Lattice Dilatation in High-Concentration Binary Solid Solutions

S. N. MAITI and D. ROY: Lattice Dilatation in High-Concentration Solutions 449

phys. stat. sol. (b) 122, 449 (1984)

Subject classification: 4; 12.2; 22.5.4

Solid State Physics Research Centre, Department of Physics, Presidency College, Calcutta1) (a) and Physics Department, Jadavpur University, Calcuttae) (b)

Lattice Dilatation in High-Concentration Binary Solid Solutions BY S. N. MAITI (a) and D. ROY (b)

A simple extension of Huang's theory of lattice dilatation for low concentration is found to reproduce Retger's law of additivity of molar volume. Deviation from the law is accounted for by considering the elastic strength of the defect to be concentration dependent and an attempt is made to calculate the elastic strength of the defect from the parameters of the perfect crystal. The method is then used to predict the concentration dependence of volume over the entire concentration range of eight alkali halide solid solutions and two metallic alloys and fairly good agreement is obtained.

Es wird gefunden, da13 eine einfache Erweiterung der Huangschen Theorie der Gitterdilatation fur niedrige Konzentration das Retgersche Gesetz der Additivitiit der Molarvolumen reproduziert. Abweichungen vom Gesetz werden dadurch erkliirt, daB die elastische Stlrke des Defekts als konzentrationsabhangig beriicksichtigt wird, und es wird versucht, die elastische Stiirke des Defekts aus den Parametern des perfekten Kristalls zu berechnen. Die Methode wird dann benutzt, um die Konzentrationsabhiingigkeit des Volumens iiber den gesamten Konzentrationsbereich fur acht Alkalihalogenidmischkristalle und zwei metallische Legierungen vorauszusagen, und es wird ziemlich gute ubereinstimmung erhalten.

1. Introduction The empirical relations of Vegard [l] and Retger [2], which propose linear variation of lattice parameter and atomic volume with the concentration of an alloy, approximately describe the experimental behaviour for the variation of lattice parameter with concentration.

Huang [3] treats the defect lattice as a mean lattice plus a fluctuation from the mean lattice and gives the uniform lattice expansion in every direction assuming the crystal to be an isotropic continuum. Eshelby [a] pointed out that the uniform lattice expansion as explicitly predicted by Huang is correct only for an infinite medium. For a finite medium the displacement field appropriate to an infinite medium is modified by the presence of a stress-free surface upon which the surface tractions vanish. The net result for a spherically symmetric displacement field in an isotropic finite medium is an additional dilatation factor y , or the image factor. Though the image field has negligible effect on diffuse intensity, it is highly important for the dilatation term [5]. Though Huang omitted the image term, his results may be taken to apply to a sphere subjected to a uniform hydrostatic pressure just sufficient to annul the image terms. Removal of this pressure evidently would affect the change of X-ray lattice constant and the geometrical dimensions in the same way, i.e. both would be multiplied by a factor y.

' l') 700073 Calcutta, India. 2, 700032 Calcutta, India.

450 S. N. MAITI and D. ROY

The X-ray diffraction from crystals with randomly distributed micro-defects in low concentration has recently been solved by several methods [6 to 91. Though, quite a large number of works on the X-ray diffraction from low-concentration solid solutions exist in the literature, their extension to high concentration is not a t all trivial. In high-concentration solid solutions it is extremely difficult to describe quantita- tively the distorted configuration, as the distortion due to each defect cannot be treated as independent and hence the net distortion of the medium cannot be taken as a simple superposition of the effects of the individual defects.

The calculation of lattice parameter in high-concentration alkali halide solid solutions was attempted, probably for the first time, by Durham and Hawkins [lo]. They computed the total energy of displacement by restricting the relaxation to common neighbours of the defects and from a knowledge of the probability of occupancy of the various configurations. Using the thermodynamic relation J(S , VIP, V ) = = J( V , PIT, P) , with the neglect of the change in thermal energy with volume, they then computed the variation of lattice parameter with concentration assuming a linear variation of compressibility and volume expansion coefficient with concentration. They used the Born-Mayer potential and numerically obtained a lattice parameter close to that predicted by Retger’s law.

