MONTE CARLO STUDIES OF CRYSTALLINE HELIUM

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MONTE CARLO STUDIES OF CRYSTALLINE HELIUM

D. Ceperley, G. Chester, M. Kalos, P. Whitlock

To cite this version:

D. Ceperley, G. Chester, M. Kalos, P. Whitlock. MONTE CARLO STUDIES OF CRYS- TALLINE HELIUM. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-1298-C6-1304.

�10.1051/jphyscol:19786562�. �jpa-00218052�

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JOURNAL DE PHYSIQUE

Colloque C6, supplPment au

no

8, Tome

39,

aoat

1978,

page

C6- 1298

D.M. c e p e r l e y + , G.V. c h e s t e r x , M.H. ~ a l o s ~ ~ and ? . A . WhitlockXX

Courant I n s t i t u t e of Mathematical Sciences, New York Uniuersity, New York, N . Y . 10012, U . S. A . Laboratory o f Atomic and Solid S t a t e Physics, ComeZZ University, Ithaca, New York 14853, U.S.A.

XX Courant I n s t i t u t e o f .kfathematicaZ Sciences.

REsum6.- On p r 6 s e n t e l e s r d s u l t a t s pour l e s p r o p r i 6 t d s d e l ' h 6 l i u m s o l i d e 5 l a t e m p d r a t u r e d e zgro.

La p l u p a r t d e s donn6es s o n t o b t e n u e s p a r une m6thode d e Monte C a r l o q u i donne une s o l u t i o n e x a c t e d e l ' d q u a t i o n de S c h r z d i n g e r . Pour l ' h d l i u m 4 , nous exposons l l d n e r g i e e n f o n c t i o n d e l a d e n s i t 6 , l e s p r o p r i d t d s de l a t r a n s i t i o n l i q u i d e - s o l i d e , l a d e n s i t 6 ?I un c o r p s , l a f o n c t i o n de s t r u c t u r e , e t l a d i s t r i b u t i o n d e s v i t e s s e s , c a l c u l 6 e s pour un r d s e a u f . c . c . . L ' a c c o r d d e s p r o p r i d t d s thermodynamiques c a l c u l 6 e s a v e c l l e x p d r i e n c e e s t m e i l l e u r que c e l l e s q u i s o n t c a l c u l d e s p a r l e s mdthodes v a r i a t i o n - n e l l e s , mais l e p o t e n t i e l de Lennard-Jones ne f o u r n i t p a s une d e s c r i p t i o n q u a n t i t a t i v e d e 1 1 h 6 1 i u m 4 . Les d i f f 6 r e n c e s e n t r e l e s p h a s e s f . c . c . e t h.c.p. n ' e x p l i q u e n t p a s l e d g s a c c o r d . P o u r 11h61ium 3 , o n donne l e s r d s u l t a t s d e s m6thodes v a r i a t i o n n e l l e s u t i l i s a n t exactement d e s f o n c t i o n s d ' e s s a i a n t i s y - m d t r i q u e s .

A b s t r a c t . - R e s u l t s f o r p r o p e r t i e s of s o l i d h e l i u m a t z e r o t e m p e r a t u r e a r e p r e s e n t e d . Yost o f t h e d a t a were o b t a i n e d by a Yonte C a r l o method which g i v e s a n e x a c t s o l u t i o n o f t h e S c h r s d i n g e r equation.

For h e l i u m 4 we e x h i b i t t h e e n e r g y a s a f u n c t i o n of t h e d e n s i t y , t h e p r o p e r t i e s o f t h e l i q u i d - s o l i d t r a n s i t i o n , t h e one-body d e n s i t y , t h e s t r u c t u r e f u n c t i o n , and t h e momentum d i s t r i b u t i o n , c a l c u l a t e d f o r an f . c . c . l a t t i c e . The agreement of t h e c a l c u l a t e d thermodynamic p r o p e r t i e s w i t h e x p e r i m e n t i s b e t t e r t h a n t h o s e c a l c u l a t e d v a r i a t i o n a l l y , b u t t h e Lennard-Jones p o t e n t i a l d o e s n o t g i v e a q u a n t i - t a t i v e d e s c r i p t i o n o f h e l i u m 4. D i f f e r e n c e s between f . c . c . and h.c.p. p h a s e s d o n o t e x p l a i n t h e d i s - c r e p a n c y . For h e l i u m 3 , r e s u l t s o f v a r i a t i o n a l c a l c u l a t i o n s u s i n g e x a c t l y a n t i s y m m e t r i c t r i a l func- t i o n s a r e g i v e n .

