Surface structure, roughness, energy and texture of crystals

Surface structure, roughness, energy and texture of crystals

J. Phys. Chem. Solids Vol. 53, No. 1, pp. 167-174, 1992 Printed in Great Britain. SURFACE 0022.3697192 S5.M)+ 0.00 Pergamon Press plc STRUCTURE, AN...

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J. Phys. Chem. Solids Vol. 53, No. 1, pp. 167-174, 1992 Printed in Great Britain.

SURFACE

0022.3697192 S5.M)+ 0.00 Pergamon Press plc

STRUCTURE, AND TEXTURE

ROUGHNESS, ENERGY OF CRYSTALS

Z. A. MATYSINAand L. M. CHUPRINA Faculty of Physics State University, Dnepropetrovsk,

U.S.S.R.

and S. Yu. ZAGINAICHENKO Metallurgical Institute, Dnepropetrovsk,

U.S.S.R.

(Received 18 August 1989; accepted in revised form 1 May 1991) Abatra&-Tbe energy of the boundary between crystal and melt is calculated in a quasi-chemical approximation taking into consideration the correlation in the substitution of lattice sites by the atoms and the atomic interaction anisotropy. The molecular roughness of the interface is determined. The influence of correlation and anisotropy on the crystal surface roughness is ascertained. The numerical analysis for bismuth permitted one to obtain some information about the atomic interaction in solid, liquid and different phases. The energy of different lattice sites planes of monocrystals was calculated using the broken bonds method and its angular relation defined. The calculations are carried out for relative surface energies of lattice site planes which make up the various angles with plane (0001) of Be, Hf, Ti, Zr, Mg, Co, Zn and Cd monocrystals. A comparison between theoretical calculations and experimental data is carried out. Keyword: Surface energy, roughness of surface, crystal-melt boundary.

1. INTRODUCTION The crystal surface structure or interface between the crystal and its surroundings which deal with the formation of texture is important in crystal growth theory [l-18]. The surface of crystal may he a flat lattice site plane or it may have specific defects, and so he rough. For

some external planes with sites, the crystal surface may have a complex steplike structure depending on the type of lattice and the indices of this plane. The energy of the crystal depends on the structure of the latter and on the energies of the atomic interactions. Knowing the surface energy allows one to predict the type of free face arising in the process of monocrystal line growth, type of texture during polycrystal formation and the possible glide plane in case of deformation of a nondefective crystal, etc. Our work is aimed at the investigation of crystalmelt interface structure for substances with anisotropic interatomic interaction, and the calculation of crystal surface energy on the crystal-vacuum boundary. The results of theoretical investigation are compared with the known experimental data. Our paper consists of two parts. The first part deals with the results of a theoretical analysis of the crystal-melt interface. The crystalmelt boundary energy is calculated in the quasichemical approximation [19-241 taking into account PCS %/l-L

the correlation in the substitution of lattice sites by atoms and atomic interaction anisotropy. We should add that we used the simplified quasi-chemical method described in [19,20]. The roughness of the interface was assessed. The effect of correlation and anisotropy on the crystal surface roughness was established. The comparison of the theoretical calculation with experimental data for bismuth permitted one to obtain some information about the atomic interaction in solid, liquid and different phases [25-271. Such interfacial characteristics as the free energy @ and atomic roughness s are discussed. The @ and s values are determined for various values of the interaction energies between atoms: V, , V,, V,,, D,, vd. vu (“s” is the atom belonging to the solid phase, “solid” atoms; “1” is the atom belonging to the liquid phase, “liquid” atom). yk, u,~ (j, k = s, 1) are the energies of strong and weak bonds, respectively. The difference in interaction energies depends on the different distances between the atoms on the crystal lattice sites [28]. In the case of bismuth these distances are equal to r, = 3.11 A, r, = 3.47 A [29,30]. But as a whole, the crystal of bismuth is isotropic. Anisotropy of the interatomic interaction may be responsible for anisotropic properties of a crystal. Figures 2-4 show the original results obtained in this work. Curves 1, 2 (Fig. 4) are the only ones representing the results of calculation for roughness 167

