On the dynamical instability in metastable liquids

Physica l17A (1983) 497-510 North-Holland Publishing Co.

ON THE DYNAMICAL INSTABILITY IN METASTABLE LIQUIDS* H.B. SINGH and A. HOLZ Fachrichtung Theoretische Physik, Universitiit des Saarlandes, D-6600 Saarbriicken, FRG Received 3 August 1982

The claim of Schneider et al. that a supercooled liquid transforms into a solid over a soft relaxational mode at the instability temperature is investigated within linear-response theory based on Zwanzig-Mori formalism. We have used two non-mean-field approximations and they confirm the view held by Schneider et al. Our studies reveal, however, that the stability limit temperature can be predicted reliably only over a knowledge of the point of divergence of the static structure factor. The experiments of Suck et al. which apparently disprove the theory of Schneider et al. are critically discussed in view of the constraint imposed by the surface coating on the free volume of the system. It is suggested that the soft relaxational mode drives the system into a state which is superplastic at the instability temperature.

1. Introduction It has been k n o w n since long that the liquids can be easily supercooled and this p r o p e r t y has been usefully exploited in m a n y branches of science, especially metallurgy. The theory of this state of matter, however, is not well understood and is presently an object of growing interest. Lately, quite a number of attempts to understand the stability limit of a supercooled liquid have been made. Questions raised in this context are, at which temperature does a metastable liquid transform into the stable solid phase, is it a first or second order transition and what type of solid state is it. These are also the questions to which we shall address ourselves in this paper. Most of the earlier attempts~), starting with Kirkwood, are based on an analysis of nonlinear integral equations for the single particle density. Within such an approximation the stability limit of a supercooled liquid is associated with the point at which such an integral equation exhibits apart from the usual liquid like, a non-uniform solid like solution. Such a bifurcation analysis leads to the result that the static structure factor S(qo) of the system must diverge at the instability temperature, TL. Here q0 is a wave number corresponding to the first maximum in S ( q ) and corresponds to a reciprocal lattice vector of the solid. Similar criteria have been obtained 2) from an analysis of the asymptotic * Work supported by Deutsche Forschungsgemeinschaft within SFB 130. 0378-4371/83/0000-0000/$03.00 O 1983 North-Holland

498

H.B. SINGH AND A. HOLZ

behavior of the pair correlation function. All these approaches are static in nature. There exist also a few attempts 3-7) to study the stability limit dynamically. Among these the work of Schneider et all) has caught much attention and has stimulated some of the experimental studies 8-~2) on this problem. These authors 5) assume that the stability limit is associated with a soft relaxational density mode. This criterium has been worked out by them using a mean field approximation (MFA) and leads to the result that S(qo) must diverge at the liquid-solid transition. This implies a narrowing of the quasi-elastic peak lt-12) in the dynamical structure factor, S(qo, to), of the system on approaching TL. Experimentally 8-9) however, S(qo) does not show any tendency to diverge near the stability limit. SjiSlander and Turski 7) have discussed the inadequacy of M F A to describe the dynamics of supercooled liquids and have convincingly stressed the role of self-diffusion in determining the width of the quasielastic peak. They seem to imply that the defect of a diverging S(qo) in the theory of Schneider et al. 5) is an artifact of M F A used by these authors. We considered it therefore worthwhile to calculate the density response function in non-mean field approximations and analyse its normal modes. In the following we study the density response function of the system using the memory function method 13) of Zwanzig and M o r i - o f which the mean field theories 5-6) are a special case. Then we go beyond M F A and determine the response function in two different non-mean field approximations. In both cases we find that the static structure factor must diverge at the stability limit as long as it is determined by a soft relaxational mode. Accordingly a diverging S(qo) at the stability limit is a consequence of a soft relaxational density mode and not due to the mean field nature of the approximation used in determining it. Furthermore, we point out some flaws in the work of earlier authors 5-6) and correct them. A discussion of the experiments conducted by Suck et al.~2) which apparently disprove the theoretical conclusions reached is given in the light of free volume concepts.

