Calculating the energy parameters for the CV and OPF methods

A. I. GUSEV and A. A. REMPEL: Energy Parameters for the CV and OPF Methods 335

phys. stat. sol. (b) 140, 335 (1987)

Subject classification: 61.50; 64.60; S1.61

Institute of Chemistry, Academy of Sciences of the USSR, Ural Scientific Centre, Sverdlovsk?)

Calculating the Energy Parameters for the CV and OPF Methods

BY A. I. GUSEV and A. A. REMPEL

A method is proposed for calculating the energies 82) (with c$) being the energy of the m-configuration of a type s figure), which are parameters of the cluster variation (CV) method and the order parameter functional (OPF) method. An analytic expression is derived which describes the energy &) in terms of power series expansion coefficients of the formation enthalpy of a disordered crystal. It is shown that a necessary condition for ordering is the presence of a no nlinear composition dependence of the formation enthalpy of a disordered crystal.

npenjromeH cnoco6 pacseTa a~eprn i i &) (cg) 3 ~ e p r ~ 1 1 m-KoHamypaqmi amypbI Tma s), HBJIHIOUXXXCH napaMeTpaMH MeToaa sapkiaum KnacTepoB (CV-MeToA) II MeToaa (pyHKunoHana napaMeTpoB nopHnHa (OPF-MeTOR). Hai‘tneHo aHanmmecKoe sbrpameme a ~ e p r m cg) sepea K O ~ ~ ~ C T ~ H ~ H T ~ I p a 3 n o m e ~ ~ a B p ~ n 3~~anbnki11 06pa30Ba~HR Hey- IlOpfIAOseHHOI’O KpXXCTanJlZt. HOKa3aH0, YTO He06XOnAMbIM YCJIOBlleM ylIOpRHOqeHHR RBJIReTCR HaJIlNlle HeJIEiHefiHOfi 3aBMCHMOCTH 3HTaJIbnMH 06pa30BaHXXR HeynOpHaOseH- Horo Kpmxanna OT ero CocTaBa.

1. Introduction

The various statistical ordering theories are based on a treatment of possible atomic arrangements in Ising lattice sites and on a calculation of the energies of those arrangements, proceeding from particular assumptions as to the character of interparticle interactions. The most common assumption is that the energy of the crystal may be represented as a sum of pairwise interatomic interaction energies, which depends on the relative position of atoms, and as some constant term (no allowance is made for geometric lattice distortions or correlations in the relative position of atoms). Such a theory was originally developed in [l to 41.

Cluster methods take into account the interaction of both nearest and remote atoms and the configurational energy of the crystal involves not only pairwise but also many-particle interaction energies 15 to 71 ; thanks to this circumstance it has been possible in a number of cases to obtain a good agreement with experiment. The use of cluster methods is most appropriate to describe structural order-disorder phase transitions which are often accompanied by the presence of sharp short-range correlations such as “blocking” (prohibition of some configurations of neighbouring particles, e.g., the absence of adjacent nearest-neighbour vacancies in ordered nonstoichiometric carbides).

I n all of the afore-mentioned methods of describing atomic ordering, the energies of pairwise or many-particle interactions are assumed to be independent of temperature, the degree of long-range order, the composition of the solid solution, or non-

1) Pervomaiskaya 91, 620 219 Sverdlovsk, USSR.

336 A. I. GUSEV and A. A. REMPEL

stoichionietric compound that undergoes ordering and are viewed as parameters of the particular model employed.

An order parameter functional (OPF) method was proposed [8] to describe structural order-disorder-type phase transitions. This technique enables one to describe the ordering in both binary and multicomponent solid solutions which, when in the ordered state, possess an arbitrary number of sublattices. I n its major points, the order parameter functional method belongs to the group of cluster methods, but differs from the latter in that it permits a detailed allowance for the symmetry of a crystal with any degree of long-range order.