2. A. MATYSINAet al.

168

by formulae described in papers [1,2]. They are given for a comparison of the new results with ones in the literature. The relationship Qi (X,) for different relations between the energies V, and V, is shown in Fig. 3, where X, is the atomic concentration in the solid phase. At the transition from V, = V, to V, < 0.968 V, and for constant values of the rest of the interaction energies one can observe the transition from an atomically rough (@ minimum corresponds to X, = 0.5) to an atomically smooth crystal-melt interface (@ minimum corresponds to X, = 0 or X, = 1) for a given substance. The variation in Vti can be attributed, for instance, to a small amount of impurity. The same di (X,) relations are given in Ref. 2, but they were obtained for the case of Vs,= I’,,, u, = I’,; curves 9 (X,) correspond to different substances. Using this calculation the author of [2] divided the substances into two groups: with an atomically smooth crystal-melt interface and with an atomically rough crystal-melt interface. The second part of our paper contains the calculation of crystal surface energy of close-packed hexagonal structure for various crystallographic planes on the crystal-vacuum interface. The results of our calculations are compared with the results of other authors and known experimental data. The energy of different lattice site planes of monocrystals was calculated using the broken bonds method and its angular relation defined. Its external dependence permitted one to distinguish the types of planes possible for the formation of free faces in monocrystals. These calculations allowed one to explain the formation of ribbed and cubical textures in transformer steel. The calculations were carried out for the relative surface energy of lattice site planes making up the different angles with plane (0001) of Be, Hf, Ti, Zr, Mg, Co,, Zn and Cd monocrystals which have the hexagonal close-packed lattice with axial ratio c/a # 1.63. It was found that an increase in the c/a value led to an increase in the surface energy. The results of the calculations agree quantitatively with the known experimental data and the theoretical computations of other authors. 2. MOLECULAR ROUGHNESS OF CRYSTAL FACE The (100) face of the rhombohedral crystal lattice characteristic for the metals of the bismuth group was examined. The crystal is assumed to be in equilibrium with the melt. The existence of an interphase transition zone between the crystal and the melt is supposed. It is assumed that atomic density and position of the atoms in the liquid phase near the boundary are the same as in the crystal. The grid-type, one-layer, model is used. The atoms belonging to the solid phase shall be named 4 atoms, and those belonging to the liquid phase atoms.

l 0e0a0a0 0* 0

o-

l*0, T*-aO ’0

& ‘-poe ’

oa6.0

l o*o*o~o~

Fig. 1. Scheme of atom location in the crystal lattice sites: (a) solid lines correspond to strong interatomic bonds; (b) location of strong and weak bonds in (100) plane.

The correlation is taken into account by the introduction of the aposteriori probabilities PM(k,j = d, G) of the nearest lattice sites pair replacement with atoms dd, de, &. Atomic interaction anisotropy is allowed for by the introduction of two types of lattice sites assuming that for the first type of sites the strong bond interacts with atoms located in front of, at the bottom and to the right of a given atom, the bond with the remaining atoms being weak (Fig. 1). The energies of the strong and weak interatomic bonds shall be labelled Vkj and vkj, respectively. Let Pkj and P$ denote the a posteriori probabilities for strong and weak bond. The calculation of the contiguration energy of the interphase zone in quasi-chemical approximation was carried out using the following formula

E,=~Nx,(v,,+o,,)+tNx,(V,+v,,) +fN(V,,+o,,)-fNtl(3W+w) + NW’,,

+ wPk),

(1)

where N is the total number of atoms in the interphase zone, x,, x, are the atomic concentrations in the solid and liquid phase in this layer, rl is the long-range atomic order in the distribution of solid and liquid atoms, and w=2v,-

v,,-

V{C,

w =2v,-v&,-v,,.

(2)

Far the smooth crystal face (absence of interphase zone) the conQuration energy is equal to E,, = $ N( I’,, + v,,).

(3)

Surface energy and roughness of crystals The difference in energies (1) and (2) determines the change in the system’s configuration energy, caused by the transition from a smooth to rough crystal surface

AE=4N[x,(V,,+o,,)+x~(V,+vtrll -fNtl(3W+w)+N(WP,,+wPk).