2. Basic formalism

The natural way to study the dynamics of density fluctuations in the system is via the density correlation function 1

F(q, t) = -~ (o(-q, t)p(q)),

(2.1)

where o(q, t) is the density fluctuation of wave number q at time t. Further the angular brackets denote a classical thermal average and N is the total

DYNAMICAL INSTABILITY IN METASTABLE LIQUIDS

499

number of particles in the system. The density response function is related to the correlation function as

x(q, t) = n/3 d F(q, t),

(2.2)

where /3 is the inverse temperature and n is the uniform number density of the system. In the memory function method it is convenient to work with the Laplace transform of the correlation function F(q, Z)

[ dt eiZtF(q, t);

z = to + ie,

(2.3)

0

where E is positively infinitesimal. By Laplace transforming eq. (2.2) and using the causality property of the response function, it is easy to see that its Fourier transform is given by r + ito F(q, to + iE) 1 x( q, to) = - n/3S( q)[1

(2.4)

where S(q) is the static structure factor of the system. In the following the unknown function F(q, to + i~) shall be calculated using the memory function method13). The central equation in this method is the generalised Langevin equation. Using this equation successively one can obtain the following continued fraction hierarchy for the Laplace transform of the density correlation function

S(q) F(q, z) = - i z + M,(q, z)'

(2.5a)

~2(q) M.(q, z) = - i z + Mn+~(q, z)'

(2.5b)

where M.(q, z) is the Laplace transform of the memory function entering at the nth stage of expansion. Furthermore, the coefficients ~n2(q) represent the initial values, M.(q, t = 0) of the corresponding memory functions and are related to the frequency moments of S(q, to). The dynamical structure factor is defined as the Fourier transform of F(q, t) and is given by

S(q, t o ) = I7r Re

F(q,z)=-

1

~rn/3to Im x(q, to).

(2.6)

The 2nth frequency moment of S(q, to) is defined as

(OJ2n) = f dto to2nS(q, to). -e¢

(2.7)

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H.B. SINGH AND A. HOLZ

The low order coefficients 82(q) can be calculated exactly and we quote them below for later use

(0)~) _ 0)~(q)_

tt E

8~(q)= S ( q ) - S(q)

(0)') ~](q) = (~

(2.8a)

- [3mS(q)'

(0):)

(2.8b)

S(q)'

and r(0) ~)

( 0)4)

2

(2.8c)

The fourth frequency moment is given by (o)4) = 0)o2(q)[30)o2(q) + I(q)]; I(q) = ~-

d3r g(r)[1 - cos q • rl(~"

(2.9)

V)2q~(r),

(2.10)

where q~(r) is the two-particle potential and g(r) is the pair correlation function. The expression 14) for the sixth moment, (0)6) is quite lengthy and can be seen in the original paper. To proceed further, the continued fraction expansion given by eq. (2.5) has to be closed. Depending on how and at which stage we do that, different approximations follow. In the following three sections we shall discuss different expressions thus obtained for the density response function from the point of view of analysing the normal modes of the system.

3. Effective field approximations Approximations which follow from the linearized Vlasvov equation can be obtained 15) within the memory function method if one stops at the second stage of the continued fraction hierarchy, i.e., F(q, z) =

S(q) 82 -iz +

(3.1)

- i z + M2(q, z)

The memory function can be determined in a free particle or renormalised free particle approximation. 3.1. Mean held approximation The usual mean field approximation follows from eq. (3.1), if the second order memory function is determined such as to give the correct free gas

DYNAMICAL INSTABILITY IN METASTABLE LIQUIDS

501

density correlation function (3.2) in an appropriate limit. In eq. (3.2), yf(q)= 2to2(q) and the function W(z) is defined by

W(z)=~

i f

e -t2 ~dt,

Imz>0.

