Pair correlations and probabilities of many-particle configurations occurring in a flat triangular lattice

ISSN 1063-7834, Physics of the Solid State, 2006, Vol. 48, No. 5, pp. 854–863. © Pleiades Publishing, Inc., 2006.Original Russian Text © A.I. Gusev, 2006, published in Fizika Tverdogo Tela, 2006, Vol. 48, No. 5, pp. 613–621.

854

In the statistical theory of atomic ordering, the long-and short-range orders in ordering systems have not yetbeen taken into account simultaneously. There are twosets of methods for solving the problems of ordering.The cluster methods (in particular, the most developedmethod of cluster variation [1, 2]) make it possible toexactly take into account the interaction of particles inclusters, that is, the short-range order and many-particlecorrelations, but do not include the interaction betweenthe cluster and its environment. Therefore, the methodof cluster variation is cannot be applied to systemswhere the long-range order appears abruptly in a first-order phase transition. The other set of methods isrelated to the mean-field approximation. Here, themethod of static concentration waves is the most devel-oped and suitable for describing order–disorder transi-tions [3]. However, a way to calculate the particle inter-action potentials has not yet been found within thisapproach; therefore, the equilibrium superstructures inreal systems cannot be calculated using this method.For the purpose of describing the order–disorder struc-tural phase transitions in the substitution compounds,the method of an order-parameter functional [4–7],which exactly takes into account the symmetry of thelattice and the interaction between particles withinclusters, is quite powerful. This method has been usedto theoretically predict the types of superstructures thatform in ordering nonstoichiometric

MX

y

�

1 –

y

com-pounds (

M

= Ti, Zr, Hf, V, Nb, Ta;

X

= C N;

�

is avacancy) and

A

y

B

1 –

y

solid solutions [6–9]. However,near the order–disorder transition temperature

T

trans

, theboundaries of the ordered phases in the calculated

phase diagrams are shifted toward the

AB

(or

MX

0.5

�

0.5

)compounds as compared to their actual positions.

This flaw appears because the order-parameter func-tional technique takes into account only the long-rangeorder and the corresponding correlations. However, inaddition to long-range correlations, the ordered phasesexhibit short-range correlations, which do not disap-pear at the order–disorder transition temperature but areretained in the disordered phase and gradually decreasewith increasing temperature.

Thus, the problem of simultaneous description ofthe short- and long-range orders in the thermodynamicpotential of an ordering system has not yet been solved.A first step in this direction could be to express theprobabilities of the occurrence of many-particle config-urations via correlations or the short-range orderparameters. However, even this particular problem hasnot been resolved in the general case. At present, prob-lems of this kind are usually approached numericallywith the use of computer models (see, for example, [10,11]). For instance, in [11], the short-range order in non-stoichiometric titanium carbide TiC

y

was calculatedusing the Monte Carlo [12] and cluster variation meth-ods [2, 7]. In order to include interatomic correlations,an octahedral cluster consisting of six sites of a nonme-tallic sublattice was used. It was demonstrated in [11]that the clusters with adjacent vacancies are energeti-cally less favorable than the clusters in which thevacancies are located further apart. This conclusionshould be taken into account in the order-parameterfunctional method, in which, in its present form, it isassumed that clusters with the same number of vacan-cies have equal energies regardless of their mutual con-

SEMICONDUCTORS AND DIELECTRICS

Pair Correlations and Probabilities of Many-Particle Configurations Occurring in a Flat Triangular Lattice

A. I. Gusev

Institute of Solid State Chemistry, Ural Division, Russian Academy of Sciences, Yekaterinburg, 620219 Russiae-mail: [email protected]

Received March 29, 2005

Abstract

—Using the example of a disordered

A

y

B

1 –

y

solid solution in which the atoms occupy sites on a reg-ular triangular lattice, it is demonstrated that the probabilities of the occurrence of many-particle configurationscan be calculated analytically with allowance for pair correlations in the first coordination shell. An analyticalsolution is obtained by simultaneously taking into account the normalization conditions for the probabilitiesand maximizing the configurational entropy. Analogous individual solutions are also obtained for

A

y

B

1 –

y

solidsolutions on square and face-centered cubic lattices. It is demonstrated that pair correlations in the first coordi-nation shell on cubic and fcc lattices give rise to pair correlations of opposite sign in the second coordinationshell.

PACS numbers: 02.50.–r, 05.50.+q, 61.43.Dq

DOI:

10.1134/S1063783406050076

PHYSICS OF THE SOLID STATE

Vol. 48

No. 5

2006

PAIR CORRELATIONS AND PROBABILITIES 855

figuration. By calculating the order-disorder transfor-mations with due regard to this difference in energy,one could correct the phase diagrams of the orderingnonstoichiometric compounds.

In spite of a certain degree of success in computermodeling of the short- and long-range order in substitu-tional solid solutions, the relation between the correla-tions and probabilities of the occurrence of many-parti-cle configurations has not been established yet. Let usconsider whether it is possible to introduce correlationsinto analytic calculations of the probabilities of many-particle configurations.

