The statistical mechanics of imperfect molecular crystals

J. Phys. Chem. Solids

Pergamon Press 1969. Vol. 30, pp. 2 187-2 199.

Printed in Great Britain.

THE STATISTICAL MECHANICS OF IMPERFECT MOLECULAR CRYSTALS A. R. ALLNATT and L. A. ROWLEY Department of Chemistry, The University, Manchester (Received

20 December

1968; in revisedform

13, England

26 March

1969)

Abstract-Equations are developed for the calculation of thermodynamic functions of imperfect molecular crystals containing one species of point defect on a cubic lattice, following a method proposed by Mayer [7]. The treatment of vibrations requires essentially only the techniques applicable to a perfect crystal, requires no static relaxation calculation, and can allow for defect-defect interactions. The free energy of solution, and the activity coefficient in an approximation similar to that of Bragg and Williams, are calculated for argon in krypton in moderate agreement with experiment. Results for vacancy concentrations in krwton are less satisfactory. The limitations of the method are <. indicated. 1. INTRODUCTION

lattice contains point defects such as impurity atoms, vacant sites, or interstitial atoms then the calculation of the thermodynamic functions is much more difficult than for the corresponding perfect crystal. The three dimensional periodicity of atomic sites in the perfect lattice allows an analysis of the thermal vibrations based on independent normal co-ordinates and the interaction between them due to anharmonicity. In the imperfect crystal the point defects destroy the three dimensional periodicity and the calculation of thermodynamic functions generally proceeds as follows. The displacement of the atoms around a single point defect is first calculated either by an approximate analytic method, or by a relaxation calculation on a fast computer. The vibrational properties of the imperfect crystal are then calculated treating the difference in the Hamiltonian of imperfect and perfect crystal as a perturbation to the vibrations of the perfect crystal. The last step is not very simple because the lattice is not triply periodic. The details have varied from calculation to calculation [l-4] but the following general remarks can be made about the programme outlined. The relaxation calculation and the use of perturbation theory appear to be practicable IF A MOLECULAR

only for very dilute defect systems in which the point defects are independent or at most interacting in pairs rather than higher aggregates. A few attempts have been made to treat solid solutions at higher concentrations [5,6] but an adequate treatment is not yet available, because it is difficult to treat both the configurational (order-disorder) problem and the vibrational problem and the interaction between them to an adequate approximation. It was pointed out by Mayer[7] that in principle there is an alternative procedure for imperfect crystals in the classical region. The order of the averaging processes in the evaluation of the partition function can be inverted. One may first average over all possible occupations of each site in the crystal for a fixed displacement of the occupant of each particular site from the site centre, to obtain a potential of mean force which is triply periodic with respect to the site centres. This function can then be expanded in powers of the displacements and the integration over displacements be performed by the same methods as for a perfect crystal. The advantages of the Mayer method are that the tedious relaxation calculation is eliminated and the vibrational properties can be treated by exactly the methods developed for the perfect crystal even when defect interactions between

2187

2188

A. R. ALLNATT

up to three or four defect sites are allowed. Further, one is forced to perform the calculation in a more self-consistent manner than is usual because calculation of the equation of state and chemical potentials must generally be performed simultaneously, thus providing a more complete check on the model used. Unlike most previous methods there is no question of adding together separate calculations of ‘defect contributions’ and the ‘perfect crystal contributions’ for the various thermodynamic functions. A price must be paid for these advantages and we do not wish to imply that the Mayer method is any easier to use than the conventional methods. A mixing of the static and vibrational aspects of the problem occurs as compared with the conventional method, and the relaxation calculation is eliminated only at the cost of both increasing the importance of and increasing the algebraic complication in the vibrational part of the theory. Furthermore, a series of iterations is sometimes necessary to solve the equations for the thermodynamic functions. The present paper describes an attempt to discover just how practicable the Mayer method is. Section 2 outlines the general equations of the method. Sections 3 and 4 give equations for the vibrational part of the calculation for a single defect type in a cubic lattice. Some illustrative calculations for rare gas solids are described in Sections 5 and 6. 2. GENERAL

EQUATIONS

We consider a solid of several species of monatomic molecules in a volume I/ at temperature T. The activity of species s, z.,, is defined by

and L. A. ROWLEY

ZN = zlNlz2NZ . . . zsNs. . .

N!

. . . N,! . . ..

