Pair correlations in the cluster variation approximation

1llA

Physica

(1982) 200-216 North-Hollund

Publishing

Co

PAIR CORRELATIONS IN THE CLUSTER APPROXIMATION SANCHEZ

J.M.

Received

The cluster system

variation

method

in an inhomogeneous

correlation

functions.

the basic cluster, temperature.

in order

In the two-dimensional

the correlation

The effect

29 July 1981

is used IO obtain field,

functions

of second-neighbor

VARIATION

approximate

to calculate

square lattice, are correctly

free energy the

Fourier

functions transform

for a binary of

using the nearest-neighbors

given up to the order

pairs and many-body

interactions

the

pair

square as

7 in the inverse

is investigated.

1. Introduction The cluster variaton method (CVM)‘) has been extensively used for the calculation of thermodynamic properties of magnetic systems and alloys”-‘). In particular, phase diagrams for order-disorder, clustering, ferromagnetic and antiferromagnetic systems have been obtained which show marked improvement over those calculated in the mean-field approximation (BraggWilliams) and which compare favorably with recent Monte Carlo simulations*). An important

characteristic

of

the

CVM

is that

higher

levels

of

ap-

proximation can be realized by including larger clusters in the expansion of the free energy. The method is limited, however, by the drastic increase in the number of variational parameters with the cluster size which, eventually, renders theless,

the problem relatively

of minimizing small

clusters

the free energy (4 to 6 points)

utterly appear

impractical. to suffice

Neverfor

an

accurate characterization of the free energy away from the critical region. For example, the critical temperature for the spin-4 Ising ferromagnet, a parameter most frequently used for ascertaining a given cluster approximation, usually falls within a few percent of the exact or best estimated values4). Additional information concerning the performance of a given cluster approximation has been obtained by explicitly calculating the coefficients of the high-temperature expansion for either the zero-field magnetic susceptibility or the specific heat. In the tetrahedron-octahedron approximation, the specific heat is exact up to fourth order in the inverse temperature4). 037%4371/82/0000-0000/$02.75

@ 1982 North-Holland

PAIR

CORRELATIONS

IN THE

CLUSTER

VARIATION

APPROXIMATION

201

The CVM also gives a detailed approximate description of the state of short-range order since pair and many-body correlations for all clusters included in the approximation are directly obtained from the minimization of the free energy. The method does not provide, however, direct information concerning correlations for clusters larger than those explicitly included in the expansion of the free energy. Thus, despite the success of the CVM in phase-diagram calculations, the method has so far not been used to calculate, for example, the Fourier transform of the pair correlation functions. Such calculations would allow direct comparison with available experimental shortrange order (SRO) data, such as X-rays or neutron scattered intensities. The object of this investigation is to perform preliminary calculations of SRO intensities in two-dimensional lattices for different CVM approximations. The intensity calculations are carried out by formally writing the CVM free energy functional in an inhomogeneous field (staggered chemical potential or magnetic field) and by expressing the correlation functions in terms of successive derivatives of the free energy with respect to the field. The organization of the paper is as follows. In section 2 a brief review of the CVM is given and those results of fluctuation theory relevant to the calculation of SRO intensities are summarized in section 3. As an example, the Fourier transform of the pair correlation functions for the one- and two-dimensional Ising models in different approximations of the CVM is calculated in section 4. As we shall see, the results of the CVM for the Ising model cannot approach the degree of accuracy presently achieved with high-temperature expansions of the pair correlation functions. It should be emphasized, however, that the main application of the CVM is not expected to be for the Ising model, for which already very accurate results are available %‘I), but for more complicated systems requiring longer range of pair and many-body interactions. Such an example is presented in section 5, where SRO intensities are calculated for the square lattice in the square approximation of the CVM.

2. The free energy functional The CVM is closely related to the fact that the free energy of a system can be formally obtained by minimizing the following functional”): 9 = c E(a)X(a) 0

+ kBT c X(a) In X(a), (r

(1)

where for a binary system with N lattice sites the sums run over the 2N configurations and where E(a) and X(u) are, respectively, the energy and

J.M.