In [13, 14], the conditional probabilities were usedfor this purpose. In the simplest version of thisapproach, the probabilities of two-particle configura-tions are considered to be known parameters, givenwith due regard to pair correlations. Assuming that arandomly chosen initial site is occupied with statisticalprobability, it is possible to calculate the probability ofthe next site being occupied by one or another kind ofatom using the known probability of the occurrence ofa two-particle configuration. Successively applying thisprocedure to the chain of atoms that form a many-par-ticle configuration, it is possible to find the conditionaloccupation probabilities for all sites and, taking intoaccount pair correlations, calculate the probability ofthe many-particle configuration. However, it isimproper to replace the independent single-particleprobabilities with conditional probabilities, because inthis case there appears an ambiguity in the choice of theinitial site, the corresponding probability, and the sitetraversal direction.

In [15, 16] it was suggested to calculate the proba-bilities of many-particle configurations using correla-

tion moments

ε

. In general, the probability of find-ing in a binary

A

y

B

1 –

y

crystal the

i

th configuration ofcluster

s

consisting of

R

(

s

)

lattice sites occupied by(

R

(

s

)

–

p

) atoms of species

A

and by

p

atoms of species

B

is given by the crystal average

(1)

where

σ

j

(

β

) is the occupation number, which is eitherunity or zero;

β

is the type of atom; and

j

is the site num-ber. Some configurations can be equivalent; i.e., each(

i

th) configuration of cluster

s

is characterized by its

multiplicity . The multiplicity is equal to the

index of the point symmetry group of the

i

th con-figuration of cluster

s

with respect to the point group

of cluster

s

, in which all sites are occupied by

atoms of the same species. Therefore, =

n

( )/

n

( ), where

n

(

G

) is the order of the group

G

Pis( )

Pis( ) σ j β( )

j 1=

Rs( )

∏ ,=

λis( ) λi

s( )

Gis( )

G0s( )

λis( )

G0s( )

Gis( )

[17]. According to [18, 19], the correlation moment

ε

s

of order

s

is given by

(2)

Using Eq. (2), the probability for the

A

y

B

1 –

y

solidsolution can be expressed as [15, 16]

(3)

Here,

p

is the number of sites of the cluster

s

occupied

by

B

atoms, is the number of

n

th subclusters in acluster

s

having the

i

th configuration (an

n

th subclusterconsists of

n

=

q

+

r

sites,

q

of which are occupied byatoms

A

and

r

, by atoms

B

), and

ε

n

,

q

,

r

is the correlationmoment of order

n

. Since the smallest subcluster con-sists of two adjacent sites, the sum in Eq. (3) runs from

q

= (2 –

r

) to

q

= (

n

–

r

), where

r

= 0, 1, or 2. If there areonly pair correlations (

ε

2,

q

,

r

≡

ε

q

,

r

) between atomsoccupying adjacent sites in the crystal, then

n

= 2 andEq. (3) becomes

(4)

For the sake of simplicity, we introduce the notation

ε

q

,

r

≡ ε. In the absence of long-range order, the meanoccupation numbers are equal to the concentrations ofthe constituents. Therefore, for the disordered AyB1 – ysolid solution, we have ⟨σj(A)⟩ = y, ⟨σj(B)⟩ = (1 – y), and

(5)

The probabilities satisfy the usual normaliza-tion condition

(6)

If is the fraction of sites occupied by atoms A in theith configuration of cluster s, then we can write the fol-

εs σ j β( ) σ j β( )⟨ ⟩–[ ]j 1=

Rs( )

∏ .=

Pis( )

Pis( ) σ j A( )⟨ ⟩ R

s( )p– σ j B( )⟨ ⟩ p

=

+ σ j A( )⟨ ⟩ Rs( )

p– q– σ j B( )⟨ ⟩ p r–aq r,

n( ) εn q r, , .n s∈∑

q 2 r–=

q = n r–

∑

aq r,n( )

Pis( ) σ j A( )⟨ ⟩ R

s( )p– σ j B( )⟨ ⟩ p

=

+ σ j A( )⟨ ⟩ Rs( )

p– q– σ j B( )⟨ ⟩ p r–aq r,

2( ) εq r, .q 2 r–=

q = 0

∑

Pis( )

yR

s( )p–( )

1 y–( ) p=

+ yR

s( )p– q–( )

1 y–( ) p r–( )aq r,

2( ) ε.q 2 r, 0= =

q = 0 r = 2,

∑

Pis( )

λis( )

Pis( )

i s∈∑ 1.=

lis( )

856

PHYSICS OF THE SOLID STATE Vol. 48 No. 5 2006

GUSEV

lowing normalization condition for the AyB1 – y solidsolution:

(7)

In general, a cluster can include sites belonging to sev-eral coordination shells (from the first to the kth shell).

If , , and are the relative numbersof the A–A, A–B, and B–B pairs (pair bonds), respec-tively, among all pair bonds in the kth coordinationshell of the ith configuration of cluster s, then we canwrite the following relations between the probabilitiesof the clusters and pair bonds in the kth coordinationshell:

(8)

The multiplicities of the A–A, A–B, and B–B pairs are

equal to = 1, = 2, and = 1 for any valueof k.