(2.3)

At temperatures at which the classical approximation is valid the grand partition function, E, is given by [8] E(Z, V, T) exP (-Pu,{N])d{N].

(2.4)

{N} denotes the 3N co-ordinates of the set N, and d(N) the 3N dimensional volume unit d-r*. . . dzN. UN{N} is the potential energy. Let the lattice sites of a unit cell be labelled by the integer h, 1 =ZA < r, including any interstitial sites occupied only in the imperfect crystal, and let the integers I,, lZ,l3 specify the positions of the unit cells in terms of primitive translation vectors a,, a,, u3. The total volume k’ of M unit cells can be subdivided into N = Mr non-overlapping cells, one cell being associated with each lattice site I = II, I,, 13,X. We call these Mayer cells. Let nsl be the number of atoms of species s in cell 1 and let noz be unity if cell 1 is vacant and zero otherwise. Denote by (I) = u,(l), u,(l), ~~(1) the displacement of the atom in cell 1from the cell centre. The occupation set it, comprising the nsl for all s and 1, and the co-ordinate set {,C}, representing the set of all co-ordinates (I), completely specify the state of the system. Provided the Mayer cells have been chosen so that we may restrict consideration to configurations with not more than one atom per cell then equation (2.4) can be written [7] in the equivalent form E=TznJ

zs = (2~-rn,kT/h~)~‘~ exp (&)

= N,!N,!

(2.2)

exp

(--PU,{Ll)d{Ll

(2.5)

(2.1) where

where p = (kT)-l, p, is the chemical potential per atom of species s, and m, the mass of one atom of s. Let N denote a set of N atoms, N, of species 1, N2 of species 2,. . . . . We use the abbreviations

zn = 9 7 ZAO =

l/%0

zsnsl

(2.6)

(2.7)

where vAois the volume of an empty cell of

IMPERFECT

MOLECULAR

kind A, and in the product s = 1,2, . . ., 10, 20,. . ., r0. In equation (2.5) the artifice of writing an ~nte~at~o~ over fictious coordinates (r) has been adopted for empty ceJfs. The factors zAocorrect for this integration. An alternative method for treating vacancies has also been used. If the occupation set n has a vacancy at I then one adds 6(uz2(1) + ~~~~~~~~~~~~~}to the potential energy. In the ~nte~rat~~~ over d~s~Jaceme~ts the range is always extended over the range --03 to *, and so this gives an additional factor (v~TIS)~‘~, We therefore define

2189

CRYSTALS

only when site 1 is occupied by a point defect of species 9 is defined by the equation T Q = %r - ~&SX%M. Comparison yields

of equations

(2.5) and (2. JO)

U,*(L) = U,@) - U,(L)*

and proceed as before. The value of 6 is arbitrary and the result of an exact calculation of B would be independent of its value. For an a~~ro~~rnat~ calcutation the best choice of 6 is that which satisfies

can read& be shown to be a true ~ot~nt~~ of average force such that -M(~)/&Y,(/) is the average force in direction ~1 on the occupant of cell I for fixed {L}, the average being over all crystal occupation sets. However, this is not the case if the ~te~at~v~ method for vacancies is used. A useful expression for 6 can be derived, Let sh, s =I: 1,2, , . ., denote the species occupying a cell of kind A in the perfect crystal and define yS by I 3%= W&i%=

@*II)

A set of numbers v,r such that v,t is non-zero

(2.14)

Next, Jet {n,fR denote a particufar subset, number a, of n cells out of the totality of N lattice cells and let Z;(nJN represent summation over all the N!/(N -6) !n! such subsets for PI= 0, 1, . *. N_ Denate by vppG an occupation set such that there are defects at every cdl of (qJN but no defect at any other ceff and let j$II denote summation over all occupation sets of this kind. Define

Let m denote the o&~u~ati~~ set for the perfect crystal, Tke function B defined by ideni~fyin~ the integrand of equation (2.5) with the integrand in

(2.12)

2190

A. R. ALLNATT

where the functions u can be calculated from equations (2.17) and (2.15). If the important terms of u are those whose occupation sets n differ little from m then it is reasonable to hope that the summation converges, the terms for ~1, greater than 3 or 4 or corresponding to spatially distant sites being negligible. The preceding results summarize the essentials of the Mayer [7] scheme in condensed form. They may be used as follows. The position of an atom in cell I= II, lZ,I,, A can be written as x f , where in this context 0 I= II, 12,I,. We define r($

