Phonon dispersion relations, effective interionic potential and liquid resistivity of Al

J. Phys. Chem. Solids, 1974, Vol. 35. pp. 669-684.

Pergamon Press.

Printed in Great Britain

PHONON DISPERSION RELATIONS, EFFECTIVE ~INTERIONIC POTENTIAL AND LIQUID RESISTIVITY OF AI* P. V. S. RAOt Department of Materials Science, Northwestern University, Evanston, Ill. 60201, U.S.A. (Received 14 June 1973; in revised [orm 7 September 1973) Abstraet--Phonon dispersion relations and effective interionic potential for solid AI are calculated using the nonlocal optimized model potential of Shaw screened with the improved self-consistent dielectric function of Vashishta and Singwi. A very good agreement is obtained between the calculated and experimental phonon frequencies. Comparison is made using three other dielectric functions. It turns out that in AI the interionic potential is very sensitive to the screening function. Also electrical resistivity of liquid AI has been calculated using the model potential when screened with the four different dielectric functions. Influence of the spatial distribution of depletion charge at the ion site is discussed. A comparative study of nonlocal and local potentials by taking two local potentials: (1) modified point-ion, and (2) empty core, screened with the self-consistent dielectric function, is also presented.

1. I N T R O D U C T I O N

Since the early calculations of Harrison [ 1] of phonon dispersion relations in aluminum using pseudopotential theory, there have been a f e w other calculations [2--8] with varied degree of success. These calculations are distinguished by their two basic ingredients: (1) pseudo- or modelpotential, and (2) electron gas screening or dielectric function. Several forms of the pseudopotential have been used, some of which include nonlocal corrections, as well as various forms of the dielectric function, ranging from simple Hartree screening to more complicated forms incorporating exchange and correlation effects. The way the two ingredients enter the calculation is through the conduction electron response to the vibrating ions. The conduction electron response has the effect of canceiling a part of the direct ion-ion coulombic repulsive interaction. In the case of the polyvalent metal, AI, the cancellation is so large that measured phonon frequencies at the zone boundaries are, on the average, only about 35 per cent of the unscreened ionic lattice frequencies, as against 80 per cent in the case of monovalent metal, Na[6]. This special feature m a k e s it essential to obtain the pseudopotential, as well as the dielectric function, to a much *This work was supported by the Advanced Research Projects Agency of the Department of Defense through the Northwestern University Materials Research Centre. tCurrent Address: Ford Motor Company, Scientific Research Staff, P.O. Box 2053, Dearbo~'n, Mich. 48121, U.S.A.

higher degree of accuracy for AI than is needed for Na. In recent years significant advances have taken place in the treatment of electron screening, as well as in the development of pseudopotentials. The most recent work on electron screening in the metallic density range is that due to Vashishta and Singwi[9], who have presented a self-consistent treatment of correlations in the homogeneous electron liquid. Their treatment is an improvement over earlier works of Singwi and his co-workers[10] in that the new dielectric function satisfies the compressibility sum rule almost exactly while giving a physically acceptable pair correlation function. To gain confidence in the theory, one w a y i s to apply it to the calculation of properties which can be compared with experiment. As a step in this direction, we report here the results of our calculation of phonon dispersion relations in AI using the new dielectric function of Vashishta and Singwi, evs. Since the phonon dispersion relations are closely related to the compressibility of a metal, the improved characteristic, i.e. the satisfaction of the compressibility sum rule by ~vs should exhibit significant ramifications on the phonon frequencies; and AI is a good system, as mentioned above, to consider for stringent testing--though not as a definitive test for reasons discussed later. F o r a pseudopotential, we employ the widely used model potential due to Shaw [ 11]. This potential is a nonlocal optimize.d model potential ( N L O P ) incorporating full nonlocality and energy dependence of

669 JPCS VOL. 35 NO. 6---A

670

P.V.S.

the potential. This potential is based on more firm theoretical foundations compared to the wellknown Heine-Abarenkov (HA) model potential. We have also studied effects of locality of the model potential by employing two local potentials, namely the empty core potential (ECP) of Ashcroft[12] and the modified point-ion potential (MPIP) of Harrison[l 3]. Unlike the nonlocal potential, which is, in a way, a 'first principles' potential depending only on the spectroscopic term energies of the free ion, the local potentials are paramaterized. We have determined their parameters to give best overall fit to the measured dispersion relations. In these local calculations also, electron screening is done by evs. In fitting or checking a pseudo- or modelpotential liquid resistivity is another property one selects frequently. We have, therefore, calculated the liquid resistivity for A1 at its melting point (660°C) using the theoretical structure factor a ( q ) determined from a solution of the Percus-Yevick equation[14] for a liquid consisting of hard spheres[15]. Using the nonlocal optimized model potential, we have investigated the effects of exchange and correlation corrections to the dielectric function on the phonon dispersion relations and liquid resistivity. As a complimentary study, we also calculated the effective interionic potential V(r), which is the starting point in the investigation of many properties of metals, such as lattice defects and liquid dynamics. The two essential ingredients entering this calculation are the same as those of the lattice dynamic calculations. As such, earlier calculations of the interionic potential[13, 16-18] suffer from discrepancies that can be due to inadequacies of either one or both the ingredients, e.g. the latest calculation of interionic potential in simple metals by Shyu et al.[18], though employing Shaw's optimized model potential, is based on ESSrL, that does not satisfy the compressibility sum rule. With the use of the N L O P and ~vs in our present calculation, we have the best available form not only for the pseudopotential but also for the dielectric function of the electron gas in accordance with the emphasis made in the recent conference on "Interatomic potentials and simulation of lattice defects" [19]. It is quite probable that our V ( r ) will give meaningful results. In Section 2, the theoretical scheme is briefly outlined. Section 3 describes the calculation of the dispersion relations and discusses the results obtained. In Section 4, we present the liquid resistivity results. In Section 5, results for the effective in-

RAO

terionic potential are presented, and the implications of significant changes in the potential that are brought about by an improved dielectric function are discussed. Finally, Section 6 summarizes our results. 2. THEORETICAL SCHEME

In the pseudopotential formulation for simple metals, it is possible, with the help of perturbation theory, to sum the total energy of the system of ions and electrons, and express it as a function of the position of ions--a feature of great importance in lattice dynamics, wherein one is interested in the change of lattice energy due to displacement of the ions from their equilibrium position. The total energy is regarded as the sum of three terms: E = E ~ + E c + E E.

(1)

E c, the electrostatic energy, represents the direct coulombic interaction between the (point) ions of charge + Ze immersed in a uniform compensating negative charge. E ~, the ion core exchange energy, arises from repulsive interaction due to overlap of core wave-functions. This is usually treated in the Born-Mayer approximation. Finally, E E, the socalled band-structure energy, is the ionic interaction via conduction electrons with which We are most concerned here. For AI, the ion core exchange (repulsive) energy E R is insignificant because of the smallness of the core size[5], and therefore, we neglect this term. Consequently, we are left with a dynamical matrix consisting of only two components, viz. DC~(q) due to electrostatic energy between ions, and DE,~(q) due to band-structure energy: DaB(q) = D ca~(q) + D oB(q).