202

probability

for configuration

SANCHEZ

cr. Thus the free energy

of the system

is given by

FcvM = min{%}, where

the minimization

to the normalization

is carried condition

2 X(u)

The first term on the right-hand energy

of the system

out over all density

matrices

= I.

side of eq. (I) is the average

and can in general

X(m) subject

be written

as a linear

configurational combination

pairs and/or many-body correlations. In the present formulation and many-body correlations are variational parameters. The configurational energy for a binary system with pairwise and in an inhomogeneous

such

of pairs

interactions

field is

(2) where e(r) is an effective rth neighbor pair interaction energy, pP is the field (staggered chemical potential or magnetic field) at lattice point p and the spin operators up take values + 1 and - 1 for A and B atoms, respectively. Thus the average energy is

(E) = 2 E(u)X(a) = f C E(r) C &(p, r

0

where 51 and given by

The

central

52 are,

idea

respectively,

of the

CVM

the point

In X(a)

C

(3)

p,,[,(p),

P

and

is to approximate

right-hand side of eq. (I). A number available that yield, for the disordered ART $J X(u) CT= I

r)+

I’

of equivalent state’.‘2.‘3):

pair

correlation

the

second

methods

C 7A.l J$, .r,(n, 1) In xJ(n, I). n.1

= NkeT

functions

term are

in the

currently

(4)

where the indices n, I enumerate a finite set of clusters of n lattice points. where x,(n, I) is the probability of finding the n, 1 cluster in the configuration J (J=l,2,..., 2”) and where the coefficients yb.1 depend on the crystal lattice and on the level of approximation chosen. For the general case of a staggered chemical potential, the cluster probabilities x,(n, 1) will also depend on the location of the cluster (n, 1) in the lattice. Eq. (4) can be easily generalized to

Kit

c X(u) In X(a) = kBT c n.1

Y”.~ (p I.

x,Ynl,z,x,(P,.

x In XJ(PI,. . . , P,),

, Pn)

(5)

PAIR

CORRELATIONS

IN THE

CLUSTER

VARIATION

APPROXIMATION

203

where the sum over the set {pr, . . . , pn} runs over all n, 1 clusters in the lattice and where

with M,, the number of n, 1 clusters per lattice point. Combining eqs. (l), (3) and (5), one can approximate functional by 9 = t c e(r) c &(P, r)+ c P(P)SI(P)+ , P P

x xJ(Pl,.

. . ,

..

kr,T 3 Y~,I(p c p ) ,$, 19

.”

pn) In xJ(pl, . . . , pd.

As shown elsewhere, the probabilities xJ(pl,. correlation functions &(p,, . . . , pn) by4,5) xJ(Pl,

the free energy

. , ~“1 = W(ph

W(P2,

(6) . . ,

i2), . . . , T(P”,

p,) are given in terms of the

id),

(7)

where i,, iz, . . . , i, denote the configuration J and where the i, take values + 1 or - 1 if lattice point pn is occupied by an A or a B atom, respectively. The operator T(p, i) is defined as T(p, i) = +[1 + iup],

(8)

and therefore it equals 1 if there is an i atom at p and zero otherwise. The equilibrium values of the correlation functions are obtained from the minimization conditions 69 &n(Pl, . . * 9Pn) =

0.

(9)

In practice, eqs. (9) are solved in three steps. First, from the knowledge of the ground-states, which are determined by the interaction parameters e(r), one singles out a set of relevant structures14). It is then assumed that the absolute minimum of 9 is to be found among such ground-states. Secondly, the infinite set &(pr,. . . , pn) is reduced to a finite set by applying the symmetry operations of the space group of each of the ground-state structures. The final step consists in explicitly carrying out the minimization with respect to the remaining variables. 3. Short-range

order intensity

The multisite correlation (UP,’* *. apn) = (- kBT)" 9

functions

can be formally written as

L z app,,a”z . . . , akp.’