(4)

The last component of this formula characterizes the roughness which is determined as follows: S =4AEJN(W+

nearest atoms in the first coordination sphere for strong and weak bonds, respectively (Fig. lb). The pair interaction energies were determined by the values of the melting heat L and the evaporation heat H: 3( V,, + v,,) ‘y 2H,

The energetic parameters W, w were, using eqn (9), obtained as follows:

(5)

w=-(2a-1)3 is absent

the roughness

So = 4 AE;/N( W + w) =4x,x,.

is

(6)

The change in crystal face free energy caused by transition from its smooth surface to the rough one is equal to -;$T(3W+w)

+ L (WP,,+ kT

WP:,) - $ (PC’) d In P(l) d

+ PC) In PC) + Pi*) In Pa) + Pi*) In Pi*) + P,. In P,. + Pit In Plc + P,( In Pd + PC, In P!, + Pi, In Pi, + Pit In PiI + P> In P> + Pi, In Pi,),

(7)

where Plf), Pf)

are the a priori probabilities of replacing the first and second types of sites with atoms. Our formulae lead to Jackson’s equation for the full energy change,

sT=2x6(l

(9)

3 1+r

= q WP, + wP:,)/( w + w).

@ =AF,NkT=

3(2 Vtt - V,, - u,,) = 2L.

w=_@_*)!y+2K

w)

When the correlation equal to

169

H-L

2 +5

-

H

I+$

The parameter a = V,/Vcc characterizes the differences between the interaction energies of atomic pairs d and of pairs M. A part of the calculations was carried out using the value of the parameters similar to the case of equilibrium for a faceted crystal of bismuth with its melt. It is known that a Bi-crystal has smooth faces. In the case of bismuth, the strong interatomic bond interacts with the neighbor at a distance r, = 3.11 A, the weak bond-at a distance r, = 3.47 A. Anisotropy with the parameter [ = 0.9 corresponds to the atomic interaction potential rp N I/r(n = 1). The numerical calculations showed that accounting for correlation and anisotropy led to a decrease in the system’s free energy. The calculated results for changes in free energy @ for cases with accounting for correlation and anisotropy and without accounting for the latter are shown in Fig. 2. The calculation by eqn (8) at V,, = 0.98 V, permits one to obtain the dependence of @ on X, which is characteristic of a smooth crystal-melt interface. Taking account of the correlation leads to the dependence of @ on x0 which separates the region of atomic-smooth and rough interface. Taking account of correlation and the anisotropy C = 0.9 permits one to obtain the dependence 9 on x,, which is characteristic of the atomic-rough interface.

-x,)W/kT+x,lnx,

+(l -x,)ln(l

-x,)

(8)

in the particular case when correlation and anisotropy are absent. Numerical analyses of these equations were carried out for cases when the interaction energies of the liquid atoms and also the liquid and solid atoms are isotropic, i.e. V, = v,,, V, = v,. Anisotropy was taken into consideration in the case of solid atom interactions by the parameter C = v,/V, (the smaller the c parameter, the more manifest is the anisotropy). Anisotropy was calculated with the selected atomic interaction potential of cp w l/r”, with (v,,/V,) = (rl/r2)” where r,, r2 are the distances between the

Fig. 2. Dependence of the change in free energy on the ‘Solid’atoms concentration at V, = 0.98 V&. (1) Solution without taking account of correlation and anisotropy, cornputation according to eq (8). (2) Solution with taking account of correlation. (3) Solution with taking account of correlation and anisotropy { = 0.9.

2. A.

170

MANSINA

a

f?t id.