(2)

Once we know DoB(q), the phonon dispersion relation between frequency v and wave vector q for a Bravais lattice can be easily obtained, in the harmonic approximation, from its eigenvalue equation: ~Da~(q)ei~ = vi2ej~, j = 1,2,3.

(3)

The term DCB(q) can be evaluated by standard methods [20]. Whereas, the calculation of the term D~B(q), which contains the dependence upon the electron-ion potential, is involved, and is governed by the type of input, viz. pseudopotential and dielectric function. In any case, it can be related [21] to a scalar function G ( q ) introduced by Cochran [22], which is the ratio of the Fourier trans-

671

Liquid resistivity of AI

form of the ion-electron-ion interaction to that of the direct Coulomb interaction. Using second-order perturbation theory, one gets[23]:

D.~,(q)=

v 2~ [(qo + h.)(q~ + hB) -

lq + hi2

p

x G(lq+ h l ) - ~

G(h)],

(4)

where h is a reciprocal-lattice vector and v, is the ion-plasma frequency given by

v,2= Z2e2/~rMOo.

(5)

Z, M and F~oare,respectively, the valence, mass of the atom, and volume per ion. In case of the nonlocal model potential theory[l 1], Z is replaced by an effective valence Z*( = Z - p) to account for the depletion hole charge p, which is taken to be a point charge at the ion site. Recognizing that the Cochran function G(q) to be identical to Shaw's [ l l ,16] normalized energy-wave-number-characteristic FN (q), we have then G(q) corresponding to Hartree or Random Phase Approximation (RPA) screening (in the nonlocal model potential theory):

and the Thomas-Fermi screening constant

k ~rF= (m */m )(4kH~ras).

Here, aB is the Bohr radius, and m * is an effective electron mass introduced, following Price et al.[23], to include band-structure effects. Effects of the lattice on the conduction electrons are thus included only in this average sense. But, in our present work, these are included only in the empty core potential case. In the other two cases, the effective mass corrections have been neglected. In this context, it is perhaps worth mentioning that whenever m * is introduced as above, one should modify, when computing the dielectric function, the parameter r~ defining the inter-electronic distance to r * = (m*/m)r,, so that the compressibility character of the electron gas in a given approximation is retained. In equation (7), F(q, k) is the nonlocal contribution to the bare ion potential wb(q, k), w~(q, k) = Vq + F(q, k),

4 wRp*(q, k) v,,., = ~---~ fk~kF d3k k 2 - Ik+ql2"

2 ,rr(~-~e)2 [(~-~)~3 Iw~,^(q, k)]2 q'-fl0. .2-I (h"12rn)(k"-Ik + ql")- ~--~ to,,.~J.

f ~ , A.~k

(6)

Here, wRp^(q, k) is the screened model potential form-factor, given by wRp^(q, k) = ~RP^(q) --"t' k) + g(q).

(7)

where vq (= -4~rZe:/D.oq 2) is the local contribution due to valence change, vdq (= 4~rpe2/D.oq') the local contribution due to the depletion hole charge p, E~r^(q) the RPA dielectric function:

(13)

Many electron effects are then incorporated into wRp^ and GRp^ by adding, respectively, to them the correction terms Aw(q) and nG(q), which are given by [24, 25]:

Aw(q)Vq + Vaq + l~'(n

(12)

and g(q) the nonlocal screening contribution. Finally, vscq in equation (6) is related to the screened form-factor by the expression:

,~oq z

GRp,(q) =

(1 I)

{

f(q) ,(q) 8w(q),

~oq2 ,~2~.,^(q)

f(q){Sw(q)} 2,

(14) (15)

and

8w(q) = WR,A(q.k) - wb(q. k).

(16)

Here, E'(q) is the electron-dielectric function reEReA(q) = 1 + Qo(q),

(8) lated to eR~^ by the relation:

where Q0, the Lindhard electron polarizability times - 41re2/q 2, is

Qo(q) = (~dq2)~;(rl), ~ = qlkp

(9)

with 4-

2

2+'0 (10)

e ' ( q ) = 1 +{ 1 - l ( q ) } { ~ R P ^ ( q ) - 1}.

(17)

While evaluating 8w(q), Shaw[25] assumed the depletion hole to have a spatial charge distribution in reciprocal space given by the relation: p ( q ) = pM(q). M ( q ) is a modulating function, taken arbitrarily to be of the form 1/{1 + (qlkF)2}, so that p(q)

672

P . V . S . RAP

decreases to zero as q goes to infinity. We adopt the same procedure. It is to be noted, however, that the results of F* and W*u presented by Shaw and Pynn [24] actually correspond to a further approximation of E(q) = 1 made by them while evaluating 6 w ( q ) . t The procedure we have adopted differs only in that for the same M ( q ) we have (correctly) left the e(q) in the calculation. The function f ( q ) in equations (14), (15) and (17) accounts for exchange and correlation effects amongst the conduction electrons, and depends on the type of treatment one adopts in deriving a particular dielectric function. In RPA or Hartree approximation, which includes neither exchange nor correlation effects, f(q) is naturally zero. Hubbard [26] made the first attempt to incorporate the effects of exchange between electrons of parallel spin, and obtained the expression f ( q ) = q2/{2(q2 + ~:k#)}.

(18)

In the original Hubbard form, which has been selected by us for the purpose of comparison, ~ = 1. Later, Geldart and Vosko[27] chose a value for such that the resulting dielectric function ~cv(q) satisfies the compressibility sum rule. But, E~v has still the unsatisfactory behavior that for small values of r, the electron-pair correlation function goes negative--a physically impossible result. More recently, Singwi et al.[10] have improved the RPA by including the exchange and Coulomb correlation effects. The latest work on electron correlations by Vashishta and Singwi [9], as mentioned in the introduction, is a further improvement. The functions fssvL(q) and fvs(q) are available in tabular form. For a local potential, which is coulombic outside the ion core and a small potential (representing an operator) inside the core, G ( q ) is given by:

Of the two local model potentials under consideration, the modified point-ion potential (MPIP) due to Harrison [13] takes a point-ion Coulomb potential, plus a repulsive exponential function which tends to cancel the Coulomb potential in the core region. The potential is: wb(r) = -Ze2+,------~. /3 e-r/P" r 8 ~'p.

where [3 and p, are the two positive adjustable parameters relating to the core part. The corresponding bare-ion form-factor is: 4 7rZe" [3 wh(q) = - DOq,. q DO(I + q ,_p . ).,2-

wb(q) 2/~P(q)-1-), -4~rZe2/(Doq 2) ~

wh(r) = ( - Z e " / r ) O ( r - r, ),

(note the absence of k-dependence), and EP(q) the proton-dielectric function:

(20)

The distinction between ~P(q) and ~ ( q ) have been discussed elsewhere [28]. tR. W. Shaw, Jr. (private communication).

