Physica 93B (1978) 59-62 0 North-Holland Publishing Company

RADIAL DISTRIBUTION FUNCTIONS OF LIQUID Na AND Cs

T. LEE *, J. BISSCHOP and W. VAN DER LUGT

Materials Science Center, Solid State Physics Laboratory, University of Groningen, The Netherlands

and

W. F. VAN GUNSTEREN

Department of Physical Chemistry, University of Groningen, The Netherlands

Received 21 July 1977

Radial distribution functions of liquid sodium and caesium at 1OO’C have been calculated by the method of molecular dynamics with interionic pair potentials derived from Heine-Abarenkov-Shaw type model potential. The results were found to be in good agreement with recent experimental data.

The liquid state of metals has been studied extensi- vely in recent years and a great deal of information is accumulating about their properties. In order to calcu- late these properties, one needs liquid structure factors, which are related to the Fourier transforms of the radi- al distribution functions. The liquid structure factors can often be determined from X-ray or neutron diffrac- tion experiments. These experimental structure factors

can then be used in the calculation of transport proper- ties, the Knight shift and others. When there are no such experimental structure factors available, one often

makes resort to the hard sphere Percus-Yevick (HSPY) structure factors [ 1,2]. This is especially so for liquid alloys, where the availability of experimental structure factors is rather limited.

The HSPY structure factors give a qualitatively cor- rect overall description of the structure of liquid metals, but they are rather inaccurate [3]. For a more quanti- tative interpretation of liquid metal data, therefore, one needs more realistic theoretical structure factors. In the case of simple metals, where the model potential theo- ry has been shown to give a rather satisfactory descrip-

* Adress after 1 October, 1977: Department of Physics, State University of New York, Albany, N.Y. 12222, U.S.A.

tion of electronic scattering phenomena [4,5], one can find an interionic pair potential which is physically more acceptable than the hard sphere model potential. This more physical pair potential can then be used for the calculation of the radial distribution function em- ploying either the molecular dynamics [6] or the Mon- te-Carlo [7] method.

In the present investigation of liquid sodium and caesium, we have used the Heine-Abarenkov-Shaw type model potential [8-l l] in deriving the interionic pair potential and subsequently employed the molecular dynamics method to find the radial distribution func- tion. We start with a brief description of the interionic potential as derived here.

A simple liquid metal or an alloy of simple liquid metals can be considered as composed of rigid ions and nearly free conduction electrons. We consider a sys- tem of N ions in a volume of a, the average volume per ion being 52e = Q/N. Let us focus our attention on a particular pair of ions in the system and study their mutual interaction in the presence of conduction elec- trons. The direct Coulomb interaction is simply Z1Z2eZ/lRr - Rzl. Our main concern here is not this direct interaction, but the indirect interaction between the two ions induced through polarization of the con- duction electrons in the system. In the simple one-body

59

60 T. Lee et aL/Radial functions o f liquid Na and Cs

perturbation theory, this interaction can be obtained by evaluating the second-order perturbation energy of the electronic system in the field of the two ions. De- noting the electron-ion potentials for the ions by w 0) and w (2), respectively, the relevant second-order energy is given by

gl2nd = 2 ~ ~ ' (klw(1)lk + q) (k + q Iw(2)lk) 1(k2 (k + q)2 ) , (1)

k q ~ - -

where the k-summation is limited to [kl <~ kf and the prime on the q-summation indicates that the q = 0 term should not be included in the sum. Converting summa- tions into integrals and using D(k, q) for the energy de- nominator, we may write eq. (1) as

Viln~= ;~;3 f d3q eiq'(R1-R2) F12(q) , (2)

where

2~2o r 3 w(1).w (2) . . q q F12(q) = ~} -~ J d k D(k, q) (3)

In the last equation, we have defined the bare ion form factor w (~) according to

(k + qlw(°OIk) = 1 f d3re -i(k+q)'r w(a)(r-Ra)e ik'r

l (a" i R e - q" . ( 4 )

In the literature F12(q) is referred to as the energy-wave- number characteristic (EWNC) [ 12]. The potential that the electrons actually see is screened by the electrons themselves. Inclusion of this screening effect modifies the EWNC. In addition to the screening effect, one has to include the effective mass correction. Both of these effects have been discussed by Shaw [13-15] and others [ 1 O, 11], and we shall give only some final results relevant to our calculations. It is convenient to write the fully corrected EWNC as a sum of three terms as follows.