Also, the calculations showed that the transition to the smooth crystal-melt interface for bismuth took place for the solution without correlation at a = 0.9912 (when 8 = XT/(W + w) < 1.24), but for the solution with correlation at a < 0.974, t; = 1 (8 G 0.722) and at a < 0.970, [ = 0.9 (@ G 0.678). Taking account of correlation and anisotropy requires the reduction of Vd in comparison with V,, for agreement of the theoretical and experimental results. The calculational results for molecular interface roughness are shown in Fig. 4. In the case of a = 0.9 and [ = 0.9 for bismuth at the melting temperature we obtain 8 = 0.33. To this value of 8 corresponds the small roughness 4 = 0.012 which agrees with the experimental data for the smooth crystal-melt interface. The value 8 = 0.33 corresponds to the small concentration x, = 0.0036, which also points to a smooth crystal-melt interface. In this case the order parameter rl and the apriori probabilities Pi’), P&) are equal to

0.4 r

Fig. 3. Dependence of change in free energy on ‘solid’ atoms concentration for solution with taking account of correlation and anisotropy [ = 0.9: (1) a = 0.88, (2) a = 0.92; (3) a = 0.95; (4) a = 0.968; (5) a = 1.0.

tl = 0.0018,

Pi’) = 0.0054,

and

Pi*) = 0.0018,

pointing to the predominant distribution of solid atoms on the first type of sites. The a posteriori probabilities and the correlation parameters became The change in free energy @depends on the V,, ,V,, equal to and Vd energies. A decrease in the energy V,,at P,, = 0.319 x 10-3, Pi,= 0.107 x 10-3 (P,,>P:,), constant values of V,, and V,,leads to the change in Pee= 0.9931, P;c=o.9929 CPU> Pit), @ values which is usual for transition from a rough to a smooth crystal-melt interface (Fig. 3). On the P,(=o.490 x 10-2, P:f=0.512 x 10-2 (P,l
8 = 2&T/(W + WI.(1) Onsager’s solution [l]; (2) Jackson’s solution [2]; (3) Retheh soltition for V, = V, [l] and (4) solution with taking accourktof correlation and anisotropy C = 0.9 at V,= 0.9 V,.

v, < 0.97 If,,

v,, = 0.9v,, ,

v&-u&

V/l= UN*

(11)

Surface energy and roughness of crystals

171

Table 1. Relative surface energy NllIllber (hkif)

a0

[l/$&l

Formula

y=O.O88

1

1 1.053 1.098 1.134 1.162 1.181 1.182 1.193 1.190 1.173 1.142 1.124 1.068 1.093 1.097 1.097 1.093 1.081 1.061

+y)lsina +eosa

(loT4) (10T3) (2025) (1oT2) (2OZ3) (3034) (lofl) (3032)

25.24 32.15 37.03 43.31 51.50 54.74 62.06 70.53 75 75.15 80 ::

(2021) -

(13 + llY)lJ137(1 + Y) (3/2J%in a + [(l - y)/3(1 + yl@os y

(GO) !

3/&p

These energy parameter values correspond to the smooth crystal-melt interface. This means that agreement of theoretical and experimental results corresponds to the weaker atomic interaction in different phases as compared with the atomic interaction in the liquid phase. 3. SURFACE ENERGY OF CRYSTALS The configurational part of the surface energy for lattice site planes of different types can be estimated using the broken bonds model. It can be determined using the sum of the atomic pair interaction energies in one or two coordinate spheres which are truncated with the free face of the crystal. The surface energy atW estimation for different crystal faces in close-packed hexagonal structures has been carried out.

y=o

6 = qW&4o,,

0

11.54 22.21 28.56 33.15 39.23 47.43 58.52 67.79 72.98 83.02 90

Formula

t@Jw

(11216) (llZ8) (1126) (1125) (1124) (1123) (1122) (33W (1 lZ1) (55m2) (1 lZ0,

*

(12)

for two particular cases: (1) lattice site plane families (~~~) which are perpendicular to the (T2TO) plane; (2) lattice site plane families (hkil) which are perpendicular to the (1TOO)plane. a is the angle between the observed plane and (0001) plane. y is the coefficient, characterizing the relative effectiveness of broken bonds for the second coordinate sphere in comparison with the first sphere. The value y = 0 means the inclusion of the atomic interaction in the first coordinate sphere only. From Tables 1 and 2 it follows that there is an angular dependence of the surface energy 6 = 6(a).

