GROUND STATE FLUCTUATIONS IN POLARIZED 3He

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GROUND STATE FLUCTUATIONS IN POLARIZED 3He

M. Ristig, P. Lam, H. Nollert

To cite this version:

M. Ristig, P. Lam, H. Nollert. GROUND STATE FLUCTUATIONS IN POLARIZED 3He. Journal de Physique Colloques, 1980, 41 (C7), pp.C7-213-C7-221. �10.1051/jphyscol:1980734�. �jpa-00220171�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n o 7 , Tome 41, juiZZet 1980, page C 7 - 2 1 3

GROUND STATE FLUCTUATIONS IN POLARIZED 3 ~ e M.L. R i s t i g , P.M. Lam and H.P. Nollert

I n s t i t u t fiir Theoretische Physik, Universittit zu Kisln, 5 K6Zn 41, R.F.A.

Resume.- On adopte une approche variationnelle pour decrire l e s fluctuations de densite de masse e t de spin dans 3He liquide. On represente d'une facon approchee l ' e t a t fondamental par une fonction d'onde de type Jastrow-Slater qui i n c l u t l e s e f f e t s de corr6lations s p a t i a l e s . On emploie- l a thgorie HNC nouvel lement developpee pour analyser l e s fonctions de s t r u c t u r e (qui comportent une p a r t i e dependant du spin) l e s fonctions de renversement de spin, l a d i s t r i b u t i o n des impulsions e t l a matrice-densit6 a u n corps. On presente des r e s u l t a t s numeriques r e l a t i f s ?I ces quantites pour l e liquide 3He non polarise e t complStement polarise. L ' e f f e t de l a polarisation sur l a s t r u c t u r e e t l a fonction de d i s t r i b u t i o n radiale e s t 6tudi@ en d e t a i l . On presente une estimation numerique prcl iminaire de 1 'equation d ' e t a t magnetique.

Abstract.- The a r i a t i o n a l approach i s adopted t o describe mass- and spin-density fluctuations in po- larized 1 iquid He. The ground s t a t e i s approximated by a s u i t a b l e Jastrow-Slater wave function which

Y

incorporates the e f f e c t s of s p a t i a l correlations. The newly developed Fermi-hypernetted chain theory i s employed t o analyse the associated theoretical p a r t i c l e

-

1. Introduction.- A macroscopic ensemble of ^' 3 He- atoms seems t o display a f u l l arsenal of i n t e r e s t - ing many-body quantum e f f e c t s . The discovery of t h e spin-ordering i n solid He a t low temperatures /1, 3 2/ and the production of t h e spin-polarized l i q u i d phase /3/ pose a major challenge t o existing many- body theories /4,5/.

3 3

I f we could turn off t h e He- He interaction the atoms - being fermions of spin 1/2

-

where k+ i s the Fermi wave number of a spin-up

(-down) atom corresponding t o t h e density _{p* =} (6r2)-'k$ the t o t a l density of t h e system, p =

,+

t i c l e o r b i t a l with momentum R1 and Ilk and spin-up and spin-down out of the bare vacuum $o. For s t a t e ( 1 ) the magnetization M of t h e system a t tempera- t u r e T = 0 i s a good quantum number,

the vector 2 being t h e t o t a l spin operator of A p a r t i c l e s , 2 ⁼ tzi. The s t a t e 4 displays only ki- nematic correlations due t o t h e exclusion princi- ple. This ideal gas wave function i s of course en- t i r e l y inadequate f o r describing t h e actual behav- i o r of t h e polarized l i q u i d phase.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980734

JOURNAL DE PHYSIQUE

To i n c o r p o r a t e t h e e f f e c t s o f dynamical corre- l a t i o n s induced by t h e 3 ~ e - 3 ~ e i n t e r a c t i o n we must g e n e r a l i z e ansatz ( 1 ) t o

The v a r i a t i o n a l approach /6-10/ t o t h e problem o f - f e r s a reasonable f i r s t choice o f c o r r e l a t i o n op- e r a t o r ,

