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B.K. Singh and Sudhir Singh / BIBECHANA 9 (2013) 7-12 : BMHSS, p.7 (Online Publication:  Nov., 2012) 
 
 

BIBECHANA 
A Multidisciplinary Journal of Science, Technology and Mathematics 

ISSN 2091-0762 (online) 

Journal homepage: http://nepjol.info/index.php/BIBECHANA 

 

 

Temperature dependence of the Helmholtz free energy of 
liquid alkali metals 

 

B. K. Singh1*,  Sudhir Singh2 
1 

University  Department of Physics, T.M. Bhagalpur University, Bhagalpur 
2 

Department of Physics, R. D. College Sheikhpura, T.M. Bhag. Univ., Bhagalpur 
*
Corresponding author: E-mail : bijay.k.singh@gmail.com 
Article history: Received 12 February, 2012; Accepted 5 July, 2012 

 

 
Abstract 

The Gibbs-Bogoliubov variational method has been considered to study the Helmholtz free energy of 

liquid alkali metals (Na, K, Rb and Cs) as a function of temperature, using Heine Abarenkov type model 

potential with Hubbard-Sham exchange and correlation function. The computed values are in very good 

agreement with experimental observations.  

 

Keywords: Helmholtz free energy; Entropy; Specific heat; Boltzman constant; Perturbation energy; Madelung energy 

 
1. Introduction 
 

The variational prescription based on the Gibbs-Bogoliubov inequality [1] has provided a convenient 

computational tool for evaluating the free energy of liquid metals with the framework of a hard-sphere 

reference system and pseudopotential theory [2,3]. Recently, many workers [4-8] have utilized 

successfully the Gibbs Bagoliubov (GB) variational technique to study the thermodynamic properties of 

liquid metals and alloys. In most of the cases the earlier studies are confined near the melting temperature 

(Tm). 

The present work is an attempt in this direction to investigate the thermal effect on the Helmholtz free 

energy of liquid metals for T > Tm, which is least understood. This will help us to examine the utility of 

hard-sphere reference system for election-ion interaction energy. We consider the Heine-Abarenkov (HA) 

form of model potential [9] in conjunction with the Hubbard- Sham [10-11] (HS) exchange and 

correlation function. For the present purpose we choose Na, K, Rb and Cs because the pseudopotential 

perturbation can be applied without committing any appreciable error [12]. 

 

2. Theory 

 
The details of the variational method used for investigating the thermodynamic properties of liquid metals 

and alloys are available in the work of Umar et  al [13] and Ashcroft and stroud [14]. In the framework of 

GB method the Helmholtz free energy, F, per ion at fixed temperature T and volume Ω can be expressed 

as  

   



B.K. Sing and Sudhir Singh / BIBECHANA 9 (2013) 7-12 : BMHSS, p.8 (Online Publication:  Nov., 2012) 

 

F = Fhs + Fps        (1) 
 

Here Fhs is the Helmholtz free energy of the hard-sphere system, which can be expressed as 
  
  Fhs = (3/2) KBT - TS       (2) 
 

where (3/2) KBT is the mean kinetic energy and S, the entropy of the system can be written as    
 

S = Sgas + Sη        (3) 
 

with      Sgas = (5/2)KB + KB ln{Ω(mKBT/2πħ2)3/2}    (4) 

 

 Sη = KB ln (1-η) + 3/2 KB{1-(1-η)2}     (5) 

 

Eq. (3) has further been improved by incorporating the low temperature specific heat contribution of 

electron gas. The resulting expression for the entropy becomes 

 

  S = Sgas + Sη + Selec      (6) 
 

with    
2 2 2/elec B FS K T Kπ=

       (7) 

where KB is the Boltzmann constant and KF stand for the Fermi wave vector 

3 2( 3 ,FK nZπ=
 n is the 

number density). 
 

Fps  =  Feg  +     F1  +     F2  +     Fm     (8) 

 

Where Feg is the free energy of the electron gas, F1 and F2 are, respectively, first and second order 

perturbation energies due to the electron-ion interaction. Fm is the Madelung contribution which takes 

into accounts for the ion-ion interaction. The expressions for these contributions for a metal have been 

worked out in detail by Harrison [2]. 