Clearly, the probabilities of cluster configurations inthe presence of correlations have to satisfy normaliza-tion conditions (6)–(8). Moreover, the probabilities

cannot be negative. Equation (5) satisfies all nor-malization conditions. In order to verify that therequirement that the probabilities be nonegative is sat-isfied, we need to find the range over which the corre-lations vary with the composition of the solid solution.

Let us consider an AyB1 – y solid solution with an

arbitrary lattice ( = y, = 1 – y). If atoms takelattice sites in the first coordination shell with the cor-relation ε (εAA = εBB ≡ ε and εAB ≡ –ε), then the proba-bilities of the nonequivalent A–A, A–B, and B–B pairsare given by

(9)

It follows from Eqs. (9) that the mathematically possi-ble range of pair correlations is

(10)

lis( )λi

s( )Pi

s( )

i s∈∑ y.=

ni AA( )k

s( )ni AB( )k

s( )ni BB( )k

s( )

ni AA( )k

s( ) λis( )

Pis( )

i s∈∑ λ0

bk( )P0

bk( ),=

ni AB( )k

s( ) λis( )

Pis( )

i s∈∑ λ1

bk( )P1

bk( ),=

ni BB( )k

s( ) λis( )

Pis( )

i s∈∑ λ2

bk( )P2

bk( ).=

λ0

bk( )λ1

bk( )λ2

bk( )

Pis( )

P0a( )

P1a( )

P0b( )

y2 ε 0, P1

b( )≥+ y 1 y–( ) ε– 0,≥= =

P2b( )

1 y–( )2 ε 0.≥+=

1 y–( )2–

y2

– ⎭⎬⎫

ε y 1 y–( ),for y 0.5≥for y 0.5.≤

≤ ≤

The positive upper limit ε = y(1 – y), which is universalfor all lattices and all coordination shells, can also befound from the following simple considerations. Themaximum positive value of the short-range orderparameter αmax (corresponding to short-range separa-tion) in any coordination shell of an AyB1 – y solid solu-tion with any lattice is equal to unity. Since the short-range order parameter and the pair correlation arerelated as

(11)

the maximum value of the pair correlation (at αmax = 1)is εmax = y(1 – y)αmax ≡ y(1 – y).

Notice that Eq. (10) gives the widest range of possi-ble values of the pair correlation ε in the first coordina-tion shell. This range is equal to the physically admis-sible ranges of ε values for AyB1 – y solid solutions withsquare and bcc lattices.

Indeed, in the general case for y < 0.5, the maximum

probability is = (where PA = y) for anycoordination shell. If the number of A atoms is small

(y 0), then = 1 and therefore = y. Thehighest short-range order is achieved in an ideal com-pletely ordered superstructure. For AyB1 – y solid solu-tions with a square lattice and y ≤ 0.5, superstructuresA3B (y = 1/4) and AB (y = 1/2) are possible. In these

superstructures, the probability for the firstcoordination shell is equal to 1/2 and 1, respectively.

Since λAB = 2, the probability ≡ is 1/4and 1/2 in these superstructures, respectively (i.e., thisprobability coincides with y). Thus, for AyB1 – y solid

solutions with a square lattice, we have = y inthe range 0 ≤ y ≤ 0.5. It can also be demonstrated in a

similar way that = (1 – y) in the range 0.5 ≤ y ≤1.0. According to [20], the short-range order parameterin the jth coordination shell of the AyB1 – y solid solutionis given by

(12)

Using Eqs. (11) and (12) and the values of indi-cated above, the limit of the correlation parameter ε ≡εAA in the first coordination shell of AyB1 – y with asquare lattice can be found to be ε = –y2 for 0 ≤ y ≤ 0.5and ε = –(1 – y)2 for 0.5 ≤ y ≤ 1.0. Therefore, we cometo the conclusion that the physical limits of the pair cor-relation ε in AyB1 – y with a square lattice coincide withthe range given in Eq. (10).

For AyB1 – y solid solutions with a bcc lattice, usingthe same considerations as for the case of a square lat-

tice, we can find that = y when the concentration

ε εAA≡ εBB y 1 y–( )α,= =

PABmax

PBmax

PA

PBmax

PABmax

λABPAB1( )max

PAB1( )max

P1

b1( )max

PAB1( )max

PAB1( )max

α j 1PAB

j( )

PABbin

---------–y 1 y–( ) PAB

j( )–

y 1 y–( )-----------------------------------.≡=

PAB1( )max

PABmax



of A atoms is small (y 0). For AyB1 – y solid solu-tions with a bcc lattice and with y ≤ 0.5, the possiblesuperstructures are A3B (at y = 1/4; for example, Fe3Alwith the D03 structure) and AB (at y = 1/2; for example,FeAl, CuBe, CuZn with the B2 (β-brass) structure). Forthese superstructures with a bcc lattice, just as for thesuperstructures with a square lattice, the probability

