Mean-square atomic displacements in f.c.c. crystals

1. Phys. Chum. Solids Vol. 41, pp. 1163-l 170 Pergamon Press Lfd.. MO. Printed in Great Britain

MEAN-SQUARE ATOMIC DISPLACEMENTS IN f.c.c. CRYSTALS A. P. G. KUTTY and S. N. VAIDYA Chemistry Division, Bhabha Atomic Research Centre, Bombay 4OOOt35,India (Received 22 October 1979: accepted in revised form 2 May 1980) Abstract-The mean-square displacements of volume atoms, surface atoms and atoms near a vacancy are calculated for f.c.c. metals and rare gas solids,using a simpleseries expansion of the Wallis formula. We employ a model with rotationally invariant bond bending forces, unlike the de Launay model used by Masri and Dobrzynski. For atoms near a vacancy and for rare gas solids this method has been applied for the first time. The calculations are done to the R4 term in the expansion.The inclusion of the R’ term together with temperature-dependent elastic constants give results which are in better agreement with experiment. For rare gas solids Lindemann ratios calculated by this method agree closely with the values obtained from elaborate computations as well as those deduced from entropy data. The present calculation also supports the view that the melting of solids is initiated at the crystal surfaces and in the neighbourhood of vacancies. 1. INTRODUCTION

The mean-square displacements (MSD) of volume and surface atoms in many crystals have been calculated from lattice dynamics[ I-31. Masri and Dobrzynski[4] adopted a well-known formula due to Wallis[5] and used Taylor series expansion of the dynamical matrix to calculate MSD in monatomic f.c.c. crystals. Singh and Sharma[6] have calculated the MSD of volume atoms in f.c.c. lattices using lattice dynamics. The MSD calculated by them are higher than those deduced from the X-ray Debye-Wailer factor. On the other hand, the MSD obtained by Masri and Dobrzynski, using the first two terms of the Taylor expansion are considerably lower than the experimental values. Lattice dynamics calculations involve elaborate computations, while the method used by Masri and Dobrzynski is simple enough and easily adaptable for atoms situated on the surfaces or near a vacancy. However, the de Launay model[7] used by Masri and Dobrzynski is not rotationally invariant and can lead to erroneous results in the surface calculations[l]. The object of this paper is (i) to calculate the MSD from the Wallis formula using rotationally invariant model (ii) to investigate the effect of including the next higher order term in the Taylor expansion (iii) to estimate the effect of temperature variation of elastic constants and (iv) to calculate MSD in rare gas solids. Rare gas solids provide a critical test for the present method, since the series has good convergence in the low temperature regime[9]. The existing MSD calculations for the rare gas solids are few and the experimental data scanty. An accurate calculation should therefore prove useful when experimental data become available. It is believed that the melting of solids is initiated at the external surfaces, grain boundaries[lO], and vacancies, which are the regions of low potential energy in a solid, On the basis of molecular dynamics calculations of two-dimensional lattices Broughton and Woodcock [ 1l] have conjectured that melting starts at the surface of the solid. The bubble raft experiment [12] and the arguments advanced by Cahn[l3] suggest that the nuclei of the liquid phase are formed around vacancies. Thus it may be expected that the MSD of the surface atoms and IPCS VOL. 41. No. II-A

atoms near a vacancy would be much larger than that of the bulk atom. In view of this and for comparison with the low energy electron diffraction (LEED) data, we calculate the MSD of surface atoms and atoms near vacancies.

2. THEORY

The MSD of an atom in a monatomic solid is given by the formula [S] (Uzi) = s[D-1’2 COth (ftD”‘/2kT)],i.ni

(I)

where n is the atom index, i =x, y or z denote the Cartesian components of the displacement vector, M is the mass of the atom, D is the dynamical matrix of the crystal, h is Planck’s constant divided by 27r, k is the Boltzmann constant and T the temperature in Kelvin. The matrix D can be split into a diagonal part, d, and an off-diagonal part, R: D=d+R.

(2)

On substituting for D on the r.h.s. of eqn (1) and expanding in Taylor series one gets[9,14], (~2,;)= &(

]fo(d)l.i..i + i

(R2),,,i[f2(d)lni.ni

+$ (R4h,ni [f4(4lni,ni + . . *}. The matrices fo, f2 and f4 are defined as follows:

1163

fo(d) = d-‘%(x) f2(d) = EM h(d)

= $$(x) x = (fid”2/2kT).

