Physica B 167 (1990) 61-70
North-Holland
THERMODYNAMICS AND STRUCTURE OF LIQUID METALS FROM THE
C H A R G E D HARD-SPHERE R E F E R E N C E FLUID
O. A K I N L A D E l
International Centre for Theoretical Physics, 1-34100 Trieste, Italy
S.K. LAI 2
Department of Physics, National Central University, Chung-li 32054, Taiwan and International Centre for Theoretical
Physics, 1-34100 Trieste, Italy
M.P. TOSI
Department of Theoretical Physics of the University of Trieste, 1-34100 Trieste, Italy
Received S March 1990
Revised manuscript received 17 May 1990
Perturbative variational calculations of thermodynamic and structural properties of liquid metals, based on the use of ab
initio and highly reliable nonlocal pseudopotentials for the electron-ion interactions and of the fluid of charged hard
spheres as a reference system, have been reported recently for the liquid alkali metals from Na to Cs near the freezing
point. We extend in this work the above-mentioned calculations in two directions. Firstly, we discuss the predicted
temperature dependence of the liquid structure factor for the same alkali metals over a limited temperature range above
the freezing point. Secondly, we examine the usefulness of the approach for metals with relatively strong electron-ion
interactions, namely Li and several polyvalent metals (Mg, Cd, AI, In, T1 and Pb). The charged hard-sphere reference
system leads to lower values of the Helmholtz free energy and to slightly improved values of the excess entropy for all the
liquid metals that we evaluate, even though polyvalent ones overall appear to be relatively close to fluids of neutral hard
spheres. For the liquid alkali metals at elevated temperatures, the calculated structure factors are of similar quality as in
our previous work, that is, they show a systematic shift in the positions of peaks and valleys to slightly larger wave numbers
and peak heights that are somewhat underestimated with increasing temperature. However, for liquid polyvalent metals,
our approach yields quite good agreement with experiment for the positions of maxima and minima in the liquid structure
factor, while it tends to overemphasize somewhat these structures.
I. Introduction
The study of thermodynamic and structural
properties of liquid metals has drawn much
theoretical attention both for their intrinsic interest and for their relevance to an understanding
of electronic properties. A widely used variational approach is based on the Gibbs-Bogoliubov
inequality [1], which states that the Helmholtz
free energy of the liquid metal is bounded from
above by the sum of (i) the free energy of a
suitably chosen reference system, and (ii) the
Permanent address: Department of Physics, University of
Ibadan, Ibadan, Nigeria.
2 Supported in part by the National Sciences Council of
Taiwan.
difference in average potential energy between
the actual and the reference system, calculated
using the distribution function of the latter. The
structure factor of the liquid metal follows as
that of the reference fluid at the variationally
optimized values of its parameters.
The choice of the reference system is clearly
very important and must satisfy two general
criteria. The reference fluid should first of all
mimic sufficiently well basic features of the real
metal and in particular its liquid structure. In
addition, its thermodynamic functions should be
available in an analytic or readily usable numerical form. Several model fluids satisfying these
criteria have been considered in the literature,
the most popular ones being the fluid of neutral
0921-4526/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
62
o. Akinlade et al. / Thermodynamics and structure of liquid metals
hard spheres (NHS) [2-4], the one-component
classical plasma (OCP) [5-7], the fluid of
charged hard spheres (CHS) [8-14], the hardsphere-Yukawa system (HSY) [15, 16] and the
soft sphere (inverse power potential) system [1719].
In our earlier work on the liquid alkali metals
from Na to Cs near freezing [14], which will
henceforth be referred to as I, we have given a
detailed study of the CHS reference system in
the variational evaluation of the free energy
from a Hamiltonian treating the electron-ion
interactions by means of the modified generalized nonlocal model pseudopotential (GNMP) of
Li et al. [20]. The reliability of the GNMP has
also been demonstrated in a number of other
calculations of metallic properties. One of the
successful applications that is relevant to the
present work concerns the study of structural
changes with increasing temperature [21]. However, such a study has been carried out by the
Monte Carlo simulation technique, which becomes increasingly impractical when one has to
deal with various metals in a variety of thermodynamic states. There has also been an increasing experimental effort devoted to the study of
the liquid structure [22] and electronic properties
such as the magnetic susceptibility [23] at temperatures far above the freezing point. Accordingly, our first concern in the present work is to
apply the same variational approach as in I and
to discuss the temperature dependence of the
liquid structure of the above alkali metals, over a
limited temperature range above the freezing
point. Similar investigations have been carried
out earlier by Umar and Young [24] in the NHS
model and also by Bratkovsky et al. [25] and by
Sinha et al. [26].