Recently, we have explained the experimentally observed deviations of the lattice parameter from Vegard’s law in alkali halide solid solutions from the boundary condition on the equation for the heat of formation [ll]. Subsequently in an attempt to make a unified study of the different properties of alkali halide solid solutions we used the idea that an alkali halide solid solution a t any concentration may be re- presented by an equivalent lattice in which the interaction potential between different particles is the same as that in the perfect lattice, except that the parameters of the potential are obtained from those of the perfect crystal by using Vegard’s law [12]. The calculated values of the lattice parameters explain the observed deviation from Vegard’s law. Moreover, this approach has successfully been used by us [13] in predicting the lattice parameter variation for the dual phase solid solution.

However, very few attempts [14 to 191 have been made to calculate the change in lattice parameter from a microscopic calculation of the elastic strength. As the elastic strength of the defect enters directly the expression for the change in the lattice parameter a reliable microscopic estimate of the elastic strength can be obtained from a comparison of the change in the lattice parameter and this in turn helps to estimate many properties of the defect lattice depending on the relaxation of the lattice, such as the entropy of formation of a defect. The methods used in calculating the relaxation of a lattice containing defects in low concentration are the force balance method and the energy minimisation method. However, the relaxations obtained by the two methods widely differ [18]. Paul and Sengupta [19] pointed out the cause of this discrepancy and suggested a consistent approach for estimating the elastic strength of the defect. Following the procedure of Paul and Sengupta we have estimated the contribution of van der Waals and second-neighbour repulsive forces to the elastic strength of the defect. It is found that the calculated values of the elastic strength fairly reproduce the change in the lattice parameter at low concentration. Moreover, using a simple extension of Huang’s theory by assuming the elastic strength to be a function of concentration satisfying the two extreme boundary conditions, we have calculated the variation of atomic volume with concentration over the entire concentration range. It may be remarked that only perfect-crystal information is sufficient for these calculations and we have not used any particular behaviour of the solid solutions as input at any stage of our calculations.

Lattice Dilatation ip High-Concent,ration Binary Solid Solutions

2. Theory

451

Assuming the crystal to be an isotropic elastic continuum and taking the radial relaxation of the ions to be of the form

K r u ( r ) -, T 3

K being the elastic strength. Huang [3] explained the uniform expansion of the lattice caused by a random distribution of foreign atoms in a dilute solid solution and his result may be written as

r(x) = r,, 1 + - ( 4;: + where r,, is the distance between any two ions in the perfect lattice, r(x) is that a t concentration x, and v is the atomic volume.

From (2), Eshelby’s expression [a] for the macroscopic volume change per unit volume in an infinite crystal with low concentration of defects readily follows, i.e.

It may be remarked here that though Vegard’s law expresses a linear dependence of the lattice parameter on concentration even at high concentration, any linear dependence does not imply Vegard’s law as it should reproduce the other end (x = 1) of the solid solution. A number of approaches [14 to 191 exists in the literature for calculating the elastic strength of the defect on a microscopic basis. When the calculated K values are substituted in (2), the other end of the crystal is, in general, not reproduced. Moreover even if one assumes that (2) is valid for high concentration, one faces the difficulty as to which values of K and v should be used in (2). As it is meaningless to identify host or defect lattice in a solid solution of finite concentration, one is left with two choices for the values of K and v corresponding to the two ends of the solid solution. Above all, Huang’s result of lattice dilatation, as indicated in (2), is basically based on a theory suited for low concentration of defects and should not be used a t finite concentration without proper justification.

3. Extension to High Concentration

Huang’s theory may be extended to high concentration on the basis of the following assumptions: After each addition of a defect the host crystal is changed to some other perfect crystal with changed lattice parameter given by Huang’s theory and K is changed to a value appropriate to the defect concentration. This means that the defect characteristics develop gradually and equally in each and every ion in the solid solution and not in discrete manner. We also assume that the lattice structure of the solid solution is not destroyed or altered a t any stage of the concentration change. On the basis of these assumptions (2) may be written as

where v = gr3.


When mediately on integration

is assumed to be independent of concentration (K(x ) = K), then we get im-

v(2) = v(0) + 4nK2. (5) When the boundary condition v(x) + v(1) as x -, 1 is used we get

- v(1) - v(0) K = - = K ,

v(2) = (1 - 5) v(0) + xv(1)

4n

and (7)

which is Retger’s law. It is found to be closer to the experimental values than Vegard’s law in lattice parameter. We have not used the microscopic K-values in the above theory; however, those may be compared with the calculated K , obtained from (6) . This is done in Table 4.