A. P. G. KUTTY and S. N. VAIDYA

1164

The matrices cpare of the form [ 151

and wo. By expanding the displacements ui. vi and Wiin eqn (5) in Taylor series about uo, v. and w. and comparing with the continuum wave equation, we obtain the relationships between the force constants and elastic constants:

p,,(x) = coth x

cpz(x)= 3 coth x t 3x sinh-’ x t 2x2 cash x sinh-’ x &(x) = 105coth x t 105x sinh-’ x t 90x* cash x sinh-3 x

a =$3c&d

t 20x’ sinh-’ x(3 coth’ x - 1) t 8x“ cash x sinhm3x(3 coth’ x - 2).

C12)

p=$c,,-2C44)

For computational purposes d is expressed in units of 104dyne/cm and M in amu. The MSD (dropping the atom label n) in (A)’ units is then given by

y = $‘l-

Cd

(6)

The elements of the matrices appearing in eqn (4)are obtained from the equations of motion. For a bulk atom one gets

R’B (u2) = B%(A) + jp2b4)

(4) with A = (2%.456/7’)(d/M)“* and B = 0.04091(dM)-“2. In this expression d, R* and R4 represent diagonal elements of the respective matrices. We consider central force interactions extending upto second nearest neighbour and angular forces involving three nearest neighbours forming a right angle at 0 (Fig. 1). The angular force is calculated in rotationally invariant form[l6] by defining potential energy arising from change in the bond angle ho as E, = ;(AQ)*

and the corresponding restoring force along OX as

where y is the angular force constant. For example, the restoring force along OX due to bond angle 203 is

d=(4a+2/3+8y) R2= (4a2t2/32t16~2)

and

R4 = (R’)*.

(7)

When atom 0 is on (100) surface the terms involving its interactions with atoms 1, 4, 5, 7 and I3 (Fig. 1) are dropped from the equations of motion. The matrix elements of d and R for motions parallel and perpendicular to the surface are derived from these equations. For (110) and (II I) surfaces the equations of motion in appropriate coordinate system were derived from eqn (5) by coordinate transformations. For studying the effect of vacancy on the MSD of the central atom, we suppose that site 5 is vacant and drop the terms involving atom 5. Matrix elements for the various cases are given in Table I. We have not taken into account changes in the force constants near a vacancy and at the surface of the crystal, since reliable estimates for them are not available. 3. RESULTS AND DISCUSSION

F?03(x) = y(-2Uo t u* t u, - w* t w3) where uir v, and w, are the components of the displacement of atom ‘i’ along OX, OY and 02 respectively. In our calculation twelve angles of this type are included. The model used here is similar to that of Clark et a!.[171 for b.c.c. lattices. However we have not included the twentyfour angles of second type such as 0, 5, 15, because it introduces an additional parameter which cannot be determined independently. The equation of motion of the central atom along OX is of the form M$=

3.1 Metals The MSD in a number of f.c.c. metals were computed from eqn (4) and the matrix elements given in Table I.

-(4at2/?t8y)uat(;ty)i$ui

where a and p are the first and second neighbour central force constants. Similar equations can be obtained for o0

18 Fig.

I

Mean-square

1165

atomic displacements in f.c.c. crystals

Table 1. Diagonal elements of the matrices d and R2

Table 2. Elastic constants and other parameters used in the calculations Elastic (UP

N

Metal

(amu)

4

Tin (K)

Tsmperatme

constants~')

0f

dYne/&

(13 Fj

B (m)

Cl,

CM

CC0

107.87

1234

0.4s

1.22

0.92

C.496

Al

26.98

933

0.4W

1.05

0.63

O.Z?