in the first coordination shell is equal to 1/4 and1/2, respectively (i.e., it coincides with y). Therefore,the limit value of the correlation ε ≡ εAA in the first coor-dination shell of the AyB1 – y compound with a bcc lat-tice is equal to –y2 for 0 ≤ y ≤ 0.5 and to –(1 – y)2 for0.5 ≤ y ≤ 1.0. Thus, the physical limits of the pair cor-relation ε in AyB1 – y with a bcc lattice likewise coincidewith the range given in Eq. (10). However, in manycases, the structure of a particular lattice puts additionalconstraints on the mathematically admissible range of εvalues and narrows it. For instance, in AyB1 – y with a

regular triangular lattice (Fig. 1), we have = y

for y ≤ 1/3 and = (1 – y) for y ≥ 2/3. The maxi-

mum probability of the A–B pair ≡ inthe range 1/3 ≤ y ≤ 2/3 is equal to 1/3 and is achieved inthe completely ordered phases of A2B, AB, and AB2. Iffollowing [6, 7, 21] we assume that the probability

is a linear function of y over the range 1/3 ≤ y ≤

2/3, then is constant and equal to 1/3 over thisrange of y values. Using this fact and Eqs. (11) and (12),we can find that the limit of the pair-correlation param-eter ε ≡ εAA in the first coordination shell of AyB1 – y witha triangular lattice is ε = y(1 – y) – 1/3 over the range1/3 ≤ y ≤ 2/3 and that the physically accessible range ofε values in a triangular lattice is given by

(13)

For AyB1 – y solid solutions with an fcc lattice, the

probability is equal to y, (1 + 2y)/6, (3 – 2y)/6,and (1 – y) over the ranges 0 ≤ y ≤ 1/4, 1/4 ≤ y ≤ 1/2,1/2 ≤ y ≤ 3/4, and 3/4 ≤ y ≤ 1, respectively [7, 21]. Tak-

PAB1( )max

PAB1( )max

PAB1( )max

PAB1( )max

P1

b1( )max

PAB1( )max

PAB1( )max

1 y–( )2–

–1/3 y 1 y–( )+

y2

– ⎭⎪⎬⎪⎫

ε y 1 y–( ),

for y 2/3≥for 1/3 y 2/3≤ ≤for y 1/3.≤

≤ ≤

PAB1( )max

ing into account these values of , the physicallyaccessible range of ε values in an fcc lattice is given by

PAB1( )max

(14)

1 y–( )2–

–1/2 y/3 y 1 y–( )+ +

–1/6 y/3– y 1 y–( )+

y2

– ⎭⎪⎪⎬⎪⎪⎫

ε y 1 y–( ),

for y 3/4≥for 1/2 y 3/4≤ ≤for 1/4 y 1/2≤ ≤for y 1/4.≤

≤ ≤

P0(a) P1

(a)

P0(b) P1

(b) P2(b)

P0(c) P1

(c) P2(c) P3

(c)

AyB1 – y

A2B

AB

A B

Fig. 1. Nonequivalent configurations and probabilities P ofpatterns a (site), b (pair bond), and c (triangular cluster) in thesequence {s} employed to describe a disordered AyB1 – ysolid solution in which the atoms are located in sites of aregular triangular lattice. Solid circles denote A atoms, andopen circles, B atoms. A2B and AB are ordered solid solu-tions in which the maximum probability of the A–B bond in

the first coordination shell ≡ = 1/3 is

reached.

PAB1( )max

P1b1( )max

858


GUSEV

0.3

0.1

0

–0.1

–0.30 0.2 0.6 1.0

AyB1 – yεmax = y(1 – y) (‡)

εmin = –y2 εmin = –(1 – y)2

y

ε0.3

0.1

0

–0.1

–0.30 0.2 0.6 1.0

AyB1 – yεmax = y(1 – y) (b)

εmin = –y2 εmin = –(1 – y)2

yε

εmin = –1/3 + y(1 – y)

0.3

0.1

0

–0.1

–0.30 0.2 0.6 1.0

AyB1 – yεmax = y(1 – y) (c)

εmin = –y2

y

ε

6+ y(1 – y)

–1 + 2yεmin = –

6+ y(1 – y)

2y – 3εmin =

εmin = –(1 – y)2

Fig. 2. Admissible ranges of pair-correlation ε values in the first coordination shell of disordered AyB1 – y solid solutions with(a) square, (b) regular triangular, and (c) fcc lattices. The physically admissible regions for square, triangular, and fcc lattices givenby Eqs. (10), (13), and (14), respectively, are shown by solid lines. The physically admissible range of ε values for the bcc lattice isthe same as that for the square lattice. The intervals of admissible ε values given by Eqs. (17), (19), and (20), which are obtainedfrom Eq. (16) (for the triangular lattice), Eq. (18) (for the square lattice), and Eqs. (3)–(5) (derived using the method of correlationmoments [15, 16]), are shaded. The physically admissible ranges of ε values are much wider than the intervals of ε values that followfrom the method of correlation moments. This result means that, if the ε values that lie in the physically admissible regions givenby Eqs. (10), (13), and (14) for the square, triangular, and fcc lattices, respectively, but are beyond the intervals given by Eqs. (19),(17), and (20), respectively, then Eqs. (18) and (16) and analogous equations for the fcc lattice following from Eqs. (3)–(5) [15, 16]produce negative probabilities for some of the nonequivalent cluster configurations, which is a physically invalid result.