However, K should depend on concentration of defects as the elastic property of the host crystal should change due to addition of defects.

We now present a theory which takes into consideration the microscopic values of K and also its concentration dependence. Using

K ( z ) = K(0) + K’x + K”x’ + ... (8) in (4) and integrating, we get

v(2) = v(0) + 4n K(0) 2 + --+ ~ [ K’x2 2 “:I. With the boundary conditions

and

we eet

K ( z ) = K(0) , w(z) = v(0) at x = 0

K ( z ) = K(1) , v(2) = v(1) at z = 1 0

(K(1) + K(0) ) - 2~ and

K’ = K ( l ) - K(0) - K” .

(9)

Here K(0) and K(1) are low-concentration values of K calculated on a microscopic basis following the approach of Paul and Sengupta [19]. ~ ( x ) is readily calculated from (9),

This model is capable of predicting the experimentally observed deviations of volume and lattice parameter from the respective Vegard’s law.

4. Paul and Sengupta’s Model to find K

For a calculation of K on a microscopic basis we have followed the method of Paul and Sengupta [19]. Following their scheme the defect crystal with single isoionic substitution is assumed to grow in two steps. First, the crystal is made to relax to the final state by application of external forces. Then the central host ion (A) is substituted by a guest ion (B) of the same charge. For an isovalent impurity this second

Lattice Dilatation in High-Concentration Binary Solid Solutions 453

step introduces only differences in non-Coulomb interactions between the guest ion and the host ions. So the energy of the defect crystal is expressed as

up to terms quadratic in relaxation (u), where is the energy of the host crystal (A) in equilibrium. The second term includes the contributions from the first step and the third term represents the contributions from the second step, with Apnc = q& - - p&, vnc and ri are the non-Coulomb interactions and the ion positions of the perfect host crystal in equilibrium, respectively. Taking nearest-neighbour (n.n.) relaxation as Era, where ro is the n.n. distance, and the relaxations of the other ions a t ri in the perfect crystal as Krilr?, and assuming all the relaxations to be radial, the defect crystal energy is expressed from (13) as

All the n.n. ions to the defect site are included in the region I and the rest of the crystal is the region 11, which is assumed to be an isotropic continuum. The relaxations in the two regions are assumed to be independent of each other; so the equilibrium equations for determining the relaxation of the ions are given by

= o . a@ - 0 and - a@ a t aK _-

5 and K are calculated from these equations and the values of K are shown in Table 4. I n our calculation we have used Coulomb (C) and van der Waals (V) interactions

in their usual form and the repulsive interaction is taken in the form

where rd, r f are the radii of i-th and j-th ion, respectively, and

where zi and ni are the valency and number of electrons on the outermost shell of the i-th ion, respectively. Repulsive interaction is taken up to second neighbours only. b' and e are calculated (Table 2) using ionic radii, van der Waals constants, harmonic n.n. distance, and bulk modulus as input data (Table 1). 30 phyaica (b) 12212

Tab

le 1

In

put

data

harm

onic

ela

stic

oon

sta.

nts (

loll

K/m

*) h

arm

onic

ra

dius

ra

dius

va

n de

r W

aals

con

stan

tsf)

d J

ma)

n.

n. d

ista

nce

of th

e of

the

c

(10-

41 J

me)

- TO

+ ve

-ion

-ve-

ion

61

61,

Ed44

B (1

0-1n

m)

rt(1

0-1n

m)

rb(1

0-ln

m)

c,, c-

- c,-

c

d,,

d--

d+-

d

NaC

l 0.

6058

) N

aBr

0.48

9 N

aI

0.38

8 K

Cl

0.50

28)

KB

r 0.

4328

) K

I 0.

342

RbC

l 0.

458

RbB

r 0.

399

RbI

0.

329

Ni

2.69

2c)

cu

1.78

0d)

Au

2.04

28)

0.12

8)

0.09

7 0.

077

0.05

88)

0.04

58)

0.02

8 0.

074

0.03

9 0.

027

1.51

4C)

1.25

9d)

1.71

6e)

0.13

78)

0.10

9 0.

079

0.06

78)

0.05

48)

0.03

8 0.

05

0.04

0.