Au

196.91

1336

0.407

l.e5

1.59

0.426

Cu

63.55

1356

0.362

1.69

1.22

0.754

Ni

58.71

1728

0.352

2.48

1.53

1.15

Pb

207.19

600

0.491

0.491

0.408

3.146

Pa

106.40

1825

5.398

2.27

1.76

0.717

Pt

195.09

2042

0.392

3.457

2.507

0.765

(a)

ad.

coeViaiwt4

dc8ti~400n3tar.t3k

d lnCll d InCa, d InC+r dT dT dT

-2.8

-2.1

-4.2

-l.R

-1.5

-3.3

-2.4

-1.6

-3.6

CiaJ

The elastic constants were taken from[l8] and are listed in Table 2. The results are shown in Figs. 2-9. Calculations were also performed using three-parameter de Launay model (41 for comparison. Bulk atom Masri and Dobrzynski[4] in their calculation used series expansion to R* term and found that the values of MSD for bulk atom, (r&, were considerably lower than the experimental values. While our calculations to R* term yield results essentially in agreement with Masri and Dobrzynski, the inclusion of the R’ term leads to a substantial increase in (u*)~ at high temperatures. This can be seen from the Figs. 2-4 for Cu, Ag and Au where results obtained by truncating the series at R*, are also included for comparison. It is evident that for calculation of MSD at finite temperatures, (u*)T, elastic constants appropriate to that temperature should be employed. For Cu, Ag and Au, elastic constant data from 300 to 800 K are available [ 181. These were used to calculate (u*)~ to R’ term. This

modification yields somewhat higher values of (I& (Figs. 24), which are in closer agreement with the experimental results. The above procedure corresponds to quasiharmonic of lattice dynamics at finite approximation temperature[l9] and is expected to yield better results for other substances as well. The values of (u’), for Al calculated in R* approximation are considerably lower than the experimental results. The inclusion of the R4 term removes some of the discrepancy. Apparently the discrepancy at high temperatures is due to the neglect of temperature variation of elastic constants. We notice a large discrepancy between the calculated and experimental values for Ni and Pb particularly at high temperatures (Figs. 6 and 7). In Ni the situation is complicated by the presence of a magnetic transition at 631 K. It has not been possible to locate the source of large discrepancy in the case of Pb[3]. For Pt and Pd experimental results are not available for comparison with calculated values.

A. P. G. KUTTY and S. N. VAIDYA

1166

GOLD /.y

4b !

“,I”

3-

i

(loo!,

Y /c

2

3

l.O-/

VAC -

(IOO)~,

0.12 f

/R4

11 L ,’

TEMPERATURE

/’

/ K

0.03

Fig. 2. The mean-square atomic displacement (u’) as a function 0 200 400 Curves 600 800 1000 1400 to of temperature for copper. labelled R2 and1200 R4 refer approximations IO RZ and R4 terms in eqn (4). The dashedcurve represents values calculated with temperature dependent elastic constants. Ex~rimentai data: c1, Owen E. A. and Williams R. W., Proc. Roy. S’OC.(~~~doff~ Al88,%9 (1947): X, FIinn P. A., McManus G. M. and Rayne J. A., Phys. Rev. 123,909 (1961). The variation of (u”)~/(u& with temperature for the surface atoms and for vibration along 05 (vacancy at 5) is shown in the upper diagram.

3

0

“,”

/c “7 v

VAC 6 1.0 -

1000

1200

1400

j-----Ez----11501~

31

*

VAC IlOO),,

I,:

wq,

,L .T

R4

0

5

I

200

I

400

I

I

600 600 TEMPERATURE

I

1000 /K

I

1200

I

Fig. 3. Variation of (I?) and (u2)s@& with temperature for silver. Experimental data: Cl, Haworth C. W., Phi!. Mug. 5, 1229 (19601; A, Simerska M., Acta Crysf. 14, 1259 (l%l). See Fig. 2 for details.

Surface

800 800 TEMPERATURE/K

v \ (loo!,

0

400

Fig. 4. Variation of (u’) and (~~)~/(~~)~ with tem~ratu~ for gold, Experimental data: 0, Owen E. A.and Williams R. W., Pruc. Roy. Sot. (London) A188,509 (1947).

x

SILVER

200

atom

The MSD of the surface atoms were calculated from the relevant matrix elements given in Table 1. The variation of the ratio Rs = (~*)~{~~)~ with tem~rature is shown in Fiis. 2-9. The numerical values of (u*)s can be obtained from Rs and the corresponding values of (u’)~ shown in the lower part of the diagrams. (Note that (ar2)*= 3{u&.) For vibrations parallel to the surface the enhancement in the amplitude is small, as seen from the curves for the representative case (100). The ratio & is large when the atoms vibrate normal to the surface and is nearly same for the three surfaces considered. Since the