More interestingly, however, we proceed to
examine the usefulness of the approach in the
evaluation of the thermodynamic properties and
the liquid structure factor for other metals, in
which the electron-ion interactions are appreciably stronger. These are Li, thus completing our
study of the alkali metals, and several polyvalent
metals. Li differs from the other alkalis in that p
states are absent from the core, with the implication that the electron-ion interactions are espe-
cially strong [27]. Thus, a proper description of.
the pseudopotential requires that one should go
beyond the conventional second-order perturbation theory. Li et al. [20, 28] have made various
attempts to incorporate higher-order perturbation effects in their pseudopotential and in particular for Li they have focussed attention on
possible state mixing effects, which may arise
from the proximity of the s and p valence electron states. Their results for the phonon spectra
and the elastic constants, as well as the recent
thermodynamic calculations by Lai [27] using the
OCP reference for the liquid, give credence to
their pseudopotential for El.
In addition to Li, we evaluate below the polyvalent metals Mg, Cd, A1, In, TI and Pb. Bertoni
et al. [29], in their extensive study of polyvalent
metals, came to the conclusion that higher-order
pseudopotential perturbation effects have to be
taken into consideration in the determination of
pseudopotentials for these metals. For the
GNMP potential that we use in the present
work, Li et al. [20] have given evidence that
higher-order effects have been better taken care
of for the polyvalent metals under present consideration than for the others (such as Ca, Sr,
etc.). Thus, the results that we shall report below
for Mg, Cd, In, T1, AI and Pb should be considered as an improvement on the usual low-order
perturbation theory. In a subsequent work we
shall demonstrate that such an improvement,
when directly brought into the theory of liquid
structure, is quite crucial to explain on basic
grounds the characteristic asymmetry shown by
the liquid structure factor in polyvalent metals
such as Cd or Zn.
The layout of the paper is briefly as follows. In
section 2 we report the essential equations for
the variational calculation of thermodynamic
properties based on the CHS as reference. The
approach involves two variational parameters,
which are the packing fraction rl and the valence
z entering the plasma parameter F = ( z e ) 2 /
R s k B T, R s being the radius of the atomic sphere.
Section 3 is devoted to the discussion of the
liquid structure and the thermodynamic functions of alkali metals above the freezing point,
while section 4 presents our results for the
O. Akinlade et al. / Thermodynamics and structure of liquid metals
thermodynamic and structural properties of Li
and of polyvalent metals at or near freezing.
Finally, section 5 summarizes our main conclusions.
63
In eqs. (2.3) and (2.4), Sid is the ideal gas
entropy, ~ : = ( 1 - ' 0 ) 3 / ( 1 + 2 r / ) 2 and K is the
Debye-H/ickel inverse screening length, K =
(12"02/3V) 1/2
Further, (F2)cu s in eq. (2.l) is the band
structure energy, given by
2. Theory
We summarize in this section the essential
equations for variational calculations based on
the CHS reference system, from the analytic
expressions derived for the direct correlation
function and the internal energy of this model
fluid by Palmer and Weeks [9] in the mean
spherical approximation. More details can be
found in I.
The Gibbs-Bogoliubov inequality allows one
to write the variational expression of the Helmholtz free energy per ion in the liquid metal as
F
F('0, F)
= FEG + FCH s + F 1 + ( F 2 ) c u s + ( A F M ) c ~ s •
(2.1)
Here, FEC is the free energy of the electron gas
(see eq. (2) of ref. [3]) and F l is the first-order
electronic term accounting for the average interactions of the valence electrons with the nonCoulombic part of the bare ion pseudopotential.