We r e a l i z e t h a t t h i s simple p i c t u r e m i g h t be s t i l l t o o crude t o p e r m i t an adequate q u a n t i t a t i v e de- s c r i p t i o n o f t h e p o l a r i z e d phase. It seems t o be a p p r o p r i a t e t o i n c l u d e e x p l i c i t l y spin-dependent e f f e c t s i n t h e c o r r e l a t i o n o p e r a t o r by adopting the

r i c h e r form F =[F ri , G

-

t i o n operators have been s t u d i e d f o r t h e unpolar- i z e d f l u i d i n a number o f r e c e n t p u b l i c a t i o n s /11, 12/. However, a t t h e present e x p l o r a t o r y stage i t i s advisable t o begin w i t h t h e modest ansatz (4) p r o v i d i n g a u s e f u l f i r s t s t e p i n t h e r i g h t d i r e c - t i o n . Equations (2) a r e a l s o v a l i d f o r t h e more complex Jastrow-Sl a t e r wave f u n c t l o n (3), ( 4 ) since t h e o p e r a t o r F commutes w i t h t h e sptn-operator _S.

I n t h i s c o n t r i b u t i o n we a r e i n t e r e s t e d t n s t u d y i n g t h e l o c a l c o r r e l a t i o n s between t h e mass- and s p i n - d e n s i t y fluctuati'ons associated w i t h a s t a t e o f form (3), (4). These fl uctua Tons a r e de-

A:

s c r i b e d by t h e s c a l a r ope a t o r

a

.

The l a t t e r q u a n t i t y m& bbk conveniently represent- ed by t h e l o n g i t u d i n a l component p Z and t h e t r a n s - verse components pX, J ^{o r p k} f ^{= 2-G2(pt t} i p i ) .

!5 k

The c o r r e l a t i o n s b e t w e e h h e above"entit?es s h a l l be analysed w i t h i n t h e hypernetted-chain (HNC) approach t o v a r i a t i o n a l many-body t h e o r y 113-16/. A numerical e v a l u a t i o n o f the s p a t i a l shape o f these c o r r e l a t i o n s i s based on t h e S c h i f f - V e r l e t form o f Jastrow f a c t o r f ( r ) which has been f r e q u e n t l y employed f o r d e s c r i b i n g unpolarized l i q u i d He /17,15,16,18/. 3 These r e s u l t s a r e u t i l - i z e d t o approximate t h e magnetic equation o f s t a t e f o r t h e p o l a r i z e d phase. We s h a l l present p r e l i m i n a r y numerical estimates o f t h i s f u n c t i o n which r e l a t e s t h e i n t e r n a l energy t o t h e magneti- z a t i o n M.

Further, t h e hypernetted-chain method i s a l s o adopted t o e x p l o r e t h e one-body d e n s i t y m a t r i x as- s o c i a t e d w i t h t h e c o r r e l a t e d s t a t e (3),(4). The m a t r i x i s given by t h e F o u r i e r i n v e r s e o f t h e ex- p e c t a t i o n value o f t h e occupation number opera-

t o r a{oaka, u = + ^{o r} +, w i t h respect t o t h e s t a t e

$ ( t i = <$I$> ^),

To d e r i v e a f i r s t numerical estimate o f both func- t i o n s we f o l l o w t h e procedure described i n Ref.

/18/. Results s h a l l be presented o n l y i n t h e l i m i t - i n g cases M = 0 and M = A.

Sec. 2 provides t h e formal frame f o r d e s c r i b - i n g t h e l o c a l d e n s i t y c o r r e l a t i o n s . We e x p l o i t t h i s i n f o r m a t i o n t o d e r i v e an estimate on t h e magnetic equation o f s t a t e . The modern HNC techniques t o evaluate various e x p e c t a t i o n values a r e sketched i n Sec. 3. Numerical r e s u l t s on t h e s t r u c t u r e func- t i o n s , d i s t r i b u t i o n f u n c t i o n s , occupation p r o b a b i l - i t i e s and r e l a t e d q u a n t i t i e s a r e presented and d i s - cussed i n Sec. 4.

