Prami0a-J. Phys., Vol. 26, No. 2, February 1986, pp. 161-170. © Printed in India.
Structure factor of liquid metals in one-component plasma system
K N KHANNA* and G SHANKER
Department of Physics, VSSD College, Kanpur 208 002, India
* Present address: Groupe De Physique des Liquides Metalliques Universite De Metz, 57000 Metz, France
MS received 17 June 1985; revised 3 December 1985
Abstract. We present the analytical expression to determine the static structure factor S(q) of liquid alkali metals based on the semi-analytic theory of Baus and Hansen. The calculations were motivated to point out the anomafies exhibited by one-component plasma system (ocp) in describing the structure factor. The numerical example of Na illustrates our results, ocp structure factor is also compared with those obtained by the hard sphere system.
Keywerds. Liquid alkali metal; one-component plasma system; random phase approxima- tion; dielectric screening; structure factor.
PACS No. 61.25; 65-50
1. Introduction
The one-component plasma system has been studied extensively in recent years (Hansen 1973; Slattery et al 1982) by using the Monte Carlo method. In this system electrons are assumed to form a rigid, uniform background neutralizing the average space charge field of the ions characterized by a dimensionless parameter F
= (Ze)Z/aKBT; where g e is the electrical charge of an ion and a represents the ion sphere radius; a = (3/4nn) 1/3; where n is the number density of ions. For a long time, it appeared that this system holds good only for small values of F i.e. very high density plasma. Recently it has proved to be fruitful for liquid metals (Galam and Hansen 1976;
Ono and Yokoyama 1984) as it supplies free energy lower than the hard sphere system ( M o n e t al 1981). Monte Carlo simulations are too tedious, time-consuming and confine to ionic composition. The analytical expression for structure factor, as conveniently available for hard sphere system, is still lacking and it appears difficult to offer an exact analytic expression that describes the fullstructure factor. However, some attempts are notable (Chaturvedi et a11981; Evans and Sluckin 1981; Baus and Hansen 1979) which remained successful in a particular domain of structure factor. For instance, Chaturvedi et al (1981) faced the well-known problem of empirical cut-offdue to large value around first peak of the structure factor. Evans and Sluckin (1981) calculated only the long wavelength limit of structure factor. At metallic densities, it is necessary to consider the electron gas screening effects of the degenerate electron liquid.
For liquid alkali metals, the coupling between the electron and ion can be assumed weak and thus random phase approximation (RPA) can be applied. In the present paper we extend the Bans and Hansen scheme by combining it with SPA to calculate the structure factor that is found successful upto the first peak lying beyond 2K F. It may be 161
162 K NKhanna and G Shanker
useful to describe properties like liquid metal resistivity where we need structure factor upto 2K e. Our main attention is focussed towards the limitations of the ocP system while some attention is also paid towards comparing the results of the ocP system with the hard sphere system. An effective reduction is observed in the ionic correlation resulting from the electron gas screening.
2. Formulation
We consider the analytical function for direct correlation function suggested by Baus and Hansen (1979)
p
C o c P (r) = ~.~ air 2~ r <~ ro, i=o
# Z 2 e 2
-- r >1 ro,
r
(1)
where the coefficients ai are determined by requiring CocP (r) and p, the first derivative is assumed to be continuous at the radius to, a sealing parameter. This theory has been extensively studied by Bretonnet and Khanna (1985) who found that p = 3 yields the best results for the structure factor of liquid alkali metals. Thus for p = 3 the direct correlation becomes
Co~. (r) = - - ~Z2e2 [ - 3 5 + 35(r/ro) 2 - 2 1 (r/ro) 4 + 5(r/ro) 6] r <~ ro, 16ro
f l Z 2 e 2 r ~> r O.
r (2)
The Fourier transform ofCocr (r) can be combined With RPA to determine the structure factor as follows
S(q) 1 (3)
1 - t,[co,:~ (q) -/~u~c (q)]'
with #sc(q) is the perturbation to the direct Coulomb repulsion due to electronic screening and can be written in terms of the normalized energy wave number characteristic F s (q )
4n Z 2e2
Use (q) = q2 F N (q), (4)
Finally we arrive at an analytical simple expression to determine the structure factor as
3r'xo 2 [- 105~ ( 15 \ sin X
P[Co~(q)-t~u~:Cq)] = - X--r-i_ X" [ ~ - - 6 ) 5(
(5)
Structure factor of liquid metals 163 where X -- qro and Xo = ro/a are two dimensionless parameters.