The admissible ranges of the values of ε ≡ εAA in the firstcoordination shell of AyB1 – y solid solutions ε withsquare, bcc, regular triangular, and fcc lattices areshown in Fig. 2. It is clear that the ranges of ε values forthe solid solutions with square and bcc lattices coincidewith the mathematically admissible range given inEq. (10) and that the physically admissible ranges (13)and (14) for solid solutions with triangular and fcc lat-tices are narrower than that given by Eq. (10).

Let us now demonstrate that the application of themethod of correlation moments for calculating theprobability of many-particle configurations analyti-cally, as suggested in [15, 16], does not allow one toinclude all possible values of the pair correlation and, atlarge absolute values of ε, leads to negative probabili-ties for some clusters, i.e., to a physically meaninglessresult.

Let the atoms of the AyB1 – y solid solution occupysites of a regular triangular lattice. To describe the tri-angular lattice, we use a cluster in the form of an equi-lateral triangle (R(s) = 3), which has four nonequivalentconfigurations (Fig.1) characterized by the probabili-

ties (all three sites are occupied by A atoms), (two sites are occupied by A atoms and one site by a B

atom), (one site is occupied by an A atom and two

sites by B atoms), and (all three sites are occupiedby B atoms). The multiplicities of these configurations

are = 1, = 3, and = 1. The probabilities

are positive and cannot exceed unity by definition;hence, in the most general case, they lie in the ranges

(15)

For different configurations of the triangular cluster,

we find that 1 ≥ ≥ 0, 1/3 ≥ ≥ 0, 1/3 ≥ ≥

0, and 1 ≥ ≥ 0. The overlap patterns for the trian-gular clusters are the A–A, A–B, and B–B pair bonds,

which have probabilities , and the overlap patternsof the bonds are sites occupied by atoms A or B with

probabilities . The triangular cluster and the over-lap patterns form a sequence of patterns {s} whichunambiguously characterizes the lattice in question.

For the triangular cluster, the correlations εq, r inEq. (5) are as follows: ε2, 0 ≡ εAA = ε, ε0.2 ≡ εBB ≡ ε, andε1, 1 ≡ εAB ≡ –ε. For the triangular cluster with configu-

ration (i.e., the cluster consisting of atoms A only),

we have p = 0, q = 2, r = 0, and = 3; so, according

to Eq. (5), = y3 + 3yε. For the triangular cluster

with configuration , we have p = 1; so r and q can

P0c( )

P1c( )

P2c( )

P3c( )

λ0c( ) λ1

c( ) λ3c( )

Pic( )

1/λic( )( ) Pi

c( )0.≥ ≥

P0c( )

P1c( )

P2c( )

P3c( )

Pib( )

Pia( )

P0c( )

a2.02( )

P0c( )

P1c( )

assume only two values: q = 2, r = 0, and = 1 and

q = 1, r = 1, and = 2. Using these values and

Eq. (5), we find = y2(1 – y) + 2yε1, 1 + (1 – y)ε2, 0 =

y2(1 – y) + (1 – 3y)ε. The probabilities and

can be found in the same way. Thus, by including paircorrelations and using Eq. (5), which was derived bythe method of correlation moments, the probabilities ofthe occurrence of different configurations of the trian-gular cluster are found to be

(16)

Using the values of and inequalities (15), we find

from Eq. (16) that ≥ 0 only if the correlation ε lies

within the intervals

(17)

These intervals define the region of admissible val-ues of the correlation ε in the disordered AyB1 – y solu-tion with a triangular lattice that result from Eqs. (5) or(15) (Fig. 2). As seen in Fig. 2b, this region is narrowerthan the mathematically admissible interval (10) andthe physically admissible interval for the triangular lat-tice (13). This circumstance prevents correct consider-ation of all values of the pair correlation ε. For this rea-

son, some of the probabilities as calculated using

Eqs. (5) or (16) for certain values of the correlationlying in range (13) prove negative, which is inadmissi-ble. This means that using Eqs. (5) and (16) and moregeneral expressions (3) and (4), suggested by authors of[15, 16], leads to a physically incorrect solution.

Let us consider now an AyB1 – y solid solution with asquare lattice. We choose a square consisting of foursites for the cluster. This cluster has six nonequivalentconfigurations. Taking into account pair correlations inthe first coordination shell only, the probabilities of thenonequivalent configurations of the square cluster canbe found to be

a2 0,2( )

a1 1,2( )

P1c( )

P2c( )

P3c( )

P0c( )

y3

3yε,+=

P1c( )

y2

1 y–( ) 1 3y–( )ε,+=

P2c( )

y 1 y–( )22 3y–( )ε,–=

P3c( )

1 y–( )33 1 y–( )ε.+=

λic( )

Pic( )

y2/3– ε y 1 y–( )2

/ 2 3y–( ), 0 y 1/2,≤ ≤ ≤ ≤

1 y–( )2/3– ε y

21 y–( )/ 3y 1–( ),≤ ≤

1/2 y 1.≤ ≤

Pic( )

860


GUSEV

(18)

Using the values of and inequality (15), whichdefines the range of probabilities, we can find fromEq. (18) that the correlation in Eq. (5) can vary withinthe interval

(19)

It is clear from Fig. 2a that the region defined byEq. (19), which follows from Eqs. (5) and (18), is muchsmaller than the physically admissible range for thesquare lattice given by Eq. (10). Therefore, for thesquare lattice, the application of Eq. (5), which wasderived using the method of correlation moments, alsodoes not allow one to consider all possible values of thepair correlation in the first coordination shell.