(3.3)

Introducing the expression for F(q, z) thus obtained in eq. (2.4) yields the familiar mean field expression for the density response function

x(q, o~) =

x0(q, O )

1 - ~bf(q)xo(q, 0 ) '

12 -

~o

X/~"

(3.4)

Here

Xo(q, D) = - n/3[1 + iX/TDW(D)],

(3.5)

is the response function of an ideal gas and the effective field is related to the structure factor as 1

S(q) - 1 + n/3~bf(q)"

(3.6)

Schneider et al. 5) have used eq. (3.4) to discuss the dynamics of the liquidsolid transition. In particular they have analysed the normal modes of the system which are given by the poles of eq. (3.4), and find a purely relaxational mode for the intermediate q region. Furthermore, the frequency of this mode has a minimum for the wave vector q0 corresponding to the first maximum in S(q). Up to first order in ~, the frequency of this mode is given by i

O(q) - X/-~

1 + n/3~bf(q)

n~bt(q)

(3.7)

Schneider et al. assume that it is this mode which leads to the dynamical instability connected with freezing and thus the stability limit is given by

kBTL + n~bf(q0, TL) = 0,

(3.8)

which applies to a system at constant density. Before commenting on the use of this equation by Schneider et al. 5) to calculate the stability limit temperature TL, we shall briefly discuss another version of mean field theory.

502

H.B. SINGH AND A. HOLZ

3.2. Renormalised mean field approximation It is well known 16) that the mean field expression given by eq. (3.4) is inadequate to describe the spectrum of density fluctuations even qualitively. Pathak and Singwi 17) have improved the situation considerably by including phenomenologically the effect of collisional damping of the collective modes which has been neglected altogether in arriving at eq. (3.4). The expression due to Pathak and Singwi can be obtained if M2(q, z) in eq. (3.1) is constrained to yield the correct renormalised free gas density correlation function

Fr(q,z)= ~ q )

W(~).

(3.9)

Thus the renormalisation consists in replacing ~f(q) in eq. (3.2) by an unknown parameter yr(q) which is related to the width of the gaussian in eq. (3.9). The expression thus obtained for the density response function if given by x ( q , to) =

Xr(q, ~) 1 -- ~r(q)Xr(q, 11)'

(3.10)

where

Xr(q,~)=-n[3 "Yf[l +i'k/-w~W(O)],

to g2 _ ~/~r'

(3.11)

~r

and ~f/~r

S(q) = 1 + n[3~kr(q)ytl%"

(3.12)

The renormalisation parameter ~r is determined such that the fourth moment sum rule is satisfied exactly. This leads to the result %(q) = 82(q).

(3.13)

Eq. (10) has been used by Mitra and Shukla 6) to discuss the stability limit just following Schneider et al. Naturally, they also find a purely relaxational density mode with frequency

i 1 + n[3~r(q)'Yf/3'r ~'~(q) = ~/-~

(3.14)

n~l~r(q)~f/~r

From this it follows that the stability limit is given by ~ 7e(qo)[ = 0, kBTL+ na,r(.. v, ~I0S%---~0)IT=TL which via eq. (3.12) again implies a divergent S(qo).

(3.15)

DYNAMICAL INSTABILITY IN METASTABLE LIQUIDS

503

3.3. Comments Eq. (3.8) has been used by Schneider et al. s) to estimate the temperature TL of the stability limit. For that purpose they calculate the value of the effective field using molecular dynamics data of Rahman 18) for S(qo) via eq. (3.6). Thus using eqs. (3.6) and (3.8), the stability limit temperature as calculated by Schneider et al. can be written as

The main purpose of the paper by Mitra and Shukla 6) is to improve the above relationship. The expression obtained by them 6) is given by TL[1

3S~q0)J = r [ 1

-

1

]

(3.17)

which is claimed to give a better estimate of TL. For example 6) for liquid sodium, eq. (3.16) yields TL = 226 K while eq. (3.17) gives TL = 260K. Unfortunately, Mitra and Shukla's 6) approach to derive eq. (3.17) suppresses some important contributions. If we use the renormalised mean field equations correctly, then instead of eq. (3.17), we get the following expression: 3S(q0) ~ - ~ ]

T

-S(~0) + ~ j .