In the cluster variation (CV) method [5 to 71 and the OPF method [S] the crystal is regarded as EL set of type s (s = a , b, c ...) figures with i configurations. The set of figures comprises a basis figure and overlap figures. I n the general case a figure of a given type is some finite group of crystal lattice sites and is specified only by the number of sites contained in it and by their spatial arrangement (geometry of the figure). Atoms of several species, which form a solid solution a t the sites of the same figure may be arranged in various fashions i. Each mode of arrangement corresponds to a determinate i-configuration of a given figure s. Therefore, any configuration of a given figure is characterized by the number of atoms of each species and by their relative position a t the sites of the figure. I n other words, any configuration of the figure is a cluster, i.e., a group of atoms of the same or several species arranged in a certain fashion a t the sites of the figure.

Each i-configuration of the figure s both in the CV and in the OPF method is specified by the probability Pi8) of its existence in the crystal and by the energy &as). I n the CV method the probabilities are variable quantities; the number of such vari- ables depends on the size of the basis figure and may be very large.

I n the OPF method the probabilities Pp) are expressed in terms of a distribution function n( r ) . The latter is directly related to the long-range order parameters q which are variable. This permits taking into account the symmetry of a crystal with any degree of order and allows the scope of calculations to be reduced considerably as compared to the CV method.

In so far as the energies EY) go, both the CV method and the OPF method treats these as model parameters.

The present paper propose a method for calculating the energy of various R- particle interactions in a crystal, i.e., the energy of the various configurations of a figure involving R sites of the lattice that undergoes ordering. A key feature of the technique proposed is that the configurational part of the free energy of the crystal is represented as a sum of the energies of individual configurations (this representation holds good in both the CV and the OPF methods).

2. Deriving a Formula to the Calculate Cluster Energy Consider a crystal whose Ising lattice breaks down into t sublattices as a result of ordering. According to [8], the free energy of this crystal may be represented as a sum of the configurational energy and the entropy term

with N being the number of sites in a lattice, y@) a coefficient that allows for the re- valuation of the number of microstates because of the overlap of figures, l y ) the multiplicity of the i-configuration of a type-s figure (number of equivalent configurations, i.e., configurations that may be brought into coincidence with a given i-configuration by applying to them disordered-crystal point group symmetry operations),

Calculating the Energy Parameters for the CV and OPF Methods 337

and v = a, B, ... z the atomic species. According to the authors of [8], the probability of the i-configuration of a type-s figure is defined as follows:

where x@) is a coefficient indicating the factor by which the number of type-s figures exceeds that of lattice sites, and l p ) , Lay“) stand for the designation and number of

type-s figure i-confiwration sites occupied by species vatoms, respectively ( C LjSy)= R@) is the total number of lattice sites comprised in a type-s figure), and nv(r9k@)) is the value of the distribution function of species v atoms a t the figure s sites that are occupied by species v atoms and correspond to the positions of any of the t sublattices of an ordered crystal. The summation in (2) is carried out over the equivalent configurations, the number of which is equal to ?$), and over all type-s figures in the crystal.

The distribution function represents the probability of finding an atom of a given species at an Ising lattice site r and, according to [9], may be expressed in terms of the quantity c, (the fraction of lattice sites occupied by species v atoms) and a super- position of static plane waves exp (ikr) with nonequivalent superstructural wave vectors k,

v=u

n,(r) = c, + + 2 q(k) [ y ( k ) exp (ikr) + y * ( k ) exp (-ikr)l , (3) k

where q(k) is a long-range order parameter, and y ( k ) is a special normalizing coefficient that allows for the symmetry of an ordered crystal.

