PAIR POTENTIAL AND STRUCTURE FACTOR OF POLYVALENT SIMPLE LIQUID METALS

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PAIR POTENTIAL AND STRUCTURE FACTOR OF

POLYVALENT SIMPLE LIQUID METALS

H. Beck, R. Oberle

To cite this version:

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JOURNAL DE PHYSIQUE Co ZZoque C8, supp Z&ent au n o 8, Tome 41, aoet 1980, page

C8-289

P A I R POTENTIAL AND STRUCTURE FACTOR OF POLYVALENT S I M P L E L I Q U I D METALS

H. Beck and R. Oberle

I n s t i t u t de Physique, Universite' de NeuchiiteZ, CH-2000 Net~chdteZ, Suisse

R6sum6. - Nous avons calculi? des potentiels effectifs B deux particules pour des metaux liquides simples polyvalents, tenant compte du libre parcours moyen fini des electrons de conduction. Les oscillations longue porti?e de ces potentiels produisent un &paule- ment dans le facteur de structure statique.

Abstract.

-

We have calculated effective pair potentials for polyvalent simple liquid metals which include the effect of the finite mean free path of the conduction electrons The long ranged oscillations of these potentials lead to a shoulder in the static structure factor.

One of the major goals of classical liquid theory is the determination of the structure factor S(k) on the basis of the interaction between particles. Here we want to discuss a specific deviation from the otherwise hard-sphere-like structure of a series of polyvalent simple liquid metals : liquid Ga, In, Sn, Ge, Si, Pb, Bi and Sb have shoulders on the high k side of the main peak of S(k) [I]. Metallic liquids are specially challenging because they are binary systems consisting of ionic cores and conduction electrons. In the adiabatic approximation effective interionic forces can be

introduced by taking the trace over electronic variables in the Hamiltonian

where p is the electronic density and V a pseudo-

m P

potential. The contribution F =

1

to the total

v=o

energy coming from Hel and the electron-ion interaction contains the energy B of the interacting

( 0 )

electron gas and a sum over v-ion terms

E ( v ) ,

v L 1, of v-th order in V see for example [ 2 ] . P'

The second oraer term, added to the Coulomb repul- sion of the ions, yields the usual. "linear

screening" ionic pair potentials with long ranged oscillations. Only few attempts have been made to evaluate

E ( g ) ,

which yields intrinsic three-body interactions [ 2 ] . For polyvalent metals it is certainly desirable to take such higher order terms into account. On the other hand the theories of structure determination of liquids 131 are usually set up to deal with pair potentials only. For this reason effective pair potentials are devised which include the effects of many particle forces (for references see [ Z ] ) . Obviously their explicit form depends on the statistical mechanical quantity one wants to express by it. We propose the following approach : the effective pair i?teraction W between two ions in the liquid should be related to the

+

amount by which the total energy E(R1, ..,R~), depending on all ionic coordinates, changes when the two particles are slightly displaced :

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JOURNAL DE PHYSIQUE

c8-290

+ 3 823 the ionic arrangement after each Coulombic inter-

V.V.W(%,%) E

-

1 3 (n # m). (2)

a%j action. Finally, exchange and correlation effects

This expression represents the variation of the _{are approximately accounted for}by inserting X(P) force on ion n, when ion m is displaced, and is

certainly correct when E contains only pair interactions. (In a solid (2) would define the spring constants between particles.) For Hamiltonian (1)

+ +

-+

-+ -1

we find W

(%,&)

= znzme2

1

Q

-

I$,,

I

+

(kt&,)

with

+ -+

I

-+

+

3 -+ + 3

+(%,%I = d3r d3r' Vp(r -Rn)V (rl - R,JX(r,r1). ( 3 )

P

This is the same form as the linear screening

-+ -+

approximation but X(r,rf) is the susceptibility of the interacting electron gas in the presence of all ions, depending on all ionic coordinates. Formally

-

X can be evaluated by performing a diagrammatic double expansion of the two-electron Green function Gp in powers of Vp of the Coulomb interaction Vc. Since the electron-ion interaction in (1) is a given external potential U for the electrons the diagram for G2 have "U-lines" attached to the free electron lines G(') 141. As a first approximation, corresponding to RPA for the susceptibility Xh of the homogeneous electron gas, we can represent X by the usual sum of "electron-hole-bubbles" and Vc-lines, replacing, however, G(') by G@), the propagator of non-interacting electrons in an ensemble of ions carrying a screened pseudopotential Wp [ S ] . At the end of the calculations eq. (3)

should be subject to a configurational average over

-+ -f +-

all R,, s # m, n; yielding a potential $(Rn

-%I.

Our approximation consists in performing this average in RPA independently for each "G(P)-bubble". This amounts to replacing ~indhard's function X ( O )

showing up in the usual forms of

Xh

by a renormalized X(P). In such a "factorized" configuration average an electron-hole pair Looses its memory for

into the Vashishta-Singwi form [ 6 ] of Xh. This result for X was then used to calculate W(R). X(p) is determined by the solution GiP) of a Bethe- Salpeter equation, found also in the evaluation of Kubo's formula for the electrical conductivity [ 7 ] .

To second order in Wp it reads

The self-energy of the renormalized one-electron propagator G(P) reads, to the same order in WP :

De Gennes 181 has calculated the effect of impuri- ties on the susceptibility of a non-interacting electron gas by using essentially these two equa- tions, replacing, however, the function W* (q) S (q)

P by a constant, which yields a k-independent self- energy.