(3.18)

The extra term I(qo)/3oo~(qo) of this equation has its origin in the contribution of the interaction potential to the fourth frequency moment sum rule given by eq. (2.9). This term is of the order of one or even larger and is much larger than 1/S(qo). Therefore, this term is very important in determining TL and thus the estimates of TL due to Mitra and Shukla cannot be reliable. For example, for liquid sodium the stability limit temperature according to eq. (3.18) should be 307 K in contrast to 260 K as obtained by them 6) from eq. (3.17). Furthermore, we wish to point out that the method given above to estimate TL cannot be reliable as it implies that the effective potential is temperature independent which is not the case. Use of eq. (3.16) to estimate TL, for example implies that with decreasing temperature the value of S(qo) should increase appropriately in order to compensate this effect which however does not happen. In order to show this explicitly we use the x-ray diffraction data for S(q) obtained by Bosio et al. 9) which cover a wide range of temperature. For instance if the temperature at which the experimental data for liquid mercury are used, is varied from 172 K to the equilibrium freezing temperature at 234 K; the stability limit temperature according to eq. (3.16) varies

504

H.B. SINGH AND A. H O L Z

from 118 K to 149 K. Thus the numbers given for TL from any of the eqs. (3.16)-(3.18) should be taken with caution. Because in the literature 4) comparisons have been made with stability limit temperatures obtained from eqs. (3.16)-(3.17), we thought it worthwhile to point out such discrepancies. Naturally, the correct way to estimate TL is to calculate S(qo) theoretically4) and to look for the temperature where it diverges.

4. Markovian approximation This approximation is similar in spirit to the one used by Lovesey~9). We first write an exact equation of motion for the second order memory function appearing in eq. (3.1). This requires the knowledge of the third order memory function which is assumed to be frequency independent and is determined by sum rule arguments. Details can be read in ref. 20. The expression obtained for the density response function is given by x ( q , oo) = nq----~z o~ + i~ol m ~o3+ io~z~Ol- w~o~- i8~o1'

(4.1)

where 00~ z = (o~4)/(o~z). The normal modes of the system are obtained from the poles of eq. (4.1) which can be calculated analytically. Two of the three poles are complex and correspond to the familiar damped propagating modes. The third pole is purely imaginary and thus corresponds to a purely relaxational density mode. Its frequency is obtained as O(q) = i ( U + V - ~°-~),

(4.2)

where U = [ - Q + ~/~r~--ff~],/3

and

V = [ - Q _~/~r~---~],/3.

(4.3)

In eq. (4.3) the abbreviations

and

(4.4) e

have been used. The variation of the frequency of the relaxational mode with wave number is shown in fig. 1 for liquid aluminium. It is seen to approach a minimum at q0. If this mode is considered to lead to the dynamical instability, associated with freezing, then the stability limit of the system is obtained by

DYNAMICAL INSTABILITY IN METASTABLE LIQUIDS

505

1.0 8.0

'T

6.0

b 4.0 C~ 2.0 i

O0

1.0

i

2.0

I. 30

i

i

4.0

5.0

q (,~-1) Fig. 1. Variation of relaxational mode frequency with wave number.

the condition that this mode gets soft. Using eqs. (4.2) and (4.3) this condition may be written as [_ Q + V ~ - - b - J ] ,, _ [Q + ~ / ~ r T ~

] ,, = o), 3"

(4.5)

Taking the third power of this equation and doing some rearrangements leads to q2 a~(q0) -

[3mS( qo) =

0

(4.6)

which implies that the static structure factor, S(qo) diverges at the stability limit.

5. Renormalised free particle approximation at the third stage Such an approximation is expected to give better results for the dynamics of the system than the earlier listed approximations as apart from retaining their essential features, it satisfies the sixth f r e q u e n c y moment sum rule. In this approximation 21) we go one stage b e y o n d that used in eq. (3.1), i.e.,

F(q, z) =

S(q) - i z 4 a~ - iz +

822 - iz + M3(q, z ) .