It is clear from (1) to (3) that in the OPF method the free energy of a crystal with any degree of ordering is determined by its composition, temperature, and long-range order parameters. At constant pressure the first term of the free energy (1) of a crystal with any degree of ordering has the meaning of enthalpy. This enables one to find the energy E?) if the probabilities Pp) and the forniation enthalpy (in describing nonstoichiometric compounds the term “formation heat” is normally used instead of “formation enthalpy”, and in describing solid solutions the “heat of mixture’’ is used) are known for some state of the crystal. As demonstrated in [8 ] , it is most convenient to choose this state to be a disordered state, for which the long-range order parameters q are equal to zero and, as a result, the probabilities n,(r) are the same for all sites r and are equal to c,. Allowing for this, it can be readily shown that the free energy of a disordered crystal has the same representation in both the OPF method and the CV method. Since it is a disordered state that is chosen for energy E?) calculation in the model under discussion, the cluster energy formula derived later is equally applicable, in terms of the approximations made, in the two methods.

For a disordered binary substitutional solid solution A,B1-, or a disordered nonstoichiometric compound MeX,, in which the substitutional solution forms interstitial atoms and structural vacancies, n,(r) = y. Accordingly, the summation in (2) over equivalent configurations and type-s figures is removed ; as a result, the probabilities of the various i-configurations in a disordered crystal are determined by the binomial distribution

m being the number of species B atoms (or structural vacancies - in the case of a nonstoichiometric MeX, compound) in the i-configuration of a type-s figure.

338 A. I. GUSEV and A. A. REMPEI.

According to (4), the probabilities Pjfi)bin depend only on the composition of the crystal, so the presence of the temperature dependence of the formation enthalpy Hdisord (y, T) of a disordered crystal indicates that the energies &(iS) of the various configurations of figures depend on temperature. I n this context, the equation relating the energy E?) to the formation enthalpy of a disordered compound will be

However, the system of linear equations (5), written for different compositions y, cannot be solved for @( T) in a straightforward manner. The poi t is that in both the

figures that include a basis figure and overlap figures. According to (Al) given in Appendix A, the probabilities of any of the subfigures of a basis figure (including overlap figures) are linear combinations of the probabilities of different basis-figure configurations, as a result of which the system of equations ( 5 ) is indeterminate. The latter can be solved only provided that the ratios between the energies of figures of different types are known. This permits the configurational energy of the crystal to be broken up into several terms, each of which allows for the configurational energy of a figure of one type alone and has the following form:

OPF method and the CV method the crystal is described by a 4 uccession of special

where dS) is a coefficient that establishes a correlation between the enthalpy and the energy of all the figures of a given type s with different configurations. The coefficients As) and y(s) are interrelated by the normalization condition C Y ( ~ ) X ( ~ ) = 1.

I n the general case (6) may be solved numerically; however, subject to certain

We expand the formation enthalpy Hdisord(y, T) in a power series of composition y,

S

assumptions, the solution of the system may be tried in analytical form.

i.e., R(.)

?Z=O (Y, T) = fl c Y n H n ( T ) . (7) Hdisord

Nonequivalent configurations with the same number m of atoms of the same species, but with different atomic arrangements will be assumed to possess equal energies, i.e., ~ y 2 ~ = &$). Inspection of (4) shows that in the disordered state the probabilities of configurations i E m with the same number m of same-species atoms are equal to each other, i.e., Py>, = PE). I n keeping with the familiar combinatorial relations, the multiplicities 22) = c A?) of the m-configurations of a type-s figure are equal to

the binomial distribution coefficients C ~ C , ) . Transforming (6) with due regard of the approximations made and substituting

into it (4) and (7), yields

i c m

(1 - y)" = c (-l)k Ciyk k = O

(8) becomes

(9)


The summation on the left-hand side of ( 1 0 ) is carried out over the rows of some triangular matrix T = [tmk] with elements

tmk = ( - 1 ) k Cg(,)CLy(R(')-m+k) Em ( 8 ) ( T ) - On the left-hand side of (10) we collect terms with equal powers of y . This collection corresponds to going from summation over the rows of the matrix T to summation over the principal diagonal and the lines parallel to it; for each of these lines the sum (R@) - m + k ) = n is a constant quantity. Collecting similar terms yields

R(S) Rf l ) R(a) C yn C ( - l ) n - R c a ) + m Cg(r )C i (a ) -n~g) (T) = As) C ynHn(T) . (11 )

Equating the coefficients like powers of y on the left-hand and right-hand sides of ( X I ) , w0 pass over to a system of (M) + I ) linear equations, written for all values of n from 0 to R@) and having the form

( 1 2 )

n=O m=o n = O

R(')

m=O C ( -l)n-RR(')+m C~(d'$a)-n~$)( T ) = d8)Hn( T ) .