(10)

204

J.M.

with

the

partition

= C

Z

given

by

the

following

sum

over

the

2N

u :

configurations z

function

SANCHEZ

e-E(d/ksT,

IT

and where

the configurational

E(u) (see eq. (2)) includes

energy

the staggered

chemical potential p,,. Thus, if the minimum of free energy functional 9, i.e., FcvM, is identified with the actual free energy F = - k,T In Z, all correlation functions could in principle be calculated via eq. (10). The correlation functions thus obtained will obviously be approximate ones reflecting all intrinsic limitations of the CVM. For the point

b,) =

$

(~,a,+,> -

and pair correlation

functions

eq. (10) yields

(11) P

a”;; .

bp>b,+~) = - kBT

P

By Fourier-transforming

the above

(12)

P+r

equations

with respect

to the positions

x,,,

one obtains (o(k))

(lo(k)12>where

(13)

= N &,

j(a(k))l’

k stands

= - kBTN2

for a reciprocal

“F +(k)W-

space

vector,

(14)

k)’ the Fourier

transform

is defined

by

and the reciprocal &

space

differential

operator

is

= & C emik’s $.. P

P

The left-hand side of eq. (14) is the kinematic short-range order intensity calculated under the assumption that the atomic scattering amplitudes k-independent, and normalized by $(f, - f2)’ 14). In the disordered state (wp constant), the integrated intensity is

T I(k) = A’ T [(a;) - (up,>*] = N*(I - s:,.

I(k), fl are

(15)

PAIR

CORRELATIONS

IN THE

CLUSTER

VARIATION

APPROXIMATION

205

or, using eq. 12:

(16) where the derivatives are to be evaluated at constant kP and where 5, = (a,). Eq. (15) (or (16)) expresses the fact that the integrated intensity is constant, and it follows from the constraint that a given lattice point is occupied by either A or B atoms (i.e., a;= 1). Note, however, that eq. (16) will not, in general, be obeyed by the cluster variation free energy. Thus eq. (16) provides a temperature sensitive measure of the performance of the CVM in a given approximation. By taking the derivative of the minimization equations for the free energy functional (see eq. (9)) with respect to the field, we obtain, in the disordered state 6(s, 1)6(p, p’) + 2 2 F&p p”

s’

- p”) y

= 0,

(17)

P’

where 6(s, 1) and 6(p, p’) take value 1 for s = 1 (point cluster) and p = p’, respectively, and zero otherwise and where

(18) the subscript zero indicating that the derivative is to be evaluated in the disordered state. Note that due to the translational symmetry of the disordered state the second derivatives depend only on the difference p - p”. Furthermore, since the second derivatives of 9 vanish for any pair of points p, p” not included in the largest cluster of the CVM approximation, the number of different terms produced by eq. (18) is finite and, thus, they can be calculated explicitly. A more useful form of eq. (17) is its Fourier transform, namely s(s, 1) * 6(k, - k’) + T F,,,(k) ,;?Tkk),) = 0,

(19)

where F,,,,(k), &(k) and p(k) are, respectively, the Fourier transforms of F,,,(p), t,(p) and pp, and where 6(k, -k’) equals one if the reciprocal vectors k and -k’ are equal and zero otherwise. Calling FE:,(k) the inverse of the finite matrix F,,,(k), we obtain from eq. (19) (disordered state) the following well-known result of fluctuation theory15): &S(k) = - 6(k, k’)F$(k), +-4k’)

(20)

206

J.M. SANCHEZ

which

for s = 1 (i.e., for the point

I(k) = (la(k)l’) The

- l(o(k))1’

conservation

expressed kaT

of

the

correlation

[r(k) = (c(k)))

becomes

= NkRTF;,‘(k).

integrated

(21)

intensity

(see

eq.

(IS))

can

now

be

as

F F;;(k)=

N(l-6:).

(22)

As we shall see in the next section, eq. (22) is not strictly obeyed in the CVM (except for the one-dimensional chain), the left-hand side being in general temperature-dependent. In fact the integrated intensity for the two-dimensional square lattice in the pair approximation diverges logarithmically near the critical point. Despite the not surprising failure of the method near critical points, eq. (22) is closely obeyed approximation of the 2-dimensional agree

with the exact

result

at high temperatures: For the square square lattice, the intensity is found to

up to order

7 in the inverse

temperature.