Number (hkil)

1.058

1.108 1.149 1.182 1.205 1.206 1.223 1.224 1.213 1.176 1.155 1.093 1.111 1.111 1.111 1.103 1.086 1.061

In Tables 1 and 2 the calculational results are presented for the relative surface energy

Table 2. Relative surface energy a0

1

($0 + 7y)~3~(1+ (13 + 9Y)/J‘wl+

Y) Y)

y =0.088

y =o

1 1.130 1.210 1.237 1.248 1.249 1.241 I.190 1.229 1.237 1.223 1.192

I 1.143 1.234 1.269 1.284 1.291 1.278 1.219 I.260 1.269 1.256 1.225

2. A. MATYSINA et al.

172

The character of the S(a) dependence for the first family of planes allows one to suggest that the (OOOl) plane can be energetically the most effective crystal face. Then the (lOT1) and (lOTO) planes are almost equally probable. Allowance for atomic interaction only in the first coordinate sphere (y = 0) increases, as in the case of crystals with a body-centered cubic structure, the value of 6 (a), retaining at the same time the dependence trend. For the second family of planes the energy 6(a) increases considerably. This is why it can be asserted that the formation of crystals, the free faces of which would be the planes of the considered family with angles a > 0 is hardly probable. It is known that monocrystals of many substances with close-packed hexagonal lattices, as a rule, have the (0001) or (lOTO) free faces for which the surface energy is minimal, and do not have the (1120) free face with a high surface energy. Knowing the surface energy in different lattice site planes of the crystals can lead to prediction of the possible glide plane under crystal deformation if the mutual orientations of the directions for the external force and the crystal’s basic vectors are known. Thus, for example, if the external force on the crystal is perpendicular to the (1270) plane, then according to the calculations it may be assumed that the glide plane is represented by (OOOl), (lOT1) or (1010) for which the surface energy is small, and not by the (lOT3) plane which has the maximal surface energy. If the external force is perpendicular to the (1100) plane, then the glide plane can only be the (0001) plane. Estimation of the surface energy for different lattice site planes can lead to a determination of the value of the crystal surface free boundary energy, which helps to study, understand and to predict the crystal growth kinetics. The surface energy plays an important role in the formation of a specific polycrystalline texture. It is known, for instance, that in the process of secondary recrystallization in ferrosilicon, the lower surface energy of planes (1 lo), (100) provides the predominant growth of the grains which are bounded by these planes, and so leads to the formation of a ribbed or cubical texture. It was found that an increase in silicon concentration decreases the surface energy. Atomic adsorption on the surface, just as an additive in the volume, also exert a pronounced effect on the surface energy value. In the case of close-packed hexagonal lattices with the a and c parameters such that c/a # 1.63, the atomic interaction was taken account of in three coordinate spheres, the radii of which were equal to rl =

a,

r~=+-m

r3=+mm

(13)

Table 3. The relative surface energy for c/a > 1.63 Plane (10T10) (lOT6) (10T4) (10T3) (2075) (1oT2) (2023) (3034) (loT1) (3032) (2021) (40W (1oTo)

*WI l2 + 15(y,+ yz)l/ 3(~, + n),/m l2 + 9(Y,+ Y2)1/ J 3(Y,+ n)~G@211+ 3(Y,+ Y2)1@(YIf Y*)JGD-l4 + 9(y, + Y*)l/J5(Y,+ Y,>J4W l8 + 15(~,+ Y&&Y, + n)
for c/a > 1.63 and

(14) for c/a<1.63.

At c/a=1.63

we have rl=r2=a,

r,=aJi.

The results of calculations

for surface energy with

c/a > 1.63 and c/a < 1.63, are shown in Tables 3 and

4, respectively. The coefficients y, and y2 characterize the relative efficiency of broken bonds for the second and third coordinate spheres in comparison with the first sphere. The crystallographic parameters [31] for such metals as Be, Hf, Ti, Zr, Mg, Co,, Zn and Cd for the determination of the relative surface energy and its angular dependence are shown in Table 5. The results of the calculations are presented in Table 6. The S(a) dependence is of the same type for all the metals: two maxima and one minimum. An increase in the c/a ratio shifts the position of the first maximum to the

Table 4. The relative surface energy for c/a < 1.63 Plane (1oT10) (lOT6) (10T4) (1oT3) (2025) (1oT2) (2023) (3034) (1oT1) (3032) (2021) (40W (1oTo)