= x(;)-x(;:)

(2.20)

and write

and L. A. ROWLEY

functions are slightly different from the perfect crystal canonical ensemble case and these are now summarized. For a crystal of M unit cells one may define a quantity f depending only on intensive variables by Mf(z, a, T) = log (I exp (-PO{LI)d{LI) (2.24) where a denotes the set of lattice parameters. Equation (2.10) can then be written log 9(z, V, T, a) = (V/A) X ( X a, log z,+f(z, s

where A = V/M is the unit cell volume, CX,= and M, is the number of sites occupied by species s in the perfect crystal. For cubic lattices one has M,/M,

r(;f,)=ro(;;,)+u(;;:) (2.21) “($) =n(:)-u(;,)

A = va3 where v is a constant and a the lattice para(the extension to other lattices is straightforward). Maximization of log 9 with respect to a at constant z, V, T to find the lattice parameter yields

(2.22) meter

where r. denotes vectors between cell centres and u denotes displacement from cell centres. 8 can be expanded as a Taylor series in the

(2.26)

-.f

1

0

s

5

From the theory of the grand partition function

I9=20i

i=O

=eo+& x L=l ll...,

(2.27)

2 li

V,T

a ,..>Y

Ala..,hi

X

(2.25)

a, 7))

&..$;;;:;i) aa (;t:). .a$)

and from these equations one finds

(2.23)

and the partition function, equation (2.10), evaluated by methods similar to those ordinarily used for the canonical partition function. Unfortunately the general formulae cumbersome. for the 13~..~are excessively Explicit formulae for a common example are given in Section 3. The details of evaluation of thermodynamic

P=

cs = (MS--N,)IM

= -

(

&

3. 0 FOR ONE KIND OF DEFECT LATTICE

(2.29) a,T,I(T+~) > r IN A CUBIC

When there is only one kind of defect equations (2.15) and (2.17) yield (+(1,(L))

=-kTlog(l+hJ

(3.1)

IMPERFECT

cr(l, k(L))

= - kT log (I+ hllr)

MOLECULAR

(3.2)

2191

CRYSTALS

the results es(/,l,)=

where Y, exp (-pul”)

hl=

(3.3)

hk = hhcfikl[(l+ Ml + h/c)1

(3.4)

fil, = ew (-P&i> - 1

(3.5)

u: = U/,*,(L)

(3.6)

l&=

u,*I,{L}-uup-l4,*

for 0, and 0 1

straightforwardly by combining equations (2.19) and (3.1) to (3.8), the function 0 being evaluated with all r = I-,,. 0, is zero for the present model.

(3.12)

c e;! (r&J = 0 12

(3.13)

0% (/I&) = -E

manipulations

(3.15)

112)= 0

(3.16)

; cr&p(lll, 112) = 0.

(3.17)

E fls(&,

The subscript zero on a bracket denotes that the functions are evaluated with all r = ro. The contribution from a(lk) functions was evaluated only in the approximation Al,, G 1 so that from equation (3.2) u( lk) = - kThllc.

(3.18)

The result is then %?(UA

=-(h/l+h)( x 2 (

8, can

(3.9)

(3.14)

where the remaining crupare defined by

-kT(h/l

8, = 8 z z e,(i,f~)u,(r,)u,(~~) 11.12 a.4

a&(&, U

I, =I= I, I2 * I

@

By straightforward be written in the form

1

- [r,(~l,)D(ff,)r,(ll,)~(~~~)u(~)l0,

0, follows

(6) Expressionfor

(&)lo,

11* 1,

(3.7)

where by previous definition U,*, and U,*,, are the change in potential energy when defects are introduced at 1and 1,k respectively in perfect crystals. In the remaining equations we assume that we have only one kind of substitutional defect in one of the cubic lattices. Mayer cells may then be chosen to be the same as crystallographic cells. We also assume that the potential energy is the sum of pairwise forces

(a) Expressions

- [~,(~l~Z)~(lllZ)~P(l11,)0(1,1,) 6

C’f) (5?111,))_ +h)*F,(1JJ

(3.19)

where FOP

where

%3(~14> = p$(&).