As the perturbation effects are important in the ocP system, particular attention is paid to construct FN(q). We therefore use Fs(q) constructed by using the Shaw- optimized model potential (Bretonnet and Khanna 1985). A novel a~pect of the calculation lies in the use of screening functions proposed by Ichimaru and Utsumi (1981 ) (lu) and Vashishta and Singwi (1972) (vs). Where appropriate, we compare our results numerically by using the Ashcroft (1966) model potential and the Hartree dielectric function.
3 . R e s u l t s a n d d i s c u s s i o n
Our interest in the present work is to study the various factors that hinder in the determination of structure factor of liquid metals in the ocP system. One knows that S(q) is well related to the effective pair potential (Bratkovsky et al 1982) which in turn depends upon the density o f the liquid metal through the pseudopotential and the dielectric screening. In the framework o f the linear response theory, the screening action may generally be described in terms of the static dielectric function e (q). The bare ocP system is linked with electron gas response function through weak coupling i.e.
random phase approximation applies. Thus we discuss the role of dimensionless coupling constant F, the model potential, the electron gas response function and the exchange and correlation functions.
An equilibrium state o f the ocp with the number density n and the temperature T may be characterized by a dimensionless parameter F. The choice of the parameter F in the ocP system is an important aspect as it governs all the thermodynamic properties including the stability of the reference liquid. There are many methods generally used to determine F e.g. (i) by fitting long-wavelength limit o f the structure factor S(0) value or compressibility data (ii) by fitting the first peak o f the structure factor and (iii) by fitting the entropy of the system (Khanna and Shanker 1985). The numerical values of F obtained by these procedures are displayed in table 1. Marked differences are seen
Table 1. Calculated values of F obtained by different methods I, II and III correspond to values obtained by fitting S(0), first peak of structure factor and entropy of the system respectively. Comparison should be made for the same model potential.
Fsa~nins
Metal Fitting Procedure F Hartree vs IU ov
N a I - - -- 210.0 188"0 185"0
II - - 193"8 185-0 184.4 --
III 124 . . . .
K I -- -- 200-0 182-0 181.0
II -- 190-3 185"8 185.3 --
III 120 . . . .
Rb I -- -- 210"0 184"0 182"0
II -- 192"0 191.5 191"7 --
III 126 . . . .
164 K N Khanna and G Shanker
between the values of F obtained by fitting the structure factor and the entropy of the system. The discrepancy is due to the following reasons. To carry out the entropy calculations outlined in the previous paper, we must have explicit knowledge of Helmholtz free energy F (F)
F = Fsa s + A Focp, (6)
with AFoc v = N K B T ( A F + B F I / * + C I n F + D ) ,
where A, B, C and D are constants (Galam and Hansen 1976). The entropy of the system is determined as $ = - ( O F / ~ T ) a . The expressions for AFoc P, proposed by different workers, for example Galam and Hansen (1976), Slattery et al (1982), are obtained by fitting the Monte Carlo data. It is well known that the Monte Carlo ocp 'exact results' are not well suited to liquid metals (Evans and Sluckin 1981); for instance MC structure factor largely deviates from experiments (Waseda 1980) in low q region. Further Monte Carlo results for specific heat and hence entropy obtained by averaging the fluctuation of the excess internal energy, have statistical uncertainties of the order of 10 9/o (Hansen 1973) for large F as for liquid metals. This discussion also remains true for determination of F by using the procedure (&/~gF)r.a = 0 (Itami and Shirnoji 1984).
Further, electron-gas screening plays a significant role in the determination of F. The screening effects decrease the scattering cross-section and enhance the density fluctuations of the ions in the intermediate wave number domain below 2Kv which is near the first peak of the structure factor.
We now compare F obtained by fitting S (0) and the first peak of the structure factor.
We use a local model potential of Ashcroft to fit S(0) for Xo 2 = 2.34 as FX~ 3 F / 2 n )
S(0) -1 = 1 - ~ - + ~ - ~ r c -~ 4~. F ~F2 , . Vo (7) where r, is the core radius and vo is a dimensionless parameter connected with local field correction. The values of vo for vs, tu and Geldart Vosko (Gv) dielectric screening functions can be calculated as
q2
% = A l l - e x p ( - B ~-~2) ] , (vs), (7a)
~'0 = 0"25 + V 1
V2/V3,
(IU), (7b)vo = 0"25 + 0"153/4nK r, (ov), (7c)
where A and B are constants (Vashishta and Singwi 1972). v~, v2 and v3 are defined by Ichimaru and Utsumi (1981). The determination of F by fitting S(0) value is very sensitive to the parameters, Xo and r,. For example, X~ = 2.34 can predict a reasonable value of F while Xo 2 = 3.2 predicts an unreasonable value of F. Thus, this method appears to be unsuitable for a correct determination ofF. This may be due to the special characteristic of the ocP system i.e. it shows the singular feature of the behaviour for q = 0 satisfying the Debye-Huckel limit. The perturbation due to the electron gas screening is also found sensitive in the determination ofF'z~ = 2.34 (Evans and Sluckin 1981) fails to predict the first peak of the structure factor.