4. Ising model The calculations outlined in section 3 will be carried (nearest-neighbor interactions) in the one-dimensional dimensional square lattice. The CVM approximations

out for the Ising model chain and in the twoto be used are the pair

(Bethe) and square approximations (Kramers-Wanniers-Kikuchi)‘). To exemplify the method, the intensity calculations are described in detail for the pair approximations. In such an approximation, the linear chain with zero magnetic field is treated exactly whereas for the two-dimensional square lattice the SRO intensity is that obtained by Elliot and Marshall’“). In the Elliot-Marshall-Bethe approximation, the intensity is correctly given to order l/T’ ‘4. The square approximation will be shown to give the correct SRO intensity

to order

l/T’.

4.1. Pair approximation The free energy 9

= iE 2

P

C

P

functional 52(P, PI + C

P

+ keT(1 - Z) where

the sums

in the pair approximation F&I(P)

+ FC

p

C

P

C

ii

is given yij(p, p)

by

In yir(p9 p)

2 C Xi(P) In -TV

are over all lattice

points

(23) p and over the z nearest-neighbor

IN THE CLUSTER

PAIR CORRELATIONS

VARIATION

vectors p, and where the point (xi) and pair (yii) probabilities Xi(P)

= :[I+

201

APPROXIMATION

are given by (24a)

&(P)l,

Yij(P, PI = a[1 + i&(P)

+ j&(p + P) +

Wb)

p)l,

ij52@,

with i, j taking values + 1 and - 1. The elements of the matrix Fs,Jr) (see eq. 18) are:

Fdr)

=

$$ cII $II[Wr, 0) + js(r,

F2p.&9 = Fz+Vr. II

-p)l,

Wb)

OMp, $1.

II

WC)

For zero chemical potential (or magnetic field), the minimization energy yields: 51=

e2

(264

0,

=

-

of the free

tanh(e/kT).

(26b)

Under the condition of zero chemical potential, the Fourier transform of the matrix of second derivatives F,,,(k) is diagonal. Thus, for calculating the SRO intensity at tl = 0, only F,,(k) need to be considered. From eq. (25a) we obtain Fll(k)

=

kBT

i1+(’ - ‘)& 1-

The short-range

s:

kBT52 $

(l-53

cos

p=l

k

. p

*

(27)

order intensity in zero field (el = 0) is given byr4)

W-5:)

r(k) = (b(k)12> = [1+(z-l)&~2C~=lcosk~p]’

w-0

4.1.1. One-dimensional chain For the linear chain (z = 2), eq. 28 becomes

J’J(l-5:) (14k)12) = l+[:-2&coska’

(29)

where a is the nearest neighbor distance and where k equals 2nIlNa with 1 integer. The pair correlation for mth neighbors are obtained by Fourier-transform-

J.M. SANCHEZ

208

ing eq. (29):

where

the last equality

Combining

follows

from

the explicit

evolution

of the integral.

eqs. (30) and (26b), we obtain

t?(m) = 57 = [- tanh(e/kHT)]“.

(31)

an exact result for the linear chain with zero chemical cular, for m = 0, eq. (31) gives the integrated intensity

F I(k) = N2[&(0) - .$:I= Thus, in the present case CVM free energy functional strictly obeyed. 4.1.2. Squure l&rice For the square lattice

(ldk,12>= where

N

(one-dimensional chain, pair approximation) the is such that eq. (16) for the integrated intensity is

(z = 4), eq. (28) reads

N(l-5;) 1 + 355 - 2&(cos k - p, + cos k . &’

/JI = u(l.O),

vectors

(32) given

by

pz= u(0, I),

with a the lattice

parameter.

the following

The summation integrated TT

I(,=$ F

In parti-

‘.

PI and p2 are the nearest-neighbor

zone yields

potential”).

