Quasielastic neutron scattering and correlated self-diffusion in crystals

Solid State Commurtications,Vol. 23, pp. 853-858, 1977.

Pergamon Press.

Printed in Great Britain

QUASIELASTIC NEUTRON SCATTERING AND CORRELATED SELF-DIFFUSION IN CRYSTALS D. Wolf Department of Physics, University of Utah, Salt Lake City, UT 84112, U.S.A. (Received 12April 1977 by M. Cardona)

The effect of correlations between successive jumps of atoms during selfdiffusion on the quasielastic neutron-scattering line width is investigated. Temporal and spatial aspects of these correlations are described in terms of the so-called "encounter model", which had originally been developed for the interpretation of nuclear-magnetic resonance studies on selfdiffusion. Owing to the rather high temperatures (i.e. large diffusivities) to which quasielastic neutron-scattering experiments on self-diffusion in crystals are restricted, the effect of both mono- and divacancies is considered. It is found that the line width associated with a correlated mechanism is smaller and in single crystals shows a different orientation dependence than expected for a random-walk diffusion mechanism. Numerical results are presented for (i) the monovacancy and two divacancy mechanisms of self-diffusion in a b.c.c, lattice, a situation likely to be encountered in sodium metal, and (ii) mono- and divacancies in the f.c.c. lattice. In the random-walk limit earlier results of Gissler and Rother and Chudley and Elliot are reproduced. All of the results derived also apply to the diffusion-induced broadening of Mibssbauer lines. The article therefore presents a generalization of a recent theory in w~ch only the effect of monovacancies on the Mt)ssbauer line broadening has been considered. QUASIELASTIC NEUTRON SCATTERING has proved to be a valuable tool for the investigation of diffusion processes in liquids and transition metal hydrides [ 1]. Owing to the requirement of very large diffusion coefficients (D ;~ 10 -7 cm 2 sec -~ ) and large incoherent scattering cross sections oine, only in few monoatomic crystals a broadening of the quasielastic line is expected to be observable and thus to yield microscopic information on self.diffusion processes in such materials [2]. Ait-Salem [3] and Goeltz [4] have shown very recently that self-diffusion in sodium metal is detectable through the corresponding broadening of the quasielastic line. Although mono- and divacancies are generally accepted as the underlying self-diffusion mechanisms in sodium [5], due to the lack of a better theoretical approach these experiments have been interpreted in terms of a much simpler random-walk diffusion mechanism. Consequently, values of the self-diffusion coefficient D determined (i) from these line broadening experiments in terms of a random-walk model, and (ii) measured directly, e.g. by means of radioactive tracers do not agree very well. The purpose of this article is to incorporate the effect of correlations between successive atomic jumps into the usual formalism which relates the mean time f between successive jumps of a given atom to the broadening of the related quasielastic line. According to Van Hove [6] the intensity of neutrons scattered into the solid angle dE and into an energy

interval dE is proportional to the differential scattering cross section d2o 1 k - - -oincSi~c(K, ~o), d~dE 4n~h ko

(l)

where h = h(ko -- k) and hto = Eo - - E = h2(k2o -- k=)/2rn denote the momentum and energy transfer during the scattering process. In terms of the self-correlation function Gs(r, t) associated with atomic motions, the quasielastic incoherent scattering cross section is given by [6] 1 t" / . Sine(k, w) = ~ .~ J Gs(r, t) exp (iKr -- ~ot) dr dr.

