APPLICATION OF A NEW PSEUDOPOTENTIAL APPROACH
TO LIQUID Na-K-Cs SYSTEM
N.E. Dubinin, T.V. Trefilova, A.A. Yuryev and N.A. Vatolin
Abstract.
This paper is devoted to a quantitative investigation of liquid Na-K-Cs system on
the basis of the thermodynamic perturbation theory using a new version of the local
Animalu-Heine model pseudopotential developed by the authors earlier. Partial pair
interactions and heat of mixing of the eutectic Na-K-Cs alloy at different temperatures
are calculated. Obtained results are in a good agreement with available experimental
data.
1. Introduction
At the present time, a great interest of researchers to ternary liquid metal
systems is observed. Although the level of theoretical studies on pure liquid metals
and binary liquid metal alloys is very high now, from the viewpoint of the
fundamental theory, investigations of n-component liquid metal mixtures (with n3 )
are in a rather unsatisfactory stage. Especially it relates to calculations of
thermodynamic properties. The works of this field are mostly phenomenological or
semi-empirical [1-6]. As an exception there are some works [7,8] based on the
microscopic theory. In [7] the variational method [9] was expanded on n-component
mixtures and applied to thermodynamic study of liquid Na-K-Cs and Na-K-Rb alloys
at different temperatures and alloy compositions. Jin et al [8] developed a selfconsistent energy-independent non-local model pseudopotential [10] for ternary metal
alloys and investigated the partial pair potentials and electrical resistivities in liquid
K-Rb-Cs system. Also, the molecular dynamics simulation of amorphous Al-Cu-Y
and Mg-Cu-Y alloys atomic structure [11] can be mentioned in this context.
Ion-ion interaction in [7] was described by using the local version [12] of the
Animalu-Heine model pseudopotential (AHMP) [13]. This pseudopotential model has
been successfully used to calculate the thermodynamic properties of liquid pure
metals [14] and binary alloys [15]. However, the results obtained for eutectic Na-KCs alloy [7] are not in a very good agreement with experiment. One of the ways to
improve the theoretical results is to correct the used pseudopotential approach.
Recently, we suggested such correction [16] for the local AHMP. The essence of this
improvement is taking account of the dependency of AHMP parameter A on alloy
composition for each component of alloy under consideration. Since the value of the
atomic density of alloy have an effect upon results of calculations, the dependency of
the named parameter on temperature of alloy takes place too. Hereafter, we shall call
theoretical formalism suggested in [16] as modified local AHMP (MLAHMP).
The aims of the present work are to combine the MLAHMP with the
expansion of the variational method from [7] and as a result to use this combination
for quantitative investigation of the liquid Na-K-Cs system.
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2. Theory
2.1. Variational method for ternary liquid metal alloys
Detailed description of the generalization of the variational method in conjunction
with the pseudopotential theory to n-component mixtures was given in [7]. Below we
write the basic expression to calculate the free energy (per atom), F, of the threecomponent liquid metal alloy:
3

F  3/2 kBT  T(SHS +Se) + Ue + 2  cicj drr2gHSij(r)ij(r)
i,j=1
ij
(1)
where T is the temperature; kB the Boltzmann’s constant; SHS the entropy of the hardsphere (HS) mixture; Se the entropy of the electron gas; Ue the structure-independent
contribution to potential energy;  the atomic density; ci the concentration of the ith
component; ij the HS diameters; gij(r) the radial distribution functions; ij(r) the
interatomic pair potentials, which are expressed in the pseudopotential theory as
follows ( in a.u.):