FcHs is the free energy of the CHS, which can be
written as
e×
FCHs = FNu s + FcH s ,
(2.2)
where the free energy of the NHS, as derived
from the virial theorem, is given by
3
6"0 + 21n( l _ "0)]_ ku TS~a '
(2.3)
FNHs=kBT 2 + 1--'0
and the excess free energy of the CHS is given
by
FeXcHs --
-ABT ~
{1 + '0- 1"02 -t-- 2(1 +K2'0)
2(1 + 2'0)
]
3~:Kz [(1 + 2K~c)3/z - 1] [ .
(2.4)
(F2)c. s _
Z~ffxrf dq ScHs( q, F, "0)GEC(q).
0
(2.5)
H e r e , Z e f f is the effective charge, which is given
for the GNMP [4] as Z,f~ -- (Z 2 - p2)1/2 in terms
of the nominal valence Z and the depletion
charge pdGEC(q) is the normalized energywavenumber characteristic, including exchange
and correlation effects, and ScHs(q, F, "0) is the
structure factor of the CHS. An explicit expression for the latter has been reported by Singh
[30], whereas for G EC(q) and F 1 these are given
in Wang and Lai [31] and Lai et al. [4], respectively.
Finally, the last term on the right hand side of
eq. (2.1) is the deviation in Madelung energy
and in the context of the GNMP theory it can be
written as
(AFM)c• s
= -k~T
(~eff ___ )
1 nt- "0
~'0
2'0
1+
K
2"0[(1 + 2x~j) 1/2 - 11}, (2.6)
where Feff = Z~ff/(RskuT). Of course, the expressions for the internal energy U and the excess entropy S ex can easily be obtained from the
above-mentioned expressions for the Helmholtz
free energy (see I).
In performing full thermodynamic variational
calculations with the CHS reference system, the
free energy in eqs. (2.1)-(2.6) is minimized according to the conditions
(2.7)
Here, n is the atomic number density. Equations
O. Akinlade et al. / Thermodynamics and structure of liquid metals
64
(2.7) will thus enable us to obtain the values of
and F at the minimum of the free energy (~m and
F,,) and hence to evaluate the thermodynamic
functions and the structure factor of the liquid
metal.
3. Thermodynamics and structure of liquid
alkali metals above the freezing point
We have used eqs. (2.7) to evaluate the variational parameters r/m and Fm at the minimum of
the free energy for liquid Na, K, Rb and Cs at a
few values of the temperature above the freezing
point, taking the corresponding values of the
atomic volume g20 from the tabulation given by
Waseda [32]. These input values* and the values
of r/m and Fm are reported in table 1, together
with the corresponding values of the Helmholtz
free energy F, the internal energy U and the
excess entropy S ex. Comparison with measured
values of these thermodynamic functions near
freezing is also reported from I in table 1.
As discussed in I, the present approach yields
for these alkali metals near freezing values of the
* The atomic volume can in fact be determined variationally within the present G N M P theory. We expect the
values thus obtained for 12o to lie within about 10% or
less of the measured values in the temperature range of
interest here (see Li et al. [20]).
packing fraction ~m which are quite insensitive to,
the value of the plasma parameter F, whereas
the values of Fm are reduced very significantly
relative to those of the nominal ionic valence. It
can be seen from table 1 that both T~m and Fm
decrease rather rapidly with increasing temperature. Indeed, the decrease of Fm is more rapid
than would be expected from the expression of
the plasma parameter.
Figure 1 compares our results for the liquid
structure factor S(q) of the above metals at
various temperatures with experimental data
from Waseda [32]. The general effect of increasing temperature in broadening the peaks of the
structure factor and in damping its oscillations is
well reproduced by our results. Quantitatively,
while our calculated positions of the peaks and
valleys are in fair agreement with the observed
ones, the height S(qm) of the main peak in the
structure factor tends to be somewhat underestimated with increasing temperature. We nevertheless find that the calculated values of the ratio
o f S(qm)T/S(qm)mp for the height of the main
peak at temperature T over that at the melting
point T,,, when plotted against reduced temperature T/T,, fall on a single curve for the four
alkali metals that we have evaluated (fig. 2).