853

(2)

G,(r, t), in essence, describes the probability of finding an atom at time t at the position r if, at t = 0, the same atom was located at the origin (r = 0). For a randomwalk diffusion mechanism in a cubic crystal lattice, Gs(r, t) was determined first by Chudley and Elliot [7] and later, using a more direct technique, by Gisster and Rother [8]. The effect of correlated relative atomic motions on the nuclear magnetic resonance (NMR) relaxation behavior has been investigated some time ago [9, 10]. In what follows the basic ideas developed for the interpretation of NMR diffusion experiments will be applied to determine the effect of correlated motions on quasielastic

854

CORRELATED SELF-DIFFUSION IN CRYSTALS

neutron scattering. At the rather high temperature (close to the melting point) to which quasielastic neutron-scattering experiments of self-diffusion in crystals are restricted, more than one diffusion mechanism (due, e.g. to mono- and divaeancies) usually contributes to diffusion. From the outset we shall therefore consider the effect of two self-diffusion mechanisms which simultaneously may induce correlated atomic jumps. To describe these correlated jumps quantitatively, an "encounter" is det'med as the sum of all interchanges of a given atom with the same point defect. Then the following two aspects of these correlations are to be distinguished [9-11 ].

O) Spatial correlations account for the phenomenon that successive jump vectors of an atom due to the same point defect are correlated. The familiar correlation factor f results from this effect. (ii) Temporal correlations appear since atomic jumps induced by the same point defect are bunched into groups ("bunching effect"). Jumps induced by different point defects (during different encounters) are to be distinguished from the jumps of an atom due to the same defect (during a given encounter). In contrast, in a random.waik diffusion mechanism all nuclear jumps are identical and assumed to occur randomly in time. As a result of this "bunching effect", within a very short time (of the order of several za, where za denotes the mean time between successive jumps of a given point defect), an atom once "visited" by a point defect is very likely to perform more than one jump. At the rather high temperatures involved, ¢a is very short (ra <~ 10 -t2 sec). Consequently, from the point of view of a scattering atom an encounter is practically experienced as a sudden displacement (eventually by more than just a nearest-neighbor vector), and the details of the,actual rearranging process are "invisible". Even at the largest thermally-created vacancy concentrations (% <~ 5 x 10 -~), interactions between different point defects are very unlikely. Different encounters may therefore be assumed to be independent of one another. The "'visible" motion of a scattering center, therefore, consists of a random sequence of encounters with single and double vacancies. If q(St, Sa, t) denotes the probability that in time t a given atom experiences St encounters with monovacancies and $2 encounters with divacancies, analagous to Gissler and Rother [8] the correlation function G,(r, t) may be written as follows [12] : G.(r, t) =

~

~

q(St,S2,t)Ps~,s~(r),

(3)

81-0 $:"0

where Ps,, s~ (r) represents the geometrical probability

Vol. 23, No. 11

that as a result of St encounters with monovacancies and $2 encounters with divacancies the atom originally at the origin has been displaced to the position r. Every atom in the crystal experiences its individual mixed random sequence of encounters of the two types. In good approximation any such sequence is determined by the probability [12] =

E

,=

[b,v(r)w,v(rm)

* b,v(r)W,v(r,,,)]

a(r -

r=)

(4)

that in a "representative encounter" (averaged over the two types of rearrangements) an atom moves from the origin to the lattice site r. The geometrical probabilities due to encounters with mono- and divacancies have been denoted by Wav(rm) (o = 1,2). Since after an encounter of either type the displaced atom occupies some lattice site, these probabilities are normalized to unity, i.e. T%n Wav(rm) = 1. bay(T) denotes the temperaturedependent probability that some encounter is caused by a single (a = 1) or double vacancy (a = 2). For normalization reasons [12] b t v ( T ) + b2v(T) = 1. With the approximation (4) the probabilities Ps,. s~ in equation (3) no longer depend on the individual sequence and numbers of encounters, S~ and $2, with the two types of vacancies, but they are only functions of the total number, S = St + $2, of encounters experienced by a given atom. Analogous to Gissler and Rother [8] or Torrey [13], Chandrasekhar's theory of random flights [14] may be applied to determine Ps(r) in terms of the probability Pt(r)---Pmc(r) due to a single "representative" encounter. With the Fourier transform of Pene(r), henc(K) = ~ Pent(r) e tKr dr,

(5)

it is thus readily found that Ps(r) = [hene(t<)] s.