(3.10)

The superscript i = 0, 1,2, . . refers to the contribution from I/,, u(f) functions, ~(1, k) One finds, using functions, . . respectively. the abbreviation

L),1a

r dr

h and f denote hl and fik functions evaluated at r = ro. For any function X we denote by

X(s) the function derived by replacing each hl by fh,. Finally we used the abbreviation

2192

A. R. ALLNATT

uaiwdk w

C’f = Z’fik k

where here and elsewhere that k = 1 is omitted.

the prime denotes

indicated by

8, = CI,+ 02.

(3.22)

The harmonic Einstein approximation, C 0,,(ff)~,~(f).

= %,+iC 1

(d) Anharmonic

HE, is

x (@,,,(W .I 01 + 20 a&P(UU)) X ua4(1) (3.24)

= -C’

11

[&YB(I; W,M+

0~/&(1111) = - 7 u&&II; f,f,f,f,)

4. EVALUATION

= -Gc&rs(f; ll,12,&,

1111) W3~4b 1, f

1

(3.29) OF f

is convenient to separatef, equation (2.24), into static and vibrational parts, f(O) and fCo), defined by It

approximations

(3.26) (1)

(a) Harmonic

(4.1)

f(V) = f-f’“‘.

(4.2)

approximations

0, may be transformed

to a sum of squares by the normal co-ordinate transformation u,(f)

= M-‘/z C, eJJ)Q(f)

exp (2rrzk. x(f)).

kj

(4.3) The Jacobian for the transformation may be shown to be unity. By steps quite analogous to those used by Bradburn [ 131 it may then be shown that the harmonic ap#-oximation obtained by combining equations (2.24), (3.9), (3.22) gives fH(z, T, a) =fto)+#log

(2nkT)

- IZ log (Pas(k) k

provided that the matrix D,(k)

1)2M

(4.4)

, defined by

DaP(k) = x O,4(11I) exp (- 2mik. r,(fl,))

103

(3.27)

f’“’ = -#80,/M

In addition to f one also requires its derivatives with respect to a and z for a self-consistent treatment of thermodynamic properties. These cumbersome expressions are easily derived from the results for f given below and will not be quoted.

(WI0 (3.25)

u&*(f;

X rdlf4)WI14)1.

(3.23)

where @!%6(W

(3.28)

f,W,)

a

The sensitivity of the calculations of defect concentrations to the anharmonic terms has been studied using the Henke1[9] approximation. An alternative would be to calculate the first order anharmonic perturbation to log z following the methods for perfect a formidable crystals[lO, 111. However, amount of additional algebra would be required for the imperfect crystal and higher order terms may also be important in the classical region. A -more promising alternative would be a variational method[ 121. Until perfect crystal calculations are more highly developed we do not feel that these more elaborate methods can be justified in the present context. The anharmonic Einstein, AE, calculations therefore use the approximation [9] @AE= Q,,f$E

L&l;

= I1.12./3./4+1 c ~~~~~~~~rI&14)

= [r,(fl,)D(II,)r,(ll,)D(ll,)r,(f1,)D(II,)

(c) Harmonic approximations The harmonic approximation, subscript H, is

8,,

and L. A. ROWLEY

(4.5) is positive

definite at all points in k space.

IMPERFECT

MOLECULAR

CRYSTALS

x [r,(ll,)r,(ll,)~‘2’(11*)lr(ll,)2

where the asterisk again denotes replacement of4 by+*. The explicit expressions for 03 and for the anharmonic contributions discussed below are very unwieldy but straightforward to derive and will not be stated.

- (r,(n,)yP(lll)Iy(I11)3

(b) Anharmonic

From equations (3.12)-(3.17) the contributions to D, from O$ andO($ are ZI$j( k) = x ’ [ 1 - exp (- 2rrik . r,(ll,))l II

-S,/r(n,))~“‘(ff1)1, =-2(/z/l -

(4.6)

fAE(z, T, a) =fco)+3 +h)D$j(k)*

[hl(kT(1t-

h)2)

X exp(- 2~~1 . r&)) where

=

log/

(4.13)

I = Jr exp (-A@* - Bpx4) d_x

(4.14)

where

1

x [ ; ’ (r,(U,)~“‘(U,>*lr(lll))

P’(4)

terms

Equations (2.24) and (3.24) yield

and, after simplification D%:(k)

2193

1

o2

(t$j$,

A = O&11)/2 B = (O,,,(ffff)

(4.7)

+22o,,,(ffff))/4!.