2. S t r u c t u r e - and d i s t r i b u t i o n functions.- The c o r r e l a t i o n s between t h e mass-density f l u c t u a t i o n s present i n t h e t r i a l s t a t e Y o f p o l a r i z e d 3 ~ e a r e described by t h e s t a t i c l i q u i d s t r u c t u r e f u n c t i o n

S i m i l a r l y , t h e s p i n - s p i n c o r r e l a t i o n t e n s o r

aB 1 a 6

S ( k ) = m < Y I ~ P- / I > , ,k k

a,B = x,y,z, represents t h e c o r r e l a t i o n s between t h e s p i n - d e n s i t y f l u c t u a t i o n s , For t h e ansatz (3), (4) t h e t e n s o r ( 7 ) i s completely c h a r a c t e r i z e d by t h e l o n g i t u d i n a l component a = B = z and t h e t r a n s - verse component a = B = x ( o r a = B = y). Both f u n c t i o n s a r e normalized t o u n i t y as k + ". I n s t e a d o f working w i t h t h e element sXY(k) we may equiva- l e n t l y employ t h e s p i n - f l i p f u n c t i o n

Q u a n t i t y ( 8 ) approaches t h e value 1 - ^{A as k +}

where 0 5 A -< 1 i s t h e p o l a r i z a t i o n parameter A = M/A. A t f i n i t e temperatures t h e f u n c t i o n s s Z Z ( k ) and s X X ( k ) a r e r e l a t e d , r e s p e c t i v e l y , t o the s t a t i c magnetic s u s c e p t i b i l i t y x and t o t h e t r a n s v e r s e suscepti b i 1 i ty /19/. More complex c o r r e l a t i o n s among t h e d e n s i t y f l u c t u a t i o n s a r e represented by s t r u c t u r e f u n c t i o n s o f h i g h e r order, f o r example by,

These q u a n t i t i e s a r e usually evaluated by the con- vol ution- o r superposition approximation /8,20/.

Fourier transforming t h e functions (6)-(9) we a r r i v e , respectively, a t t h e associated radial d i s - t r i bution function g ( r ) , the spin-dependent d i s t r i - bution functions y Z Z ( r ) , g X X ( r ) , the s p i n - f l i p dis- t r i b u t i o n function g + ( r ) and t h e three-body d i s t r i - bution function g(_'lr12913). For non-interacting fermions these functions a r e given by simple analy- t i c expressions,

where 1 (x) i s t h e Slater-exchange function 1 (x) = 3x-3 ( s i n x - x cos x ) . The liquid s t r u c t u r e func- t i o n of non-interacting fermions then follows as

with

and unity otherwise.

To evaluate t h e d i s t r i b u t i o n functions f o r a s p a t i a l l y correlated wave function such as ansatz ( 3 ) , ( 4 ) we must r e s o r t t o t h e powerful tools pro- vided by HNC theory. These procedures will be out- lined in the following section.

Based on ansatz ( 3 ) , ( 4 ) we may derive a f i r s t estimate of the internal energy as a function of magnetization. I t i s known'that the expectation value of the energy with respect t o the wave func- t i o n (3),(4) can be written in the so-called Clark- Westhaus form 1151,

involving only the d i s t r i b u t i o n functions g ( r ) and g ( l l ,r,,t-,) . The e f f e c t i v e potentials a r e given by

*

v ( r ) = v ( r ) + - ( 1 f ) f ( r ) , m

where the potential v ( r ) represents t h e bare in- teratomic interaction.

3. Hypernetted-chain approximation.- The expecta- tion values ( 5 ) - ( 9 ) with respect t o the Jastrow- S l a t e r wave function (3),(4) may be analysed by using standard c l u s t e r developments and Ursell- Mayer diagrammatic techniques /15,20/. Applying re- cently developed HNC treatments of Fermi systems we a r e then able t o achieve p a r t i a l summations of these expansions t o any order in the correlation f a c t o r f ( r ) - 1 or f ( r ) - 1 and t h e exchange f a c t o r s 2 1 (rk*)

To get a preliminary estimate of quantity (6) we may employ a p a r t i c u l a r l y simple version of HNC theory suited f o r t h e s t a t i c s t r u c t u r e function /21,16/. This procedure t r e a t s approximately the exchange portion

+ *

1g B 1J

multi-dimensional integral thus generated from eq.