029

1.36

0C)

0.83

5d)

0.46

28)

0.28

28)

0.22

8 0.

181

0.20

68)

0.17

48)

0.13

3 0.

202

0.15

9 0.

128

1.90

7C)

1.43

3d)

1.82

59

2.79

1 2.

9526

3.

195

3.10

4 3.

248

3.51

05

3.26

24

3.41

28

3.63

26

2.48

17C)

2.

5420

6)

2.87

09e)

0.98

0.

98

0.98

1.

33

1.33

1.

33

1.48

1.

48

1.48

1.81

1.

95

2.16

1.

81

1.95

2.

16

1.81

1.

95

2.16

1.68

116

1.

68 1

96

1.68

392

24

.3

124.

5 24

.3

206

24.3

40

3 59

.4

130

59.4

21

5 59

.4

428

11.2

18

0 0.

8 14

27

1 0.

8 19

.1

482

0.8

48

451

24

60

604

24

82

927

24

79

692

82

99

901

82

135

1331

82

233

450

1100

25

0 47

0 11

30

260

490

1200

13.9

17

9 19

29

7 31

63

1 73

55

8 99

80

6 15

6 14

20

134

960

180

1335

28

0 22

34

") S

lagl

e an

d M

cKin

stry

[22

], b)

Kit

tel

[21]

, oth

er d

ata - S

imm

ons a

nd W

ang

[23]

, c)

Ale

rs e

t al.

[24]

, d)

Ove

rtone

and

Gaf

fney

[25

], e)

Nei

gh-

bour

s an

d A

lers

[26]

, f)

May

er [

27]

F

3 5: 2 fi

Lattice Dilatation in High-Concentration Binary Solid Solutions

Table 2

Parameters used

455

CRl CRQ CR&,

b' e 6' e v e (10-19 J) (10-1 nm) J) (10-I nm) ( J) ( 10-1 nm)

NaCl NaBr NaI KC1 KBr KI RbCl RbBr RbI

3837.8 0.2895 3275.5 0.30099 0.201 1 3956.6 0.3032 3521.1 0.3139 0.1880

0.1609 7 749.3 0.29455 5266.9 0.3154 0.1741 8 166.7 0.305 1 5327.8 0.3284 0.1646

10595 0.3186 6299.6 0.345 75 0.1673 0.1497 0.1553 0.1444

0.323 2 0.345 9 0.3595 0.3036 0.3178 0.337 7 0.2700 0.2990 0.3130

CR, CRIV : Coulomb + n.n. repulsion + van der Waals interaction CR,R, : Coulomb + n.n. and s.n.n. repulsive interaction

: Coulomb + n.n. repulsive interaction

The lattice sums in (15) have been evaluated for NaCl structure with the positive ion at the origin (defect site),

+ 84.992 % + 676.64% + 73.511 + ro r0 "-1 r0

+ 36r: (q&-. +

d--l ril d+- + 6.1064 + 38.624 11 + 5.7818 - -

r0 r0

1.5&+- + 8.4852 &)-$(3.22994'-- + 5.6921 ___ r0

+ 223.9 '$ + 14.79 %+ r0 ro

d++ d+- + 29.3 y + 389.5 14 + 12.26 6.233~;'+- + 8.034 ~

r0 r0

13.73&'++ + 10.98 &)+; (5.84p$-- + 5.61 ~ v"-)). (19) r0 ro

Lattice sums due to Coulomb, van der Waals n.n. repulsion, and 8.n.n. repulsive interactions are shown in (19), y1+- represents n.n. repulsive interaction between positive and negative ions. 30'


+ 216- +

g2 = - 126- [ A;;- To"

+ 3 4 6 exp i" + ?+ - ") _- ; exp [$ -.---___ +;- - To)) 9

e B A

+ 2.1568 Ad+- + 0.1808 11 + 3.3126?

TO TO TO 11

+

AC,'

+ 3.3912 14 +

Ad,- + 3 . 9 8 1 6 1 2 "+'+ 0.144 __ TO rA4 70

Ternis in square brackets are due to van der Waals interaction and the rest for repulsive interaction. g, and g2 contain n.n. contributions and g3 and g4 contain second and higher neighbour contributions. For negative ion substitutions '+ +' and '- -' are to be interchanged, e.g. Ac++ + Ac--, p-- + y2++, etc. in the above sums (19) and (201, b' = i b x and ij = (@A + @ B ) / 2 represent approximately the parameters for interaction between guest ion (say K+) and the host ions (say Na+) a t s.n.n. to the defect site, when A = NaCl and B = KC1 (say). Evaluation of go is not necessary for finding K and 6 from (16).