(23)

where ® is the unit step function, and r,. the only one adjustable parameter representing an effective radius of the ion core. Fourier transformation of equation (23) gives the bare ion form-factor: 4 zrZe "-

DOq2 cos (rcq).

wh(q) -

(24)

Unlike the nonlocal model potential case, the screened form-factors in case of local potentials are easily obtained by dividing the bare ion form-factor by the electron-dielectric function: =

wb(q)/E~(q).

(25)

(19)

where wb(q) is the bare-ion local model potential

Q,,(q ) ~°(q) = 1 -~ I - f ( q ) Q o ( q ) "

(22)

Whereas, the empty core potential (ECP) due to Ashcroft [12] implies that the effective repulsive potential arising from the orthogonalization of the conduction-electron wavefunction to the core wave wavefunctions exactly cancels the Coulombic potential inside the core, and is given by:

w(q)

G(q)= [

(21)

Once we know the screened form-factor of an individual ion in either type of model, local or nonlocal, the liquid resistivity can be satisfactorily determined by following Ziman's pseudo-atom approach [29]. In this approach the problem is formulated in terms of an individual pseudopotential associated with the properly screened ion and a structure factor a (q) describing the ionic configuration in the liquid metal. The electrical resistivity pH~ of a liquid metal is given by the formula: 3win*Do (2k~ P""= ~ Jo a(q)l(k +qlwlk)l~q-'dq,

(26)

673

Liquid resistivity of AI where m* is the effective mass of the electron (taken here to be the real electron mass), E~ the Fermi energy, a(q) the liquid structure factor, and ( k + q l w l k ) [or w(q,k) in the nonlocal case, and w(q) in the local case] the screened individual pseudopotential form-factor. 3. PHONON DISPERSION RELATIONS: RESULTS AND DISCUSSION

We have calculated the phonon dispersion curves for three symmetry directions [ 100], [ I I0] and [ 111] using the three model potentials, N L O P , M P I P and ECP, screened by Evs. In the N L O P case, dispersion curves have also been obtained with three other dielectric functions, E., ecv, and essr,. The relevant physical data on which the present calculations are based are given below: M(amu) = 26.98; l%(aB3) = 111.4; kF(aB-') = 0"9273; Z = 3"0;

Z * = 3"197.

Here, we have selected the values corresponding to T = 0°K rather than the temperature of measurement. So, our results represent harmonic values as against the experimental data containing some tTabulated values of .f(q) in SSTL and VS approximations are kindly furnished by Prof. K. S. Singwi and Dr. P. Vashishta.

anharmonic effects. F o r the N L O P case, we have used the optimized model potential parameters given by S h a w [ l l ] , and calculated with R P A screening the form-factor WRpA(q),and the normalized e n e r g y - w a v e - n u m b e r characteristic GRPA(q) for values of q extending upto 10 kF using equations (6)-(13). Then, the many-electron corrections, Aw and AG, in the four different screening approximations are obtained using the respective f(q) through equations (14)-(19). In case of .f6v(q), the parameter ~: was obtained, following Price et al.[23], from the interpolation scheme of Nozieres and Pines[30]. For ~SSrL and Evs we have used the tabulated values of the corresponding f(q).t The four forms of f(q) are compared in Fig. 1. The functional form [f('O)/-O 2] is selected, since the differences of f ( q ) of the different theories in the important small q range are well demonstrated in this form, and further in the limit q equal to zero it becomes equal to v, where v is the parameter that defines the ratio of compressibility of a noninteracting electron gas to that of an interacting electron gas through the compressibility sum rule. F o r the sake of assessment of, significance of many-electron correction terms, we have plotted in Fig. 2 results of Aw for the four dielectric functions, and results of AG in Fig. 3. It is seen that Aw, as well as AG, peak around q = 0.75 to 0.8 kF, and gradually taper off to zero around q = 4.0 kF, with the first node in Aw occurring at q --- 1.53 kF. A

0.5

o4

f

A P,

STL\ -~~~~..

I'r/I

. .

O0

1 0.5

Fig. I. Comparison of the function

I 1.0

I I I I 1.5 2.0 2.5 3.0

?/

I I I 3.5 4.0 4.5 5.0

f(q) for AI in different screening approximations:

Hubbard

(I-I);--,--Geldart and Vosko (GV); . . . . Singwi et aL (SSTL); .... Vashishta and Singwi (VS); ---Geldart and Taylor (GT); .... Toigo and Woodruff (TW).

674

P . V . S . RAO 0.05 XlO 0.04 0.03 0.02 0.01

-

,,f

\j¢'

0 3

-0.01 - 0.02

i

I

i

I 2.5

I 3.0

I 3.5

- 0.03 - 0.04 - 0.05

0

I

I

I

I

0.5

1.0

1.5

2.0

I 4.0

I 4.5

5.0

Fig. 2. Many-electron correlation correction Aw to nonlocal optimized model potential form-factor w(q, k~) for A1 as a function of scattering momentum q. Results for four types of electron screening: (1) Hubbard, (2) Geldart and Vosko, (3) Singwi et al., (4) Vashishta and Singwi. Labeling as for Fig. 1. 0.08 /

A.t ( WNLoP )

0.071"0.06

~

0.05 ,XG

0.04

H SSTL

0.03

GV

0.02

XIO

t

0.01 0

\\

0

vs I • 0

I 4.5

5.0

Fig. 3. Many-electron correlation correction AG to the normalized energy-wave-numbercharacteristic G (q) of nonlocal optimized model potential for AI as a function of the scattering momentum q. Results for four types of screening are shown, Labeling as for Fig. 1. further discussion of Aw and AG will be taken up later. In case of MPIP, the parameters,/3 and pn were adjusted to give the best fit to the experimental data. The parameters so determined are: /3 = 49.4 Ry a~3; pH = 0.262a~. In the ECP case, we have rather assumed for r,. the value = 1-117 a8 obtained

by Wallace[5]. As adopted by Price et al.[23] in their dispersion calculations for alkali metals, we have introduced here an effective electron mass m * = 0 . 9 4 . This, becoming in effect a second parameter, did help in bringing the calculated values in better agreement with experimental data. As opposed to the case of MPIP, we did not try to

Liquid resistivity of AI

675

0.250

At o.~25- (~vs) 0 o O3 i,-

t

r_.~MPIP

t

.®.~.

-:-.~.;EcP (~

-0.125 -

.

..~.