By expanding the exponential

(4.9)

(4.17)

n=o

1%= j,a x4n exp (-pAx2) zo=J,m exp (-pAx2)

and this may be converted for log Z

and DL!(B(k)* is Z&!j(k) with C#J replaced by C#I *. The harmonic Einstein approximation yields

log Z =

logI,+

i

(4.16)

one obtains

I = Z. 2 (-_PB)V,/n!

(4.8)

and $I (11,)* denotes the potential function such that U? from equations (3.6) and (3.7) can be written in the form U:: =-c’+(Z&)* 11

(4.15)

dx/Zo

dx

(4.18) (4.19)

to a power series

(-P)“w,Jn!

(4.20)

n=1

fHE =f(O)+ $log(27rkT) -~logO,,(ff)

(4.10)

for cubic lattices, and one finds Ok

= xc’ [r,(ff,)rp(ff,)C$‘*‘(ff,)/r(ff,)*

n-1

” (r,(ffl)rfl(ffI)lr(ff1)3

-SaB/T(111))~(l)(111)10 O$(ff)

w,

xn

[

(4n)!/(2n)! -(n - l)! x

((4(n -m))!/

m=1

((n-m)!(m-l)!))

X (w,/xm)]

(4.21)

where x = B/( 16A2/3*).

rNw-(1+h)*)l

X [ ~‘ra(4)r~(K) X (~‘2’(~~,)*)2/r(ll,)2]o

=

(4.11)

=-2(h/lS_h)@~(ff)” -

From the general relation between W, and Z functions [ 141 combined with reduction of the integral I, by integration by parts one can obtain the following general expression for calculation of w, as a function of A and B:

(4.12)

(4.22)

Equations (4.13), (4.20), and (4.2 1) combine to give fAEas a function of A, B and p. The successive partial sums of the series in (4.20)

2194

A. R. ALLNATT

oscillate with increasing amplitude in the regions of interest below. The series was summed by the method of Pade approximants [15], which form a convergent sequence generally constant to 0.1 per cent after the fourth approximant.

and L. A. ROWLEY

2400

5. RESULTS FOR RARE GAS SOLID SOL~~ONS The conventional -form for the chemical potential of an impurity, s, dissolved in a host, t, is CL,= ps’O’+ kT log c,y,

(5.1)

where ys -+ 1 as c, -+ 0. From equations (2.29), (3.11, (4.1) and (4.2) and using the fact that zs * 0 as cs + 0 one finds

I I

/

80

90

/

100

T,OK

-kTlog(l-~(~)Z~Zj

~~=(l-$(~),,=,)/(f-~(~))

(5.2)

(5.3)

where ut is u*, equation (3.6), evaluated with all r = r,. The first term in brackets in equation (5.2) is the static contribution but it is clearly different from the static approximation in conventional treatments since the energy u* is calculated without lattice relaxation. The effect of the latter is therefore wholly in the vibrational term. In the limit zs + 0 only the cr functions for single sites contribute to (?$‘U’/az,) as expected physically. Figure I shows results for p(o) at zero pressure in the approximations H, HE, AE for argon in krypton at three temperatures, and also values calculated from the experimental results of Fender and Halsey [16]. The result from using experimental values of zt [ 161 and a [ 171 for the AE approximation is also shown. Ar and Kr potential parameters for the 6-12 Lennard-Jones potential given by Horton and Leech [ 181 were used and the Ar-Kr parameters calculated from the com-

Fig. 1. Free energy of solution, p(O), as a function of temperature, T, for Ar in Kr at zero pressure. X, experiment[lb]; 0, H, S.5;0, HE, 5.5;A, AE, 5.5; +, H, 5.6; 0, HE, 5.6;V, AE, 5.6;A, AE, 5.5, using experimental ZK? and a; ‘I, AE, 5-6, using experimental zxr and a. (Letters refer to approximation and numbers to the equation for the potential parameter combining rule).

bining rules (5.4) (5.5)

and from the rule [ 191 which replaces (5.5) by EAR

=

~AA%3/

(EAA + %3)-

(5.6)

The procedure followed was to first calculate the equation of state of pure krypton to find the lattice spacing and chemical potential corresponding to zero pressure at a fixed T. The values were used to calculate pCo’ from (5.2). Some lattice summations additional to those required for the perfect crystal[l3] were calculated on a computer. The numerical integrations over k used 30 points in the irreducible element of the Brillouin zone [ 131. increasing the density to 149 points has a negligible effect on the very similar calculations of Section 6.