Since C?&)Ctff(')-n = Cg(a)Cf(')-m, one obtains

.'( 1 3 )

where the quantity anm = (- l)n-RR(')+m C;(')-" is an element of the triangular matrix A = [a,,]. This allows the system consisting of (13) written for all values of n to be represented in matrix form

whence

The element of the reciprocal matrix A-l may be found using (B7), which is

X C f ( ' ) - m c R ( ~ ) - m = d,,, and the condition AA-1 = C ( - l ) n - R ( * ) + m n cn R(r)-m -1 -d - km; then it follows that for k = m the element agl is equal to CnR(i)-m. Substituting the value of into (15 ) yields

furnished in Appendix B, and should in this case be transcribed into C ( -l)n-R(s)+m X

n

With allowance for the transformation c R ( a ) - m / c g ( a ) = Cg(a)-n/Cg(,), the final expression for the energy ~ g ) has the form

3. Transformation of the Configurational Energy of the Crystal It follows from ( 1 ) that in the general case the configurational energy of a crystal with any degree of order is

23 physica (h) 140/2


To the energy equality approxiinat,ion, = .$, substitution of (17) and (18) yields

or, taking into account (A2) in Appendix A, R(S)

S n = O E = N C y( ' )dS) L; H n ( T ) P t ) ,

Since the probabilities PIp) = 1 and series expansion of configurational energy may be represented as

0 = y (see (A3) and (A4) in Appendix A) the

As is clear from (21), the terms H,,(T) and yH,(T) do not depend on the degree of order in the crystal. Therefore, the difference of the free energies of the ordered and disordered phases, A F = Ford - Fdisord, is equal to

Xord and Sdisord being the entropy of an ordered and a disordered crystal, respectivelj . It follows from (22) that the free-energy difference AF does not depend on the

coefficients H,,(T) and &(T) of the linear expansion terms ( 7 ) . I n the general case (Xord - Sdisord) < 0; let the composition dependence of the formation enthalpy be linear (Hdisord(y, T) = Ha( T) + yH,( T)). Then the difference A F is a positive value (AF = Ford - Fdisord > 0) and no ordered phase will form. Thus, in terms of the approximations made, a necessary condition for the ordering of tc nonstoichiometric compound is the presence of a nonlinear dependence of the formation enthalpy on MeX, composition. Similarly, a necessary condition for the ordering in a substitutional solid solution is the presence of a nonlinear dependence of the heat of mixture on A,B1-, composition, i.e., an unideal character of the solution.

To calculate the configurational energy of the crystal by (18), it is required to define the probabilities Py) of all configurations of the basis figure and overlap figures in terms of the distribution function. Modifying the configurational energy into the form of (21) permits a considerable reduction in the scope of calculations, since in this case i t is only the probabilities Pp) of several complete subfigures of the basis figure that need to be expressed in terms of the distribution function. The number of complete subfigures taken into account is determined by the degree of the polynomial ( 7 ) which approximates the dependence of the formation enthalpy of the crystal on its composition. Most of the available experimental data on the formation heat of nonstoichiometric compounds or on the heat of mixture of solid solutions (alloys) may be described to an adequate accuracy using low-degree polynomials. I n this context, the use of (21) allows the scope of numerical calculations indispensable for the determination of thermodynamic equilibrium parameters to be reduced to a minimum.