(5.1)

506

H.B. SINGH AND A. HOLZ

The third order memory function is determined in the renormalised free particle approximation given by eq. (3.9). The renormalisation parameter is now determined by satisfying the 6th frequency moment sum rule exactly so that "Yr(q) = 2 332(q),

(5.2)

where the coefficient 632(q) is given by eq. (2.8c). The result obtained for the density response function can be written in the form

Xr(q, O) O x(q, to) = 1 - to(q, $'~)Xr(q, $~)'

=

to

~/'YE'

(5.3)

where 2to 2

Xr(q, O) = -n~8 ~

[1 + i'k/-~DW(D )],

(5.4)

is the response in renormalised free particle approximation. In addition the local field in contrast to the mean field theories 5'6''7) is frequency dependent and is given by to(q, ~'~) = ~ 1

[~'] 2( 6 2 - 'Yr) -t- 62 _{ 62].

(5.5)

The normal modes of the system are obtained as the solution of 1

-

t0(q, g])Xr(q, 12) = 0.

(5.6)

This equation can be simplified using eqs. (5.4) and (5.5) and rewritten as

iX/~DW(O)[02+ a] + DE+ b = 0,

(5.7)

where

(5.8) and b = 621/(622 -- "}/r)"

(5.9)

Eq. (5.7) has solutions corresponding to propagating and relaxational modes. The smallest frequency of the relaxational mode is obtained for the wave number corresponding to the first maximum in S(q). Taking in eq. (5.7) the small D limit, the frequency of the relaxational mode is obtained as 2(q) D(q) = V i~ a,~(q)a ,-~ a~(q)"

(5.10)

DYNAMICAL INSTABILITY IN METASTABLE LIQUIDS

507

If the stability limit is again associated with a softening of this mode, it obviously implies a divergence in S(qo) at this limit.

6. Discussion and conclusion

The modern approach to the theoretical study of the stability limit TL of a supercooled liquid has been initiated by Schneider et al)). According to their theory the transition into the solid state should be driven by a soft relaxational mode leading to a divergence of the static structure factor S(qo). Because such behavior is not observed experimentally 8-9) we have treated this problem going beyond the mean field approach of these authors. Using the renormalised mean field approach within the Zwanzig-Mori formalism, we derive eq. (3.18) for the instability temperature which contains important additional terms apparently overlooked by Mitra and Shukla 6) in a similar approach. This result, however, does not contradict the conclusions reached by Schneider et al:) that the transition if driven by a soft relaxational mode leads to a divergence of S(qo). In contrast to Schneider et al)) and Mitra and Shukla 6) we find that TL cannot be determined unambiguously with data of the static structure factor at T > TL, but that a theoretical calculation of the temperature where S(qo) diverges is necessary. The reason for that is that the effective potential entering the theory cannot be consistently assumed to be temperature independent. Two further improved (non mean field) approximations applied to study the instability behavior of the supercooled liquid are the Markovian approximation and the renormalised free particle approximation at the third stage of the continued fraction expansion. The results obtained confirm again the view of Schneider et al:) that the instability is approached continuously. From this we conclude that within linear response theory, presumably also in higher orders of approximation, no qualitative change of the conclusions reached by Schneider et al. is achieved. In order to get an explanation of the discrepancy between theory and experiment pointed out by Suck et al)2), it may be useful to have a closer look at the experiments conducted by these authors. The first point to note is that the surface coating of their droplets prevents the free volume (in the form of vacancies for example) to leave the system on lowering the temperature. This constraint has been discussed in ref. 22 in connection with superheating and supercooling phenomena. The argument used there is that the annihilation of free volume with decreasing temperature, if it cannot be expelled through the surface due to the surface coating, creates dilational strain in the system which cannot be screened out. This then may lead to a mechanical instability