F12(q) = m.__kk F~2(q) + AF~2c(q) + 2utz~(q) , (5) , mE

where F~Z(q) denotes the EWNC calculated in the Hartree approximation, AF12c(q) the correction due to the exchange and correlation effects, and A F ~ ( q ) and the factor mk/mE in front of FlZ(q) arise from the ef- fective mass correction. The two effective masses mk and m E are those given by Appapillai and Williams [ 10],

and they have shown that these effective masses may be taken to be constant. The explicit forms of the three terms on the right-hand side of eq. (5) have been given by Shaw [13-15] for pure metals (w (1) = w (2)) and will not be repeated here. In evaluating the second term, the exchange and correlation effects on the EWNC, we have used the screening function calculated by Toigo and Woodruff [16].

From eqs. (2) and (5), the indirect potential is ob- tained and combining it with the direct Coulomb poten- tial, one finds the effective pair potential in metals. One should note, however, that owing to the appear- ance of the so-called depletion hole charge in the ener- gy-dependent model potential theory, it is convenient to use a modified direct Coulomb interaction, namely,

i t 2 P Z 1 Z 2 e / ]R 1 - R21 with the effective valence Z~ given by Z,~ - Oa, Pa being the depletion hole charge. Thus we may write the effective pair potential in simple metals as

12 - Z'lZ'2. e 2 12 Veff I-R~lTR~2[ + Vind(lR 1 - R2]). (6)

In actual calculations, we should further add to this effective pair potential the Born-Mayer type repulsive part to take care of the hard core of the ions.

In the present investigation, we have used the meth- od of molecular dynamics to calculate the radial distri- bution functions (RDF) of sodium and caesium metals, both at 100°C. At this temperature, we find a density of 0.0243 atoms/A 3 for sodium and 0.0081 atoms/A a for caesium. The pair potentials for these metals are shown in fig. 1. These potentials include the Born- Mayer repulsive part [17]. The vertical arrows indicate the nearest neighbour distances obtained from the ex- perimental RDF's [ 18]. The potentials were cut off at r = 8.7 A and r = 13.8 ,~ for sodium and caesium, re- spectively. This cut-off radius corresponds to the fourth node of each potential. In the molecular dynamics calcu- lation, 216 atoms were randomly placed in a cubical box and periodic boundary conditions were imposed to eliminate the effects arising from a surface. The time step used for integrating the equations of motion were chosen to be about 1.0 X 10 - 1 4 S for sodium and 0.5 )< 10 - 1 4 S for caesium. With these time steps, it took some 250 steps to reach equilibrium in both cases. After equilibrium had been reached, we have run 3000 steps more to obtain the RDF. The results are shown in fig. 2 and compared with the experimental RDF's.

T. Lee et al./Radial functions o f liquid Na and Cs 61

/ * . 0 - -

T 2.0

}00 -2.0

Na Cs

- o0 8'o 120 160 r i l l

Fig'. 1. Interionic potentials for Na and Cs at 100°C. The verti- cal arrows indicate the nearest neighbour distances obtained from the experimental radial distribution functions.

The agreement was found to be very good.

In order to find the liquid structure factors, we have to carry out the Fourier transformation of our calcu- lated RDF's. For this purpose, however, the range of

r for which the RDF's were evaluated is too much limit. ed. To increase the range of r, we have to increase the number of particles used in the calculation and this requires too much of computation time. Instead of do- ing that, we have fitted the tail part of the calculated RDF to an asymptotic form [19]

A e - a r 1 + cos(~3r + ~i)

r

and then carried out the required "Fourier transforma- tion. Of course this is a crude approximation and one cannot expect to obtain accurate structure factors, es- pecially for low q values, in this way. The results for caesium are presented in fig. 3 together with the exper- imental structure factor. As expected, the results for low q values are not reliable at al~l, but those for high q values are quite reasonable. Although the low-q re- suits are not reliable, it is still interesting to observe that the theoretical results are oscillating about the smooth experimental ones.