(6) may now be viewed formally a s an expectation- value of A bosons described by t h e symmetric wave functi0n:n.f ( r . A . ) f B ( r . ⁾ . Appropriate construction

19 1 J 1 J

of t h e e f f e c t i v e exchange correlation f a c t o r f B ( r ) leads t o the e x p l i c i t expression /21,15/

Thereupon, we may evaluate t h e HNC approximant of the s t r u c t u r e function S(k) by solving the time- honored s e t of (BHNC) equations f o r the radial d i s t r i b u t i o n function of a Bose f l u i d /8/,

which involves t h e nodal portion G(r) and i t s Fourier inverse G(k) suitably normalized.

I f we adopt t h e superposition approximation 181 f o r the l a s t term in eq. ( 1 3 ) ,

the BHNC solution g ( r ) of eq. (16) provides us with a simple estimate of t h e magnetic equation of s t a t e .

More sophisticated approximations f o r the s t r u c t u r e function S(k) as well as f o r the other

J O U R N A L D E PHYSIQUE

q u a n t i t i e s i n q u e s t i o n may be d e r i v e d by employing t h e newly developed- FHNC-schemes /13,14,22/. They i n v o l v e c e r t a i n s e t s o f FHNC equations which con- s t i t u t e t h e Fermi analogs o f HNC equations such as eqs. (16). The e x p l i c i t form o f these equations f o r t h e unpol a r i zed (and completely p o l a r i z e d ) s t a t e o f l i q u i d He may be taken from Ref. /15/. T h e i r gen- 3 e r a l i z a t i o n t o d e s c r i b e a l s o p a r t i a l l y p o l a r i z e d s t a t e s v i a eqs. ( 3 ) ,(4) i s s t r a i g h t f o r w a r d .

To e x p l a i n t h e main f e a t u r e s we sketch one FHNC-procedure which determines f u n c t i o n s (6)-(8) and t h e corresponding d i s t r i b u t i o n f u n c t i o n s . We r e s t r i c t ourselves t o t h e l i m i t i n g cases A = 0 and A = 1. F o r completely p o l a r i z e d fermion m a t t e r t h e f o l l o w i n g s t r u c t u r a l decomposition holds f o r t h e r a d i a l d i s t r i b u t i o n f u n c t i o n 1151

w i t h v = 1 and p = v(6m2)-lk;. The c o n s t i t u e n t s Gkl(r) represent d i f f e r i n g nodal p o r t i o n s , t h e i r i n t e r p r e t a t i o n and e x p l i c i t r e p r e s e n t a t i o n may be taken from Ref. /15/. The q u a n t i t y E ( r ) c o l l e c t s c o n t r i b u t i o n s which i n v o l v e elementary diagrams.

A complementary s t r u c t u r a l study may be per- formed f o r t h e l i q u i d s t r u c t u r e f u n c t i o n y i e l d i n g I 1 5 1

I . . . } = 1 + [xl1(k) + Gll(k)] [l + X2,(k)] . ⁽¹⁹⁾

The q u a n t i t i e s Xi ( k ) are .non-nodal p o r t i o n s which correspond, r e s p e c t i v e l y , t o t h e nodal pieces G. . ( r ) and t h e associated F o u r i e r i n v e r s e Gi ( k ) .