•

FERMI SURFACE DATA

-0.250 Z

- 0.375

• _.OSo| E

- 0.500

I

~..A, ~-NLOP

"~

5.0

, I0.0

,

I

15.0

20.0

- 0.625

-0.750

0

i

i

1

0.5

1.0

1.5

2.0

1

I

I

I

I

2.5

3.0

3.5

4.0

4.5

5.0

Fig. 4. Form-factor w for AI computed for three forms of model potential screened with VS screening function: .... Nonlocal optimized model potential (NLOP); -.-Modified point-ion potential (MPIP); . . . . Empty core potential (ECP). It is given in units of Fermi energy E~. adjust the parameters of ECP for obtaining the best possible fit. The screened form-factor curves, as well as the normalized e n e r g y - w a v e - n u m b e r characteristics, for the two local potentials are compared with that of the nonlocal potential in Figs. 4 and 5 respectively. In Fig. 4 we also note, as noticed before by Wallace, that for the two local potentials, the w(q) are close to each other for 0 -< 71 -< 1.8. It is also interesting to see the close agreement between them and that of the nonlocal potential in the region of 0 -< "0 -< 1-4, except for a small difference around "0 = 0.15. Though we notice in Fig. 5 the same close agreement amongst the G(q)s of the two local potentials, they both are distinctly different from the G(q) of the nonlocal potential. Furthermore, the two form-factors corresponding to the local potentials are in good agreement with the Fermi-surface data[3I], as shown in Fig. 4. But, such an agreement is absent in the noniocal case. In Fig. 6 we have plotted the dispersion curves calculated with the N L O P screened by the four dielectric functions. The dispersion relations obtained with the N L O P screened by Evs are compared in Fig. 7 against those obtained with the two local potentials, M P I P and ECP, screened by the same dielectric function. F o r the purpose of comparison we have also plotted in both the figures the experimental points of Stedman and Nilsson[32]. Here, we have followed the labelling of Bouckaert et al.[33] for symmetry points and irreducible representations.

I.O

o,\ xoi 0.8

0.7

!I

0.6 0.5 G

o.4

.~'\

0.3

f

~.P,,

~,'\.

°2

, /

.\

o,

,,/_./N,OP

--..

o

o

t a5

I ,.o

~/,"r'--~_

,.5 2.0 2.s */

3.0

.~ ,~..-,

: "~

3.5 ,.o ,~

.5.0

Fig. 5. Normalized energy-wave-number characteristic G(q) for AI computed from the three model potentials using VS screening function. Labeling as for Fig. 4. An inspection of Fig. 6 reveals that the screening by Evs results in phonon frequencies which are in better agreement with the measured frequencies than the frequencies obtained with the other three dielectric functions. Even though the longitudinal branches (A,, ~, and A~) are close to experiment, the transverse branches (As, E~, E, and A3) are consistently 4 - 6 per cent too high, with the largest discrepancy between calculation and experiment occur-

676

P . V . S . RAO II r

X'

,o~ /

I

]

,,-7.i~,.!,

t~oo] ~ vs~_/,,>;.'/

_~2

t.ol

/

t-~], ~'

I K,~;.¢'-<>,'.

/

!//,

ol-

i ,!t/

I-

/ ;L

7V/"<21i o o, O0 r

,

02. 0!4 0.6 0.8 ~--~

I

I

,

_

I

1.0 0.8 016 0.4 O12 0.0 0.2 0.4 X ~ r ~-L

Fig. 6. Comparison of calculated and experimental phonon dispersion relations for AI. Open points represent the data of Stedman and Nilsson. Results for the nonlocal optimized model potential are shown with the four different types of electron screening. Labeling as for Fig. 1. II

I

I

I

I

[

x,~,

I0

I I I

I

I

I

[~oo] ~. ~,f*'~-L [~;o]

/,/

I

,

E~]

~

1

I K~ '~

/

7 6 0

5

2 I

°oL

¢~/ I

(.vs) I

I

I

0.2 0.4 0.6 0.8

I

I

I

I

1.0 0.8 0.6 0A 0.2 X ~

o[' 0'2 o!, ~--~

Fig. 7. Comparison of experimental phonon dispersion relations for AI with the calculated results for the three model potentials screened with VS screening function are shown. Labeling as for Fig. 4. ring at the point X~. It is interesting to note the strong correlation between the phonon dispersion curves (Fig. 6) and the AG curves (Fig. 3) for the four dielectric functions. One can see that the longitudinal phonons are sensitive to ZIG of the q-

region, 0 < q ~ 1.2kv, while the transverse phonons are dependent on ziG of large q-region, i.e. q > 1-2kF. F o r instance, the near coincidence of AGssTL and AGvs curves of Fig. 3 for values of q > 1.2kF led, as shown in Fig. 6, to almost equal transverse

Liquid resistivity of AI phonon frequencies (lower branch in case of [110]), even though the AGssTL and AGvs are quite different from each other in the region 0 < q < l-2k~. The small q-region differences have, of course, shown up in the longitudinal frequencies. This is somewhat quite analogous to an observation made by Bjorkman et al. [34] in connection with the damping of phonons in AI due to the electron-phonon interaction. They noticed 0-1 kr to be the main region for giving damping of longitudinal phonons, while the region of 1.2-2.0kF being of significance to transverse phonons. Granted the nonlocal optimized model potential due to Shaw is quite reliable we may (tentatively) make the following observations: The improved agreement between the experimental frequencies and the frequencies obtained with ~vs as opposed to those obtained with CH, err and ~SSTL,re-emphasize the importance of satisfying the two basic criteria of a reliable dielectric function. One may say that the satisfaction of the two criteria assures the reliability of the dielectric function only for small- and large-q regions, leaving the intermediate range still uncertain. It is also in this range that our Gvs(q) calls for improvement to get the transverse branches closer to experiment. Unlike the interpolated or phenomenological dielectric functions (for example, dielectric function due to Geldart and Vosko[27] or that due to Shaw[25]), which are designed to satisfy certain criteria, the evs is based on a self-consistent theory and is considered to be reliable over the entire qrange. In the context of intermediate q-range behaviour of E it should be pointed that the two recent dielectric functions, one due to Geldart and Taylor[35] and the other due to Toigo and Woodruff[36], as can be seen in Fig. 1, are significantly different from the other functions, especially in the region kF < q < 2kv. A critical look at Figs. 1 and 2 regarding the relationship between f ( q ) and G(q) reveals that these two dielectric functions may very well give rise to larger AG values in the q-region of significance to the transverse branch and thus, a closer agreement. But, unfortunately this will be at the expense of the agreement in longitudinal branches. Also, we may have to examine the model potential, and reevaluate the approximations that have gone into its formulation. One such approximation is that introduced by Shaw[25] in connection with the spatial distribution of the depletion hole charge to obtain the correct short wavelength limit of AG(q). This non-uniqueness of the optimized model potential as a result of our ignorance about the

677

exact nature of the depletion hole charge distribution is well discussed by Pynn [37] in his work on bulk modulus of simple metals. As against the Shaw's modulating function M ( q ) = l / ( l + ~ 1 2 ) , Pynn considered a different form for M(q):

M ( q ) = e -q't~,

(27)

where

= alrL

(28)

Here, r~ is the ion core radius and a is a factor whose value lies between 1 and 4. In real space, they are: psha,(r) = pk~ e-~" 4~- r '

(29)

and f O ' ~ 312

p~,on(r) = p ~ , ~ )

e-

o-r2

.