Similar reasoning when applied to the fcc latticeusing the octahedral cluster, which has ten nonequiva-lent configurations [6, 7, 20], shows that Eq. (5) leadsto the following range of values of the correlation ε:

(20)

It is seen in Fig. 2c that the range defined by Eq. (20) ismuch narrower than the range given by Eq. (14), whichis physically admissible for an fcc lattice.

The estimations of the admissible ranges of valuesof the pair correlation in the first coordination shellobtained for triangular, square, and fcc lattices and sim-ilar considerations for other lattices demonstrate that, inall cases, the application of Eq. (5) following frommore general expressions (3) and (4) [15, 16] leads tophysically incorrect solutions for which all probabili-ties are positive only over a very narrow interval of cor-relation values. Therefore, the method of correlationmoments proposed in [15, 16] for calculating the prob-abilities of many-particle configurations does not allowone to include all physically admissible values of thepair correlation.

Thus, using the conditional probabilities [13, 14] oran expansion in terms of correlation moments [15, 16]to calculate the probabilities of many-particle configu-

P0c( )

y4

4y2ε,+=

P1c( )

y3

1 y–( ) 2y 1 2y–( )ε,+=

P2c( )

y2

1 y–( )24y 1 y–( )ε,–=

P3c( )

y2

1 y–( )21 2y–( )2ε,+=

P4c( )

y 1 y–( )32 1 y–( ) 2y 1–( )ε,+=

P5c( )

1 y–( )44 1 y–( )2ε.+=

λic( )

y2/4–

1 y–( )2/4– ⎭

⎬⎫

ε y 1 y–( )/4,for 0 y 1/2≤ ≤for 1/2 y 1.≤ ≤

≤ ≤

y2/12– ε y 1 y–( )2

/ 4 2 3y–( )[ ], 0 y 1/2,≤ ≤ ≤ ≤

1 y–( )2/3– ε y

21 y–( )/ 4 3y 1–( )[ ],≤ ≤

1/2 y 1.≤ ≤

rations leads to erroneous results, especially for largeabsolute values of the pair correlation. However, theprobabilities of many-particle configurations can becalculated with allowance for correlations by maximiz-ing the configurational entropy.

Indeed, in the limit of high temperatures T ∞ or

small cluster energies 0, the free energy F ofa solid solution is proportional to the negative of itsentropy, F ~ –S. In other words, a minimum in freeenergy corresponds to a maximum in entropy. The statein which only pair correlation is present and correla-tions of higher orders are zero corresponds to a disor-dered distribution of pairs in the crystal and the maxi-mum value of the configurational entropy.

Let us consider again an AyB1 – y solid solution inwhich the atoms are distributed over the sites of a regu-lar triangular lattice (Fig. 1) with pair correlation ε inthe first coordination shell. The probabilities of the

occurrence of pairs are given by Eq. (9). Let us

find the probabilities of three-particle configura-

tions (triangular clusters). The probabilities haveto satisfy normalization conditions (6)–(8), which canbe written in the form

(21)

Following the method of cluster variation [2, 7], theconfigurational entropy Sc of a macroscopic state of theAyB1 – y solid solution can be written as

(22)

For the selected succession of patterns (site a, pair bondb, triangular cluster c of three nearest neighbor sites)that describe the triangular lattice, the renormalizationfactors y(s) [7, 22] are y(a) = 1, y(b) = –3, and y(c) = 2.

Using these values and the multiplicity factors , wecan rewrite Eq. (22) as

(23)

eis( )

Pib( )

Pic( )

Pic( )

P0c( )

3P1c( )

3P2c( )

P3c( )

+ + + 1,=

P0c( )

2P1c( )

P2c( )

+ + y,=

P0c( )

P1c( )

+ P0b( )

y2 ε.+= =

Sc y ε,( ) Sc Pis( )( )≡

= –kBNA ys( ) λi

s( )Pi

s( )Pi

s( ).ln

i s=

∑s a=

c

∑

λis( )

Sc y ε,( ) kB P0a( )

P0a( )

ln P1a( )

P1a( )

ln+( )[=

– 3 P0b( )

2P1b( )

P1b( )

ln P2b( )

P2b( )

ln+ +( ) 2 P0c( )

P0c( )

ln(+

+ 3P1c( )

P1c( )

ln 3P2c( )

P2c( )

ln P3c( )

P3c( )

ln+ + ) ].



