

To obtain the diffusion constant, a measurementof the scattering intensity versus energy transferof the scattered particle, for fixed momentumtransfer and temperature is performed, and thefull width at half-maximum (FWHM} is determined.

The above expression is appropriate for small k

values, and is useful for analysis of the diffusivebehavior from QNS measurements. Study of de-tails of the motion in localized regions such asthe vicinity of defects requires an analysis ofS(k,u&) for large k values, which can be accom-plished by employing in the derivation of S(k,e)the full propagator, which contains contributionsfrom k's lying in all the Brillouin zones of thedefect superlattice [see Eqs. (I2.21)-(I2.23)j.

As an application of Eq. (6.10), consider theQNS from an interstitial whose motion is on asc lattice with periodic defects, such as has beendiscussed for the case of hydrogen diffusion inniobium containing nitrogen traps. ' ' We as-sume that the interstitial leaves normal siteswith a rate g and defect sites with g. rate d, andthat the defect superlattice has a unit-cellvolume of 0 '. Then

@"'(i&a)=(a+i~) ', 4(i~) =d+i~,D(k, i~) = d/(d+ i(u) —g/(g+ i&a),

t) "'(k, (u)

= a(a+ iv) '—,'(cosk„f„+cosk, f, + cosk, f,) .

Substituting into Eq. (6.10), we find as a -0S(k, (u)

= v 'Q (k, u))([Q(a —d) +d](ad[1 —C(k)] -u }+&a'fa+ d —C(k) [a —n(a —d)]}),

(6.11)

where

Q(k, (u) = (ad[1 —C(k)] —&u }+~'(a[1 —C(k) (1 —0)] + d [1—C(k) 0]}'

l

Obtaining such data for a number of k values, wecan make a plot of.I WHM vs O'. Subsequently,one attempts to best fit the experimental observa-tions by choosing the optimal set of parameters inthe theoretical expression (for the simple Lorent-zian diffusion case a plot of X' vs k' yields astraight line with slope D). Also, measurementsat different temperatures could be used in order toobtain the activation energy E, and frequencyfactor p, for the diffusion. In the case of a single-state diffusion mechanism in a defect-free medi-um such measurements yield a straight line (ofslope E, and-intercept v, ) in a plot of ln(1'/k')vs I/ksT. However, for multistate diffusion, cor-related motion, and in the presence of defects, theabove simple method of analysis is not adequate tofit the data and determine the characteristic dif-fusion pa, rameters. In these circumstances onemust best fit the experimental results with theo-retical expressions derived from complex diffusionmechanisms.

The detailed analysis of the incoherent scatteringlaw S(k, co) for diffusion systems where the diffu-sants may be dimers or other clusters, may havevarious internal states (e.g. , mobile and immo-bile), may have transitions correlated to previousones, and in the presence of various types of de-fects, will be reported by us.

VII. TRANSITION RATES

In all the above examples from FIM, vacancydiffusion, correlated motion, and QNS we havecalculated experimentally observable quantities(diffusion distances, occupation probabilities ofdifferent internal states, and line shapes) in termsof assigned transition rates. We have given sev-eral explicit examples and indications of how toinvert the data [Eqs. (3.2), (3.30), (6.3), and (6.12)]to find the individual transition rates between theinternal states of the system under study. Ingeneral, if there are N transition rates then oneneeds N equations to fully determine the individualrates. If there are M internal states then M -1detailed balance relations can be found. The otherN -M+1 equations may be found by consideringdiffusion distances or line-shape parameters.Other independent equations may be used, whenappropriate, to describe the data for NMR, Moss-bauer, and various reaction experiments. Ifthere are more possible equations than transitionrates then consistency checks can be made. It isalso possible that there are more transition rates

6236 UZI LANDMAN AND MICHAEL F. SHLESINGER 19

than can be determined from the data. For ex-ample, consider a dimer which can exist in threestates' (straight, staggered, and extended stag-gered) undergoing a 1D motion on a surface. Thereare four transition rates connecting the threestates. Detailed balance relations between thethree states yield two equations, and the position-.al variance of the dimer centroid a third. It isreadily seen that an experiment with an electricalbias to polarize the adatoms and favor motion inone direction is necessary to yield a fourth equa-tion for the centroid mean position, which thenallows a determination of the four transition rates.Thus the degree of microscopicity one wishes toobtain will determine the type of experiment thatshould be performed. In addition, the data mustbe plotted in a proper manner. For example, ifone, say for FIM data, simply plots the logarithmof the variance of the centroid position of a clustervs llksT, the individual transition rates cannotbe determined and the plot will yield a curved non-Arrhenius line because more than one transitionrate is involved. Although in a limited temperaturerange the curve may appear to be straight, itsslope and intercept do not characterize the indi-vidual transition rates between the states of thedimer. Thus further information, such as detailedbalance relationships as a function of temperatureand perhaps the mean motion from experimentsunder electric basis will be necessary to calculateindividual transition rates.

VIII. PROSPECTIVES

In the present study we have employed mappingsof diverse transport systems and mechanisms ontorandom-walk lattices with internal states and de-fects. The systems and phenomena which we havediscussed are: (i) single-particle and dimer mo-tion in one and two dimensions on periodically de-fective lattices, as observed in field-ion-micro-scopy experiments. (ii) Multistate diffusion of asingle particle in a two-dimensional defective lat-tice. (iii) The lattice-structure dependence of thediffusion constant for single particles on defective

lattices. (iv) The non-Arrhenius behavior of thevacancy diffusion constant in certain metals, asobserved in radio-tracer experiments. (v) Theeffect of temporal correlation on the diffusion con-stant. (vi) The effect of defects, multistate mech-anisms, and temporal and spatial correlations onthe quasielastic neutron scattering law S(k, &u).

The solutions of the time development of the sys-tems considered were obtained in a unified fashionvia a matrix Green's-function propagator (to in-clude internal states) which is renormalized bythe presence of defects.

Qther systems in which transport is an essentialstep and to which our formalism can be appliedare certain unimolecular and bimolecular reactionsystems. Examples include annealing of point de-fects,"electron scavenging in aqueous radiationchemistry, 44 one-way flux of tracer ions throughnerve membrane during action potentials, "posi-tron ' and muon ' trapping in solids, exciton trap-ping and fusion, ' and bimolecular heterogeneousreactions of the Langmuir-Hinshelwood mecha-nism, catalyzed by active sites."

In the latter reactions, adsorbed reactants mi-grate on a surface and reaction occurs upon thecoincidence of two reactants at an active site. Thereactants may possess internal states (energetic,spin, orientational, etc. ) which may determine thereaction probability upon coincidence. The sur-face may contain defects other than the activesites, such as migration promotors and inhibitors.The calculation of the rate of reaction proceedsvia a stochastic formalism, yielding expressionsfor the probability distribution of reactive coinci-dences at active sites. The random-walk propaga-tors for the reactants were discussed in Paper I.Our results show surface structural dependenceof the reaction rates and their variation with ac-tive-site concentration. We will publish these re-sults' and other reaction schemes in due course.

ACKNOWLEDGMENT

Work supported by U.S. DOE Contract No. EG-77-S-05-5489.

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