5. Image Correction

Equation (16) gives the value of K in case of an infinitely large isotropic continuum solid. In actual experiments crystals should require a correction to describe the actual behaviour of a finite crystal. This correction - called image correction - is introduced usually in either of the two forms

4 c44

3 B K(1 + y ) , where y = --

3(1 - U ) Ky' , where y' = (1 + 4 '

u is Poisson's ratio, Cl2/(C, + C12). These two forms of image corrections are equivalent in an isotropic solid. In our theory we have assumed the crystal to be an elastic isotropic continuum. So for a consistent theory it is more reasonable to use the isotropic image correction in our theory, rather than the single crystal value. We have used polycrystal average values of y as isotropic value. In polycrystal averaging from the single crystal value the bulk-modulus remains almost unaltered, whereas C,, is


replaced by

where C,”, and c,”, are the Voigt and Reuss averages. Values of ysc and ypc are shown in Table 3 and the corresponding K-values and K, and Kexp are shown in Table 4.

644 = + (CL + c,”,, 9

Table 3 Image correction for single crystal and isotropic solids

-

NaCl NaBr Na I KC1 KBr KI RbCl RbBr RbI Ni Cu AU

average harmonic elastic constants in 1011 N/m2 polycrystal single crystal -

CGV C4GR c44 ypc = + C,lB” ysc

0.179 0.166 0.173 0.818 0.648 0.144 0.133 0.139 0.813 0.637 0.11 0.098 0.104 0.766 0.582 0.129 0.093 0.111 0.718 0.434 0.11 0.076 0.093 0.713 0.414 0.086 0.055 0.071 0.712 0.381 0.107 0.071 0.089 0.587 0.330 0.096 0.058 0.077 0.646 0.335 0.078 0.043 0.061 0.635 0.302 1.052 0.893 0.973 0.68 0.951 0.605 0.444 0.524 0.488 0.777 0.342 0.266 0.304 0.222 0.338

6. Results and Discussion

The models developed in the previous section have been used to calculate the concentration dependence of lattice parameters over the entire concentration range for as many as eight alkali halide solid solutions and two metallic alloys. We have also compared the calculated values with the experimental ones in the associated figures. K-values for metallic alloys are taken from the paper of Das et al. [20]. They used Morse potential functions in Paul and Sengupta’s framework [19] for calculation of these K-values.

Our isotropic picture of a solid solution does not represent exactly the real situation. This should account to some extent for the discrepancy between calculated and experimental values. I n a high-concentration theory defect-defect interaction should not be neglected. However, we have taken the major part of this effect through our basic assumption for high concentration embodied in (4).

Table 4 clearly shows that the major contribution in K comes from Coulomb and nearest-neighbour repulsive interactions ; second-neighbour repulsive interaction is also found to be important in this respect, whereas van der Waals interaction contributes insignificantly. The K value for a finite isotropic solid is obtained by using (21) after introducing the image correction given in (21) and (23). The isotropic image correction is always found to be greater than its single crystal value in case of alkali halide solid solutions, whereas in case of metallic alloys the effect is opposite. The isotropic image term contributes about 60 to 80% in K for alkali halide solid solutions, whereas in metallic alloys it is 22 to 68%.

In our final calculation we have used Coulomb and n.n. and s.n.n. repulsive interactions neglecting van der Waals interaction for its insignificant contributions. Lattice sums involved in our process of calculation ale shown jp (19) and (20). The calculated variations of volume with concentration over the entire concentration range are shown

Tab

le 4

Cal

cula

ted

valu

es o

f de

fect

str

engt

h (R

) usi

ng d

iffe

rent

mod

els

and

thei

r co

mpa

riso

n w

ith e

xper

imen

tal

resu

lts

K (1

0-30

m3)

K

(1 + y

Sc)

~-3

om

3)

K(1

+ Y

PC)

m3)