4. Numerical Calculation for Niobium Carbide

Nonstoichiometric MeC, (Me = V, Nb, Ta) carbides possess a NaC1-type structure with statistical distribution of carbon atoms and structural vacancies a t nonmetallic sublattice sites. I n investigating the atomic order in VC, and NbC, carbides, ordered V,C, [lo], V,C, [ll, 121, and Nb,C, [13 to 151 phases were detected. An OPF calculation [16] has shown that an ordered Ta,C, phase is liable to arise in tantaluni


carbide TaC,. One cannot rule out the possibility that ordered carbides such as Me& may be produced, although no ordering of this type has thus far been detected ex- perimentally. Thus superstructures such as Me&,, Me6C5, and Me&,, or, in general form, MeztCet-l with t = 2, 3 , 4 may form in nonstoichiometric vanadium, niobium, and tantalum carbides.

Earlier [8] the sequence of figures was determined which is used in the OPF method to describe nonstoichiometric MeX, compounds with NaCl structure. This sequence includes a basis figure of type c (a metallic atom in an octahedral environment of six filled or vacant nonmetallic sublattice sites) and overlap figures: a figure of type b (a pair of adjacent nonmetallic sublattice sites, hereafter referred to as a bond) and a figure of type a (a nonmetallic sublattice site). Using as a basis figure a regular octahedron with the metallic atom a t the centre permits one to take into account all crystal lattice sites; apart from this, the energy of the basis figures, E?), implicitly involves the Me-Me and Me-X interaction energies.

The results of numerous investigations, generalized by the authors of [17, 181, indicate that the energies $1 and &ib) are verysmall in magnitude as compared with eMe-T; and &Me-3fe. Hence it follows that E?) and ,sib) are negligibly small in comparison with EJC), and therefore da) = db) = 0. The coefficient dC) may be found from the condition y(s)x(s) = I ; for y(c) = 1 [8] and da) = db) = 0 the coefficient dC) is equal to unity. Allowing for this fact, the energies #(T) of octahedral clusters in niobium carbide (Fig. I) were calculated using (17). In the calculation the data report- ed in [IS to 211 were used.

Analysis of the available [17, 18, 201 experimental data on the formation heat of disordered MeC, carbides has shown the expansion of Hdisord(y, T) in the power series (7) to be a quadratic function for these compounds, i.e.,

s -

(Y, T) = “HrAT) + YWT) + Y2H2(T)I * (23) Hdisord

Therefore, in keeping with (21) and in view of da) = db) = 0 and y(c)x(c) = I, the configurational energy may be represented as

E = “dT) + yH,(T) + qj2’H2(T)1 (24) with P r ) being the probability of a complete type n = 2 (bond) subfigure in a crystal with an arbitrary degree of order.

The ordering of a MeztC2t-1 structure is described by distribution functions that depend on several long-range order parameters [15, 161. If the latter are equal in magnitude, the distribution function n(r) assumes, a t all nonmetallic sublattice sites r,

018

L?4

* c c Z E

w-c14

-08

-?.Z

-16

23’

Fig. 1. Temperature dependences of the energies .$) of octahedral clusters with different numbers of vacancies, m, for niobium carbide

1 , , , , LLLu €0

500 iooa 1500 20110 T f K l -


two values: n, = y - (2t - 1 ) q/(2t) (the probability of detecting a carbon atom a t a vacancy sublattice site) and n2 = y + q/(2t) (the probability of detecting a carbon atom a t a carbon sublattice site). The probability Pi2) can be determined from (2); for the MeztCzt-l superstructures concerned ( t = 3; 4),

(25) 1 t

Pp = -- [nln, + ( t - I) n;] .

Substituting the values of n, and n2 into (25) yields

In keeping with the approximation adopted, the free energy of a nonstoichiometric MeC, carbide with an arbitrary degree of order is

p = “&(TI + Y W T ) + Pb2’H2(T) - TSI 9 (27) where

I 2t

S = - - kB {n, In n, + (1 - n,) In (1 - n,) + + (2t - I) [n, In n, + (1 - n,) In (1 - n2)l) . (28)

On reaching the equilibrium value of the long-range order parameter qeq, the free energy P will be minimum and, consequently,

Transcribing (29) into a form more convenient for analysis,

manifests that ordering is liable to occur only provided that H2( T ) > 0. I n the general case (30) may have one, two, or three roots, one of them always being q = 0.