I(k)=(14-$)

of eq. (32) over the first Brillouin

intensity: n

d8, d& j- 1 1 + 3675- 2&(cos 8, + cos 0,)’ 77 77

The double integral can be reduced to a complete kind, yielding for the integrated intensity

elliptical

integral

of the first

(33)

where

K(x) = 1 0

[I - x’sin’

01 -“‘d0,

PAIR CORRELATIONS

IN THE CLUSTER

VARIATION

APPROXIMATION

209

and where & is given by eq. (26b). Eq. (33) is valid for temperatures higher than the critical temperature T, (i.e., (&I
(IU(k)12)sw= [I - (&,ksT)(cosNk . p, + cos k . &I’ Integration

of the intensity in the Bragg-Williams

2 Iosw=;K

(

approximation

+J kBT .

yields (35)

J

4.1.3. Square approximation The square approximation for the two-dimensional square lattice (KramersWannier-Kikuchi) follows that of Bethe in the CVM hierarchy. For zero magnetic field, the SRO intensity (eq. (21)) is found to be (36) where w = - tanh(dkeT) Q(k)

=

c

eik.P =

and Q(k) is the lattice generating function:

z[cosk

. p1

+

cos k . ~21.

P

Eqs. (32), (34) and (36) for the SRO intensity in different CVM approximations are all of the form proposed by Fisher and Burford as a generalization of the mean-field approximation’@). Fisher and Burford found that, in the square lattice, the direct correlation function H(k) defined by I(k) = 1 + H(k)I(k)

(37)

is, correct to order 09, of the form H(k) = Ho(o) + Hdo)Q(k),

(38)

where H,,(w) = - 4w2 - 12w4- 44wh - 188~’ + 6’(w lo)

(39a)

H,(o) = w + w3 + 50~ + 210’+ 930~ + O(o”).

(39b)

and

As mentioned before, the Elliot-Marshall-Bethe approximation is correct to order w3. Expansion of eq. (36) for the direct correlation function in the

2 IO

J.M.

square

approximation

H;(w)=

SANCHEZ

gives

-4w*-

12w4-44wh-

178wR+B(w”‘)

(40a)

85w’f

(40b)

and H;(w)

= w + w3 + SW” + 2lw’+

0(0”),

thus giving the correct SRO intensity to order w’ (see eq. (39)). A comparison of integrated intensities in the Bragg-Williams, square

approximations

for the square

lattice

is shown

in fig. 1. Although

is, as expected, a marked improvement on the temperature integrated intensity with the cluster size, the approximation down at the critical integrated intensity, proached. The divergencies normalized intensity

SRO is given

IN(k) = 1;

Bethe

behavior clearly

and there

of the breaks

point. In fact the real space pair correlations, like the will diverge logarithmically as the critical point is apin question

intensity.

can be easily In

the

square

avoided

by defining

approximation,

the

a properly normalized

by

K(lxl)[l - bQ(k,l)

(4la)

‘3

where K(x) is the complete elliptical integral of the first kind and where Q(k) is the lattice generating function. In terms of w = - tanh(e/kT). x in eq. (41a) is given by x = 4w( I - 202)‘/[ 1 - w?(l + 4&J?)].

(a) ‘a

1.0

(4lb)

BRAGG WILLIAMS

(b)

PAIR APPROXIMATION

(c)

SQUARE APPROXIMATION

1.5

20

25

T/T, Fig.

1. The integrated

intensity

vs. temperature

sional Ising model with nearest-neighbor

for different

interactions.

approximation\

of the two-dimen-

PAIR

CORRELATIONS

IN THE

CLUSTER

VARIATION

Temperature expansion of the direct correlation with IN(k) yields (see eqs. (37) and (38)) H,N= -4w2-

12w4-44w6-

APPROXIMATION

function

211

HN(k) associated

18808+ O(o”‘),

HY = w + w3 + 50~ + 21w’+ 10109+ O(w”). Thus the normalization of the SRO intensity results in a slight improvement on the coefficients of the temperature expansion for the inverse correlation function, which is now given correctly to order w’. The normalization of the SRO intensity introduced in eq. (41) changes the usual critical behavior of the specific heat calculated by the CVM. By Fourier transforming TN(k), one obtains for the nearest-neighbor pair correlation (at zero field) 52=++(1-~)-‘K-‘(~) -