(30)

Using Pynn's modulating function and Cvs we have calculated Aw and AG for a = 1, 2, 3 and 4, and plotted AG in Fig. 8 along with AG obtained with the Shaw's M ( q ) and ~vs. We can see therein AG P with Pynn's function (for all a values) is almost the same as AG s with Shaw's function in the longitudinal range of q. Whereas, AG e is greater in magnitude than AG s in the transverse range of q with the difference ( A G e - A G s) increasing with the increase of a, which has the effect of decreasing the transverse frequencies. This result is very interesting, because with a proper selection of a we can bring the transverse branches also into good agreement with experiment without affecting much the longitudinal branches. In view of the uncertainties involved, as discussed below, with the nonlocal potential we have deferred the optimization of a to get the best fit. Further, such a fitting adds little to our understanding of the fundamentals of the situation. Whatever improved agreement we have seen with the new dielectric function in the nonlocal case does not, unfortunately, provide a definitive judgement on either the dielectric function or the model potential. The reasons for this are mainly related to the limitations of our presently adopted formulation of lattice dynamics. Among the most important are: (i) Neglect of higher-order terms in the electron-phonon interaction and anharmonic effects. Though known to be quite significant not much attention has been paid to this aspect, except for the recent calculations of Sandstrom and Hogberg[38], and that of Koehler et a/.[39]. Plans

678

P . V . S . RAO 0.08

0.07

-

0.06

--

("vs & WNLOP )

•G

0.05 -0.04 -0.03

/

\

0.02 0.01 0

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

~7 Fig. 8. Many-electron correlation correction AG to the normalized energy-wave-number characteristic of the nonlocal optimized model potential screened with VS screening function for AI as a function of the scattering momentum q. Results for four types of modulation function M(q) are shown: .... Shaw; - - P y n n (a = 1); ----Pynn (a = 2); Pynn (a = 4).

G(q)

are underway to investigate these effects with evs. (ii) Neglect of non-sphericity of Fermi-surface: contrary to the case of Na, the Fermi-surface of AI is distorted from the free electron sphere. From the results of Hartmann and Mitbrodt[7]. on the sensitivity of the phonon energies to the Fermi wave vector, one can see that the phonon frequencies are likely to be appreciably altered with the effects of Fermi-surface distortions. Even if one is prepared to neglect these effects, there is the non-uniqueness of the nonlocal potential that forbids us to make any conclusive observations on the new dielectric function. Besides, it is likely that other approximations, which have been made with respect to well depths, and treatment of exchange and correlation corrections in constructing the model potential, introduce uncertainties, which may be big enough to make the accuracy of our results somewhat questionable. Unfortunately it is, as reported earlier by Shaw[16], extremely difficult to estimate accurately the errors due to these approximations, and even more difficult to follow these errors through to give their estimate in lattice dynamic calculations. Finally, our calculations have not considered the two k-dependent effective masses me and ink, introduced by Shaw in his modified theory [40]. These masses, which appear as re-normalization factors in the dielectric function and in the depletion hole, have been found to give rise to corrections in the

form-factor and e n e r g y - w a v e - n u m b e r characteristic of a magnitude comparable to the exchange and correlation corrections. It will be interesting to investigate how much effect they will have on our results. Now, coming to the question of relative performance of the three different bare potentials when screened with Evs, we notice in Fig. 7 the phonon frequencies calculated with the M P I P are in the closest agreement with the measured frequencies. This is true with the longitudinal, as well as the transverse, branches albeit the [111] longitudinal branch. This is not surprising with a model potential that has adjustable parameters. Our best-fit parameters for MPIP, /3 = 49.4Ry aB3 and p , = 0.262a~, are quite close to the values, /3 = 49.0 Ry aB3 and OH =0"27a~, obtained by Hartmann and Milbrodt [7] with a dielectric function due to Toigo and Woodruff[36], and comparable to the values, /3 =47.5 Ry a , 3 and On =0-24as, reported by Wallace [5] for the screening approximation due to Geldart and Vosko[27]. Whatever differences we notice in the values of/3 and p , obtained in different screening approximations, they are indeed a direct consequence of differences in their respective [(q). It is not surprising to note the close agreement between our values and those of Hartmann and Milbrodt for the simple reason that the two dielectric functions, Evs and evw, satisfy the two basic criteria very satisfactorily in the electron

Liquid resistivity of A1 density range close to AI, whereas the e~v used by Wallace leads to a negative (unphysical) electronpair correlation function, while being made, though arbitrarily, to satisfy the compressibility sum rule. The other local potential, ECP with r~ = 1.117as and m*/m =0-94, results in dispersion curves which agree less well with experiment. As mentioned earlier, no serious effort has been put to optimize the values of r,. and m * in order to get the best fit. Nevertheless, we noted that better agreement, i.e. higher phonon frequencies, would be obtained either by increasing r,. (1 per cent increase in r~ brings about 3 per cent increase in the magnitude of the frequencies), or by decreasing m*/m further (1 per cent decrease in m*/m leads only to about l per cent increase). Further decrease of m*/m may be unphysical as our present value ( = 0-94) is much less than 1.03 obtained by Segall[41] from K K R band-structure calculations, and 1-04 calculated by Weaire[42] using the original form of the H e i n e - A b a r e n k o v model potential. Our present value is, however, interestingly equal to energy dependent component mE of Weaire's m*. On the other hand, one can bring closer agreement with an increase of rc by I to 1.5 per cent for m*/m = 0.94, or by 3 to 4 per cent for m*/m = I. 4. LIQUID RESISTIVITY: RESULTS AND DISCUSSION Here, we are concerned with the scattering of electrons only on the Fermi surface, and consequently we no longer have any problems of convergence, because the range of the modulus of scattering wave vector q is limited to O-2k~. F o r the

679

liquid structure factor a ( q ) , we have theoretically evaluated it using the hard-sphere model[14, 15] with a packing parameter ~ = 0.45. Its first peak occurs at q = 1.608kF. All density-dependent parameters except the screened form-factor w(q, k) in case of the nonlocal model potential, are evaluated at the appropriate liquid density (2.38g/cm3). F o r the w(q, k), we adopted the values obtained for solid aluminum. This approximation will only affect the final result by a few per cent. Table 1 gives the present computed values together with the other theoretical and experimental values [44]. As a result of the occurrence of a peak in the structure factor at q = 1.608k~, which is within 0-1 k~ (to the right) of the first node of the screened model potentials under consideration, the relative magnitude of calculated resistivity depends on the exact location of first node of the potential and its derivative at the nodal point in q-space. Since the location of the node for a given model potential does not get affected by the type of screening it is then only the slope that controls the relative magnitude of the calculated value of resistivity of liquid AI. Accordingly, in case of the nonlocal model potential we observe that the screening in S S T L or VS approximation that leads to comparatively steeper slope, yielding values of resistivity higher than that obtained with either eH or ~cv. Surprisingly, the values of resistivity calculated with essrL and ~vs differ insignificantly from each other. Whereas, the corresponding p,honon dispersion relations, as noticed in Fig. 6, are significantly different. This observation is in striking contrast to our experience in alkali metals[45]. Although the calculated phonon fre-