250

400 TEMPERATURE/K

600

600

Fig. 5. Variation of (u*) and (u*)~/(u~)~ with temperature for aluminium. Experimental data: A, Owen E. A. and Williams R. W., Proc. Roy. Sot. (London) Al88, 509 (1947); q, Chipman D. R., J. Appl. Phys. 31, 2012 (l%O); X, Flinn P. A. and McManus C. M.. Phys. Rec. 132, 2458 (1%3). See Fig. 2 for details.

differences are too small, the ratio Rs for (100) alone are shown in the figures. R, rises steeply with temperature and attains a saturation value, a feature first reported for chromium by Wallis and Cheng[20]. Dehye-Wailer factor for surface atoms is deduced from the LEED experiments wherein the intensity of the diffratted beam for various electron energies is measured over a temperature range[21]. The surface debye temperature is found to be constant over the temperature range used in the experiments. Therefore the experimentally determined MSD ratios should correspond to the saturation value obtained in our calculations. The experimental and calculated values are compared in Table 3. We find that calculated values are generally

Mean-square

1167

atomic displacements in f.c.c. crystals I

1

I

NICKEL

PLATINUM (loo)1

ClOOl~ x

3-

“3” Y K

*b

VAC

*f=+=-

3

VAC

(lOOI,, 1’

dOOI,,

* 0,06F

0 I 0

I

200

1

400

600

600

I

I

1000

TEMPERATURE

1200

1400

600

400

I

1

1600

/K

1600

1200

TEMPERATURE/K

Fig. 8. Variation

of (u2) and (u~)~/(u~)~ with temperature platinum. See Fig. 2 for details.

Fig. 6. Variation of (u2) and (u’).,-/(v& with temperature for nickel. Experimental data: 0, Simerska M., Czech. J. Phys. 812, 858 (1%2); A, Wilson R. H., Skelton E. F. and Katz J. L., Acta Cryst. 21, 635 (1966). See Fig. 2 for details.

for

PALLAW

LEAD (loo!,

VAC

II

200

400

600

800

1000

1200

1400

1600

TEMPERATURE/K

Fig. 9. Variation

I

0

200

I

400

I

600

TEMPERATURE/K

Fig. 7. Variation of (u*) and (u’)~/(u:)~ with temperature for lead. Experimental data: 0, Killean R. C. C. and Lisher E. J., 1. Phys. FS, 1107 (1975). See Fig. 2 for details.

higher than those based on lattice dynamics[5] and are in better agreement with the experiments. The de Launay model, on the other hand, overestimates the surface atom amplitudes (u’),. Morever (u*), are considerably different for the three surfaces. The model also gives abnormally large values of (u*(l 11))1 for Pb, Pt, Pd and AU.

Vacancy When the atom 0 is adjacent to a vacancy the calculations show that there is an enhancement in the vibrational amplitude along the line joining it with the vacancy. But its amplitude in the perpendicular direction remains practically unchanged.

of (u’) and (u*)&u~)~ with temperature palladium. See Fig. 2 for details.

for

3.2 Rare gas solids For the low melting rare gas solids, eqn (4) together with appropriate elastic constants is expected to yield good estimates of the MSD close to melting. The elastic constants data at different temperatures have recently become available[221. As in the case of Cu, Ag and Au, the MSD for the various cases and R, at different temperatures were computed from the corresponding elastic constants data (Table 4). From the results given in Table 5 it is seen that R, tends to a saturation value at high temperatures. In general, the R, for rare gas solids are somewhat lower than the corresponding ratios for metals. Lindemann ratios for the rare gas solids have been estimated from the entropy Debye temperatures[23], lattice dynamics in quasi-harmonic approximation[24), molecular and dynamics [25] Monte Carlo calculations[26]. The results obtained by various methods are compared with our calculations in Table 6. It is seen that the simple method employed in the present work provides a good estimate of the Lindemann ratio and of (u*) for the rare gas solids.