If (30) has one root q = 0, the disordered state is the equilibrium state of the crystal a t a given composition and temperature. If (30) has two or three roots, it is necessary to find the values of the free energies P(ql), F(q2), and P(q3) that correspond to these roots; the value of q a t which the minimum value of F is reached corresponds to the equilibrium state of the crystal.

According to [13 to 151, continuous annealing of a nonstoichiometric niobium carbide NbC, at T < 1300 K results in the formation of an ordered Nb,C, phase in that compound. Using (30), we wish to consider the effect of temperature and composition on the formation of this ordered phase in NbC,. Fig. 2 shows the influence of temperature on the function

for y = 516. a t which f ( y , T , q ) goes to zero are roots of (30). As is seen from

Fig. 2, at a temperature of 1000 K (30) has two roots ql = 0 and q2 = 0.884; a t 1200 K it has three roots q, = 0, q, = 0.19, and q3 = 0.738; a t 1310 K there are two roots ql = 0 and q, = 0.48; a t 1425 K (30) has one root ql = 0. A t 1000 K the equili-

The values of


. Fig. 2. Effect of temperature on the formation of the Nb,C, superstructure in niobium carbide NbC,,,,. (1) T = 1000 K, qeq = q, = 0.884; (2) (3) T = = Ttrans = 1271 K, q z y d = q - 0, qOrd =

= q3 = 0.634; (4) T 1310 K, qeq = qz = = 0.480; (5)

T = 1200K, qeq = q3 = 0.738; 1 - eq

T = 1425 K, qeq = ql = 0 s5- t.'

x s

-5 0 02 04 Qfi 0.8 7.0

-7-

brium value of the long-range order parameter that corresponds to the minimum value of the free energy is qeq = q2 = 0.884; for a temperature of 1200 K qeq = q3 = = 0.738. The temperature of 1271 K (q:id = q - - 0.634) is the order-disorder transition temperature for NbC,.,, carbide, since a t higher temperatures qeq = 0 and the equilibrium state of the crystal is the disordered state. The value Ttrans = 1271 K gives a good f i t to experimental data [lS], according to which the transition temperature for NbCo.83 is equal to 1304 K.

The effect of composition y on the function f(y, T, q) a t T = 1200 K is depicted in Fig. 3. For NbC,.,, carbide qeq = q2 = 0.834; for NbCU.83 qeq = q3 = 0.738; at a temperature of 1200 K NbC,.,, carbide is in the disordered state.

Thus, analysis of solutions to (30) permits a sufficiently easy determination of the equilibrium values of q corresponding to a given composition and a given temperature.

5. Conclusion

The present paper proposes a method of calculating the energies &), which are parameters of the CV and OPF methods. An analytical formula has been derived to calculate the cluster energy in terms of the coefficients of a polynomial that approximates the concentration dependence of the formation enthalpy of a disordered crystal. In contrast to the available tedious and approximate computational techniques, the method proposed allows the absolute values of the cluster energy to be readily cal-

x

Fig. 3. Effect of composition on the formation of the Nb,C, superstructure in niobium carbide NbC, at T = 1200 K. (1 ) y = 3/4, qeq = = q2 = 0.834; (2) y = 516, qeq = q3 = 0.738;

-5 - I I I I (3) Y = 7/89 qeq = 71 ,= 0 0 012 0.4 0.6 0.8 I@

7 7 -

-


culated by taking into account its temperature dependence. Using the energy values found with the help of the method proposed enables one to determine the absolute values of order-disorder transition temperatures with high accuracy. The utilization of this method for phase diagram calculation requires no fitting to experimental phase diagrams by means of energy parameters. Let it be emphasized that transforming the free energy of the crystal with allowance for the cluster energy formula ob- tained permits a considerable reduction in ths scape of the numerical calculations,