1 J

A l l these f u n c t i o n s a r e r e l a t e d by a s e t o f 2 x 4 FHNC-equations o f t h e f u n c t i o n a l form

Eqs. (20) c o n t a i n t h e Jastrow f a c t o r f ( r ) and t h e exchange f u n c t i o n l ( r k F ) as i n p u t . The e x p l i c i t ex- pressions o f eqs. (20) a r e given i n Ref. /15/. Sys- t e m a t i c approximation schemes a r e a v a i l a b l e t o

s o l v e these equations. Since sZZ(k) = S(k) and

~ + ( k ) = 0 a t A = 1 t h e FHNC s o l u t i o n s o f s e t (20) determine t h e s p a t i a l behavior o f t h e c o r r e l a t i o n s

( 6 ) - ( 8 ) compl e t e l y .

For u n p o l a r i z e d l i q u i d 3 ~ e ( A = 0) we must s e t v = 2 t o determine t h e d i s t r i b u t i o n f u n c t i o n from eq. (18). In t h i s case t h e spin-dependent f u n c t i o n s gZZ(r) = g X X ( r ) d i f f e r from f u n c t i o n g ( r ) and a r e given by

w i t h ~ ' ( r ) i n v o l v i n g elementary c o n t r i b u t i o n s . I n t h e general case 0 < A < 1 we may proceed i n t h e same f a s h i o n provided we t a k e account o f a number o f a d d i t i o n a l c h a r a c t e r i s t i c elements Xi and G and e n l a r g e a p p r o p r i a t e l y t h e s e t of FHNC

i j

equations (20). For p a r t i a l l y p o l a r i z e d matter t h e f u n c t i o n s S(k), sZZ(k) and sXX(k) d i f f e r from each o t h e r . I n t h i s case t h e s p i n - f l i p f u n c t i o n s ~ + ( k ) and g+(r) f u r n i s h an a d d i t i o n a l p i e c e o f informa- t i o n on t h e spin-behavior o f t h e He-system. A 3 s t r a i g h t f o r w a r d c l u s t e r development o f t h e expecta- t i o n value (8) y i e l d s an expansion which may be r e - presented , i n low c l u s t e r order, by t h e graphs o f F i g . 1.

F i g . 1.- Graphic r e p r e s e n t a t i o n o f t h e s p i n - f l i p f u n c t i o n gi(r) i n l e a d i n g c 7 u s t e r order.

Here, t h e s t r a i g h t o r i e n t e d l i n e s d e p i c t t h e ex- P +

change f u n c t i o n s - 4 ( r k + ) and a dashed l i n e repre-

P 2

sents t h e dynamical element f ( r ) - 1 . I t can be e a s i l y recognized t h a t f o r wave numbers k < k+ - ^k_

t h e F o u r i e r i n v e r s e o f t h e f o u r t h ( f i f t h ) diagram reduces t o t h e F o u r i e r i n v e r s e o f t h e second ( t h i r d ) graph, bht w i t h o p p o s i t e sign. The same p r o p e r t y

holds also in higher c l u s t e r order. We may there- fore conclude

The behavior of the s p i n - f l i p function f o r wave num- bers k > k+ + k- becomes also simple. In the l i m i t

A ⁺ 1, k- ⁺ 0 we need only t h e portion

where q u a n t i t i e s X33 and G3, have t o be calculated a t A = 1. The r e s u l t (23) may be expressed i n _k- space by t h e i n t e r e s t i n g r e l a t i o n

f o r k > k+ + k- which provides a d i r e c t physical i n t e r p r e t a t i o n of the non-nodal function X3,.

4. Numerical r e s u l t s and discussion.- To learn some- thing about the quantitative e f f e c t s of spin-orien- t a t i ~ n in polarized l i q u i d He we have performed 3 calculations on the structure-, distribution- and s p i n - f l i p functions a s well a s on the energy expec- t a t i o n value, t h e one-body density matrix and the occupation probability of single p a r t i c l e o r b i t a l s . The calculations a r e based on the model wave func- tion (3) ,(4) where we adopt Schiff-Yerlet's choice of correlation function /17/,

l a 5 f ( r ) = exp [ - -(-) ]

2 r (25)

with a = 2.888 8 a t a l l values of polarization and p a r t i c l e density considered. No attempt has been made t o employ t h e optimal value of the parameter a a t each density I91 and magnetization. We have solved eqs. (16) t o determine q u a n t i t i e s S(k) and g ( r ) . The BHNC r e s u l t s are then used in eq. (13) in conjunction with t h e additional approximation (17) t o estimate t h e magnetic equation of s t a t e . In eq.