By solving the set of equations (16), we can express the

probabilities , , and in terms of =

y2 + ε, y, and the probability of the triangular clus-ter consisting of atoms A only:

(24)

Then, we substitute Eqs. (24) into Eq. (23) and use thecondition that the configurational entropy reach a max-imum

(25)

We get a cubic equation after differentiation of

Eq. (23); substitution of Eqs. (24) for , , and

; and some mathematical manipulation involvingEq. (25). By solving this equation, we find the probabil-

ity and then, using Eq.(24), the probabilities of theother configurations of the triangular cluster:

(26)

where

P1c( )

P2c( )

P3c( )

P0b( )

P0c( )

P1c( )

P0b( )

P0c( )

,–=

P2c( )

y P0c( )

2P0c( )

,–+=

P3c( )

1 3P0b( )

3y– P0c( )

.–+=

∂Sc Pis( )( )/∂P0

c( ) ∂Sc y ε P0c( ), ,( )/∂P0

c( )≡ 0.=

P1c( )

P2c( )

P3c( )

P0c( )

P0c( )

y y2 ε+( ) ε y 1 y–( ) ε–[ ]A,–+=

P1c( )

y y 1 y–( ) ε–[ ] y 1 y–( ) ε–[ ]A,+=

P2c( )

1 y–( ) y 1 y–( ) ε–[ ] y 1 y–( ) ε–[ ]A,–=

P3c( )

1 y–( ) 1 y–( )2 ε+[ ] ε y 1 y–( ) ε–[ ]A,+ +=

(27)

The obtained solutions are valid for all values of y andε satisfying the boundary conditions (13) (Fig. 2).

Actually, these are solutions that give a nonzero pos-itive probability for any configuration of the triangularcluster. As an example, Fig. 3 demonstrates the proba-

bilities as functions of ε for y = 1/4 and 1/3.

In the particular case where ε = –(1 – y)2 and y > 2/3,

there are no B–B bonds; so the and configu-

rations are absent in the lattice. Since = 0 and

= 0, the normalization conditions involve only

and , from which we find the solution =

3y – 2 and = 1 – y. In the case where ε = –y2 andy < 1/3, i.e., where there are no A–A bonds in the trian-

gular lattice, we get an analogous solution = 0,

= 0, = y, and = 1 – 3y.

By maximizing the configurational entropy, we canfind solutions for the particular cases of AyB1 – y withsquare and fcc lattices. In the case where y > 0.5 and ε =

Aε 1 2y–( )

2----------------------

118------ 81ε2

1 2y–( )2[+⎩⎨⎧

=

---+ 12 y y2

– 2ε+( )3]

1/2

⎭⎬⎫

1/3ε 1 2y–( )

2----------------------

⎩⎨⎧

+

–118------ 81ε2

1 2y–( )212 y y

2– 2ε+( )

3+[ ]

1/2

⎭⎬⎫

1/3

.

λic( )

Pic( )

P2c( )

P3c( )

P2c( )

P3c( )

P0c( )

P1c( )

P0c( )

P1c( )

P0c( )

P1c( )

P2c( )

P3c( )

y = 1/4

λ3P3

λ2P2

λ1P1

λ0P0

1.0

0.6

0.2

0–0.1 0 0.1 0.2

λ iP

i

ε

y = 1/3

λ3P3

λ2P2

λ1P1

λ0P0

–0.1 0 0.1 0.2ε

Fig. 3. Probabilities , , , and of the configurations of the triangular cluster as functions of the

correlation ε in disordered AyB1 – y solid solutions (y = 1/4, 1/3) with a triangular lattice. The limiting values of the correlation ε areshown by dashed lines.

λ0c( )

P0c( ) λ1

c( )P1

c( ) λ2c( )

P2c( ) λ3

c( )P3

c( )

862


GUSEV

–(1 – y)2, these lattices do not contain B–B bonds; so thenumber of configurations of the basic cluster is limitedto three. In the square lattice, these configurations are as

follows: a square consisting of four A atoms ( = 1),a square consisting of three A atoms and one B atom

( = 4), and a square consisting of two nonadjacent

A atoms and two nonadjacent B atoms ( = 2). Therenormalization factors for the square lattice are equalto y(a) = 1, y(b) = –2, and y(c) = 1. In the fcc lattice, theseconfigurations are as follows: an octahedral cluster con-

sisting of A atoms only ( = 1), an octahedron that

includes one B atom ( = 6), and an octahedron that

includes two B atoms located on a diagonal ( = 3).For the fcc lattice, y(a) = 7, y(b) = –6, and y(c) = 1.

For the square lattice in the case where ε = –(1 – y)2

and y > 0.5, the solution is given by

(28)

For the fcc lattice in the case where ε = –(1 – y)2 andy > 0.5, the solution is

(29)

For AyB1 – y solid solutions with square and fcc latticesin the case where y < 0.5 and ε = –y2 and, hence, thereare no A–A bonds in the lattices, the solutions areobtained from Eqs. (28) and (29) by replacing y2 with(1 – y)2.

All sites of the triangular cluster of a regular trian-gular lattice are within the same (first) coordinationshell. Therefore, solution (26) does not allow one toascertain whether the correlation within the first coor-dination shell causes correlations in the second andmore distant coordination shells. The square and octa-hedral clusters contain sites located both in the first andsecond coordination shells.