(6)

Since encounters of the different types are independent of one-another q(S l , $2, t) = Ws,(t , "rea¢)ws~(t, iv :v r~e), where Wsa(t, ¢ame¢) denotes the probability (a Poisson distribution) that Sa encounters of type a(= I, 2) occur in time t. r~vc represents the mean time between two encounters of the same type a. The total number of encounters per second of a given atom may therefore be written as follows: 1

1

1

rene : 'r-q~-"~ v + "r2,,nvc" As shown in reference [12], for Poisson distributions ws(t, r) the following relationship holds:

(7)

and equation (11) becomes:

~ q(S,,S2t)

1 - - f ( 0 ) - - Re ~tz,.a>of(rm)e #trm b,v(r)Z,v(O) + b2v(r)Z2v(O)

2h

s,=o %=o

A r = --

s~o ,%~o w&(t'rLv-)ws'(t'rLvO

:

855

CORRELATED SELF-DIFFUSION IN CRYSTALS

Vol. 23, No. 11

(14a)

~th ~.. Ws(t, r ~ ) ;

=

(8)

8-,0

f(rm) = bsv(T)Wlv(rm) + b2v(T)W2v(rm).

(14b)

Here the orientation-independent probabilities/'(0) -i.e. the double sum over S1 and S2 reduces to a single summation involving S -- S~ + $2 and the Poisson distribution Ws(t, rme) with rmc defined by equation (7). Combining equations (8) and (6) with (3), we obtain

f(rm ffi O) = blv(T)Wlv(O) + b=v(T)W=v(O) that after

G,(K, t) = f G,(r, t) ¢flgz dr = exp

[ 1 -- h.,c(K

(9)

,

the Fourier transform of which is the following Lorentzian [see also equation (2)] :

1 [1 - h ~ ( ~ ) l r = . ~2~.= + [I-h.~(~)]

S~., ( . , w) = -

~"

~r

=

[l -

[bw(T)W,v(rm)

lrrrl

b~v(T)W2v(r m)] e m'~

.

=

~[b,v(r)Z,v(O)

+

b2v(r)Z2v(O)],

(15)

WIv(r,.)eWam].

J

Equation (I 5) was, in essence, derived recently [ 11 ] in connection with Mbssbauer experiments of diffusion in crystals. In the random-walk limit an "encounter" is reduced to a single random jump of an atom. Then Z1v(O) = I and Wlv(O) = O, and all probabilities Wtv(rm) vanish except those for which rm represents one of the G nearest-neighbor vectors re (g = 1,2 . . . . G) in the crystal. Equation (15) then yields:

(l2)

The summation involving r m includes the origin (rm = 0) and all lattice sites in its vicinity which may be reached by an atom in a single encounter. The remaining task is to determine reac as a function of the mean time ¢ between successive atomic jumps. (Remember that it is f and not rent which yields the random-walk self-diffusion coefficient D = a2/6f from the measurement of Air'.) Because of the bunching effect the mean number of jumps of a given atom, ZIv(O) or Z2v(O), in its encounter with a single or double vacancy, respectively, exceeds unity [9, 12, 15]. In a "representative encounter" (averaged over the two kinds) an atom performs b tv(T)ZIv(O) + b2v(T)Z2v(O) jumps on the average. The number of encounters per second, r ~ , of a certain atom is equal to the number of its random jumps per second, ~-~, divided by its average number of jumps per encounter [10, 15]. Therefore [ 12]

r,,~

WIv(O)-- Re

Iz m I > 0

where [see equations (5) and (4)]

+

2h [

fZlv(O) 1 --

x ~

(11)

henc(K) = Re ~

=

(10)

The width at half intensity of the quasielastic line (1) therefore becomes: 2h

a "'representative encounter" the atom sits at the origin (rm = 0) as before the encounter, have been separated from the remaining sum over r m in equation (12). These contributions for Ir,, I> 0 lead to an orientation dependence of AF in single crystals. According to equations (14), the magnitude of this anisotropy, via the probabilities b~v(T), is temperature dependent. In the special case in which only monovacancies are responsible for diffusion, b iv(T) - 1, b2v(T ) =-0 and equations (14) yield