Appendix A

Consider an A,B1-, crystal made up of atoms of two species, A and B. To describe the crystal, we choose a type-s basis figure that comprises R@) crystal lattice sites. Any figure that is formed by n sites of the basis figure (n < RO) is its n-subfigure. The smallest subfigure possible is a site (a type-a figure), for which n = 1; the next subfigure is a bond (a type-b figure), for which n = 2, etc. Note that the sites of the basis figure may be located in different coordination spheres relative to each other. For this reason, the sites that form an n-subfigure may belong to the same or different coordination spheres of the basis figure, so an n-subfigure may be of various geometric shapes. The total number of all configurations of a given n-subfigure (taken into account all the geometric shapes possible for it) in the basis figure is equal to the number of combinations of R(s) sites taken n a t a time, i.e., N ( @ @ ) = C$(8,.

Since each configuration of the n-subfigure is involved in one or several configurations of the basis figure, the probability P',") of some 6-configuration of the n-subfigure may be expressed in terms of the probabilities Pi'") of the different i-configurations of the basis figure,

?$op',") z c k p ) j p p P ) , (Al l i € S

with A?) and At) being t,he multiplicities, kc,"a)(S) = N(,$(s)/N(n)(s) the relative share of a-configurations of the n-subfigure among all of its configurations involved in the i-configuration of the basis figure, N%)(s) the number of n-subfigure @-configurations involved in the i-configuration of the basis figure.

Consider the case when all the sites of the n-subfigure are occupied by species A atoms alone. This complete (G = 0) configuration of the n-subfigure has the multiplicity At) = 1. Let each i-configuration of the basis figure contain (RO - mi) species A atoms and mi species B atoms. Then Nb'l,)@) (the number of complete n- subfigure configurations involved in the i-configuration of the basis figure) is equal to the number of combinations of (R@) - mi) species A atoms taken n a t a time, i.e., N%)(s) = C'&8)--mi. Accordingly, I~b'l,)(~) = C~(.)L~,/C'&) = C ~ L , J C ~ & ) and therefore the final expression for the probability of a complete n-subfigure has the form

For 92 = 0, (A2) reduces to the usual probability normalization condition

(A3) p(0) = c q)p!s) 1 . 0 i C S

With n = I, we determine the probability of a site (type-a figure) occupied by a species A atom, i.e., obtain another normalization condition, which associates the probabilities P?) with the composition of the crystal. For a solid A,BI-, solution or a nonstoichiometric MeX, compound, in which the substitutional solution forms


interstitial atoms and structural vacancies, Pg) = y. Allowing for Cg:8)-l =

= CgtZ) (R@) - m,)/R@), where (R@) - mi)/R@) is the relative share of basis figure sites occupied by species A atoms in the i-configuration of the basis figure, we obtain

Appendix B

Prove a lemma according to which

z (- 1)l-m cyc; = dm, , 1

where CT and Ci are binomial coefficients, dmr being the Kronecker symbol. Transform the product of the binomial coefficients as follows

m l m I - m Cl c, = c,c,-, . Therefore

2 (- 1)l--m cyc; = c,m c (- 1)l-m c : z .

c (- l )z-m CLIZ =

I I

According to the familiar combinatorial relations, 1 if k = m , { 0 if k + m . 1

For k = m the coefficient C y = C z = 1. Therefore, 1 if k = m ,

cp c 1 ( - l f m c;:; = 0 if k + m

or

cp (- 1 y c:z; = d,r . 1

Allowance for (B2) yields (- 1 ) l - m cyc; = dmlG

1

which was to be proved. I n the particular case, k = m,

(- I)l--m cyci = d,, . I

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346 A. I. GUSEV and A. A. REMPEL: Energy Parameters for the CV and OPF Methods

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New York 1983.

J. Phys. C 18, 809 (1985).

Institute of Chemistry, Sverdlovsk 1985 (p. 15, 16).

(Received November 10, 1986)

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Calculating the energy parameters for the CV and OPF methods