λ0c( )

λ1c( )

λ2c( )

λ0c( )

λ1c( )

λ2c( )

P0c( )

1 y2

– ε–( )– 1 y2

– ε–( )2

y2 ε+( )

2+[ ]

1/2,+=

P1c( ) 1

2---

12--- 1 y

2– ε–( )

2y

2 ε+( )2

+[ ]1/2

,–=

P2c( ) 1

2--- y

2 ε+( )–12--- 1 y

2– ε–( )

2y

2 ε+( )2

+[ ]1/2

.+=

P0c( ) 1

4--- 6 y

2 ε+( ) 5–[ ]=

+34--- 6 y

2 ε+( )2

8 y2 ε+( )– 3+[ ]

1/2,

P1c( ) 1

4---

14--- 6 y

2 ε+( )2

8 y2 ε+( )– 3+[ ]

1/2,–=

P2c( ) 1

4--- 1 2 y

2 ε+( )–[ ]=

+14--- 6 y

2 ε+( )2

8 y2 ε+( )– 3+[ ]

1/2.

From Eq. (8) and probabilities (28) and (29), it ispossible to find the probabilities of the pair bonds in thesecond coordination shell and, consequently, the paircorrelation and the short-range order parameter for this

shell using the coefficients , , and .For the square cluster in the square lattice, we have

= 0, = 1/2, and = 0, and for the

octahedral cluster in the fcc lattice, we have =

0, = 1/3, and = 0. Calculations show

that, for the square lattice in the case where ε = –(1 – y)2

and y > 0.5, the probability of the A–B pair bond in thesecond coordination shell differs from the binomialvalue and is equal to

(30)

The pair correlation εAB in the jth coordination shell

is εj = – . Using the relation = – ≡

−εj and Eq. (30) for ≡ for the square latticein the case of ε = –(1 – y)2 < 0 and y > 0.5, we get thepair correlation in the second coordination shell ε2 ≡

:

(31)

In much the same way, for the square lattice in thecase where ε = –(1 – y)2 < 0 and y > 0.5, the probabilityof the A–B pair bond and the pair correlation ε2 ≡ in the second coordination shell can be found to be

(32)

(33)

So, in square and fcc lattices, the pair correlation(i.e., the short-range order) in the first coordinationshell gives rise to correlation at least in the second coor-dination shell. The correlations in the first and secondcoordination shells are opposite in sign. This conclu-sion is in agreement with numerical simulations of theshort-range order and ordering processes in the fcc car-bon sublattice of titanium carbide TiCy�1 – y [11].

ni AA( )2

s( )ni BB( )2

s( )ni AB( )2

s( )

n0 AB( )2

c( )n1 AB( )2

c( )n2 AB( )2

c( )

n2 AB( )2

c( )

n1 AB( )2

c( )n2 AB( )2

c( )

λ1

b2( )P1

b2( )2P1

c( )=

= 1 1 y2

– ε–( )2

y2 ε+( )

2+[ ]

1/2.–

PABj( )

PABbin εAB j

εAA j

P1

b2( )PAB

2( )

εAA2

ε2 y 1 y–( ) 12---–

12--- 1 y

2– ε–( )

2y

2 ε+( )2

+[ ]1/2

+=

= y 1 y–( ) 12---–

12--- 8y

212y– 5+( )

1/20.>+

εAA2

λ1

b2( )P1

b2( )2P1

c( )=

= 12--- 1

2--- 6 y

2 ε+( )2

8 y2 ε+( )– 3+[ ]

1/2,–

ε2 y 1 y–( ) 14---–

14--- 6 y

2 ε+( )2

8 y2 ε+( )– 3+[ ]

1/2+=

= y 1 y–( ) 14---–

14--- 24y

240y– 17+( )

1/20.>+



Indeed, it was demonstrated in [11] that the short-rangeorder in the first coordination shell of the fcc lattice(α1 < 0) gives rise to short-range order in the second tofifth coordination shells and that the short-range orderparameters α2, α3, and α4 are positive; i.e., they areopposite in sign to α1.

Notice that the formulas derived above for the prob-abilities of certain configurations of the square andoctahedral clusters in square and fcc lattices, respec-tively, do not contain powers of y higher than 2 (i.e., y2

or [a + k(y2 + ε)2 + …]1/2). The reason for this is that, inthe particular cases considered, the probabilities of theconfigurations of the basic clusters can be expressed interms of the probabilities of pair bonds only. However,if we use a binomial distribution, then the probabilitiesof any configurations of the square (in the square lat-tice) and octahedral (in the fcc lattice) basic clusterswill be functions of the fourth and sixth powers of y,respectively.

The method proposed here for calculating the prob-abilities of many-particle configurations with allow-ance for the correlation between atomic locations canbe applied not only to two-dimensional but also tothree-dimensional lattices. Its development will make itpossible to move on to a more complicated probleminvolving simultaneous account for the short- and long-range orders in ordering systems.

ACKNOWLEDGMENTSThis work was supported by the Russian Foundation

for Basic Research, project no. 03-03-32031.

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Translated by G. Tsydynzhapov

Documents

Pair correlations and probabilities of many-particle configurations occurring in a flat triangular lattice