IMPERFECT

MOLECULAR

CRYSTALS

From the straight line drawn through the points the corresponding enthalpies and entropies of solution were calculated:

1.4 -

I.2 -

Some results are summarized in Table 1 together with calculations for other mixtures. For solutions of Xe in Kr the H and HE approximations gave

zs V’“’ h,

(az,>

>

F ->

IOl

z$=o

R

so that (5.2) cannot be evaluated or, equivalently, (2.29) predicts c, < 0. Also included in Table 1 are volumes of solution defined by Vs(O)=

2195

(y),= (~)~I(~)~

60

90

loo

T.-K

(5.8)

calculated at 90°K the differentiations being performed analytically. The volume of solution is slightly temperature dependent as is shown in Fig. 2 for calculations using the simple combining rules (5.4) and (5.5). The si~ifi~ance of these results is as follows. For moderate misfit in size, as for Ar in Kr, the predictions of the three approximations span a range of about 10 per cent in h, and 14 per cent in s,. This is quite com-

Fig. 2. Volume of solution, I/(O)(at. vol.), as a function of temperature, T, for impurities in Krypton at zero pressure using the combining rule of equation (5.5) for the 3 approximations. -I-, Ne (H): 0, Ne (HE); V, Ne (AE); 0, Ar (H); CI, Ar (HE);A, Ar WE); X, Xe

(A-0

parable to the errors arising from uncertainty in potential parameters, as is illustrated by using the different combining rules for l(ArKr). The values used were 193.04 x IO-” ergs from equation (5.5) and 190*56 x lo-l6 ergs from equation (5.6). The agreement with experiment is not unreasonable considering the expected accuracy of calculations and

Table 1. Mean en~hal~hy (hsf”)cal. mol.-‘) and enrropy (s,(O)R ~~j~s) of solution in the temperature range SO-lOO”K, and volume of solution (V,‘O’at. vol.) at 90°K for impurities in Krypton at zero pressure Combining rule equation

(5.5)

(5.6)

Approximation

H HE AE

1960 6.84 2010 6.62 1680 5.06

1.13 1.14 1.04

1110 5.13 1130 4.81 1250 4.33

H HE AE

2420 2490 2120

I.19 1.21 1.08

IO50 1060 1180 1074 k25

Experimental [ 161

7.14 7.32 5-43

0.93 0.92 0.90

5.19 0.96 4.88 0.93 4.38 0.91 5.86 -co.14

2930 10.01

1.40

2840

1.39

9.94

2196

A. R. ALLNATT

experiment. As the misfit in size becomes greater the inadequacy of the H and HE approximations must increase since the bigger relaxation about each defect throws greater weight on the vibrational calculation (cf. remarks after (5.3)) and on the anharmonic terms-compare Ne and Ar results. Indeed for Xe only the AE method yields a prediction. For such large anharmonic effects the Henkel approximation may presumably be seriously wrong, whereas for Ar this error is probably no worse than the uncertainty arising from the potential parameters. Finally we note that Fig. I seems to suggest that taking account of the anharmonic terms without the Einstein approximation would lead to a better agreement with experiment. In consequence of these conclusions calculations of activity coefficients have been limited to Ar in Kr at zero pressure for concentrations up to 4 X 10-2. The HE approximation was used retaining u functions up to two sites only. As the concentration is increased the lattice parameter at zero pressure becomes smaller. The method adopted was to calculate the value of ‘u’ for which p = 0 for fixed (z,/z~, T) from equations (2.28). yI was then calculated at these values (a, z,/z~, T) from (5.3) at a known concentration found from equation (2.29). Results are summarized in Fig. 3. The calculations are for approximations retaining one, and for one and two sites, for the two combining rules using the HE approximation and a rigid static lattice approximation (equation (5.9) below). The experimental results shown are actually extrapolations from measurements at higher concentrations. The most striking feature is the sensitivity to change in the potential parameters; even the relative order of the approx.imations changes with change in combining rule for eAB. It may be shown that for the static rigid lattice approximation retaining o functions for single sites corresponds to an ideal solution. Retaining (+ functions up to two sites yields, for h 4 1,

and

L. A. ROWLEY

IQ0

0.96 -

\

I

k

I

\

‘r

l\

lal

096

-

0

4

2

C102 Fig. 3. Activity coefficient, y, for Ar in Kr as a function of Ar concentrations, c, at 90°K and zero pressure; (a) using combining rule (5.5); (b) using combining rule (5.6). --(r for single sites; - (T for up to pairs of sites; - - static approximations; -- experiment [ 161.

ys= ew

(~C’fc,(c,-2)).