E(xF(P,

I

;+W[;-K(x)E(/3,x’)+K(x)F(P,x’) X

x7]},

(42)

where F@, x’) and E(P, x’) are, respectively, elliptic integrals of the first and second kind, where x is given by eq. (41b) and where x’

= (1 -

x2)-y

sin’P=(l+x)-‘,

E(x)=E

From eq. (42) it follows that the specific heat diverges at T, with a critical exponent of (Y= 1. Such a behavior for the specific heat differs from the classical finite discontinuity obtained with the CVM, and it coincides with that resulting from the treatment of fluctuations in the Ginzburg-Landau model by the Gaussian approximatior?). Z(k = 0) = The zero-field magnetic susceptibility, given by (7~/2)[(1- x)K(x)]-‘, diverges at T, with a critical exponent y = 1, as in the usual CVM and Gaussian approximations. 4.2. Many-body

interactions

The study of the effect of long-range and many-body interactions on the correlation functions is of fundamental importance for the analysis of SRO intensity data in alloys. Such systems cannot, of course, be properly described by simple Ising models for which pair correlations are very accurately characterized. Thus, one must resort to either analytical approximations or computer simulations. Regarding computer simulations, the long correlation range commonly observed in real crystals should require prohibitively large crystals, thus making such calculations impractical.

J.M. SANCHEZ

212

At the present

time the most

the SRO intensity variations principle

commonly

used analytical

is the Krivoglaz-Clapp-Moss

of it2’). Although

the

approximations

any range of pair interactions,

formula

approximation

for

(eq. (34)) or slight

in question

can

treat

in

they do not allow for the introduction

of many-body interactions. More importantly, and contrary to the experimental evidence in CU~AU~~), the Krivoglaz-Clapp-Moss formula predicts a temperature-independent shape for the isointensity profiles in k-space. Thus, one normally finds that in order to account for the measured intensities, the interaction parameters, in fact their ratios, should change appreciably with temperature. The elucidation of whether this unsatisfactory result is due to the poor level of approximation used or to the presence of many-body interactions will require the use of more reliable thermodynamic approximations. Within the mean-field theories, the CVM appears to be a likely candidate for such a purpose. The use of the CVM usually requires a considerable amount of computational work. However, for the two-dimensional lattice in the square approximation a closed form solution can be found for both the minimization of the free energy and for the short-range order intensity given by eq. (21). An extension of the solution to the free energy minimization found by Kikuchi and Brush23) for the case of first-neighbor interactions is given in the appendix. We have included, in addition to the first-neighbor interaction l2, a second-neighbor pair interaction E$ and a four-body interaction l4 associated with the square. The SRO intensity for zero field is found to be of the form (la(k)12) = N[u + ;bQ,(k) where

+ fcQz(k)l-‘,

Q,(k) and Q2(k) are given

(43)

by

Q,(k) = 2[cos k . pI + cos k . p?],

(44a)

Q2(k) = 4 cos k . pl cos k - pz,

tub)

with pi first neighbor

vectors.

The temperature-dependent coefficients u(T), b(T) and c(T) are given in the appendix in terms of the three types of interaction used: first-neighbor pair (~3, second-neighbor pair (E;) and four-body (Ed) interactions. For the purpose of normalization, the integrated intensity is given by lo=

2

7r(a - 2c)

A parameter is the ratio

of interest, which controls the shape c(T)/b(T) (see eq. (43)). By analogy

of the isointensity contours, with the Krivoglaz-Clapp-

PAIR

CORRELATIONS

IN THE

CLUSTER

VARIATION

Icl/lbl

APPROXIMATION

213

a -0.5

06 -

1.0

I.5

2.0

T/ Tc

Fig. 2. Absolute lattice

value of the effective

with pair and many-body

ratio of pair interactions

vs. temperature

for the square

interactions.