In conclusion, we may say that a highly reliable ra- dial distribution function can be obtained by the meth-

I

2.1+

1.6

c~

0.8

Na

Cs

O.C , 8~.0 t , 0.0 4.0 12.0 r ( A }

Fig. 2. Radial distribution functions of Na and Cs at 100°C. The theoretical results are given by solid curves and the exper- imental results are given by crosses and empty circles for Na and Cs, respectively. We have ignored small spurious oscilla- tions occurring near r = 0 in the experimental RDF's due to the use of truncated structure factors for the Fourier trans- formation.

2.~ -

2.0 -

] . 6 -

1.2-

0.8 -

. 0 . 4

0.0 :%. . / I I

o.o 2'.o 3.0 4 .o q (~,-1)

60

Fig. 3. The structure factors of liquid caesium at 100°C. The solid curve represents the theoretical results and the broken curve the experimental results.

62 T. Lee et al./Radial functions o f liquid Na and Cs

od of molecular dynamics with the pair potential de- rived from the Heine-Abarenkov-Shaw type model

potential. It should be emphasized that this pair poten- tial is derived from the basic properties of the constit- uent ions and electrons such as term values of free ions and screening functions of an electron gas. As such they

may almost be considered as being obtained from a first principles calculation and they can be generated for al- loys as well as for pure metals so long as these basic properties of the constituents are available. There are of course other types of pair potentials often used in

the study of liquids, but they are of more empirical character and contain adjustable parameters which have to be determined from available experimental data for each case. In view of the good agreement we have ob- tained between the theoretical and the experimental RDF's, we feel that one does not need the latter types of empirical potentials at least for simple liquid metals and their alloys. Application of the same procedure to Na-Cs alloys is in progress.

Acknowledgements

We would like to thank professor H.J.C. Berendsen for many useful discussions on the method of molec-

ular dynamics, and Dr. M.J. Huijben for making avail- able to us his recent diffraction data on liquid Na and Cs and also for valuable discussions about the structure of these metals.

This work is part of the programme of the "Stich- ting voor Fundamenteel Onderzoek der Materie" (FOM) and has been made possible by financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschap- elijk Onderzoek" (ZWO).

References

[1] N.W. Ashcroft and J. Lekner, Phys. Rev. 145 (1966) 83. [2] N.W. Ashcroft and D.C. Langreth, Phys. Rev. 156 (1967)

685. [3} A.J. Greenfield, N. Wiser, M.R. Leenstra and W. van der

Lugt, Physica 59 (1972) 571. [4J J.F. Devlin and W. van der Lugt, Phys. Rev. B6 (1972)

4462. [5] For a review see W. van der Lugt and A.J. Dekker, Physi-

ca 69 (1973) 148. [6] A. Paskin and A. Rahman, Phys. Rev. Letters 16 (1966)

300. [7] R.D. Murphy, Phys. Rev. A15 (1977) 1188. [8] V. Heine and I. Abarenkov, Phil. Mag. 9 (1964) 451. [9] R.W. Shaw, Phys. Rev. 174 (1968) 769.

[10] M. Appapillai and A.R. Williams, J. Phys. F: Metal Phys. 3 (1973) 759.

[ 11] J.J. Hailers, T. Marien and W. van der Lugt, Physica 78 (1974) 259.

[12] See for example W.A. Harrison, Pseudo-Potentials in the Theory of Metals (Benjamin, New York, 1966).

[13] R.W. Shaw, J. Phys. C2 (1969) 2335. [14] R.W. Shaw, J. Phys. C2 (1969) 2350. [15] R.W. Shaw, J. Plays. C3 (1970) 1140. [16] F. Toigo and T.O. Woodruff, Phys. Rev. B2 (1970) 3958. [ 17] The parameters for the Born-Mayer repulsive part for

Na and Cs have respectively been taken from W. Cochran, Proc. Roy. Soc. (London) A276 (1963) 308; A.A. Abrahamson, Phys. Rev. 178 (1969) 76.

[18] These experimental radial distribution functions have been obtained by Huijben from his recent X-ray diffrac- tion measurements. See also M.J. Huijben and W. van der Lugt in Liquid Metals 1976: Inst. Phys. Conf. Ser. No. 30, R. Evans and D.A. Greenwood, eds. (Bristol, 1977) p. 141

[ 191 I.Z. Fisher, Statistical Theory of Liquids (Univ. of Chica- go Press, Chicago, 1964) p. 140.