(5.9)

With the approximations (i) f= 0 except nearest neighbour separations and (ii) f=exp

(-u*p)-1

= -u*p

at

(5.10)

this reduces to the Bragg-Williams approximation. Figure 4 shows that whereas (ii) is not too unreasonable (i) is not a good approximation. Extension to higher concentrations would require using the complete expression for hlk instead of the approximation of (3.18), and probably allowance for the three site u function. Without a more certain potential and inclusion of anharmonicity this is scarcely worthwhile at present. We note that the summations over file functions in calculations allowing for defect interactions cannot be

IMPERFECT

-005-

MOLECULAR

2197

CRYSTALS

I I

I

A

I

1

266

R/a Fig. 4. Mayer f function and -Up (the lower curve) [equation (5.10)] as functions of distance r for Ar in Kr at 90°K and zero pressure.

parametrized to a functions of ‘a’ times a dimensionless lattice sum and so must be calculated for each ‘a’ and T, leading to extra numerical work. 6. RESULTS FOR VACANCIES IN RARE GAS SOLIDS

To determine the pressure of the pure solid with vacancies from equation (2.28) it is necessary to determine z from equation (2.26) at the selected (a, T). Equation (2.26) was solved by iteration taking the perfect crystal value of z, at given (a, T), as the starting point. The function f is independent of z for the perfect crystal and (3.26) can be solved explicitly in this case. In the alternative method for vacancies defined by equations (2.8) and (2.9) the partition function was maximised numerically at each stage of the iteration with respect to 6 at constant (z, a, T). By these methods the lattice parameter corresponding to zero pressure for Kr was found retaining (+ functions for single sites. The concentration of vacancies, c,,, was then calculated from equation (2.27) and the results are summarized in Fig. 5. From least squares lines the enthalpy and entropy of formation in the equation

,

I

09

I

I

I

I.0

(II

I.2

IOZ/T,"K-' Fig. 5. Vacancy concentrations, c,, as a function of temperature, T, for Krypton at zero pressure. A prime on the approximation means the alternative vacancy method was used, 0, H; 0, HE; a, AE; +, H’; A, AE, using experimental z and a; V, Glyde[4]; --Nardelli [3]; ----experiment[l7]. To the scale of the graph the approximations H and AE coincide at 80” and 90°K and the approximations H and HE coincide at 90” and 100°K.

log c, = -phv + s,lk

(6.1)

were found. The results are summarized in Table 2. They are mean values since the slope increases slightly as l/T decreases. It may be shown by applying the method of corresponding states that c, depends only on the reduced temperature T = kT/c at zero pressure. Calculations [3,4] for Ar were converted in this way. For the AE approximation using the alternative method for vacancies the iteration for z did not converge, whereas for the other approximations three iterations suffice for 0.01 per cent constancy. For the H approximation 6 varied montatonically from 2408 to 1205 dyn cm-l in the range 80- 110°K.

2198

A. R. ALLNATT

Table 2. Mean enthalpy (h, Cal. mol.-‘) and entropy (s, R units) offormation of vacancies in krypton at zero pressure in the temperature range 80-110°K. A prime on the approximation means the alternative vacancy method was used (equations (2.8), (2.9)) Approximation

H

h,

s,

3870 3720 3290 3180 3130

HE AE H’ HE’ AE’

Glyde [4] Nardelli [3] Experimental [ 171

2.23 1.21 - 1.25 3.30 3.56

2900 3620 1780 -t 200

3.21 8.1 21’. 05

The effect of the u functions for pairs of sites was negligible in the HE approximation. Figure 6 shows the volume of formation V,=

(6.2)

I

80

ment with experiment [20] than those of previous calculations [3,4]. Since the approximate inclusion of anharmonicity does not have a very marked effect on the results one might be tempted to join previous workers in ascribing the failure of the calculation to neglect of three-body forces [2 1,221 and other many body effects[23]. However, there remains a slight doubt as to the validity of the method whereby the sum of integrals is converted to an integral of a sum in the partition function using the method described after equation (2.7). As Mayer[71 has remarked, this method would break down for large c,. This objection does not apply to the alternative method. The failure of the latter might possibly be ascribed to inadequate treatment of the lattice dynamics, but we remain uncertain. The status of the method for systems with vacancies is therefore much less clear than for solid solutions. 7. CONCLUSIONS

kT

as a function of temperature and another calculated [3] value. The results for c, are in no better agree-

I

and L. A. ROWLEY

so

loo

110

T,OK

Fig. 6. Volume of vacancy formation, I/,(atomic volumes), as a function of temperature, T, at zero pressure for Krypton. 0, H; 0, HE; A, AE; 0, Nardelli [3].