We have evaluated G and G ( ~ ) self-consistently for an empty-core pseudopotential. For G2 the main effect of the deviation of G(~) from G(') comes from

T,

which introduces a finite mean free path .P

for the electrons. The k-dependence of

r

is weak and may be neglected, and its frequency dependence is roughly the same as if WpS were a constant

( T ( w ) is then proportional to the density of states N(w)). We therefore used two methods to solve (4) approximately for G ~ P ) :

(a) W$S is directly replaced by a constant as in ref. [ 8 ] . The effect of Wp finally shows up in the parameter F! = (29-kF)-l [ 8 ] .

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as stated above. The integral in (41, which should be a slowly varying function of

i,i'

and w, is

replaced by a constant, chosen such that X(g) has the correct behavior for q -t 0 [8].

The corresponding pair potential (eq. 3) for Sb (Z = 5, core radius r = 1.1 a-u., p = 0.08) are compared with the linear screening result (i.e. y = 0) in Fig. 1.

Fig. I. - Effective pair potentials for liquid Sb. Full curves : using methods (a) and (b) described in the text; dashed curve : linear screening.

A general quantitative discussion of the influence of finite L on W(R) is not straightforward, since already for u = 0 W(R) strongly depends on the pseudopotential and screening function chosen [91. The following points can be made, however :

(i) y is roughly determined by the electrical resistivity. Neglecting variations between elements of the same charge Z, one finds u

-

(0.02) (Z = 3 ) ,

y 2 0.05 (Z = 4),

u

1: 0.08 (Z = 5 ) ' showing an approximate z2-dependence, as expected for quanti- ties of second order in W

P'

(ii) The general trend [9] for the amplitudes of the oscillations, when calculated for

e,

seems to be a similar increase with Z : typically the first maximum is about 0.5

-

1 mRy (Z = 3 ) ,

1

-

2mRy (2 = 4).

(iii) Finite p seems to have two main effects : a general reduction of the amplitude of the long range oscillations, increasing with R (see Fig. 1 for the "worst" case of

z

= 5 and y = 0.081, qnd specific modifications in the region of the first minimum, which depend quite sensitively on the relative values of rc and 2kF. Since large Z yield large oscillations for p = 0 and at the same time a short mean free path, taking account of the latter (y > 0) is expected to yield oscillations which in the end are of similar magnitude for the various polyvalent simple metals.

Let us compare our W with other work : Hasegawa and Young [lo] extracted an effective pair potential from S(k) and found

-

in comparison with linear screening

-

large changes near the first minimum and reduced oscillations. Hasegawa's 121 W, including three-body effects, show oscillations hardly modified if not enhanced. Harrison [Ill found oscillatory behavior in the asymptotic form of all v-body terms, but it is difficult1 to see how they would interfere in forming ap average pair potential.

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JOURNAL DE PHYSIQUE

a1 [12] by ORPA and by Monte Carlo techniques. On intensity of the shoulder decreases with increasing the other hand Mon et a1 1131 obtained such shoul- temperature

-

a simple consequence of the fact that ders by taking into account core polarizability. the ratio Wosc/kBT enters the calculations. The

position ks of the shoulder is somewhat smaller than 2kF (for free electrons), again in agreement with experiment. The occurrence of the shoulder can be understood as follows : the general structure of these liquids is dictated by the strongly repulsive part of W. The first peak of the hard-sphere-like S(k) at kp characterizes the main Fourier components of the structure (kp = 2n/AN

-

2n/RM, where

AN = mean neighbor distance and RM = position of the first minimum of W). However, the secondary

minima of W, distant from each other by about

AF = 2n/2kF, provide further preferred distances between next-nearest etc. neighbors. This introduces important Fourier components at ks s 2kF into the structure (a detailed analysis of the ORPA

1.2 1.6

k

[ Q U ~ calculations shows that ks j 2kF, since the oscillations decay for large R). In Ga, at low T (super- Fig. 2.

-

First peak of S(k) for liquid Ga for

T = 2 5 1 K and T = 928K.

cooled and amorphous state), the shoulder even grows to a subsidiary maximum of S(k) 1121. The following comments should be made concerning

This work was supported by the Swiss National our calculations : first, as in experiment, the

Science Foundation.

REFERENCES

[I] R. OBERLE, H. BECK, Sol. State CoIDm.

32

(1979) 161 P. VASHISFfTA,K.S. SINGWI, Phys. Rev. B

6

959, This paper contains references of measured (1972) 875.

S (k) for these liquid metals.

[7] J. RUBIO, J. Phys. C _-2 (1969) 288. [2] M. HASEGAWA, J. Phys. F

6

(1976) 649.

[81 P.G. DE GENNES, JournaldePhys.

23

(1962) 630. [3] H.C. ANDERSEN, D. CHANDLER, J.D. WEEKS, Adv.

Chem. Phys.

2

(1976) 105. [9] R. KUMARAVADIVEL, R. EVANS, J. Phys. C

2

(1976) 3877.

[4] R. BROUT, P. CARRUTHERS "Lectures on the many

[lo] M. HASEGAWA, W.H. YOUNG, J. Phys. F

8

(1978) electron problem", 1963, Interscience Publ.

L81. N. York.

[ll] W.A. HARRISON, Phys. Rev. B

1

(1973) 2408. f5] For the introduction of a "screened" external

potential see e.g. L.P. KADANOFF, G. BAYM [12] C. REGNAUT, J.P. BADIALI, M. DUPONT, Phys. "Quantum statistical mechanics", 1962, W.A. Lett.

74A

(1979) 245.

Benjamin Inc., chapter 12.