A. P. G. KIJITY and S. N, VAIDYA

1168 Table

4

3. Calculated

and experimental

root-mean-square

displacement

(111)

1.11

2.27

1.46

(100)

1.73

2.m

2.16

ratios

(110)

I.?2

1.97

1.49

Ni

(110)

1.66

1.68

1.77

Ph

(111)

1.76

3.01

1.64

Pd

(111)

1.73

2.59

1.95

(ISOj

1.75

2.23

1.95

(110)

1.74

1.96

(111)

1.75

3.19

2.12

1.41

(13Oj

1.79

2.52

2.12

1.41

(1lSj

1.77

2.06

2.12

1.41

Pt

Table

4. Elastic

constants

and other parameters

for rare gas solids

Slastiz

Tie

24.55

23.133

constnnts(a)

0.0035

0.0095

0.0090

0.0093

3.0073

0.0063

n.nC03

3.006D

r.0233

".';225

^.Cl53

S.Cl24

*.~~156 0.0112 i:r

llj.??

e3.a

?.0284

?.0268

,*.0135 i.ClQi

X@

161.39 131.3

n*c173

?.a26

0.%32

r.ve95

3.0241

'G.0211

^.013c! 0.0156 2.0173

:a) hf.

^ .x50

122-J

Using lattice dynamics in quasiharmonic approximation Allen and DeWette[27] have calculated the MSD of surface atoms in rare gas crystals. Their results show that (u2fi is nearly same for the surfaces (Ill), (100) and (110). Our calculated values also exhibit a dynamics molecular similar behaviour. The calculations[28] give generally higher values for (u’),. To our knowledge, experimental results needed for comparison are not available.

(1) We observe that the inclusion of the R4 term in the expansion of the dynamical matrix and use of high temperature elastic constant data lead to a better esti-

mate of the MSD for bulk atoms. This has been demonstrated in the case of Cu, Ag and Au and the rare gas solids for which the necessary data are available. The numerical values of (u*)~ computed from the de Launay model are not appreciably different. (2) At all temperatures (u*), for (KIO), (1 IO) and (Ill) surfaces are same within 1%. This trend is in agreement with experimental observation as well as with the lattice dynamics[S] and molecular dynamics the calculations[28]. In contrast, de Launay model leads to the following inequality

Mean-square

Table 5. Calculated mean square displacement, (u*) (in A*), and Rs = (u*)&& Rs are given in parentheses

Be

&

Kr

Xe

1169

atomic displacements in f.c.c. crystals

(ucj

T(K)

(u2) B

5 6

.025 .0254

.0304(1.215) .0310(1.220)

23.7

.0471

24.3

.05OC

10

11

mJ) 11

(W1

for rare gas solids. The values of

(llO),

(111)1

.0306(1.224) .0310(1.220)

.0405(1.62) .04as(1.62) .04C6(1.62) .0420(1.654) .0420(1.654) &X20(1.654)

.0665(1.412)

.0651(1.382)

.1156(2.456) .1156(2.456) .1155(2.455)

.0709(1.417)

J688h.376)

.1246(2.492) .1246(2.492) .1245(2.490)

.Olul

.0132(1.233)

.0131(1.227)

.0187(1.747) .0187(1.747) .0187(1.747)

82

.0580

.C@56(1.475)

.0326(1.424)

.1581(2.724) .1581(2.724) .1580(2.722)

82.3

.c643

.0944(1.469)

X890(1.385)

.178 (2.770) .1778(2.766) .1775(2.762)

10

.oc67

.OCB4(1.256j

.0033(1.249)

.0124(1.855) .0124(1.855) .0124(1.855)

114

.0646

.0959(1.485)

.0918(1.421)

.1789(2.770) .1787(2.768) .1786(2.766)

115.6

.0742

.1095(1.476)

.1036(1.396)

.2067(2.787) .2065(2.784) .2062(2.780)

10

.0050

.0065(1.282)

.OC6 (1.288)

.0097(1.926) .0097(1.926) .CO97(1.926)

111

.0395

.0584(1.480)

.0572(1.449)

.1012(2.717) .1072(2.717) .lU72(2.717)

151

.0725

.1091(1.492)

.1045(1.442)

.2003(2.762) .2002(2.761) .2001(2.760)

159.6

.0796

.1191(1.496)

.1163(1.460)

.2188(2.748j .2188(2.748) .2187(2.747)