Moss formula, c(T)/b(T) will be referred to as the “effective” ratio of second-to-first-neighbors pair interactions. Contrary to the predictions of the Krivoglaz-Clapp-Moss approximation, the effective ratio of pair interactions is in general temperature-dependent. The temperature dependence of the magnitude of c(T)/b(T) is shown in fig. 2 for different values of the ratios (Y= li/lezl and /? = Q/)E~(. For negative values of first-neighbor pair interactions (E*(0) the system clusters at low temperature (ferromagnet) and thus the SRO intensity peaks at the center of the Brillouin zone. As seen in fig. (2), the “effective” ratio of second-to-firstneighbor pair interactions becomes equal to (Yonly at very high temperatures. Furthermore, the temperature dependence of c(T)/ b (T) is considerably enhanced by many-body interactions. If ~2 is taken to be positive, the low-temperature phase corresponds to an ordered structure generated by a concentration wave of wave vector (i, i). Therefore, the SRO intensity in the disordered phase is maximum at the Brillouin zone boundary position ($, 4). As in the clustering case, the effective pair interaction ratio c(T)/b(T) (negative) attains the value - (Yonly at high temperatures.

5. Summary The Fourier transform of the pair correlation function has been calculated from the approximate free energy functions generated by the CVM. In order to avoid most of the numerical work characteristic of the CVM, the calculations were carried out for the one- and two-dimensional lattices. In

J.M.

214

particular,

we investigated

first-neighbor

interactions

The results not compete

the one-

lattice

temperature. As expected, two dimensions,

data.

yields

the

two-dimensional lattice

mean-field

although

expansions,

approximations

In fact, the four-point correct

SRO

Ising

with many-body

interactions,

high temperature

over other

pret SRO intensity

and

and the square

for nearest-neighbor with exact

improvement square

SANCHEZ

they obviously

represent commonly

cluster

intensity

model

can-

a remarkable used to inter-

approximation

to order

with

interactions.

for the

7 in the

inverse

the approximation fails near the critical point, predicting, in a logarithmic divergency for the integrated intensity. The

divergency can be eliminated by properly normalizing the correlation functions, which improve slightly the coefficients of the temperature expansion of the SRO intensity. As a consequence of the normalization of the correlation functions, the specific heat shows a divergency at T,, with critical exponent a = 1, instead of the finite discontinuity predicted by the CVM. The critical behavior is thus identical to that obtained from the treatment of fluctuations in the Gaussian approximation. In view of the very accurate results available for the two- and threedimensional Ising model, the usefulness of the CVM for determining pair correlations is quite limited. The approximation may provide, however, a valuable tool for the interpretation of SRO intensity data in complex systems with long range of pair and many-body interactions. For such systems a more rigorous approach is not yet at hand and the most commonly used mean-field approximations fail to explain properly the experimental notably the change in shape of the isointensity contours observed in the Cu3Au system”). A preliminary study of the effect

of many-body

evidence: most with temperature

interactions

on the SRO

intensity was carried out for the two-dimensional square lattice. In the terminology of the Krivoglaz-Clapp-Moss formula, it was found that the ratio of effective second-to-first-neighbor pair interactions is temperature-dependent. This result at least,

Appendix

is particularly

a qualitative

encouraging

understanding

since it may provide

of the experimental

the basis for,

data in Cu3Au.

A

In the disordered state, the free energy the square approximation is given by’) 4 = (E) -t keT

Ii

functional

C xi ln xi - 2 C Yij In yii + C ij

ijkl

Wijk[

for the square

In

wijkl

,

lattice

in

PAIR

CORRELATIONS

IN THE

CLUSTER

VARIATION

APPROXIMATION

215

where Xi, yij and are, respectively, the point, nearest-neighbor pairs and square probabilities. In terms of the pair correlations for first (&) and second (5;) neighbors, and of the square multisite correlation (&J, the configurational energy is Wijkj

(E) = 2~252+ 2&5 +

~454,

with ~2, li and ~4 the corresponding interaction The cluster probabilities are related by xi

=

c jkl

wijkf,

Yij

=

c

kl

energy parameters.

wijkl,

where the indices i, j, k, 1 take value + 1 and - 1 for A and B atoms respectively. In the probability Wijkl,species i and k (or j and I) are located on the diagonal of the square. Minimization in the disordered state for zero field (x, = xi= 3 yields Wijk,= A~I~~+jk+kl+if)/4H~(ik+jl)H~kl, where i, j, k, 1 take values + 1, where A is determined condition I, and where xijkr

Wijkl

Ho = [H:(2Hi2+

H:)-

lJk,T],

Hz = exp[-

by the normalization

=

The unnormalized

Hi4H:]/[H:+

2Hi2- H:H;4H$],

Hi = exp[- ei/kgT],

H4 = exp[- EJkeTl.