173

Surface energy and roughness of crystals Table 5. The crystallographic parameters of the elements Element

a (nm)

(nG

cla

Ete

0.2286

0.3684

1.5678

0.2226

Yif Zr Mg Co, Zn ccl

0.3197 0.2951 0.3223 0.3202 0.2505 0.2664 0.2960

0.5057 0.4679 0.5147 0.5199 0.4090 0.4946 0.5630

1.5856 1.5818 1.5970 1.6235 1.6325 1.8566 1.9020

0.2894 0.3131 0.3176 0.3190 025048 0.2664 0.2960

r3

Table 6. The values S,,,,/a(depree) Element (loTlo) (lOT6) (1oT4) (1oT3) (2025) (1oT2) (2023) (3054) (loTI) (3032) (2021) (4041) (loTo)

(nm)

YI

Y2

0.2286

0.3191

0.83

0.0804

0.2951 0.3197 0.3223 0.3202 0.2505 0.2912 0.3290

0.4133 0.4775 0.4525 0.4520 0.3542 0.3947 0.4430

0.87 0.86 0.90 0.97 1.00 0.54 0.47

0.0825 0.0821 0.0839 0.0872 0.0884 0.0638 0.0600

of indicated planes

Be 1.085

Hf 1.088

Ti 1.089

Zr 1.093

Mg 1.100

Co, 1.103

Zn 1.195

Cd 1.219

10.3 1.121

10.4 1.126

10.4 1.128

10.5 1.133

10.6 1.145

10.7 1.149

12.1 1.290

12.4 1.328

16.8 1.144

16.9 1.152

17.0 1.154

17.1 1.160

17.4 1.176

17.4 1.181

19.7 1.371

20.1 1.421

24.3 1.145

24.5 1.137

24.6 1.159

24.8 1.167

25.1 1.186

25.2 1.192

28.2 1.416

28.8 1.475

31.1 1.142

31.3 1.151

31.4 1.154

31.6 1.162

32.0 1.182

32.1 1.189

35.5 1.434

35.9 1.121

36.2 1.131

36.2 1.134

36.4 1.144

36.9 1.165

37.0 1.173

40.6 1.440

36.2 1.498 41.3 1.510

42.2 1.084

42.4 1.096

42.5 1.098

42.7 1.109

43.2 1.134

43.3 1.142

47.0 1.436

47.7 1.512

50.4 1.063

50.6 1.075

50.7 1.078

50.9 1.089

51.3 1.114

55.01.427

55.7 1.505

53.6 1.003

53.9 1.016

53.9 1.019

54.1 1.031

54.6 1.058

51.5 1.123 54.7 1.068

58.1 1.392

58.7 1.475

61.6 1.026

61.3 1.039

61.4 1.072

61.9 1.083

62.1 1.093

65.0 1.427

69.8 1.029 74.6 1.018

69.9 1.042 74.7 1.032 82.2 1.003

69.7 1.045 74.7 1.035 82.2 1.007

61.5 1.054 70.1 1.058 74.8 1.048

70.4 1.086 75.1 1.078

70.5 1.097 75.2 1.088

72.3 1.436 76.9 1.434

65.6 1.513 73.1 1.522 77.2 1.522

82.3 1.019

82.4 1.050

82.5 1.061

83.4 1.413

I .502

90

X-90

82.1 0.989

-

90

-

__

90

-

90

90

side of the bigger angles u. For all the metals, the minimum of the 6 (a) dependence occurs at the (lOT1) plane. Besides, an increase in the c/a ratio leads to an increase in the 6(a) value. For Mg, Co,, Zn and Cd, the surface energy values for the basal (0001) plane

~

-

90

83.6

are the smallest, which is in agreement with the known experimental data. For Ti, Zn, Mg and Cd, the surface energy values for different lattice site planes which qualitatively agree with the experimental data and with the results

Table 7. The experimental and theoretical values of 6,, a(nm) Metal Ti

Zn

c(nm) 0.2951 0.4679

0.2664 0.4946

c/a 1.5856

1.8586

Plane

Surface energy

&i/J exp.