CR

, C

R,V

C

R,R

, C

R,

CR,V

C

R,R

, C

R,

CR

,V

CR

,R,

K+

in N

aCl

Na+

in K

CI

K+

in N

aI

Na+

in K

I R

b+ in

KC

l K

+ in

RbC

l B

r- i

n N

aCl

C1-

in N

aBr

Br-

in

KCI

C1

- in

KB

r I-

in K

Br

Br-

in

KI

Br-

in

RbC

l C1

- in

RbB

r I-

in R

bBr

Br-

in

RbI

A

u in

Cu

Cu i

n A

u Cu

in

Ni

Ni i

n Cu

0.58

5 -0

.693

0.30

6

0.32

9

0.63

9

-0.3

59

-0.3

73

-0.7

65

0.56

8 -0

.702

0.27

6

0.31

0.59

4

-0.3

24

-0.3

54

-0.7

30

0.69

9

0.93

4

0.41

4

0.38

0.36

5

0.74

7

0.36

3

0.62

8

-0.7

29

-0.9

16

-0.3

67

-0.4

41

-0.4

14

-0.8

67

-0.4

55

-0.7

16

0.21

2*)

- 0.1

98*)

-0.0

66*)

0.

053*

)

0.96

3 0.

936

-0.9

94

-1.0

06

0.50

4 0.

454

0.47

2 0.

445

0.90

4 0.

831

-0.5

88

-0.5

31

-0.5

27

-0.5

01

-1.0

56

-1.0

08

1.15

1.47

8

0.59

4

0.62

6

0.52

3 -0

.585

1.

06

- 1.

20

0.48

3 -0

.607

0.

838

0.37

7

0.10

3

-1.0

5

-1.2

65

-0.4

88

-0.7

22

-0.9

32

-0.2

65

-0.1

17

1.06

4 -1

.191

0.55

6

0.56

5

1.09

5

-0.6

51

-0.6

39

-1.3

1

1.03

3 - 1.

206

0.50

2

0.53

3

1.01

8

-0.5

87

-0.6

06

- 1.

25

1.27

-1

.25

1.65

0

0.71

1

0.69

1

0.62

7

1.28

0.57

6

1.03

3

0.31

5 -0

.242

0.

090

-0.0

98

- 1.5

68

-0.5

82

-0.8

0

-0.7

09

- 1.4

8

-0.7

49

-1.1

70

uz ?: 0.

436

0.05

1 -0

.379

-0.0

78

F

K,

(10-

30 m

3)

1.30

1.69

5

0.76

7

0.63

7

0.69

4

1.43

2

0.80

0

1.30

3

0.40

7

0.06

4

-1.3

0

-1.6

95

-0.7

67

-0.6

37

-0.6

94

- 1.4

32

-0.8

00

- 1.

303

-0.4

07

-0.0

64

Kex

p

( 10d

30 m

3)

1.37

1.75

1

0.77

7

0.66

9

0.70

5

1.40

0.82

4

1.30

-1.2

3

- 1.

639

-0.7

56

-0.6

05

-0.6

83

-1.4

7

-0.7

77

-1.3

06

*) D

as e

t al.

[20]


'' 84

Fig. 1. Calculated values of the variation of volume with concentration along with experimental values ;

theoretical (9), [33], A [28], 0 [29], [30], +[311, x ~321

graphically in Fig. 1 with the respective experimental values in case of the following solid solutions: (Na, K)C1, (Na, K)I, (K, Rb)Cl, Na(C1, Br), K(C1, Br), K(Br, I), Rb(C1, Br), Rb(Br,I) , Au-Cu, and Cu-Ni. The straight line joining the two ends of the graph of volume represents Retger's law.

The agreement with experimental results may be somewhat improved by adjusting the input parameters of the perfect crystals within experimental error limits (/I & 5% and r, & f O.Olyo). When the harmonic bulk modulus of NaCl is decreased by 5 % ) K for Na+ in KCl is decreased by l.2y0, and that for Br- in NaCl and C1- in NaBr are increased by 12.2 and0.7yo, respectively. So & 5% variation in bulk modulus causes K to vary about 5 7 % (maximum) in case of positive ion substitution and about

-f13y0 (maximum) in case of negative ion substitution. The analysis of different experimental values of elastic constants shows an average variation in y by about &6.5y0. The range of variation in calculated values shown in our graphs are estimated on the basis of the above variations. The variation in r,, being relatively negligible, is however not taken into account. Fig. 1 shows that all the experimental values lie well within this range.