03)

Ar = 2h[l_Re

T L

t; eZKre] , ~1 E I?'1

(16)

J

which is the result first obtained by Chudley and EUiol [7] and later confirmed by Gissler and Rother [8]. Finally the explicit temperature dependence of the parameters b w ( T ) and b2v(T ) is specified. As discussed in reference [ 12], in terms of the four parameters Do1, Do:, E~v and E2v which characterize the temperature dependence of the random-walk self-diffusion coefficient,

D = Do1 e °E'V/hT +Do: e -E2v/kT,

(17)

associated with the random migration of mono- and divacancies, b w ( T ) and b2v(T ) may be expressed as follows (see equation (2.28) of reference [12] ):

b•v(T) = [1 + g a y ~ D o p e~v

~v(O) Do.

e_(E#v_E,,v)IkT ] -1. (~ ~ ~)

] (18)

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CORRELATED SELF-DIFFUSION IN CRYSTALS

Vol. 23, No. 11

Values o f Z(0) and W(I rn I) for a monovacancy- and two divacancy mechanisms o f self-diffusion in the b.c.c. • tice 1.which b [ra [around thelorigin n = 1. IrteI = 0). The so-called 1 N - 2 N - 1 N wastsubdivided into tshells n with radii

mechanism [ 16] involves alternate lumps o f the two monovaeancies forming a divacancy between nearest (1N) and second-nearest neighbor (2N) eonfiffurations. In the 1N-2N--4N mechanism 116] the dimcancy is assumed to dissociate partly by lumping from a 2N conp~uration either into a 4N or a 1N configuration (with probabilities v2, and v,l, respectively) [ 16]. The above values were determined with v2t/v24 = 1 (see reference [ 17] }. "a" denotes the distance between nearest-neighboring atoms

Shell number n

Radius t rn I of shell n

1

0

2

Number N of atoms in shell n

Diffusion mechanism

W(t r, I) for atoms in shell n

N × W(Lr. I)

Z~v(O)

1

1V 2 V(1N-2N- IN) 2 V(1N- 2N-4N)

0.129 0.133 0.111

0.129 0.133 0.111

1.36 2.47 2.19

a

8

1V 2 V ( I N - 2 N - 1N) 2 V(1N- 2N-4N)

0.096 0.073 0.092

0.768 0.584 0.736

3

]x/3a

6

1V 2 V(IN- 2N- IN) 2V(IN-2N-4N)

0.010 0.027 0.020

0.060 0.162 0.120

4

]v~a

12

lV 2 V ( I N - 2 N - 1N) 2 V(1N- 2N-4N)

0.003 0.004 0.002

0.036 0.048 0.024

1V

5

x/11/3a

24

2 V(IN- 2N- 1N) 2 V(1N- 2N-4N)

0.000 0.002 0.000

0.000 0.048 0.000

6

2a

1v " 2V(1N-2N-1N) 2 V(1N- 2N-4N)

o.oo I 0.003 0.001

0.008 0.024 0.008

8

Table 2. Diffusion parameters for the mono-and divacancy mechanisms o f self~tiffusion in the f.c.c, lattice. The latter was assumed to involve only nearest-neighbor configurations o f the two monovacancies which form the divacancy [12, 16].All other parameters are the same as in Table I

Shell number n

Radius Irn I of shell n

1

0

2

a

3

Number N of atoms in shell n

Diffusion mechanism

W(I rn I) for atoms in shell n

N x W(lrn 1)

Z~v(O)