This 'corresponding states' behaviour for alkali
metals was demonstrated experimentally by Winter et al. [22].
Table 1
Atomic volume g20, temperature T, plasma p a r a m e t e r Fm, packing fraction r/m, Helmholtz flee energy F, internal energy U and
excess entropy S °x (in units of kB) for liquid alkali metals at various temperatures. The experimental data are taken from Waseda
[32] for ~o and from Lai et al. [14] for the t h e r m o d y n a m i c functions. All in atomic units.
Metal
~,
T( °C )
~n
nm
F
Fexpt
U
U ....
-SeX
- S~p,
Na
277.942
285.264
292.388
100
200
300
45
30
21
0.466
0.431
0.401
-0.2356
-0.2380
-0.2407
-0.236
-0.2272
-0.2259
-0.2247
-0.2320
4.12
3.53
3.09
3.45
K
528.500
534.921
549.686
64
105
200
50
41
28
0.461
0.444
0.409
-0.2009
-0.2021
-0.2048
-0.201
-0.1920
-0.1914
-0.1901
-0.1956
4.06
3.76
3.24
3.45
Rb
654.600
661.366
685.510
39
200
300
47
37
25
0.456
0.431
0.392
-0.1927
-0.1947
-0.1980
-0.193
-1/.1830
-1/.1822
-0.1809
-0.1870
3.96
3.56
3.01
3.63
Cs
810.800
829.178
856.356
29
100
200
59
43
30
0.460
0.428
0.391
-0.1809
-0.1833
-0.1869
-0.182
-I).1708
-0.1698
-0.1685
-0.1757
4.08
3.56
3.03
3.56
65
O. Akinlade et al. / Thermodynamics and structure of liquid metals
t
'
6 3
'
'
0
I
'
o
0
'
'
~
2
I
'
'
'
:
4
q (l/t)
6
o0
2
q (1/A_)
(a)
8~
,
,
,
I
'
'
'
I
'
'
'
'
r
,
6~ ~
~1_~~~~
oo]126
I ....
2
6
(t,)
u I
0
i
4
q (1/A.)
t
'
'
~
I
'
'
'
..........[c2oo1
s
.....Rb2?ol
....
4
6
0
2
q (~/~)
4
6
(d)
(c)
Fig. 1 (a-d). Liquid structure factor S(q) versus wavenumber q for Na, K, Rb and Cs at various temperatures. The dashed
curves give our theoretical results (Na: T = 100, 200, 300°C; K: T = 63.5, 105,200°C; Rb: T = 39, 100, 200°C; Cs: T = 29, 100,
200°C) and the full curves are experimental data from Waseda [32] at the temperatures shown in the figures.
4. Thermodynamics and structure of lithium
and of polyvalent metals
4.1. L i q u i d lithium
T h e results of o u r variational calculations for
liquid Li n e a r the freezing p o i n t are r e p o r t e d a n d
c o m p a r e d to the e x p e r i m e n t a l e v i d e n c e [20, 3 2 -
34] in the t o p r o w s of tables 2 a n d 3 a n d in fig. 3.
A s can be seen f r o m table 2, the values of the
free e n e r g y a n d the internal e n e r g y are in v e r y
g o o d a g r e e m e n t with e x p e r i m e n t , T h e m a g n i t u d e of the excess e n t r o p y is o v e r e s t i m a t e d ,
b u t of c o m p a r a b l e quality as t h o s e o b t a i n e d for
the o t h e r alkali metals in the s a m e t h e r m o d y n a m i c state (table 1). C o m p a r i s o n with N H S
66
O. Akinlade et al. / Thermodynamics and structure o f liquid metals
,o I
x
0.9
O
X+
0.8 r
0 X
0.7 I
0.6
k
o.5 I
1.00
~
1.50 . . . .
2.00
T/T~
Fig. 2. Calculated reduced height S(qm)r/S(qm)mp of the
main peak in the structure factor versus reduced temperature
T / T m for Na ( + ) , K (*), Rb (©) and Cs ( x ) .