The model may be used in case of any solid solution including two-phase solid solutions with heterostructure solutes approximated as isotropic solid. Considering the fact that no particular behaviour of the solid solution is used as input and allowing for the uncertainties in experimental and input data, the agreement of our present work with the experimental results may be said to be quite satisfactory.

70

8

0 02 04 06 08 10 x i rno l% ofsecond component)---

Acknowledgement

The authors express their sincerest thanks to Prof. S. Sengupta of the Physics Depart- ment, Presidency College, Calcutta, for many helpful discussions.

References

[l] L. VEOARD, Z. Phys. 6, 17 (1921). [2] J. RETOERS, Z. phys. Chem. 3, 497 (1889). [3] K. HUANG, Proc. Roy. SOC. A 190, 102 (1947). [4] J. D. ESHELBY, J. appl. Phys. 26, 255 (1954); Solid State Phys. 3, 79 (1956).

460 S. N. MAITI and D. ROY: Lattice Dilatation in High-Concentration Solutions

151 D. T. KEATINC, J. Phys. Chem. Solids 29, 771 (1968). [6] P. H. DEDERICHS, P h p . Rev. B 4, 1041 (1971). 171 P.H.DEDERICHS, Solid State Phys. 27, 136 (1972); Proc. Internat. Summer School of

[8] N. KATO, Acta Cryst. A36, 763 (1980); A36, 770 (1980). [9] V. HOLY, phys. stat. sol. (b) 111, 341 (1982).

X-Ray Dynamical Theory and Topography, Limoges (France) 1975.

[lo] G. S. DURHAM and J. A. HAWRINS, J. chem. Phys. 19, 149 (1951). [ l l ] S. N. MAITI, D. ROY, and S. SENCUPTA, phys. stat. sol. (b) 99, 327 (1980). [I21 S. N. MAITI and D. ROY, Indian. J. pure appl. Phys. 19, 580 (1981). [13] S. N. MAITI and D. ROY, Proc. Nuclear Physics and Solid State Physics Symp., New Delhi

[14] P. BRAUER, Z. Naturf. ia, 372 (1952). [15] B. G. DICK and T. P. DAS, Phys. Rev. 127, 1053 (1962). [16] J. R. HARDY, J. Phys. Chem. Solids 29, 2009 (1968). 1171 T. B. DOUGLAS, J. chem. Phys. 46, 4571 (1966). [18] J. H. SIMPSON, Canad. J. Phys. 50, 729 (1972). [19] S. PAUL and S. SENGUPTA, phys. stat. sol. (b) 68, 703 (1975). [20] S. K. DAS, D. ROY, and S. SENOUPTA, J. Phys. F 7, 5 (1977). [21] C. KITTEL, Introduction to Solid State Physics, Wiley Eastern University Edition, 1974

[22] 0. D. SLACLE and H. A. MCKINSTRY, J. appl. Phys. 38,437 (1967). 1231 G. S ~ M O N S and H. WANC, Temperature Variation of Elastic Constants and Calculated

[24] G. A. ALERS, J. R. NEICHBOURS, and H. SATO, J. Phys. Chem. Solids 13, 40 (1960). [25] W. C. OVERTONE and J. GAFFNEY, Phys. Rev. 98, 969 (1955). [26] J. R. NEICHBOURS and G. A. ALERS, Phys. Rev. 111, 707 (1958). 1271 J. E. MAYER, 3. chem. Phys. 1, 270 (1933). [28] 0. D. SLACE and H. A. RICKINSTRY, J. appl. Phys. 38, 446 (1967). [29] E. T. TEATUM and N. 0. SNITH, J. phys. Chem. 61, 697 (1957). 1301 W. T. BARRET and W. E. WALLACE, J. Amer. Chem. SOC. i6, 336 (1954). [31] N. B. CHANH, J. chem. Phys. 61, 1428 (1964). [32] W. B. PEARSON, Handbook of Lattice Spacings and Structure of Metals, Pergamon Press,

[33] M. AHTEE, Ann. Acad. Sci. Fennicae AVI, No. 313, 1 (1969).

1980.

(p. 129).

Aggregate Properties, MIT Press, Cambridge (Mass.) 1971.

1958.

(Received June 23, 1983)

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Lattice Dilatation in High-Concentration Binary Solid Solutions