To obtain F by fitting the first peak of structure factor we use the non-local model
Structure factor o f liquid metals 165 potential of Shaw, combined with Hartree, vs and [u dielectric functions. Table 1 reveals that this procedure leads to a small effect of density and dielectric screening on F values than those obtained by fitting S (0) or compressibility data. This is due to the fact that exchange and correlation and hence F N (q) dominantly affects the low q values rather than the first peak of the structure factor. This can be seen by plotting p [CocP (q) -flPsc(q)] in figure 1. Numerically figure 2, drawn for the same F and Z0, estimates ASVS-H(q) (q/K~ -- 0.1) ten times larger than ASVS-H(q) ( q / K r = 2.3). We further notice that F (Hartree) is smaller than F bare O C P ( l " ~--- 200) and F true (F = 210). It is further reduced by exchange and correlation corrections. This implies that the electron gas response and the local field correction reduce the ionic correlation. It can be accounted for in the following way: when the electrons are attracted to the vicinity of an ion, both the kinetic energy and Coulomb energy between those electrons generally increase. The screening action and thereby the effective reduction of the ionic charge will be completed when those energy increments are balanced by the energy decrement due to the electron ion attraction. Moreover, the first peak of the structure factor attains a value of 2.71 (Ferraz and March 1980) which may correspond to F = 178 (Slattery et al 1982) which is lower than our values and suggests reduction of ionic charges. Thus we find inconsistent F values from different methods described above.
However, a reasonable value of F in the ocp system can be expected to lie between the values obtained by fitting the entropy and the structure factor.
Another important point of our discussion is the choice of the model potential. Thus, we construct F s (q) by using the local model potential of Ashcroft (1966) and the non- local model potential of Shaw (1969) separately. Table 2 shows that for the Ashcroft model potential (!", = 1.7), an empirical cut-off of the F N (q) around 2K~ is necessary to restore the first peak height. This was also discussed by Chaturvedi et al (1981) and Senatore and Tosi (1982). If r¢ is chosen to fit the first node of Shaw's form factor (r c
8'C
4 ' 0
0"0
0 " -8"0
u
::1.
- 16-C
I
u -2z,.C
o~
- 3 2 ' (
-40.(
. =
t ~¢" " d
10 3 0
.. ~ / q /KF
/ /
/ x
7 x /
.,"
// /x f
Figure 1. The variation of term P [ C o c p ( q ) - p # s c ( q ) ] for different dielectric functions.
Curve connected by × sign denote bare ocp values, solid circles denote Hartree values and dashes denote IU values.
166 K N Khanna and G Shanker
3.0
2.0
o - L/")
1.0
0"0
6
? o
2 -I.0
- 2 " 0
/: ."..,,•~
['.
I,. . . .
Zt / "'
~1 / .•
• I / •
°
• ~ i •
"•. • . "
Figure 2. The structure factor S(q) and the interionic potential u(r) for different dielectric functions for the same F = 184.4• Curve connected by solid circles denote Hartree values, dots for vs values and dashes for lu values.
= 1.61), we find a large change in the height of the first peak. It reflects that a small core is !more suited to the ocP model, as OCl, consists of point charge ions. The selection of non-local model potential estimates the accurate height and position without implying any empirical cut-off, indicating that non-locality may play a dominant role in reproducing the structure factor. This may also be due to the fact that the main peak of the structure factor is sensitive to the effective pair potential as seen in the hard sphere system (Bratkovsky et al 1982). However, electron gas screening plays a dominant role in the determination of structure factor particularly in the low q region.
Now we compare our results with the structure factor obtained by using the hard sphere theory. We present the structure factor upto the first peak for Na in table 2.