Table 6. Lindemann ratios for rare gas solids Present aa1cul3tions

"xwford L233

Coldsen

Hansen

c243

r26J

Dickey and Paiin f253

lie

0.121

0.127

0.109

0.14

pr

0.113

0.113

0.101

0.14

Kr

0.114

0.115

0.10

0.14

xe

0.103

0.114

0.099

0.14

This inequality and the abnormally large values of (u*), in some cases are probably due to the lack of rotational invariance in this model[7], which makes it unsuitable for surface calculations[8]. (3) A number of theories on melting of solids are based on the assumption that the process of melting is associated with the softening of the transverse acoustic modes[29,30]. Close to the melting temperature this should lead to a pronounced increase in (u’)~, (I.?)~ and (u*)v_ Since (u*)~ > (~~)v,~ > (u*)~, the atoms near the surface and in the vicinity of vacancies will be the first to attain very high amplitudes of vibration with mode softening. In most cases, the root-mean-square amplitude of vibration is 20-25% higher than that of bulk atoms for atoms near a vacancy and 5670% higher for atoms on the surfaces. This tends to support the view that the process of melting is initiated at the external surfaces and near vacancies[lO, 111, the regions of low potential energy in a solid. (4) The low melting rare gas solids provide a crucial test for the matrix expansion method employed in the present work. Detailed comparison with the existing

c.11

theoretical and experimental results indicates that this simple technique is sufficiently accurate for MSD calculation in the low temperature regime. Acknowledgements-The authors thank the referee for his critical comments and suggestions.

REFERENCES I. Maradudin A. A., Montroll E. W., Weiss G. H. and Ipatova, I. P., Theory of lattice dynamics. In Harmonic Approximotion, 2nd Edn. Academic Press, New York (1971). Wallis R. F., In Progress in Surfuce Science (Edited by S. G. Davison), Vol. 4. Pergamon Press, New York (1974). Shapiro J. N., Phys. Rev. Bl, 3982 (1970). Masri P. and Dobrzynski L., Surf Sci. 32,623 (1972). Wallis R. F. In The Structure and Chemistry of Solid Surfaces (Edited by G. A. Somorjai). Wiley, New York (1968). 6. Singh N. and Sharma P. K., Phys. Rev. B3, 1141 (1971). 7. de Launav J.. In So/id State Physics (Edited by F. Seitz and D. Turnbull),~Vol. 2. Academic Press, New York (1956). 8. Lax M. In Lattice Dynamics (Edited by R. F. Wallis), p. 593. Pergamon Press, New York (1%5). 9 Theeten J. B. and Dobrzynski L. Phys. Rev. B5, 1529 (1972).

1170

A. P. G. Kurrv and S. N. VAIDYA

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ww.

17. Clark B. C., Gazis D. C. and Wallis R. F., Phys. Reu. 134, 1486 (l!XA). 18. Landolt-Bomstein New Series Group Ill, Vol. I. SpringerVerlag, Berlin (1966); Ibid, Vol. 2 (1%9). 19. Leibfried G. and Ludwig W., In Solid State Physics (Edited by F. Seitz and D. Turnbull), Vol. 12. Academic Press (l%l).

20. Wallis R. F. and Cheng D. J., Solid State Commun. 11. 221 (1972). 21. Morabito Jr. J. M., Steiger R. F. and Somorjai G. A., Phys. Reu. 179,638 (1%9). _ 22. Koroium P. and Luscher E. In Rare Gas Solids (Edited bv M. L. Klein and J. A. Venables), Vol. 2, p. 815:Academic Press, New York (1977). 23. Crawford R. K. In Ref.[22, p. 71 I]. 24. Goldman V. V., 1. Phys. Chem. Solids Jo, 1019 (1%8). 25. Dickev J. M. and Paskin A.. Phys. Rev. 188, 1407 (1%9). 26. Hansen J., Phys. Rev. AZ, 221 (1970). 27. Allen R. E. and De Wette F. W., Phys. Rev. 179,873 (1%9). 28. Allen R. E., De Wette F. W. and Rahman A., Phys. Rev. 179, 887 (1%9). 29. May A. N., Nature 228,990 (1970). 30. Fukuyama H. and Platzman P. M., So/id State Commun. 15, 677 (1974).