 
2[(3 /10) (3 / 4 ) 0.0474 0.0155ln ]eg F F FF nz K K Kπ= − − −

  (9) 

 

 

2

1
0

lim  { ( ) 4 / }
q

F n z v q z qπ
→

= −
     (10) 

 

3 2 4

2

0

(1/16 ) ( ) ( )[{1/ *( )} 1]F v q s q q q dqπ
∞

= − ∈ −∫
  (11) 

 

2

0

( / ) { ( ) 1}mF Z s q dqπ
∞

= −∫
     (12) 

Where q is the phonon wave vector V(q) stands for unscreened from factor, which can be obtained using 

the HA potential, 
 

2 3( ) (4 / )cos (4 / )(sin cos )m m m mV q zn q qr An q qr qr qrπ π= − − −
 (13) 

 

 

 



B.K. Singh and Sudhir Singh / BIBECHANA 9 (2013) 7-12 : BMHSS, p.9 (Online Publication:  Nov., 2012) 

 

Here A is the well depth and rm is the model radius. These are obtained quantum mechanically by 

matching the wave function at 
*( )mr r q= ⋅∈

 in Eq. (11) is the modified Hartree dielectric screening 

function which takes into account of the conducting electron interaction, 

 

 ∈*(q) = 1 + {Є(q) - 1}{1 – G(q)}     (14) 

 

∈(q) is the Hartree dielectric function and  G(q) is the correction factor for the exchange and correlated 

motion of the conducting electron. Presently we consider G(q) prescribed by Hubbard and Sham 

 

 

2 2 31
( ) /[ {2 /(0.153 )}]

2
F FG q q q k kπ π= + +

   (15)  

 

The structure factor, s(q), for liquid  metals appearing in Eq. (11) and (12) can be calculated from the 

Parcus-Yevick  approximation for hard–sphere potential, which is characterized by the hard-sphere 

diameter(σ),or equivalently, by the packing fraction  
3( /6)Mη πσ=

 

 

3. Results and Discussion 
 

The general expression for the Helmholtz free energy of pure liquid metals can be written as  

 

F = Feg + F1 + F2 + FM  + (3/2)KBT - TS   (16) 

 

The different terms occurring in Eq. (16) are already defined in Section 2. It is well known that hard-

sphere potential serves as an effective reference system for liquid metals. The best hard sphere reference 

system is obtained by selecting those diameters which minimize the Helmholtz free energy, F, at the same 

temperature T and volume Ω through 

 

   ,

0
T

F

σ Ω

∂  = ∂ 
      (17) 

 

The optimum values of the parameters used in the calculation of the Helmholtz free energy of liquid 

alkali metals as a function of temperature are compiled in Table 1. The volume of the liquid metals at 

different temperatures required in the calculation is determined from the relation provided in the work of 

Huijben [17]. The experimental values of the entropies at appropriate temperature are obtained from 

tables found in Hultgren et al [18-21]. 

Using Table 1, we can calculate easily the various terms of Eq. (16). The computed values of   Feg, F1, F2, 

Fm and Fhs are listed in Table 2. A perusal of Table 2 show that  the major contribution  to the theoretical 

determined Helmholtz free energies comes from the Madelung terms Fm, the other terms contribute a 

relatively small amount and in order of magnitude falls as F1, Feg,   Fhs  and F2. Table 2 shows that the 

magnitude of F1 and Fm decreases with the rise of temperature where as the magnitude of Feg, F2, Fhs 

increases. The computed values of F for liquid alkali metals are displayed in Fig.1 as a function of 

temperature along with the experimental observation of Hultgren et al [18]. The best agreement is 

obtained for Cs followed by Rb, K and Na. It may be noted that the Helmholtz free energy of liquid alkali 

metals depend sensitively on temperature above the melting point. The rise of temperature  



B.K. Singh and Sudhir Singh / BIBECHANA 9 (2013) 7-12 : BMHSS, p.10 (Online Publication:  Nov., 2012) 

 

experiences a decrease in the Helmholtz free energy. Our results reveal that the decrease of magnitude of 

the Helmholtz free energy  due to first order energy and Madelung energy is compensated due to 

increasing values of F2, Feg and Fhs as a result F only depends considerably on temperature above the 

melting point. 
 

Table 1: Input Parameters in the Calculation of the Helmholtz free energy of Na, K, Rb and Cs at different 
temperatures 

 
Liquid Metals T(°°°°K) Ω(a.u) σ(a.u) S/KB 

371.0 278.187 6.2579 7.1043 

473.0 285.722 6.1659 8.0953 

573.0 293.427 6.0922 8.8416 

673.0 301.495 6.0293 9.4533 

Na 

773.0 310.077 5.9741 9.9751 

     

336.6 530.504 7.6633 8.7493 

423.0 543.360 7.5449 9.6486 

523.0 559.190 7.4313 10.4569 

623.0 576.037 7.3349 11.1095 

K 

723.0 593.824 7.2502 11.6587 

     

312.6 667.200 7.2179 10.1972 

373.0 681.292 8.1174 10.9056 

473.0 705.516 7.9788 11.8184 

573.0 731.796 7.8625 12.5406 

Rb 

673.0 759.994 7.7624 13.1400 

     

301.8 804.117 8.7922 10.8790 

373.0 824.402 3.6678 11.7315 

473.0 852.587 8.5192 12.6487 

573.0 884.556 8.3973 13.3825 

Cs 

673.0 919.709 8.2915 13.9931 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
Fig. 1 : Helmholtz free energy of Na, K, Rb and Cs as a function of temperature. Full and broken curve refer to 
theoretical and experimental values respectively 