Though the calculations have been made for all alkalis, we present our results of Na
Structure factor of liquid metals 167
Table 2. Comparison of various results for the structure factor of liquid sodium.
q/K~
Ashcroft Ashcrofi Exp MC
r, = 1"70 r, = 1 . 6 0 Shaw (H) Shaw (vs) Shaw (]u) Waseda Galam and F = 187 F = 187 F = 193'8 F = 185"0 F = 184"4 (1980) Hansen (1976)
ffl 0.015 0.018 if017 0-053 0-046 0-025 - -
if2 0-018 0.022 0-020 0.086 0.067 0-025 - -
0-3 0-019 0-023 0-021 0-084 0.069 0-026 - -
if4 0-020 0.024 0.023 0.082 0.071 0-027 - -
0-5 0-022 0-026 0.025 0-079 0-074 0-028 0-002
0-6 0.024 0.029 0.027 0.078 0.077 0-031 - -
0-8 0-030 0-036 0.035 ff078 0-086 0-037 0-008
1.0 0-040 0.049 0-047 0.084 0-100 0-047 - -
1.2 0-058 0-071 0.069 0.099 0.120 0-060 0-026
1.4 0-091 0-111 0-111 0.134 0-155 0-094 0-055
1.6 0.158 0.190 0.199 0-208 0"225 0.165 0-081
1.8 0-320 0.366 0-405 0-389 0-396 0-312 0-325
2.0 0-841 0.843 0.972 0-909 0-910 0-992 1.105
2.2 3.591 2.350 2.280 2.201 2.207 2.260 2-720
2.3 9-683 3-619 2.705 2.704 2.707 2.707 2.530
2.4 20-323 4.340 2.550 2.617 2.613 2.152 1.858
2-5 13.259 3.952 1.798 1.860 1.850 1.293 1.350
F N (q) is constructed by the pseudopotential described in the table. H, vs and lu correspond to the dielectric screening. The parameter Xo 2 = 3.2 remains constant for all the calculations.
only as we concentrate only on the anomalies of the ocP system and compare our results with the hard sphere system. A high value of S (0) corresponding to the vs local field correction is noticed while a lower value of S(0) is favoured by the iu local field correction. Similar results were observed in the hard sphere structure factor (Dharma- Wardana and Alers 1983; Bretonnet and Regnault 1985). It is difficult to indicate the pair potential which is most suitable in determining the form of S (q) in the oce system unless the full structure factor is predicted and the pair potential is formed based on the oce system. The hard sphere system seems to prefer the soft core part whereas a limited role is reserved for the Fridel oscillations. Also, for a 'soft' system like Na governed by a R - 4 potential, the soft part of the potential mainly determines the structure factor. The situation seems to be reversed for the OCl, system which is too soft to reproduce the full structure factor and hence rapidly damped oscillations are observed beyond the first peak of the structure factor and it is necessary to have a system sufficiently hard to reproduce the full structure factor. In the reference of inverse power potential V (r)
= e(a/r)~; we have to search a higher value of n rather than n = 1 (the oce system) for the exact description of the liquid alkali metals. To determine whether the structure factor depends on the well depth of the pair potential, interionic potential for Na (figure 2) was constructed by using the Shaw non-local model potential with Hartree, vs and Iv dielectric functions. To illustrate the ion arrangement relative to ~b(r), we compare our r~, (7.2 a.u.) with those calculated close to the nearest neighbour position r,~ by Kumaravadivel and Evans (1976) (7.15 a.u.), who used the Shaw model potential with vs dielectric screening. Thus the minimum of the pair potential shifts towards the first nearest neighbour by including vs or Iu exchange and correlation corrections in the Hartree dielectric screening.
168 K N Khanna and G Shanker
We now relate the structure factor of the ocP system with pair potential. Bratkovsky et al (1982) conclude that the height of the first peak of the structure factor is sensitive to the well depth ofinterionic potential in the hard sphere system. On the other hand Day et al (1979) believed S (q) to be rather insensitive to the form of pair potential in the hard sphere system. We have made calculations for the same F = 184.4 as well as the height of the first peak for the three dielectric constants. In figure 2 we find that the height of the first peak changes with the interionic potential but not too sensitive to the well depth. For example, we get a clear indication of this dependence by comparing Hartree and vs or iu but not by comparing vs to Iu. This may be due to the fact that in the hard sphere model, the interionic potential can be separated into the repulsive and attractive forces while in the ocv system it is difficult to break the pair potential. This may also be due to the fact that structure factor depends upon the well depth as well as the softness of the core (Bratkovsky et al 1982).
We now extend these lines of approach to investigate the exchange and correlation effects of the degenerate electron liquid on the structure factor. A close agreement with experiments for Hartree dielectric function in low q region (as shown in table 2) can be assigned neither to the failure of vs or iu dielectric functions, as they are thoroughly investigated by several workers, nor to the RPA which has been tested by the condition ppS°CV(q)lZsc (q) < 1 (Hansen and McDonald 1976; Bretonnet and Khanna 1985). We assign this discrepancy to the softness of pair potential in the ocP system. Bratkovsky et al (1982) have shown in their calculations that a hard core increases the height of the first peak and consequently a decrement in the low q region. Thus by allowing a little hard potential the structure factor obtained by vs or Iv corrections will be close to the experiment than those obtained by Hartree dielectric screening in the low q region. At the same time the value o f f obtained by fitting the first peak of the structure factor will be close to Ffreezing.