1

1V 2V

0.086 0.143

0.086 0.143

1.32 2.44

12

1V 2V

0.071 0.059

0.852 0.708

6

iv

0.003

o.o18

2V

0.006

0.036

4

x,/3a

24

1V 2V

0.0015 0.004

0.036 0.096

5

2a

12

1V 2V

0.0007 0.0015

0.008 0.018

Vol. 23, No. 11

CORRELATED SELF-DIFFUSION IN CRYSTALS

g~v and g2v denote geometrical constants to be determined by directly evaluating the EinsteinSmoluchovsky relation for the two vacancy mechanisms. As shown by Mehrer [16], in a b.c.c, lattice gav = 1 while g2v = 2 for both the I N - 2 N - 1 N and the 1 N - 2 N - 4 N mechanism of divacancy migration [12, 16]. In the f.c.c, lattice [12, 16] g l v = 1 and g2v = 2/3. For the actual evaluation of experiments, values of Zav(O) and Way(tin) in equations (14)have to be determined. As described in detail in reference [12], such numbers may be obtained from the computer simulation of the random vacancy migration in a cubic crystal lattice [ 17]. Corresponding results for the monovacancy mechanism in a b.c.c, and a f.c.c, lattice were presented in reference [ 11 ]. These have been included in Tables 1 and 2 which also show results for two mechanisms of divacancy migration in the b.c.c, lattice and for the usual divacancy mechanism in the f.c.c, lattice. In concluding, it is pointed out that, although both correlated and uncorrelated self-diffusion yield a Lorentzian quasielastic line, in the following two respects the line width (14a) associated with correlated diffusion differs from its random-walk value (16) (compare also reference [I 1] ): (i) For the same value of ~, AI" is substantially smaller for a correlated diffusion mechanism. (ii) Since in an encounter an atom may be displaced by more than just a nearest-neighbor vector, the anisotropy of the line width in a single crystal is a function of the underlying diffusion mechanism. With the values in Tables 1 and 2 it is observed that the orientation dependence of AF for vacancy-induced self-

857

diffusion does not differ very much from its behavior for random-walk diffusion. Therefore, the main difference between equations (14a) and (16) is the approximate replacement of f - t by C f -1 to account for correlations, where C is an orientation4n~pendent constant determined by

C = 1 -- b,v(T)W,v(O) -- bav(T)W2v(O)

(19)

b,v(r)Z,v(O) + 1,2v(T)Z~v(O) As examples, with the values in Tables 1 and 2, for bl v ( T ) = b=v(T) = 0.5, C = 0.47 for the £c.c. lattice while C = 0.45 for the 1 N - 2 N - I N divacancy mechanism in the b.c.c, lattice and C = 0.50 for the 1 N - 2 N - 4 N mechanism, respectively. In contrast, if monovacancies alone are responsible for diffusion [ 11 ], C = 0.69 for the f.c.c, lattice and C = 0.64 for the b.c.c. lattice. It is thus seen that the deviation of AV from its random-walk value (16) (with C - 1) is even more pronounced when divacancies contribute to the usually dominant self-diffusion via monovacancies. The main reason therefore is that Z2v(O) > Zlv(O). Finally it should be mentioned that by identifying ~¢ with the wave vector of the absorbed 3' quantum, all • of the above results equally well apply to the diffusioninduced broadening of Misssbauer lines. In this respect the above theory may be considered as a generalization of the results presented in reference [ 11 ] to account for the effect of divacancies.

Acknowledgement - I am indebted to Dr. H. Mehrer for suggesting the application of NMR methods to the quasielastic neutron scattering associated with selfdiffusion.

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2.

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3.

AIT-SALEMM, U_n_pub!ish~d thesis, Technische Hochsehule, Aachen (t976).

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GOLTZ G. (private communication).

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7.

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10.

WOLF D.,Phys. Rev. BIO, 2710 (1974).

I 1.

WOLF D.,Appl. Phys. Lett. 30,617 (1977).

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WOLFD.,Phys. Rev. B15,37(1977).

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858

CORRELATED SELF-DIFFUSION IN CRYSTALS

Vol. 23, No. 11

CHANDRASEKHARS.,Rev. Mod. Phys. 15, 1 (1943). EISENSTADT M. & REDFIELD A.G.,Phys. Rev. 132,635 (1963). 16. MEHRER H., Habilitationsschrift, Universitat Stuttgart (1974) (unpublished); J. Phys. F3,543 (1973). 14. 15.

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