(13) the interatomic interaction v ( r ) i s assumed t o be described by a Lennard-Jones potential

.

A t A = 0 , l the s t r u c t u r e function S(k) and d i s t r i b u t i o n function g ( r ) have been also calculat- ed by adopting one version of FHNC approximation scheme which i s based on eqs. (19) and (20). This approximation has been called FHNC-1-2 and i s de- scribed in Refs. /11,15,18/. We also present, a t A = 0,1, FHNC-1-2 r e s u l t s on t h e functions s Z Z ( k ) ,

~ + ( k ) / ( l - A ) , the one-body density matrix and rela-

ted q u a n t i t i e s such a s t h e q u a s i p a r t i c l e strength.

Hoflever, t h e numerical data presented should be considered a s preliminary r e s u l t s . What we need i s a careful and detailed comparison with r e s u l t s of d i f f e r i n g FHNC approximation procedures I151 and of Fenni Monte Carlo calculations.

Table 1 l i s t s our numerical r e s u l t s on the ra- dial d i s t r i b u t i o n function and t h e liquid s t r u c t u r e Table 1

Table 1.- Theoretical d i s t r i b u t i o n function g ( r ) and s t r u c t u r e function S(k) f o r unpolarized l i q u i d 3 ~ e a t density p = 0.0142 8-3, i n BHNC-approxima- tion, eq. (16).

function f o r the unpolarized s t a t e a t density P = 0.0142 8-3, in BHNC approximation. The variation of these functions with respect t o t h e polariza- tion A a t denstiy ^P = 0.0142 8-3 i s depicted i n Figs. 2 and 3. W i t h increasing polarization t h e s p a t i a l correlations become more repulsive a t re- l a t i v e distances r ^% 2.6 8. Simultaneously, t h e

JOURNAL DE P H Y S I Q U E

006 I I I >

1 (26)

f SA=o(k) - SA(k) 1 d_k = 0 (2s) 3~

which f o l l o w from t h e p r o p e r t i e s S(0) = 0 and g(0) = 0. Eqs. (26) a r e n u m e r i c a l l y w e l l f u l f i l l e d - by o u r approximate BHNC-resul t s , Figs. 2,3. The

dependence o f f u n c t i o n g k o ( r ) - gk ,(r) on t h e p a r t i c l e d e n s i t y v i s represented i n F i g . 4. AS

I I I I

-002 -

- L

1 I I I - -

2 3 4 5 m2 8 004

-

r I W l - _-

m 9 Fig. 2.- I n f l u e n c e o f p o l a r i z a t i o n A on t h e r a d i a l

d i s t r i b u t i o n f u n c t i o n g ( r ) a t dens& P = 0.01428-~, i n BHNC approximation.

- 002 -

Fig. 3.- Same as Fig. 2, b u t f o r t h e s t r u c t u r e f u n c t i o n S ( k ) . Dashed curve (NIF) represents r e - s u l t f o r n o n - i n t e r a c t i n g He-atoms. 3

s p i n - f l i pped He-atoms generate an increased 1 oca- 3 1 iz a t i o n a t c 4 8. However, t h e magnitude o f t h i s displacement e f f e c t i s q u i t e small being < 0.05 compared w i t h u n i t y . The v a r i a t i o n of t h e s t r u c - t u r e f u n c t i o n S(k) w i t h p o l a r i z a t i o n A i s o f com- p a r a b l e magnitude. I t develops a d i s t i n c t peak a t wave numbers k c 1.8 8-I. T h i s behavior i s e n t i r e l y caused by t h e dynainical c o r r e l a t i o n s f ( r ) s i n c e i t i s absent i n t h e case o f n o n - i n t e r a c t i n g fermions (curve NIF). The appearance o f p o s i t i v e and nega- t i v e r e g i o n s i n b o t h f u n c t i o n s i s connected w i t h t h e sum r u l e s

F i g . 4.- Density dependence o f t h e p o l a r i z a t i o n e f - f e c t on t h e r a d i a l d i s t r i b u t i o n f u n c t i o n , i n BHNC- approximation.

expected we f i n d no change i n t h e r e g i o n o f <trong r e p u l s i o n . The s h i f t a t l a r g e distances r may be e x p l a i n e d by t h e decrease o f t h e l e n g t h v - ' I 3 which provides t h e a p p r o p r i a t e s c a l e i n which d i s - tances should be measured.