In order to characterize the differences between the liquid metal and ocv structure factor, it is convenient to distinguish three parts in q space;
(i) The low q region below 2K~: The main disparity is observed between the experimental results (Waseda 1980) and the bare ocv or Monte Carlo calculations. This region can be improved by combining bare ocP with electron gas screening as discussed in the present work. The role of F s (q) is clearly visible in figure 1.
(ii) Around the main peak i.e. between 2K r to 3Kv: This region is characterized by F and ~o that govern the height and position of the first peak of the structure factor respectively. However, F and Xo are weakly coupled as shown in figure 3.
(iii) Corresponding to the large q region i.e. beyond the first peak. In the region q / K v
= 2.3 to 3.0, bare ocv should decrease rapidly to get the first minimum at its correct position while it remained sufficiently positive. This may be due to the fact that the internal energy of the system (see figure 1 of Baus and Hansen 1979 for p = 3) largely deviates from Mc data for F beyond 150, as in liquid metals.
4. Conclusion
The present study consists of ingredients viz the Shaw model potential, vs or Iu dielectric functions, which had been stringciatly tested by many workers. Thus the reliability of the present calculations is confirmed. We are led to the conclusion that while describing even alkali metals, where a high density plasma may be assumed, the
Structure factor of liquid metals 169
c _
20~
200
196
192 186
182
178
174
\
g
%.
,%
170 1.78 1.79 1,80 1.81 1,82 1.83 '
% 0
Figure 3. The variation of F with Xo after the first peak position is restored. The curve connected by x sign denotes bare ocv values and dashes for screened o c v 0 u ) values.
ocP model may be something of an over-simplification. It seems necessary to consider a small hard core or a harder effective potential in a bare ocp system and this will be reported separately. Our discussions also describe the failure of the Bans and Hansen model. The role o f Xo is not known to date. However, in our opinion Xo is a fixed quantity as described by Baus and Hansen (1979) and cannot be treated as a fitting parameter (Evans and Sluckin 1981). It is doubtful if X o z = 2.34 can satisfy the internal energy of the system to MC data for a reasonable choice of F.
Acknowledgement
One of us (KNK) is grateful to Dr J L Bretonnet for useful discussion. Financial support from uoc, New Delhi, is gratefully acknowledged.
References
Ashcroft N W 1966 Phys. Lett. 23 48
Baus M and Hansen J P 1979 J. Phys. C12 L55
Bratkovsky A M, Vaks V G, Kravchuk S P and Trefilov A V 1982 J. Phys. FI2 1293 Bretonnet J L and Khanna K N 1985 Phys. Lett. A l l 0 420
Bretonnet J L and Regnault C 1985 Phys. Rev. B31 5071
Chaturvedi D IL Rovere M, Scnatore G and Tosi M P 1981 Ph.vsica B i l l 11 Day R S, Sun F and Cutler P H 1979 Phys. Rev. AI9 328
Dharma-Wardana M W C and Alers G C 1983 Phys. Rev. B28 1701
170 K N Khanna and G Shanker Evans R and Sluckin T J 1981 J. Phys. C14 3137
Ferraz A and March N H 1980 Solid State Commun. 36 977 Galam S and Hansen J P 1976 Phys. Rev. AI4 816 Hansen J P 1973 Phys. Rev. AS 3096
Hansen J P and McDonald J R 1976 Theory of simple metals (New York: Academic Press) p. 167 Ichimaru S and Utsumi K 1981 Phys. Rev. B24 7385
Itami T and Shimoji M 1984 J. Phys. FI4 L15
Khanna K N and Shanker G 1985 Phys. Chert Liq. 15 69 Kumaravadivel R and Evans R 1976 J. Phys. C9 3877 Mon K K, Gann R and Stroud D 1981 Phys. Rev. A24 2145 Scnatore G and Tosi M P 1982 Phys. Chem. Liq. 11 365 Shaw R W 1969 J. Phys. C2 2335
Slattery W L, Doolen G D and Dewitt H.E 1982 Phys. Rev. A26 2255 Ono S and Yokoyama I 1984 J. Phys. F14 2909
Vashishta P and Singwi K S 1972 Phys. Rev. B6 875
Waseda Y 1980 The structure of non-crystalline material (New York: McGraw Hill).