ij(r)=zizj/r+(2/)  dqq2Fij(q)sin(qr)/(qr)
(2)
0
where zi is the valence of the ith component; Fij(q) the energy wave-number
characteristics, which are evaluated here in the framework of the MLAHMP.
To compute F the right side of (1) is minimized with respect to 11, 22, 33
and . Analytical expressions for entropy [7] and partial structure factors [17] of ncomponent (with n=3) HS mixture are used. Exchange-correlation correction to
Hartree dielectric function is considered in Vashishta-Singwi approximation [18].
Nozieres-Pines interpolation formula [19] is used to estimate the electron gas
correlation energy.
2.2. MLAHMP
The formfactor of the i-kind bare ion in the local AHMP theory [12] is expressed
as:
i(q)= 4 [(zAiRMi) cos(qRMi) + AiRMi sin(qRMi)/qRMi](q)/q2
(3)
where z=c1z1+c2z2+c3z3 is the mean average valence; (q)=exp[-0.03(q/2kF)4/16]; kF
the free electron Fermi wave index; RMi and Ai the ith-component parameters such as
model radius and well depth, respectively.
In MLAHMP [16] the values of the parameter Ai are determined as follows:
 A0i, n=1
Ai = 
 A′i=A0i+(E′F E0Fi)(dAi (E)/dE), n 1
(4)
where n is the number of components in alloy; A0i the initial values of the parameters
Ai obtained by Vaks and Trefilov [12] at T=0 K for pure metals; E′F and E0Fi the
Fermi energy in alloy and the Fermi energy in pure metal (of the ith kind),
respectively; Ai(E) the dependency of the well depth on the electron energy E;
(dAi(E)/dE) the constant for each alloy component.
We evaluate EF according to Animalu and Heine [13]. Due to dependency of
EF on atomic density the calculation procedure using equation (1)-(4) is selfconsistent.
266
3. Results and discussion
The changes of Ai values with alloy composition are demonstrated in Table1. A
great deviation of A′Na in 0.5Na-0.25K-0.25Cs alloy from average between A′Na in
0.5Na-0.5K alloy and A′Na in 0.5Na-0.5Cs alloy is observed. The similar tendency is
exist for A′Cs but not for A′K.
Further we investigate the liquid Na-K-Cs alloy of eutectic composition
(cNa=0.139, cK=0.435 cCs=0.426). Obtained values of the parameter A′i for each alloy
component at three temperatures under consideration are summarised in Table 2. The
MLAHMP calculated partial pair potentials in comparison with ones obtained in the
framework of the local AHMP at T=400 K and T=800 K are shown in Figure1.
As it follows from presented curves the introduction of the concentration
dependency of Ai leads to decrease of the first minimum depth for all pair potentials
under consideration except for Cs-Cs type of interaction for which the first minimum
of potential is removed to the left. Consequently, slight deviation from additivity of
pair potentials is observed. Character of changes is the same at different temperatures.
With increase of temperature the first minimum of the each pair potential shifts
downwards that is associated with decreasing of atomic density. On can assume that
these changes must positively effect on the further calculations of the thermodynamic
properties.
Heat of mixing, H, of the eutectic Na-K-Cs alloy calculated at different
temperatures in comparison with earlier obtained results [7] and experimental data
[20] is represented in Table.3. It is clear that transition to MLAHMP from local
AHMP permits to achieve a better agreement with experiment.
4. Conclusion
This work shows that the modified local AHMP applied to ternary liquid simple
metal alloys gives a good quantitative results even in the framework of the variational
method which is the simplest form of the thermodynamic perturbation theory. It is
defined that even a small non-additivity in pair interaction leads to more accurate
calculated values of thermodynamic properties.
Development of the reported approach can be realized by introduction of the
temperature dependency of the parameter Ai in pure metal i.e. by replacing equation
(4) with the following:
 ATi = f (A0i, T) , n=1
Ai = 
 A′i=ATi+(E′F E0Fi)(dAi (E)/dE), n 1
(5)
Acknowledgements
The authors are grateful to the Russian Foundation of Fundamental Research (Grant
N99-03-32317a) for financial support.
267
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20. Kagan D.N., Krechetova G.A., Shpilrain E.E.: Gibbs energy and enthalpy of
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268
Table 1
Values of Ai (a.u.) at Т = 400 K at different Na-K-Cs alloy compositions
Concentrations of
components in alloy
CNa
0,5
0,25
0,25
0,5
0,5

1,0


CK
0,25
0,5
0,25
0,5

0,5

1,0

CCs
0,25
0,25
0,5

0,5
0,5


1,0
ANa
AK
ACs
-0,2033
-0,2050
-0,2069
-0,2090
-0,2054

-0,2136


-0,1929
-0,1951
-0,1979
-0,2006

-0,1906

-0,1949

-0,1943
-0,1972
-0,2006

-0,1978
-0,1915


-0,1866
Table 2
Values of A′i (a.u.) of the eutectic Na-K-Cs alloy at different temperatures
Temperature, К
400
600
800
ANa
-0.2030
-0.2029
-0.2028
AK
-0.1925
-0.1924
-0.1922
ACs
-0.1938
-0.1942
-0.1932
Table 3
Heat of mixing (eV) of the eutectic Na-K-Cs alloy at different temperatures
Temperature, K
Experiment [20]
Local AHMP [7]
MLAHMP
400
600
800
0,0056
0,0064
0,0079
0,0170
0,0173
0,0141
0,0065
0,0096
0,0076
269
Fig. 1 Partial pair potentials (a.u.) of the eutectic Na-K-Cs alloy at T=400 K and
T=800 K: solid line – local AHMP, dashed line – MLAHMP
270