 



B.K. Singh and Sudhir Singh. / BIBECHANA 9 (2013) 7-12 : BMHSS, p.11 (Online Publication:  Nov., 2012) 

 
Table 2: Contribution to Helmholtz free energy of liquid alkali metals as function of temperature 

 

Liquid 

Metals 
T(°K) -Feg F1 -F2 -Fm -Fhs 

371.0 0.08162 0.07734 0.00479 0.21371 0.00656 

473.0 0.08168 0.07530 0.00619 0.20968 0.00988 

573.0 0.08175 0.07332 0.00761 0.20575 0.01333 

673.0 0.08180 0.07136 0.00910 0.20181 0.01695 

Na 

773.0 0.08185 0.06938 0.01066 0.19781 0.02075 

 

336.6 0.07995 0.06419 0.00465 0.17144 0.00773 

423.0 0.07982 0.06267 -0.00587 0.16825 0.01092 

523.0 0.07966 0.06089 0.00737 0.16454 0.01434 

623.0 0.07949 0.05912 0.00894 0.16079 0.01896 

K 

723.0 0.07933 0.05734 0.01057 0.15702 0.02326 

       

312.6 0.07826 0.05782 0.00476 0.15837 0.00859 

373.0 0.07841 0.05779 0.00543 0.15827 0.01111 

473.0 0.07784 0.05467 0.00747 0.15172 0.01546 

573.0 0.07757 0.05272 0.00929 0.14753 0.02004 

Rb 

673.0 0.07720 0.05076 0.01122 0.14327 0.02481 

       

301.8 0.07660 0.05795 0.00419 0.14917 0.00896 

373.0 0.07638 0.05653 0.00521 0.14641 0.01209 

473.0 0.07610 0.05466 0.00672 0.14265 0.01670 

573.0 0.07578 0.05265 0.00835 0.13868 0.02157 

Ca 

673.0 0.07545 0.05067 0.01009 0.13466 0.02663 

 
Acknowledgement 
We would like to thank Prof. R. N. Singh, Department of Physics, Bhagalpur University for his useful 

suggestions and encouragement in this work. I am also thankful to the U.G.C., New Delhi, for the 

financial support under grant no. F10-109/90(RBBII). 

 

 

References 
[1]    A. Isihare, J. Phy A: Gen. Phys, 1 (1968) 53. 

[2]    W.A. Harrison,  Pseudopotential in the theory of metals ,W A Benjamin Inc., New York (1966). 

[3]    T.E. Feber, Introduction to the theory of liquid metals ,Cambridge: CUP, (1972). 

[4]    H. Jones , J Chem Phys, 55(1971)2640, Phys Rev A, 133 (1973) 3215. 

[5]    D. Stroud and N.W. Ashcroft, Phys. Rev. B, 5 (1972) 371. 

[6]    I.H. Umar and W.H. Young, J Phys F, 4 (1974) 525. 

[7]    R.N. Singh , J Phys F, 10 (1980) 1411. 

[8]    R.N. Singh and S. Singh, Physica B, 128 (1985) 304. 

[9]    V. Heine and I.V. Abarenkov, Philos Mag, 9 (1964) 451. 

[10]   J. Hubbard , Proc. Roy. Soc. A, 240 (1957) 539. 

[11]   L.J.Sham, Proc. Roy Soc. A, 283 (1965) 33. 

[12]   V. Heine and D.L Weare, Solid State Physics, 24 (1970) 249. 

[13]   I.A. Umar , A. Meyer, M. Watabe   and  W.H. Young, J. Phys. F, 4(1974) 1691. 

[14]   N.W. Ashcroft and D.  Stroud, Solid State Physics, 33 (1978) 1. 



B.K. Singh and Sudhir Singh / BIBECHANA 9 (2013) 7-12 : BMHSS, p.12 (Online Publication:  Nov., 2012) 
 

[15]   O. Ese and J.A.  Reissland , J Phys F, 3 (1963) 2066. 

[16]   N.W. Ashcroft and J. Lekner , Phys Rev, 145 (1966) 83. 

[17]   M.J. Huijben  Ph.D. Thesis, University of Groningen, The Netherlands (1978). 

[18]   R. Hultgren,P.D. Desai, D.T. Hawkin and M. Gleiser, Kelley K K & Wagmann D D, Selected values of the   

          thermodynamic properties of the elements ,American Society of metals, Metal Park , Ohio, (1973). 

[19]   D. Adhikari,  B.P. Singh,  I.S. Jha, and B.K. Singh, J. Non-cryst. Solids, 357 (2011)  2892. 

[20]   D. Adhikari , I.S.Jha,   and B.P.Singh, Philos. Mag. 90 (2010) 2687. 

[21]   D. Adhikari, B.P. Singh , I.S. Jha, and  B.K. Singh , J. Mol. Liq. 156 (2010) 115.