(1) The Mayer method appears promising for substitutional solid solutions provided the misfit in size and potential curves is not too great. (2) The adequate treatment of the anharmonic terms becomes increasingly important as the misfit increases because the relaxation calculation of conventional methods has been replaced by a modified vibrational calculation. However, the methods for perfect crystal lattic dynamics can be applied, the only essential difference being a rise in algebraic complexity. (3) These calculations are often very sensitive to errors in the potential functions assumed and this is one of the major inhibitions to undertaking the large amount of work required by more adequate treatment of vibrations e.g. by a self-consistent phonon method. (4) For interstitial defects the method does not appear particularly promising because (a) the number of elements in the dynamical matrix increases, (b) the misfit will often be large, (c) the alternative method for vacancies

IMPERFECT

MOLECULAR

would have to be used for the vacant interstitial sites, and this method has not been very successful in the present work. (5) Mayer-[71 expressed the view that the method might prove adequate for calculating the solubility of sparingly soluble impurities and for phase transitions in general. Since these impurities tend to have large misfits the considerations of (2) and (3) above apply.

8. 9. 10.

11. 12.

13. Acknowledgement-One the receipt of the S.R.C.

of us (L.A.R.) acknowledges maintenance grant.

REFERENCES 1. KANZAKI H.,J. Phys. Chem. Solids2,24 (1957). 2. BURTON J. J. and JURA G., J. Phys. Chem. Solid. 27,961 (1966). 3. NARDELLI G. F. and TERZl N., J. Phys. Chem Solids 25,8 15 (1964). 4. GLYDE H. R. and VENABLES J. A., J. Phys. Chem. Solids 29, 1093 (1968). 5. WOJTOWICZ P. J. and KIRKWOOD J. G., J. them. Phys. 33, 1299 (1960). 6. MARADUDIN A. A., MONTROLL E. W. and WEISS G. H., In Solid State Physics (Edited by F. Seitz and D. Turnbull), Suppl. 3. Academic Press, New York (1963). 7. MAYER J. E., In Phase Transformations in Solids

14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

CRYSTALS

2199

(Edited by R. Smoluchowski, J. E. Mayer and H. Weyl), p. 38. Wiley, New York (195 1). HILL T. L., Statistical Mechanics. McGraw-Hill, (1956). HENKELJ. H.,J.chem.Phys.23,681(1955). LEIBFRIED G. and LUDWlG W.. In So/id State Physics (Edited by F. Seitz and D. Turnbull), Vol. 12, p. 276. Academic Press, New York (1961). FELDMAN J. L. and HORTON G. K., Proc. .phvs. _ Sot. 92,227 (1967). GILLIS N. S.. WERTHAMER N. R. and KOEHLER T. R., Phys. Rev. 165, 95 1 (1968); and references therein. BRADBURN M., Proc. Camb. Phil. Sot. 40, 1 I3 (1943). HILL T. L., In Statistical Mechanics, p. 135. McGraw-Hill, New York (1956). WALL H. S., Analytic Theory of Continued Fractions. van Nostrand, Princeton, N.J. (1948). FENDER B. E. F. and HALSEY G. D., J. c/rem. Phys. 42, 127 (1965). LOSEE D. L. and SlMMONS R. O., Phys. Rev. 172,944 (1968). HORTON G. K. and LEECH J. W., Proc. phys. Soc.82,816(1963). FENDER B. E. F. and HALSEY G. D., J. them. Phys. 36,188 I (1962). LOSEE D. L. and SIMMONS R. O., Phys. Rev. 172,934 (1968). JANSEN L., Phil. Mag. 8, 1305 (1963). FOREMANA.J.E.,PhilMag.8,1211(1963). DONIACH S. and HUGGINS R., Phil. Mag. 12, 393 (1965).