/
i
Li
v
~o
,,--,..
i
o
o
E
4
q (1/~)
6
8
Fig. 3. Liquid structure factor S(q) versus wavenumber q for
Li near freezing. The meaning of the curves is as in fig. 1.
results for the thermodynamic properties, which
are also reported in table 2 shows that only
limited improvement has been obtained, this
qualitative feature being also common with the
other alkalis (see I). As discussed in I, it suggests
an important role of thermodynamic self-consistency in the theory of the reference system.
From the structural point of view, the predicted locations of maxima and minima in the
structure factor of liquid Li are in good agree-
ment with experiment (see table 3). In contrast
to the other alkali metals, however, the height of
the main peak in S(q) is somewhat overestimated (figs. 1 and 3). This difference appears to
be an indirect consequence of the appreciably
stronger electron-ion interactions in Li as compared to the other alkali metals, leading, within
the present choice of a reference liquid, to somewhat larger values for the variationally optimized
values of the packing fraction and the plasma
parameter. In spite of these differences of detail,
however, we may conclude that the present approach works about as well for all the alkali
metals.
4.2. Liquid polyvalent metals
Our calculations for polyvalent liquid metals,
involving accurate nonlocal pseudopotentials
[20], should be considered to represent an improvement on similar calculations by Iwamatsu
[13], who used a local pseudopotential. Our
results for the thermodynamic and structural
properties of these metals close to freezing are
shown in the remaining parts of tables 2 and 3
and in figs. 4 and 5. Table 2 also compares CHS
and NHS values for the free energy, the internal
energy and the excess entropy with each other
and with experiment [20, 33, 34]. The following
observations can be drawn from our results:
(i) The CHS yields values for the free energy
that are lower than those obtained with the
NHS. The latter is, in turn, a better reference
system than the OCP for polyvalent metals such
as A1, as has been shown in earlier studies
[6, 13].
(ii) It is clear that the polyvalent metals can
be modelled already to a reasonable extent by
using the NHS reference system. This is particularly so for liquid Mg and Al, where the predicted values of F m are extremely small. Indeed,
a plot of the repulsive part of the pair potentials
for the polyvalent metals of present interest, as
reported in fig. 6, shows that the interionic repulsion is significantly stiffer in A1, Mg and Cd than
in In, T1 and Pb. This explains the values of F m
for the various polyvalent metals in table 2.
(iii) In relation to the points above, however,
one should note from table 2 that the difference
O. Akinlade et al. / Thermodynamics and structure o f liquid metals
67
Table 2
T e m p e r a t u r e T, plasma p a r a m e t e r F m, packing fraction "Ore, Helmholtz free energy F, internal energy U and excess entropy S ~x (in
units of kB) for liquid Li and polyvalent metals close to freezing. The experimental data for the excess entropy are taken from
Faber [33] and Hultgren et al. [34] (denoted by a and b, respectively) and from Li et al. [20] for the other t h e r m o d y n a m i c
functions, All in atomic units.
Metal
T(°C)
Li CHS
NH8
190
Mg CHS
NHS
nm
F
Fexp,
U
Uexp,
- S ex
-S~Xp,
76
0
0.478
0.492
-0.26555
-0.26543
-0.265
-0.25881
-0.25875
-0.254
4.42
4.46
3.7 ~, 3.54 b
680
0.03
0
0.476
0.477
-0.89778
-0.89778
-0.905
-0.87332
-0.87332
-0.879
4.15
4.15
3.45 a, 3.37 b
Cd CHS
NHS
350
8
0
0.457
0.459
-0.97028
-0.97027
- 1.020
-0.95061
-0.95063
- 1.000
3.85
3.86
4.15 ~, 3.68 b
AI CHS
NHS
670
0.04
0
0.482
0,484
-2.07848
-2.07848
-2.035
-2.05517
-2.05518
-2.035
4.27
4.27
3.6 ~. 3.61 h
In CHS
NHS
160
36
0
0.508
0.513
-2.00085
-2.00080
-2.042
- 1.98903
-1.98905
-2.030
4.83
4.88
4.4 ", 4.61 b
TI CHS
NHS
315
72
0
0.454
0.474
-2.08667
-2.08636
-2.165
-2.06651
-2.06633
-2.145
4.05
4.12
4.05 ", 4.13 b
Pb CHS
NHS
340
90
0
0.475
0.493
-3.59145
-3.59109
-3.656
-3.57086
-3.57063
-3.634
4.42
4.49
4.1", 3.69 ~
'
~
~
I
rm
'fi
'
'
I
'
'
'
I
'
~
~~
'
O3
0
2
4
q (l/A)
6
8
Fig. 4. Liquid structure factor S(q) versus w a v e n u m b e r q for
Mg and AI near freezing. The m e a n i n g of the curves is as in
fig. 1.