SRO intensity is

(la(k)(*) = N la + tbQ,(k) + kQzW1, where Q,(k) and Q2(k) are given by eqs. (44a) and (44b), and where a=1-4Yo+4Wo-l6AW2(Wi+ b =4W2-2Y,-4A(4W;+ c =

2W$-8AW2(W$+

W4)-8B(W:+

W$‘+ W,2+2W$Wq)-8(B+C)(W;+ W4)-2B(2W:+

W;*+ W:)-4C(WZ+

with W0 = (16)-*

% ij I

~$1,

W2 = (16))* 8 ii w$,

W$ = (16))’ & ik w$, Yo = (4)_2 c

ij

y;‘,

W?W4) - 4C(2W:+

W4 = ( 16)-2 8

Y, = (4))* C ij y;‘, ij

ijkf w&

W;*+ W:), w4)w2,

W$W4),

‘I6

J.M.

A = - Wz/[(Wo+

SANCHEZ

W$-4W;],

B = - [w”w;+

w;‘-

2W3/{(W,,-

w$)[(w,,+

w~+4w:]},

c = 1w; + W”Wi - 2 W3/{( w,, - W$)[( w,, + Wi)’ - 4Wi]}.

References I) R. Kikuchi. 2) C.M.

Van

( 195I) 988.

Phys. Rev. 81 Baa], Physica

3) D. de Fontaine

64 (1973)

and R. Kikuchi.

571.

NBS

Publication

SP-496

(1978) 999.

4) J.M.

Sanchez

and D. de Fontaine.

Phys. Rev.

B 17 (1978)

5) J.M.

Sanchez

and D. de Fontaine,

Phys. Rev.

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6) R. Kikuchi,

J.M.

7) T. Tanaka, 8) M.K.

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L. Libelo

Phani. J.L.

Lebowitz

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H. Yamauchi

and R. Kligman.

York,

Fisher and R.J. Burford.

Phys. Rev. B 21 (1980) 4027.

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156 (1967)

Phys. Rev. B 6 (1972)

Proc. R. Sot.

Schwartz

28 (1980) 651.

Vol. 3, C. Domh and M.S. Green.

eds. (Academic

583.

3426.

J. Phys. Sot. Jpn. 12 (1957) 723. 1060: J. Math.

13) J.A. Barker, 14) L.H.

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1974).

M. Ferer and M. Wortis.

12) T. Morita,

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Phys.

13 (1972)

11.

A 216 (1953) 45. Cohen.

Diffraction

from

Materials

(Academic

Press.

New

York

1977). IS) See for example,

H.B.

Callen.

16) R.J. Elliot and W. Marshall, 17) C.J. Thompson,

Mathematical

18) M.A.

Theory

New

Krivoglaz, York,

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Thermodynamics

Rev. Mod. Statistical

of X-ray

(Wiley,

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Phys. 30 (1958) 75. Mechanics

and Thermal

(Macmillan.

Neutron

Scattering

New

York,

1972).

by Real Crystals

(Plenum.

1969). and SC.

Modern

Moss,

Theory

Phys. Rev.

of Critical

142 (1966) 418.

Phenomena

(Benjamin/Cummings,

pp. 72- 102. 21) See for example

S. Wilkins.

Phys. Rev. B 2 (1970)

22) P. Bardhan

and J.B. Cohen.

Acta Crys.

23) R. Kikuchi

and S.G. Brush, J. Chem.

3935.

A 32 (1976)

Phys. 47 (1967)

597. 195.

Reading

Mass..

1976)