508

H.B. SINGH AND A. HOLZ

of the system before TL is reached. The counter example presented by Suck et al. 12) that the full width at half maximum (FWHM) scales with the diffusivity and not with the homogeneous nucleation temperature TN is from that point of view less striking. Since decreased diffusivity in the Snl-xPbx alloys with increasing x may not be a consequence of quicker annihilation of free volume with decreasing temperature (implying that TN increases with x) but simply a consequence of a less favourable internal structure of their liquid state. In fact, Suck et al. use diffusivity data calculated from Stokes-Einstein relation as a function of particle radius. Because build up of internal strain is essentially accompanied by the annihilation of free volume, it is the latter quantity which should be brought into relation with the instability temperature if universality arguments are used as done by Suck et al.l~). Next we like to point out that the extrapolation of the diffusivity and FWHM data to the temperatures where they vanish is not unambiguous. Because if it is assumed that their end points mark a continuous transition their supposedly linear behavior may be modified in the critical region due to long range fluctuations. Considering next the possibility that T N a p p r o a c h e s T L in carefully conducted experiments, we feel that it seems to be more reasonable that the continuous transition postulated by Schneider et al. 5) leads into a highly disordered solid phase. The reason for that is that the finite long range order parameter and the finite shear modulus of an ordinary solid cannot be approached at TL in a continuous fashion. It is also hard to imagine why the vanishing free energy difference between supercooled liquid and crystalline solid on approaching T L should be accompanied by such a large free energy barrier that it cannot be overcome by thermal excitations. Because if the latter is true it must be assumed that the structural changes mobilised by the critical fluctuations cover a huge range of possibilities ranging from highly ordered to disordered configurations. If it is assumed that T L marks the transition into a highly disordered solid with superplastic behavior then the difficulties mentioned above do not arise. Within the picture of a liquid as a plasma of dislocations and free volume in the form of finely dispersed vacancies2:), such a possibility can be visualized. It must only be assumed that the free volume is not expelled from the system on approaching T L and that no mechanical instability of the surface coating occurs. A finite shear modulus below T L is t h e n a consequence of a frozen in dislocation network with mobility properties at short distance scales. In contrast to what is assumed in the homogeneous nucleation theory where the transition is supposed to lead into the ordinary solid state, the continuous transition driven by the relaxational mode may terminate in a metastable solid state. The latter is presumably mechanically unstable22). The relative successful predictions of homogeneous nucleation theory as demonstrated in ref. 23

DYNAMICAL INSTABILITY IN METASTABLE LIQUIDS

509

with respect to a lower bound on the h o m o g e n e o u s nucleation temperature TN m a y also be u n d e r s t o o d from that point of view. Since the supercooled liquid is always unstable against decay into the ordinary solid state, and that the state reached eventually by the continuous transition as postulated by Schneider et al. 5) is still metastable and presumably mechanically unstable, a discontinuous transition is observed. Because in either case the transition is driven by the free energy difference between the supercooled liquid and ordinary solid state, it is plausible that if this quantity is calculated in a reliable manner that a lower bound on TN may be predicted. The problem that no divergence of S(qo) is observed experimentally so far is then a consequence of the fact that for the systems studied experimentally apparently TN > TL holds. A more detailed study of the diffusivity 7) in the postulated superplastic state and its effect on the nucleation rate entering the homogeneous nucleation theory may reveal perhaps that TN> TL must always hold. This point has not been studied so far.

Acknowledgment The authors would like to thank Prof. H. T h o m a s for stimulating discussions.

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1972).

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t4) D. Forster, P.C. Martin and S. Yip, Phys. Rev. 170 (1968) 155. R. Bansal and K.N. Pathak, Phys. Rev. A 15 (1977) 2519. 15) K.N. Pathak, Nucl. Phys. and Solid State Phys. (India) 18A (1975) 265. 16) A.A. Kugler, J. Stat. Phys. 8 (1973) 107. 17) K.N. Pathak and K.S. Singwi, Phys. Rev. A 2 (1970) 2427. 18) A. Rahman, Phys. Rev. Lett. 19 (1967) 420. 19) J.R.D. Copley and S.W. Lovesey, Rep. Prog. Phys. 38 (1975) 461. 20) H. De Raedt and B. De Raedt, Phys. Rev. B 15 (1977) 5379; H.B. Singh, Phys. Rev. A 21 (1980) 2166. 21) H.B. Singh, Aruna Sharma and K.N. Pathak, Phys. Rev. A 19 (1979) 899; P.K. Kahol, D.K. Chaturvedi and K.N. Pathak, Physica 81A (1977) 192. 22) A. Holz and H. Gleiter, to be published in Phys. Rev. B. 23) H.B. Singh and A. Holz, to be published in Solid State Comm..