in the free energy values obtained with the CHS
and with the NHS tends to increase with increasing atomic number (notice in particular the re-
suits for T1 and Pb).
(iv) In relation to liquid structure, earlier
work using the NHS reference [20] has shown
that higher-order pseudopotential perturbation
(HOPP) effects are relatively more important for
Cd, In, T1 and Pb. Specifically, the use of an
energy independent nonlocal model potential
(EINMP), which does not include HOPP effects,
predicts somewhat smaller or unphysical values
of the packing fraction for these metals (see table
2 in ref. [20]), whereas for Mg and AI the
EINMP yields values of r/ which are about as
good as the ones from GNMP. Nevertheless, one
should not be misled to think that the HOPP
corrections are essentially unimportant for the
latter metals, since only their inclusion allows
reasonable predictions for their surface tension
[35]. The latter quantity is particularly sensitive
to the treatment of the electronic part in the free
energy.
For what concerns the comparison of our CHS
results for the thermodynamic functions with
experiment in table 2, we may remark that the
values of the free energy and the internal energy
are reasonably satisfactory, especially consider-
68
O. Akinlade et al. / Thermodynamics and structure of liquid metals
Table 3
Comparison of the experimental and theoretical locations of
the maxima and minima of the liquid structure factor for Li
and polyvalent metals near the freezing point. Numerical
data for the first three columns refer to the maxima while the
last two to the minima. Experimental data [32] are given in
the first row, while the second row reports the present
calculations. Units are in A ~.
Metal
Maxima
2nd
3rd
1st
2.50
2.50
4.65
4.63
6.90
6.90
3.50
3.45
5.75
5.75
Mg
2.40
2.40
4.40
4.50
6.60
6.60
3.20
3.33
5.60
5.50
Cd
2.60
2.49
4.60
4.66
6.80
6.92
3.40
3.49
5.60
5.77
AI
2.70
2.68
4.90
4.94
7.40
7.32
3.70
3.70
6.20
6.12
In
2.30
2.36
4.30
4.30
6.40
6.38
3.20
3.22
5.30
5.33
TI
2.30
2.27
4.20
4.26
6.40
6.35
3.10
3.17
5.30
5.29
Pb
2.30
2.20
4.20
4.12
6.20
6.12
3.30
3.05
5.20
5.11
6
X,O
~:
8.0 }
0.oi0.50
+
"~
', ,
0
O
9( o
~x O
* x O©
,X o
", ",
',,
, , \'
%%O0oo
"\'s
', '
"'"
0.75
4
:,
~
+
1.00
r/r~
Fig. 6. Interionic pair potential V(r) relative to its value V,,,
at its minimum r = r m, in units of kBT, versus r/r m for AI
(+), Mg (short-long dashes), Cd (short dashes), TI (O), Pb
( x ) and In (*).
,J
4
q (i/~i)
,~
X
4.0 (
~4
2
-...E" 12.0 i
2nd
Li
0
16.0 I
>
Minima
1st
20.0 r
8
Fig. 5. Liquid structure factor S(q) versus wavenumber q for
Cd, In, TI and Pb near freezing, The meaning of the curves is
as in fig. 1.
ing that no fit of parameters to liquid metal data
is involved in our approach. The magnitude of
the excess entropy for Mg, A1 and Pb is again
overestimated as compared to the measured values, by amounts which are essentially similar to
those for the alkali metals, whereas for Cd, In
and T1 the agreement with the experimental data
is remarkably close. These results cannot be
reconciled systematically with the structure functions shown in figs. 4 and 5.