P r e l i m i n a r y r e s u l t s on t h e energy e x p e c t a t i o n v a l u e f o r t h e u n p o l a r i z e d and completely p o l a r i z e d s t a t e a t t h r e e d i f f e r i n g d e n s i t i e s a r e l i s t e d i n t a b l e 2. The r e s u l t s a t A = 0 may be b e s t compared

Table 2

Table 2.- Estimate o f energy p e r p a r t i c l e o f ' ~ e i n u n p o l a r i z e d and completely p o l a r i z e d s t a i e a t d i f f e r i n g d e n s i t i e s , i n BHNC-approximation.

w i t h FHNC data r e p o r t e d i n Ref. /16/. The ener- gies, i .e. q u a n t i t y

E E ! ~ ~

1161 are g e n e r a l l y ^Q 0.3K - ^0.4K l e s s a t t r a c t i v e than t h e BHNC r e s u l t s g i v e n here. Fig. 5 represents t h e spin-dependent e f f e c t s on the energy p e r p a r t i - c l e as a f u n c t i o n o f p o l a r i z a t i o n . I n BHNC approxi-

t r a r y r e s u l t s .

Our r e s u l t s on t h e l o n g i t u d i n a l and t r a n s v e r s e q u a n t i t i e s s Z Z ( k ) and st(k) a r e represented i n Figs.

6,7. The c a l c u l a t i o n s have been done a t A = 0 and

F i g . 5.- Dependence o f energy p e r p a r t i c l e o f He- 3 m a t t e r on p o l a r i z a t i o n o, i n BHNC-approximation.

The n o n - i n t e r a c t i n g system i s described by curve NIF.

mation we f i n d t h a t t h e u n p o l a r i z e d s t a t e i s pre- f e r r e d , t h e g a i n i n energy being t y p i c a l l y o f t h e o r d e r 0.2K - 0.3K compared t o t h e energy per p a r t i - c l e i n t h e completely p o l a r i z e d s t a t e . A p a r a b o l i c f i t t o q u a n t i t y E(A) - E(0) a t A = 0 gives a crude e s t i m a t e o f t h e s t a t i c magnetic s u s c e p t i b i l i t y x.

Table 3

Table 3.- Estimate o f s t a t i c s u s c e p t i b i l i t y x o f l i q u i d He a t d i f f e r i n g d e n s i t i e s 3 P, compared w i t h t h e suscepti b i 1 i t y xo o f independent He-atoms 3 .

The e s t i m a t e agrees s u r p r i s i n g l y w e l l w i t h t h e ex- perimental data 123,241. We stress,however, t h a t a t present t h e t h e o r e t i c a l r e s u l t s on t h e magnetic equation o f s t a t e (13) should be taken w i t h g r e a t c a u t i o n s i n c e i t c o u l d happen t h a t improved FHNC- c a l c u l a t i o n s l e a d t o q u i t e d i f f e r e n t , even con-

Fig. 6.- Spin-dependent s t r u c t u r e f u n c t i o n o f He 3 i n t h e u n p o l a r i z e d and completely p o l a r i z e d s t a t e a t d e n s i t y 0 = 0.0142 g-3, i n FHNC-1-2 approxima- t i o n .

Fig. 7.- Same as Fig. 6, b u t f o r t h e s p i n - f l i p f u n c t i o n s + ( k ) / j l - A ) .