Table 1. Electrical resistivity of liquid AI in units of/~ l~cm Experiment Calculations:

Model Potential NLOP NLOP NLOP NLOP MPIP (13 = 49.4; p. = 0.262)

f(q)

H GV SSTL VS VS

24.2

Cusack t'j

22.35 21-36 23.40 23.35 21.28

Present Present Present Present Present

ECP (re = 1.117; - ~ = 0.94)

VS

22.05 Present

MPIP (13 = 49-0; pH = 0-27) MPIP (/3 = 47.5; pH = 0"24) ECP (re = 1.117) NLOP (Pynn t~ = 1) NLOP (Pynn tx = 2) NLOP (Pynn a = 4) NLOP (M(q) = 1)

TW GV GV VS VS VS VS

20-3 21.2 20.2 24.02 24.87 25.47 26-24

1"JSee Ref. [44]. Ib~See Ref. [7].

Hartmann and Milbrodt~b~ Hartmann and Milbrodtr*~ Hartmann and Milbrodt~b~ Present Present Present Present

P. V. S. RAO

680

quencies in case of N a and K did not change much, the calculated values of resistivity varied significantly. But it is not difficult to see why. On a closer examination of Fig. l one can easily see that the concurrence between the values of p,~q with essrc and evs in case of A! is a direct consequence of the nature of f(q) of the two dielectric functions in the q-range of importance to the resistivity calculation, i.e. around the nodal point. In that q-region, the two functions, fSSTL(q) and fvs(q), are close to one another, and also double-cross each other, with the first cross-over occurring at q ~ 1.5k~, and the second at q -~ 2.0kF. Unlike the resistivity case, the calculation of phonon frequencies encompasses the full range of q-space, at either end of which, i.e. small-q and large-q regions, the two dielectric functions, eSSrLand evs, are indeed noticeably different from each other. In Table 1 we have also presented the resistivity values obtained using N L O P with its depletion hole charge distribution as assumed by Pynn. We can notice the resistivity increasing from a value of 24.0 to 25.5 as a is changed from 1 to 4. This trend is towards the value ( = 26.2) one gets without the introduction of modulation function, i.e. M(q)= 1. As far as the agreement between theory and experiment goes, it is quite good, particularly in view of the approximate nature of the structure factor. Besides the structure factor approximation, there are possible changes with Fermi energy in the model potential parameters as the density is decreased by 12 per cent of the solid value between zero degrees and the melting point [7]. Then there is the standard question of the validity of expansion in the pseudopotential because the electrical transport phenomenon are extremely sensitive to the pseudopotential as compared to the phenomena of lattice vibrations. In view of these approximations, we have not attempted to obtain t:omplete agreement in case of the local potentials by adjusting their parameters. If one desires, energy dependence of the parameters might easily be introduced by knowing it from the calculations of liquid thermoelectric power [45, 46]. 5. EFFECTIVE INTERIONIC POTENTIAL: RESULTS AND DISCUSSION

Neglecting the repulsive interaction due to core wave function overlap, the effective interionic potential is simply given in terms of the Cochran function G(q) by the expression: Z2e 2

V(r)-T

2 Z 2 e 2 ~'~

~

J0

sin qr G(q) q_~dq. r

(31)

PA

i

( WNLOP )

II ~ rr_w_\ ~-~

/

-'r/-°v ./

I

o

-'F\/" -44 I

5n

n

6I

7I

i 8i

I .T]"

9i

I

I ;0

,[o...] Fig. 9. Effective interionic potential V(r) as a function of the interionic separation r in AI metal calculated with the nonlocal optimized model potential (NLOP). Results for four types of screening are shown. Labeling as for Fig. 1. As has been done in lattice dynamic calculations, the valence, Z, is replaced here also b y Z * for the nonlocal case. In Fig. 9 we present our numerical results of the calculation of V(r) for solid AI. Several interesting points are to be noted in this figure. By comparing the potentials VRPA(r) and VH(r) corresponding respectively to the R P A and Hubbard dielectric functions, one can see that the inclusion of exchange effects changes significantly the character of the potential in the region near the first neighbor, i.e. from a potential having no minimum to a potential having a deep minimum. This can be easily seen to be due to cancellation of the strong repulsive part of vRPA(r). F o r this potential, VH, the nearest neighbor (n.n.) distance lies beyond the first minimum, implying a positive value for the slope of the potential V'~(r) at the n.n. position. Inclusion of the correlation as well as the exchange effects through the use of eSSTL has improved the situation in that the V;(r) is now negative due to the shift of the minimum to the right of the n.n. position. Besides, its depth is considerably less, suggesting that the addition of exchange effects alone would over-cancel, the repulsive part of VRP^(r). Around 2nd n.n. position the oscilla-

