Volume 118, number 9 PHYSICS LETTERS A 17 November 1986

H E L I U M AT H I G H DENSITY

Marvin ROSS and David A. YOUNG Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA

Received 7 August 1986; revised manuscript received 12 September 1986; accepted for publication 18 September 1986

We show that all available high pressure equation of state and melting data for helium can be fitted with an exponential-six potential. This potential is slightly softer than the best theoretical pair potentials and thus implies the existence of attractive many- body forces. The theory also predicts a zero Kelvin fcc to bcc phase transition at 10.6 Mbar, and a maximum temperature of 690 K for the fcc phase.

The properties of hydrogen and helium at high density are topics of continuing interest. These are the simplest elements and they constitute 95% of the matter in the solar system. Recent advances in high pressure experimental methods have provided important new data for these elements. In the case of hydrogen, measurements of solid and liquid iso- therms [ 1,2], the melting curve [3], and the shock hugoniot [4] have been studied theoretically and have been used to determine a simple intermolecular potential function [ 5 ]. With well-established statis- tical mechanical methods, this potential can be used to reproduce all of the experimental data [ 5 ]. Simi- lar data for helium have now become available and although some theoretical studies have been reported, it does not appear that a single intermolecular poten- tial has been determined for calculating all of the high pressure properties of this element.

Besson and coworkers have in recent years found evidence of a new solid phase in helium at about 300 K along the melting curve [ 6,7]. This evidence occurs in the form of a cusp in the melting curve, which indicates a triple point between two solid phases and the liquid. Since it is known that sufficiently soft pair potentials stabilize bcc [ 8 ], this structure is a likely candidate for the new phase. Several calculations in the past few years have reinforced this conclusion [ 9,10 ], even though there is as yet no direct evidence for the structure of the new phase. Our main interest in this regard is in the trajectory of the fcc-bcc phase

boundary at very high pressures. In a very different range of conditions, recent

advances in our understanding of solar system evo- lution have resulted in a renewed interest in the modeling of the giant planets and of "brown dwarfs". This theoretical effort demands accurate equations of state of the component elements at very high den- sity and temperature. In the case of Jupiter, for example, hydrogen and helium are believed to be subjected to temperatures as high as 20000 K at the fluid-solid core boundary. Shock-wave data are well suited to provide this information.

In an earlier paper on helium published in 1981 [ 11 ], we showed that all of the then-available data on the high pressure solid and liquid equation of state could be fitted with an exponential-six (exp-6) potential,

6 1 r ~ ( r , = {~--~_6exp[o~(-~--~)] (1,

6 o

o~-6

with the parameters ot=13.1, e /k=10.8 K, and rm=2.9673 A.

Recently, single and reflected shock compression data for liquid helium have been reported [ 12 ]. The shock compression curve is calculated by solving the hugoniot equation:

E-Eo = ½ ( P + e o ) ( 1Io - V) , (2)

0375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

463

Volume 118, number 9 PHYSICS LETTERS A 17 November 1986

4 6 8 10 12 600 ' .I / ' I ' 1 ~ I

| l | - -C-P Reflected . ~ / / ~- ~f-Aziz Hugoniot

4 1 ,ooo ¼11 '

150 k T = 300 K ' $i$

\,sotherrn 14,000 K "-~,.~/i /

~k ~/~ Principal

100 Exp-6 6

(expt, ~ ~ 800 K ~

4 6 8 10 12

V(cm3/mole)

Fig. 1, Comparison of experimental and theoretical helium equa- tions of state. The principal hugoniot is shown on the right; the reflected shock point is shown on the upper segment (note the broken scale above 150 kbar); and the 300 K isotherm is shown on the left. Some calculated temperatures along the hugoniot are indicated. The shaded area refers to the uncertainty in the cal- culated reflected shock which arises from experimental uncer- tainties in the reflected shock conditions. The labels for the theoretical curves refer to the potentials shown in fig. 2.

where E, P, and V refer to the energy, pressure, and molar volume, and the subscripted variables refer to the initial conditions. The pressure and energy were calculated using the "soft sphere" variational fluid theory that we used in our 1981 paper. As shown in fig. 1, the calculated principal and reflected hugoniot curves are in good agreement with experiment. We have also made hugoniot calculations with the widely used Aziz potential [ 13 ] and the recent quantum Monte Carlo potential of Ceperley and Partridge (C-P) [14 ]. Calculations made with the Aziz pair potential predict pressures that are too high. The C-P hugoniot is intermediate between Aziz and exp-6 and in fair overall agreement with the shock data. This suggests that the many-body forces in helium are weakly attractive.

Listed in fig. 1 are several temperatures calculated at the points indicated. Their high values relative to the He-He attractive well depth demonstrate that the

important contributions to the hugoniot must come from the repulsive interaction at small interatomic separations.

Recently Polian and Grimsditch, using Brillouin scattering, have measured the velocity of sound in fluid helium at room temperature up to 120 kbar [ 15 ]. These data were integrated to obtain the den- sities up to the freezing point. The resulting isotherm is shown in fig. 1 compared with the exp-6 and Aziz potential calculations. The calculated pressures are too high. We believe that at least a part of the dis- agreement is due to the accumulation of errors in the integration of the experimental data. Polian and Grimsditch assign an uncertainty not greater than 9% to the quantities in the integrand. This leads to a 5% error in the volume. In the present case this amounts to +0.21 cm3/mole at 120 kbar or just enough to include the exp-6 calculation. It would be preferable ifa direct measurement of the density could be made which would be independent of model approxima- tions for its determination.