(mJ m-‘)

(0001)

924

[151 -

(10T2)

1750

1.894

1.134

(loT1)

1150

1.245

1.019

(0001)

600

(loTo) (loT1)

-

I

1

1000

1.67

1.413

1000

1.67

1.392

Z. A. MATYSINA et al,

174

10. Van der Eerden J. P., J. Appl. Phys. 48, 2124 (1977). 11. Hartman P. and Bennema P. J., J. Crysr. Growth, 42,

Table 8. The values of Soo,,, ours and from [16] Metal

Plane

Mg Zn

:z;;

Cd

of other theo~tica~ Tables 7 and 8.

C&0 (erg cm-3

&I

WI

hw,

593 1031 1247 1484

1.74 1.19

I.05 1.41

686 1320

1.92

1.50

~lculations

are presented

in

PEFERENCJB 1. Barton W., Cabrera N. and Frank F., Phil. Trans. A243, 299 (1951). 2. Jackson K. A., in Liquid Metals and Solidifiation, p. 174. American Society for Metals, Cleveland, Ohio (1958). 3. Weeks J. D. and Gilmer G. H., J. Chem. Phys. 63,3136 (1975). 4. Klupsch Th, Phys. Stat. Sol. BtO9, 539 (1982). 5. Yumoto H.. Kaneko T. and Hasiuuti R. R. .I. Crvstal Growth 82,459 (1987). 1 ’ ’ 6. Tyomkin D. E., in Mekhanizm i Kinetika Kristallizatsii. Nauka i tekhnika. Minsk (1964). 7. Chemov A. A. and Tvomkin D. E.. Izvestiva no khimii. Bolg. Akad. Naak. Ii, 643 (1978): - 8. Van der Eerden J., Bennema P. and Cherepanova T. A., Progr. Crystal Growth Charaet. 1, 219 (1978). 9. Ecin V. 0. and Tarabaev L. P., Phys. Stat. Sol. A90,425 (1985).

145 (1980). 12. Box&sent A. and Abraham F. F., J. Chem. Phys. 74, 1306 (1981). 13. Hartman P. and Perdok W. G., Acta Cryst. 49, 521 (1955). 14. Sundquist B. E., Actu Me?. 12, 67 (1964). 15. The Solid-Gas Interfuce, Vol. 1 (Edited by E. Alison Flood). Marcel Dekker, New York (1967). 16. Williams F. L. and Nason D., Surf. Sci. 45,377 (1974). 17. Cracker A. G., Doreghan M. and Ingle K. W., Phil. Msg. A41, 21 (1980). 18. Heyraud J. C. and Metois J. J., Surf. Sci. 128,334 (1983). 19. Fowler R. H. and Guggenheim E. A., Statistical Thermodynamics. Cambridge (1952). 20. Guggenheim E. A., Mixtures. Oxford (1952). 21. Yang C. N., J. Chem. Phys. 13, 66 (1945). 22. Li Y. Y., J. Chem. Phys. 17, 447 (1949). 23. Kikuchi R., Phys. Rev: 81, 988 (1951). 24. Kikuchi R.. J. Chem. Phvs. 19. 1230 11951). 25. Matysina Z. A., Ovrutski A. M. and ehup’;ina L. M., I.v. vuzovSSSR. Fiziku 10, 124 (1985); Dep. VZPUTI. 3584-3585 (1985). 26. Matysina Z. A., Ovrutskii A. M. and Chuprina L. M., Phys. Star. Sol. (a) 92, K93 (1985). 27. Zaginaichenko S. Yu, Matysina Z. A. and Ryxhkov V. I., Atomy na poverkhnosti i v obyome kristalla. Dep. VINITI 6202-B87 Moskva (1987). 28. Physical Metallurgy (Edited by R. W. Cahn and P. Haasen). North-Holland. Amsterdam (1983). 29. Cucka P. &rd Barett C. S., Acta Cryst. l$ 865 (1962). 30. Krebs H., Grundzt?ge o’er Anorg~ische~ Kris~a~ic~rn~, F. Enke, Stuttgart (1968). 31. Svoistua Elemenrov (Reference book) (Edited by M. E. D&s). Metallurgiya, Moskow (1985).