Liquid resistivity of AI

681

tions, which are present but masked by the strong decrease in the magnitude of potentials at the n.n. repulsive potential in the case of RPA, and by the position by about 25 per cent. attractive potential in the case of Hubbard approxiWe have seen, therefore, that the nature of the mation, have now become apparent. Use of the VS interionic potential in the range of small r, when obdielectric function, which is an improvement over tained in the model potential (local or nonlocal) the SSTL dielectric function gives rise to a further scheme, is largely governed by the nature of the reduction in the cancellation of the repulsive part of dielectric function. The drastic changes in V(r) the potential with its oscillation around the n.n. po- when one goes from R P A to other approximations sition tending to be masked. The potential vVS(r) is to include exchange and correlation effects are, as surprisingly unconventional, with the minimum Shaw and Heine already observed [47], due to varyaround the n.n. distance, and V,(r) are now posi- ing amounts of cancellation of the strong repulsive tive. Nevertheless, the sign of V; is unaffected. part of V ~*A.They were also interpreted as changes Comparing the potentials corresponding to ~H and in screening range of the electron gas. Whatever the EGv, which do not account for Coulomb correlation reasons and interpretations may be, the main result effects, one notices a similar upward shift of the of our calculations is that the effective interionic potential. But, he.re the minimum around the n.n. potential V(r) between nearest neighbors is posiand V~(r) are still negative, while V'dr) changed tive rather than negative, and it is repulsive. correctly its sign from positive to negative. Even though the two basic ingredients, model poResults similar to that of VS dielectric function tential and dielectric function, that went into the are obtained with the use of Geldart-Taylor [35] and construction of the potential V vs are the best availToigo-Woodruff[36] dielectric functions. The latter able ones, they are still approximate. To increase two functions also satisfy the compressibility sum our confidence in the present potential it is worthrule.T As far as the electron-pair correlation func- while, then, to demonstrate how well different tion is concerned, it is positive for AI in the T W physical properties are predicted in terms of the poapproximation also--though at large r it is more tential. One such property that one considers at the negative than that in the VS approximation, and •very outset is the stability of crystal structure behence less physically acceptable. Unfortunately, no cause different crystal structures involve different such comparison can be made in case of the GT atomic arrangemments, and thus collectively test approximation. We notice the potentials, V eT and the potential at different points in r-space. But, V Tw, to be close to each other, with V rw slightly surprisingly it turns out to be not such a sensitive greater than V eT in magnitude, and both are shifted test. As the results of Shaw[16, 47] indicate, the refurther up as compared to V vs. Around the n.n. lative values of structure-dependent energies are position, both V ~T and V Tw are approximately not much affected by the changes in potential. For twice V vs, and no minimum is perceptible in V rw. smaller atomic arrangements what really matters Using evs we have also calculated interionic po- are the derivatives of the potential, which are seen tentials in the local pseudopotential scheme using to be relatively stable. Then, for larger atomic arthe ECP and MPIP model potentials. The poten- rangements it is critical where the sharp rise in the tials thus obtained are similar to V vs, but they are potential occurs relative to the nearest neighbors in moved further up as V Gr and V rw. These differ- the two structures under consideration [47]. For the ences can be traced back to the differences in G(q). two potentials, V ssrL and V vs, the sharp rise does Particularly, the local model potentials are charac- not occur at significantly different positions to influterized by a second peak in G(q) of a larger mag- ence the relative stability of the structure. For exnitude and slowly decaying in comparison to that of ample, we considered below the stability of f.c.c. the nonlocal model potentials. vs. diamond structures. In order to see whether there would be any major Given an effective interionic potential one can change by considering the spatial distribution of de- easily determine the more stable structure by calpletion charge different from that adopted here, we culating cohesive energy with its expression written also considered the distribution assumed by by Price [48] in real-space. In this way, one has to Pynn [37], and found the effect to be small. In gen- evaluate only the structure dependent part E,. Our eral, the potential has shifted downward with a calculations of E, for the different potentials, V", V cv, V ssr" and V vs, given in Fig. 9 have shown the fActually, the compressibility result of Toigo-Woodruff same result, namely the f.c.c, structure is more stawith v =" has no effect of Coulomb correlations in it [9]. ble than the diamond structure--Hubbard model,

682

P.V. S. RAO

however, being relatively the most stable one. One can rather draw this conclusion qualitatively by noting that the n.n. distance is 4.2 Bohrs for diamond structure as against a value of 5.4 Bohrs in case of f.c.c, structure, and as a consequence the potential at the n.n. distance in case of the f.c.c. structure is less than that of diamond structure by an order of magnitude. As a matter of fact, the stability calculations are usually carried out in reciprocal space, for reasons of convergence. What goes into these calculations then is the Fourier transform of the potential in momentum space V(q), or indirectly G(q). And we have found earlier that the G(q) obtained by using evs has given us phonon dispersion relations in much closer agreement with the experimental data than those obtained by the other dielectric functions. En passant it is quite interesting to note that one also gets from experimental liquid structure factor data a potential with a minimum, in some cases less pronounced, on the positive side of the energy axis. Recently Ruppersberg and Wehr[49] for liquid AI and North, et a/.[50] for liquid Pb have obtained potentials of the above nature from the experimental structure factor data in the Percus-Yevick, as well as the hyper-netted chain model theories. They, however, concluded that the two model theories are not applicable to liquid metals as the potentials derived by them are not similar to the Harrison's potential for AI[13], which has a negative minimum around the n.n. But, their conclusion is to be questioned in view of the fact Harrison's potential, though based on the pseudopotential theory, does not include the exchange, as well as the correlation effects, which have, indeed, a very significant influence on the potential. Another point that is worth mentioning in this connection is that the minimum in the Harrison's potential occurs to the left of the n.n. positions, making the first derivative positive, as obtained by us in case of Hubbard dielectric function. But, this is not right. Although it is comforting to notice the agreement between our potential with evs and the potential derived by Ruppersberg and Wehr, one cannot, unfortunately, put too much faith in it, because of the fact that the potential thus derived from the liquid structure factor data suffer from inaccuracies of the measurement. It is rather preferable to compute theoretically the liquid structure factor from the given interionic potential and compare it with the measured data. Such a program is underway using molecular dynamic techniques[51], as well as the

simplified procedure Weeks [52].

due

to

Chandler

and

~. $UMM~d~Y Using the conventional reciprocal-space method we have seen that with the nonlocal optimized model potential the improved self-consistent treatment of many-electron correlation effects due to Vashishta and Singwi brings the calculated phonon frequencies in good agreement with the experimental data, errors greater than 3 per cent being present in the phonon frequencies at small-q. Very good agreement is obtained for longitudinal branches, but the results for transverse branches leave something to be desired. A re-consideration of the spatial distribution of depletion hole charge seems to be one way of improving the agreement. By considering three other screening approximations due to (1) Hubbard[26], (2) Geldart and Vosko[27], (3) Singwi et al.[10] we have clearly noticed the indispensability of including the exchange and correlation corrections in screening and the importance of the precise form of these corrections. The liquid resistivity results show that the nonlocal potential when screened with the new dielectric function ~vs predicts a value in a better agreement with the experimental value than those obtained hitherto. These results rather suggest that with an approximation for exchange and correlation, second order perturbation theory in the pseudo-potential is much more reliable than one would have thought. Perhaps it is too early to be certain, but our findings fit together nicely and are very encouraging---especially when we have obtained this close agreement with the experimental data using the 'first principles' forms for the basic ingredients of the calculation. In regard to the comparison of nonlocal and local potentials we fitted two local potentials: (I) Modified point-ion, (2) Empty core, using the new dielectric function Evs. We noticed the MPIP with two adjustable parameters giving phonon frequencies in better agreement with experimental data as compared with the nonlocal optimized model potential. This, of course, has to be seen in the context of the fact that the nonlocal potential is based on first principle arguments and do not represent an attempt to obtain agreement with experimental resuits. Our complimentary study of effective interionic

Liquid resistivity of AI potential for solid AI has shown that the nature of the potential in the range of small r, when obtained in the model potential, local or nonlocal, scheme, is largely governed by screening function. With a dielectric function Evs that satisfies the two basic criteria, namely the satisfaction of the compressibility sum rule and giving simultaneously a positive pair correlation function for electrons, the effective interionic potential between nearest neighbors turns out to be positive rather than negative, and it is repulsive. It is no doubt true that we employed in our calculation of the potential the same G(q) as the one which was found to give very satisfactory but not exact fit to the phonon dispersion data. But, the remnant disparities, especially in the transverse branches, a n d ' a l s o the approximations that went into the theory of lattice dynamics on which the present calculations are based, make it imperative to explore further other physical phenomena that are sensitive to the potential in the small-r range. Phenomena of lattice defects falls into this category. Our defect calculations on Na [53] and also of others, e.g. Flocken and Hardy[54], indicate that defect energetics are very strongly influenced by the form of the potential around the first two neighbors. We are currently engaged in a calculation of energetics of a single vacancy in A[. Especially, the results of the formation energy calculation can help in judging the potential.