A = 1 o n l y . I n t h e l a t t e r case we f i n d t h a t t h e am- p l i t u d e o f t h e s p i n - f l i p f u n c t i o n ~ + ( k ) / ( l - A ) i s s t r o n g l y enhanced a t t h e Fermi surface. Unfortun-

15

JOURNAL DE PHYSIQUE

a t e l y , t h e FHNC-1-2 approximation adopted f o r these q u a n t i t i e s provides o n l y a rough estimate o f ex- p r e s s i o n (21), the corresponding s t r u c t u r e f u n c t i o n SZZ(k) and t h e s p i n - f l i p f u n c t i o n a t A 1. A quan- t i t a t i v e l y more r e l i a b l e c a l c u l a t i o n a t A = 1 and f o r t h e general case A < 1 should be based on more s o p h i s t i c a t e d FHNC-approximations. Such c a l c u l a - t i o n s a r e i n progress.

We conclude our numerical a n a l y s i s o f t h e po- l a r i z e d system w i t h a b r i e f c o l l e c t i o n o f FHNC-1-2 r e s u l t s on t h e momentum d i s t r i b u t i o n ( 5 ) and t h e associated d e n s i t y m a t r i x a t A = 1. These q u a n t i t i e s are c a l c u l ated by employing t h e s t r u c t u r a l r e s u l t s /20,18/

The c o n s t i t u e n t s appearing i n eqs. (27), (28) a r e analyzed i n d e t a i l i n Ref. /20/. They a r e

evaluated by f o l l o w i n g t h e FHNC-1-2 t r e a t n ~ e n t o f Ref. /18/. Tables 4,5 c o l l e c t o u r t h e o r e t i c a l re-

Table 4

Table 4.- S t r e n g t h f a c t o r n, f r a c t i o n o f p a r t i c l e s i n s i d e Fermi sea nc, q u a n t i t y M(kF) and s t r e n g t h o f t h e q u a s i p a r t i c l e p o l e a t t h e Fermi s u r f a c e f o r He 3 i n u n p o l a r i z e d and completely p o l a r i z e d s t a t e , i n FHNC-1-2 approximation. L a s t two 1 ines g i v e r e s u l t s on t h e sum r u l e check a t r = 0.

s u l t s on t h e s t r e n g t h f a c t o r n, tb,e number o f He- 3 atoms i n s i d e t h e Fermi sea n, and t h e s t r e n g t h o f t h e quasi p a r t i c l e p o l e z(kF) a t A = 0 , l and a t two d i f f e r i n g d e n s i t i e s . The f u n c t i o n s Q ( r ) , etc., i n - volved i n eqs. (27),(28) should f u l f i l l t h e sum r u l e s /20/ ne-Q(r) = 1 and Nl(r) + N 2 ( r ) = 1 a t r = 0. We l e a r n from t h e l a s t two l i n e s o f Table 4

Table 5

Table 5.- Same as t a b l e 4 b u t f o r completely pola- r i z e d s t a t e a t two d i f f e r i n g d e n s i t i e s .

t h a t these r u l e s a r e w e l l f u l f i l l e d by t h e FHNC-1-2 approximants a t A = 0. However, f o r A = 1, we ob- serve t h a t t h e approximate q u a n t i t i e s l e a d t o a r a t h e r d i s t u r b i n g v i o l a t i o n by about 20-30 %. The r e s u l t s i n d i c a t e t h a t we should improve t h e adopted FHNC-approximation b e f o r e doing a c a l c u l a t i o n o f q u a n t i t i e s n ( r ) and n ( k ) a t a r b i t r a r y p o l a r i z a -

4 4

t i o n . Fig. 8 d e p i c t s t h e one-body d e n s i t y m a t r i x a t A = 0 and A = 1 and a t d e n s i t y p = 0.0142 8-3.

F i g . 8.- Same as Fig. 6, b u t f o r t h e one-body densi- ty matrix, eq. (28).

Acknowledgement. - We g r a t e f u l l y acknowledge f inan- c i a 1 support from t h e Deutsche Forschungsgemein- s c h a f t under Grant No. R i 26715.

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