An interesting feature of this isotherm is that the volume achieved at highest static pressure is consid- erably smaller than that achieved at much higher shock pressures. This is the result of the very rapid rise of temperature along the shock hugoniot, which prevents further compression. In fact, since the lim- iting compression of a monatomic fluid is Vo/V= 4, the smallest final volume achievable with a single shock in liquid helium is about 8.1 cm3/mole.

The exp-6 potential is obtained from data fitting and includes contributions from many-body inter- actions. The Aziz potential includes short range self- consistent field Hartree-Fock calculations and long range dispersion energy terms. The C-P potential is computed from quantum Monte Carlo and is pre- sumed to be a very accurate representation of the two- body potential at short interatomic separations. Since our main concern in this paper is with the short range repulsion region, we show a comparison of the three potentials in this region in fig. 2.

The exp-6 potential has very nearly the same slope as the Aziz and C-P potentials, but it is significantly lower at all interatomic separations. This is consis- tent with recent studies on argon [ 16 ] and hydrogen [ 17 ] which conclude that the short range many-body interactions are attractive. From figs. 1 and 2 we conclude that the Aziz potential is too stiff to give

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Volume 118, number 9 PHYSICS LETTERS A 17 November 1986

60 X 103

40 X 103

20 X 103

=Z lO×1O3 8 X 103

6 X 103

4X 103

Aziz

2 X 103

1 X 10 3 ~ 1.0 1.2 1.4 1.6 1.8 2.0

r(A)

Fig. 2. Comparison of theoretical interatomic potentials for helium. Shown are the Aziz potential, the Ceperley-Partridge (C-P) quantum Monte Carlo potential, and the exp-6 potential

good agreement with experimental data on liquid and solid helium.

The calculation of the phase diagram of a material from first principles theory is difficult because of the extreme sensitivity of the location of phase bounda- ries to small differences in the thermodynamic prop- erties of the phases. Obtaining theoretical agreement with the experimental helium phase diagram will thus be a more stringent test of the adequacy of the pair potential and of the statistical mechanical models.

In our 1981 paper on helium, we computed the melting curve to the highest experimental tempera- ture then achieved, about 300 K. The agreement was satisfactory. Since 1981, new measurements have extended the melting curve to about 400 K [6,7]. These new data are compared with calculations in fig. 3. In numerous calculations of melting curves it has been our experience that a potential which fits the equation of state will also yield a good fit to the melt- ing curve. It is therefore not surprising that the agree- ment between theory and experiment to 400 K for helium is still very good.

200 I I I I / ' /

__ Theory /

150 c: Experiment /

"~ Solid / 100

50 -

~ I I I I O0 " 100 200 300 400 500

T(K)

Fig. 3. C o m p a r i s o n o f e x p e r i m e n t a l a n d theo re t i ca l m e l t i n g cu rves

for h e l i u m .

Although melting curves can be calculated with confidence, this cannot be said of solid-solid phase transitions. Here the volume changes may be several orders of magnitude smaller than for melting, requir- ing extreme accuracy in the solid models. In the pres- ent study we compute the fcc and bcc free energies by quasi-harmonic lattice dynamics corrected by anharmonic terms from an Einstein cell model [ 11 ]. The phase boundary is shown in fig. 4. The bcc phase occupies a larger field than indicated by the experi- mental result. Whereas the experiment suggests that bcc first appears on the melting curve at about 300 K, our calculations show bee as the stable phase along the melting curve to temperatures below 200 K. For volumes larger than 4.3 cmVmole, we find that the bcc phase is mechanically unstable in the quasihar- monic approximation, giving imaginary frequencies. This suggests that a more realistic treatment of the anharmonic motion is necessary for accurate solid-solid phase boundary calculations [ 9,10 ]. Here the discrepancy between calculation and the inter- pretation of experiment is not due to the potential but to the statistical mechanics model.

At 0 K and 10.7 Mbar, the theory predicts an fcc to bcc phase transition. Since the static lattice energy is always higher for the bcc phase, the stabilization of

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Volume 118, number 9 PHYSICS LETTERS A 17 November 1986

1 0 -

8

4

2

0 200 400 600 800

T(K)

Fig. 4. Theoreticalfcc-bcc phaseboundarycomputedfromquasi- harmoniclattice dynamics.

bcc must be due to its lower zero-point energy. The fcc phase also is predicted to have a max imum tem- perature of 690 K at 3.7 Mbar. At higher tempera- tures, bcc is stabilized by its higher entropy.

Since all of the rare gases have effective potentials with nearly equal ot values, the principle of corre- sponding states holds to good accuracy, and the fcc-bcc phase transi t ion found in He will occur in the other elements in the series. For Ne, correspond-

ing states predicts that the fcc-bcc l iquid triple point will occur at approximately 1000 K and 320 kbar, which is within reach of contemporary d iamond anvil technology. The heavier rare gases will have the tran- sition at still higher temperatures and pressures.

In summary, we find that a simple exp-6 potential adequately fits the available experimental data for

high density helium.

This work was performed under the auspices of the US Depar tment of Energy by Lawrence Livermore

National Laboratory under Contract No. W-7405- Eng-48.

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