Acknowledgements--It is with great pleasure the author would like to thank Prof. K. S. Singwi and Dr. D. L. Price for their valuable advice during the course of this investigation. He is also grateful to them for a critical reading of the manuscript and suggestions.

REFERENCES

I. Harrison W. A., Phys. Rev. 136, A 1107 (1964).

2. Vosko S. H., Taylor R. and Keech G. H., Can. J. Phys. 43, 1187 (1965). 3. Animalu A. O. E., Bonsignori F. and Bortolani V., Nuovo Cim. B44. 159 (1966). 4. Schneider T. and Stoll E., Phys. Kondens. Mater. 5, 364 (1966). 5. Wallace D. C., Phys. Rev. 187, 991 (1969). 6. Coulthard M. A., J. Phys. C: Solid State Phys. 3, 820 (1970). 7. Hartmann W. M. and Milbrodt T. O., Phys. Rev. B3, 4133 (1971). 8. Ho P. S., In: Interatomic Potentials and Simulation of Lattice Defects, p. 321. Plenum Press, New York (1972). 9. Vashishta P. and Singwi K. S., Phys. Rev. B6, 875 (1972).

683

10. Singwi K. S., Tosi M. P., Land R. H. and Sjolander A., Phys. Rev. 176, 589 (1968); Singwi K. S., Sjolander A., Tosi M. P. and Land R. H., Phys. Rev. B1, 1044 (1970). 11. Shaw R. W., Jr., Phys. Rev. 174, 769 (1968). 12. Ashcroft N. W., Phys. Lett. 23, 48 (1966). 13. Harrison W. A., Pseudopotentials in the Theory of Metals. Benjamin, New York (1966). 14. Percus J. K. and Yevick G. J., Phys. Rev. II0, I (1958); Percus J. K., Phys. Rev. Lett. 8, 462 (1962). 15. Ashcroft N. W. and Lekner J., Phys. Rev. 145, 83 (1966). 16. Shaw R. W., Jr., J. Phys. C: Solid State Phys. 2, 2335 (1969). 17. Ashcroft N. W., In: Fundamental Aspects of Dislocation Theory Vol. 1, p. 179. United States National Bureau of Standards Special Publication 317 (1970).' 18. Shyu W. M., Wehling J. H., Cordes H. R. and Gaspari G., Phys. Rev. B4, 1802 (1971). 19. Gehlen P. C., Beeler J. R., Jr., and Jaffee R. I. (Eds.) Interatomic Potentials and Simulation of Lattice Defects. Plenum Press, New York (1972). 20. Kellermann E. W., Phil. Trans. R. Soc. Lond. A238, 513 (1940). 21. See, for example, Cochran W., Inelastic Scattering of Neutrons, Vol. I, p. 3. IAEA, Vienna (1965). 22. Cochran W., Proc. R. Soc. Lond. A276, 308 (1963). 23. Price D. L., Singwi K. S. and Tosi M. P., Phys. Rev. B2, 2983 (1970). A computer program (written for IBM 360 machine) to calculate the phonon dispersion relations was kindly furnished by Dr. D. L. Price. 24. Sha w R. W., Jr., and Pynn R., J. Phys. C: SolidState Phys. 2, 2071 (1969). 25. Shaw R. W., Jr., J. Phys. C3, 1140 (1970). 26. Hubbard J., Proc. R. Soc. Lond. A240, 539 (1957); A243, 336 (1958). 27. Geldart D. J. W. and Vosko S. H., Can. J. Phys. 44, 2137 (1966). 28. Heine V. and Abarenkov I., Phil. Mag. 9, 451 (1964). 29. Ziman J. M., Phil. Mag. 6, 1013 (1961). 30. Nozieres P. and Pines D., Phys. Rev. III, 442 (1958). 31. Ashcroft N. W., Phil. Mag. 8, 2055 (1963). 32. Stedman R., AImqvist L. and Nilsson G., Phys. Rev. 162, 549 (1967). A table of measured and interpolated frequencies for AI at 80°K was kindly sent by R. Stedman. 33. Bouckaert L. P., Smoluchowski R. and Wigner E., Phys. Rev. 50, 58 (1936). 34. Bjorkman G., Lundqvist B. I. and Sjolander A., Phys. Rev. 159, 551 (1967). 35. Gedlart D. J. W. and Taylor R., Can. J. Phys. 48, 155 (1970). 36. Toigo F. and Woodruff T. O., Phys. Rev. B2, 3958 ( 1970). 37. Pynn R., Phys. Rev. B5, 4826 (I972). 38. Sandstrom R. and Hogberg T., J. Phys. Chem. Solids 31, 1595 (1970). 39. Koehler T. R., Gillis N. S. and Wallace D. C., Phys. Rev. I, 4521 (1970). 40. Shaw R. W., Jr., J. Phys. C: Solid State Phys. 2, 2350 (1969). 41. Segall B., Phys. Rev. 124, 1797 (1961). 42. Weaire D., Proc. Phys. Soc. Lond. 92, 956 (1967). 43. Shaw R. W., Jr. and Smith N. V., Phys. Rev. 178, 985 (1969).

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P. V. S. RAO

44. Cusack N. E., Rep. Prog. Phys. 26, 361 (1963). 45. Rao P. V. S., Phys. Status Solidi (b) 55, 629 (1973). 46. Ashcroft N. W., J. Phys. C: (Proc. Phys. Soc.) 1,232 (1968). 47. Shaw R. W., Jr. and Heine V., Phys. Rev. B5, 1646 (1972). 48. Price D. L., Phys. Rev. A4, 358 (1971). 49. Ruppersberg H. and Wehr H., Phys. Lett. 40A, 31 (1972).

50. North D. M., Enderby J. E. and Egelstaff P. A., J. Phys. C (Proc. Phys. Soc.) 1, 1075 (1968). 51. Paskin A. and Rahman A., Phys. Rev. Lett. 16, 300 (1966). 52. Chandler D. and Weeks J. D., Phys. Rev. Lett. 25, 149 (1970). 53. Rao P. V. S., (Unpublished results). 54. Flocken J. W. and Hardy J. R., Phys. Rev. 177, 1054 (1969).