In order to look into this point, let us examine
in some detail our results for the liquid structure
factors of polyvalent metals. As can be seen
from the comparison with data from Waseda [32]
in figs. 4 and 5, the quality of our predictions for
the detailed shape of S(q) in these metals is only
moderately good. In particular, the structures in
S(q) are overemphasized, apparently as a consequence of the fact that our variationally determined values for the packing fraction are
rather too large compared to the "canonical"
value rI -~ 0.45. This is particularly true for liquid
AI, for which two other independent measurements of S(q) [36, 37] confirm that our approach
overemphasizes its structures. On the contrary,
for liquid Pb the recent experimental determination of S(q) by Reijers et al. [38], as well as the
older measurements by Dahlborg et al. [39] and
Steffen [40], yield results which differ somewhat
O. Akinlade et al. / Thermodynamics and structure of liquid metals
3.0
2.4
1.8
1.2
~"
.
0.6 f
0.0
0.00
J
5J.00
~
J
10.00
q(l/A)
Fig. 7. Liquid structure factor S(q) versus wave number q
for liquid Pb close to freezingfrom theory (dashed curve) and
from the measurements of Dahlborg et al. [39] (full curve).
from Waseda's and agree quite well with our
CHS results (see fig. 7). We emphasize that in
this liquid metal the discrepancy of our calculated excess entropy with experiment appears to
be quite large (see table 2). Although it is difficult to draw firm conclusions, our results for the
excess entropy and the structure of liquid Pb
suggest that thermodynamic consistency in the
reference system may again be important even in
systems where the appropriate CHS has a rather
small value of F m and is therefore approaching
the NHS fluid.
Bearing in mind the fact that the locations of
maxima and minima in S ( q ) are experimentally
more reliable than its detailed shape [41], we
also compare our predictions for these locations
with experiment in table 3. The agreement appears to be quite reasonable.
69
elevated temperatures. As shown by our results
for Li, this usefulness is not crucially limited to
systems where the electron-ion interactions are
weak.
For the polyvalent metals, our results indicate
that the CHS reference system is in principle
preferable to the NHS since it yields lower values for the free energy. However, the NHS
already provides a modelling of these systems
which is quite reasonable in quality. Specifically,
using the CHS we have found only marginal
improvements in the calculated thermodynamic
functions and still quantitatively unsatisfactory
results for the detailed shapes of the liquid structure factor. These features of our results are
clearly a consequence of the stiffness of the pair
potential in polyvalent metals being higher than
in the alkalis, thus making the excluded-volume
effects dominant (see fig. 6).
Underlying all our results is the question of
the accuracy with which the CHS reference system, including as a special limit the NHS treated
in the Percus-Yevick approximation, is being
described both thermodynamically and structurally by the mean spherical approximation. In
particular, the role of thermodynamic self-consistency between the various routes to the equation
of state of the reference fluid appears to be
important especially in relation to the evaluation
of the excess entropy. This question has been
extensively discussed in I, where we have shown
that discrepancies with the measured values of
this quantity, of the amount found there and
again in the present work, can be appreciably
reduced by reducing the degree of thermodynamic inconsistency in the theory of the reference system.
5. C o n c l u s i o n s
Acknowledgements
We have presented variational calculations of
thermodynamics and structure for liquid Li and
polyvalent liquid metals near the freezing point
and for the other liquid alkali metals over a
limited range of temperature above the freezing
point. Our results for the alkalis show that the
CHS reference system, combined with a refined
nonlocal pseudopotential, provides a useful approach for their evaluation even at moderately
Two of us (O.A. and S.K.L.) wish to thank
Professor Abdus Salam, the International
Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics in Trieste. M.P.T. acknowledges support by the Ministero della Pubblica Istruzione
and the Consiglio Nazionale delle Ricerche of
Italy.
70
O. Akinlade et al. / Thermodynamics and structure o f liquid metals
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