SPATIAL CORRELATIONS IN SPIN-POLARIZED 3He LIQUID

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SPATIAL CORRELATIONS IN SPIN-POLARIZED 3He LIQUID

L. Lantto

To cite this version:

L. Lantto. SPATIAL CORRELATIONS IN SPIN-POLARIZED 3He LIQUID. Journal de Physique

Colloques, 1980, 41 (C7), pp.C7-223-C7-225. �10.1051/jphyscol:1980735�. �jpa-00220172�

JOURNAL DE PHYSIQUE CoZZoque C7, supp26ment au no 7 , Tome 41, juiZZet 1980, page C7-223

SPATIAL CORRELATIONS I N SPI N-POLAR IZED 3 ~ e LIQUID

L.J. Lantto *

Department of Physics, S.U.N.Y. a t Stony Brook, Stony Brook, N.Y. 12794, USA.

Resume.- On pr6sente des calculs utilisant la m6thode variationnelle FHNC appliqu6s 2 3 ~ e orients 2 basse temperature. La fonction d'essai est optimisee en resolvant une equation dlEuler-Lagrange dans l'approximation FHNC. On compare les resultats obtenus pour la fonction de structure du liquide et la distribution d'impulsion 3 une particule avec les m6mes quantitss pour le liquide sans orientation de spin.

Abstract.- Variational Fermi HNC calculations are performed for spin-polarized 3 ~ e liquid at zero temperature. The trial wave function is optimized by solving an

Euler-Lagrange equation in the Fermi HNC approximation. Results for liquid structure function and one-particle momentum distribution are compared with the corresponding quantities of the spin-saturated liquid.

In this report we present results of a theo- retical evaluation of liquid structure func- tion and one-particle momentum distribution for spin-polarized and normal 3 ~ e liquids.

The calculations are based on the varia- tional method. The ground state many-body trial wave function is chosen to be of stan- dard Jastrow form

where

is the ideal gas wave function and the correlation factor f is a function of radial dependence only. The expectation value of the energy is determined by em- ploying the Fermi hypernetted-chain (FHNC) method /1-4/ and the minimization of the energy is carried out by solving the Euler- Lagrange equation / 5 , 6 /

in which v is the usual Lennard-Jones poten- tial and w is an induced potential due to dynarnical and exchange correlations. The induced potential is a functional of the radial distribution function g and it is calculated self-consistently as described in ref. /5/. The liquid structure function S(k) is related to g by a Fourier transfor- mation

- -

~ ( k ) = 1 + p Irg(r)-l] eikar IF ( 3 )

* On leave from

Dept. of Theoretical Physics,

University of Oulu, Finland.

where

is the average density of the liquid.

Once the distribution function g and the correlation factor f are known one can calculate the one-particle density matrix n(rl,r;) by a method similar to FHNC treat- ment of the radial distribution function /7,8/. The density matrix exhibits the following functional form

where r

Irl-r;l

nc is a strength fac- tor and

(kFr) is the inverse Fourier transform of the ideal gas momentum dis- tribution function. The functions GDD and GWW represent infinite partial summations of the contributing diagrams : see ref./7/

for details. The one-particle momentum distribution function is obtained as a Fourier transform of the density matrix given by eq. ( 4 ) .

The present method is an extension to Ferni systems of the variational method which we have recently applied for a calculation of ground state properties of the condensed spin-aligned atomic hydrogen /9/. In the Bose case the momentum distribution shows singular low momentum behavior

n(k)

nc/k as k

0, where nc is the Bose-Eins- tein condensate fraction. The calculated condensate fraction of Hf at T = 0 K va- ried from 95% to 29% as the density of hydrogen atoms was increased from 2 x 1 0 ~ 0

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980735

JOURNAL DE PHYSIQUE

atoms/cm3 to a thirty times higher density.

For comparison the condensate fraction of the ground state '+He liquid was 12% at the saturation density.

The simple trial wave function of eq. (1) is not adequate for accurate determination of the ground state energy for normal and spin-polarized 3 ~ e liquids. This wave func- tion provides a better description for the polarized phase because due to the exclusion principle the dynamical correlations are slightly weaker in the polarized system.

Therefore the energy obtained by solving the Euler eq. (2) in the FHNC approximation is lower for the polarized than for the spin- saturated system. The optimization of the correlation factor does not lower the energy considerably compared to standard simply parametrized choices of the Jastrow factor f(r). However, the optimization may improve the resulting structure functions noticea- bly.

In figure 1 we show the calculated liquid structure functions for spin-polarized

(dashed line) and normal (solid line) 3 ~ e liquids at the saturation density

0.0164 atoms/A3 ofthe normal liquid. The correspon- ding experimental quantity measured at T = 0.56 K /lo/ is also shown. One might expect that due to larger Fermi motion the spatial order reflected by the peak in the structure function should be decreased in the polarized phase. However, the calcula- ted functions show opposite behavior. This arises from the fact that the exchange part of the induced potential of eq. (2) is more repulsive at short distances for the polarized liquid than for the normal one.

Therefore, at fixed density the oscilla- tions in the radial distribution function are slightly enhanced. This may be reasona- ble since, on the other hand, one expects that the polarized liquid shows stronger tendency towards solidification. We also note that in the polarized system the maxi- mum of S(k) occurs very nearly at k = 2kF whereas it is at k - ^2.6 kF in the spin- satu;.at~;. 1 iquid. The difference between

C

the t$.o csiculated structure functions is qulte snall, however, and improvements of

Fig.

The liquid structure function S(k) for nor- mal (solid line) and spin-polarized (dashed line)

liquid. The dots show the experimental quantity of the normal phase at T

0.56 K.

of the variational wave function may pro- duce larger corrections to the calculated quantities of the spin-saturated system.

Figure 2 shows a typical picture of the one particle momentum distribution functions for the polarized 3 ~ e liquid and for a gi- ven spin projection of the spin-saturated system. Note that the corresponding ideal gas distribution functions are normalized to unity inside the Fermi momentum. The density

= 0.022 a t o m s / ~ ~ is near the so- lidification density of the normal liquid.

In the normal phase only 30% of the parti- cles have momentum less than the Fermi mo- mentum. In the polarized phase the corres- ponding number is about 408. Respectively, the amount of discontinuity at the Fermi momentum is 0.15 and 0.24 in the two phases.

The calculated discontinuity increases al-

most linearly with decreasing density such

that it becomes 0.30 at the saturation den-

sity of the normal liquid and the change

in the polarized phase is very nearly pa-

rallel.

References

Fig. 2

The one-particle momentum distribution / 9/

function n(k) for spin-polarized and spin-saturated 3 ~ e liquids at the density p

0.022 atoms/i3.

/lo/

In conclusion we note that the spatial cor- relations in the two phases of the 3 ~ e liquid are very similar and they are domi- nated by strong dynamical correlations in- duced by Lennard-Jones interatomic potential.

Although the simple Jastrow wave function is not adequate to reproduce the observed structure the present variational calcula- tion may give some hints of possible small differences between the two phases. As ex- pected, the momentum distributions show the effects of strong correlations, too, and they deviate largely from their free gas counterparts.

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