I.INTR33UCTION.-The i s o t o p e s of h e l i u m form c r y s t a l s t i c s i s a method f o r t h e n u m e r i c a l i n t e g r a t i o n of which e x h i b i t quantum e f f e c t s i n c o n s p i c u o u s ways. t h e ~ c h r i i d i n g e r e q u a t i o n which h a s been named t h e T h e i r s t u d y p r o v i d e s s t r i n g e n t t e s t s o f o u r under- G r e e n ' s F u n c t i o n Monte C a r l o (GFMC) method. Space s t a n d i n g o f s u c h quantum e f f e c t s i n t h e s o l i d s t a t e . p r e c l u d e s any d i s c u s s i o n of t h e t e c h n i q u e i t s e l f , U n f o r t u n a t e l y a q u a n t i t a t i v e t h e o r y of t h e s e s y s - b u t i s h a s b e e n d e s c r i b e d e l s e w h e r e i n t h e l i t e r a - tems i s impeded by t h e f a c t t h e y a r e s t r o n g l y i n t e - t u r e / 1 , 2 , 3 / . I t u s e s Monte C a r l o methods b o t h t o r a c t i n g . No p u r e l y t h e o r e t i c a l a p p r o a c h h a s been d e v e l o p a G r e e n ' s f u n c t i o n f o r t h e ~ c h r o d i n g e r equa- found a d e q u a t e t o t r e a t them. On t h e o t h e r hand, t i o n and t o i t e r a t e t h e r e s u l t i n g i n t e g r a l e q u a t i o n . t h e y have been t h e s u b j e c t of n u m e r i c a l s t u d i e s , One t e c h n i c a l d e t a i l i s s i g n i f i c a n t : an importance most p a r t i c u l a r l y o f Monte C a r l o s i m u l a t i o n s . s a m p l i n g t r a n s f o r m a t i o n i s p o s s i b l e w h i c h , i n p r i n -

We w i s h t o p r e s e n t r e c e n t m e t h o d o l o g i c a l ad- c i p l e , p e r m i t s t h e c a l c u l a t i o n t o b e c a r r i e d o u t v a n c e s i n t r e a t i n g z e r o t e m p e r a t u r e c r y s t a l s t a t e s w i t h no Monte C a r l o s a m p l i n g e r r o r . T h i s r e q u i r e s of b o t h j ~ e and 4 ~ e a l o n g w i t h some o f t h e r e s u l t s t h e i n t r o d u c t i o n of t h e e x a c t ground s t a t e wavefunc- o b t a i n e d t h e r e b y . The most i m p o r t a n t c o n c l u s i o n i s t i o n . I t i s found i n p r a c t i c e t h a t t h e u s e of appro- t h a t n u m e r i c a l c a l c u l a t i o n s of t h e Bose c r y s t a l w i t h x i m a t i o n s t o t h e ground s t a t e w a v e f u n c t i o n s

-

^{i n}

e s s e n t i a l l y no a p p r o x i m a t i o n s a r e p o s s i b l e and t h a t f a c t t h o s e t r i a l f u n c t i o n s used i n v a r i a t i o n a l s t u - q u a n t i t i e s n o t y e t measured f o r 4 ~ e w a r r a n t s t u d y d i e s

-

r e d u c e s t h e c o m p u t a t i o n a l e f f o r t by o r d e r s s i n c e d e t a i l e d comparisons w i t h m i c r o s c o p i c c a l c u l a - o f magnitude. T h i s h a s a mixed e f f e c t . We g e n e r a l l y t i o n s c a n now p r o v i d e c r i t i c a l t e s t s of t h e o r y and u s e t r i a l f u n c t i o n s o f t h e form

e x p e r i m e n t .

The b e s t c o m p u t a t i o n a l t o o l a v a i l a b l e f o r t h e

e

( 1 ) t r e a t m e n t of c r y s t a l s w i t h Bose o r Boltzmann s t a t i s - where s e a r e t h e s i t e s of a p r e d e t e r m i n e d l a t t i c e .

T h i s kind of t r i a l f u n c t i o n s i n g l e s o u t a p a r t i c u -

*

S u p p o r t e d by t h e U.S. Department of Energy under

C o n t r a c t No. EY-76-C-02-307~*000 and by the N a t i o - l a r ' V e t r y and permits i t s in me- n a l S c i e n c e F o u n d a t i o n under G r a n t No. DMR-77-18329. t a s t a b l e r e g i m e s . On t h e o t h e r hand, i t l e a v e s open

+ P r e s e n t a d d r e s s , NRCC, Lawrence B e r k e l e y Labora- t h e p o s s i b i l i t y t h a t r e s u l t s a r e v i a s e d by t h e spe- t o r y , B e r k e l e y , Ck, USA.

c i f i c form o f JIT. We have d e v o t e d much e f f o r t i n

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786562

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showing t h a t t y p i c a l r e s u l t s a r e i n s e n s i t i v e t o r e a - s o n a b l e c h a n g e s i n t h e d e t a i l s of Q T . Another e f f e c t of t h e u s e o f

JIT

a s g i v e n i n Eq. ( I ) i s t h a t we t h e n t r e a t a s y s t e m w i t h Boltzmann s t a t i s t i c s . T h a t i s , we i g n o r e exchange e f f e c t s . 1.n e a r l i e r s t u d i e s of s o f t - c o r e s y s t e m s / 4 , 5 / , we used f u l l y symmetric forms of

JIT

i n b o t h v a r i a t i o n a l and GFMC c a l c u l a - t i o n s and Cound t h a t t h e e n e r g y was n e v e r lowered by symmetrizing JIT. I n a d d i t i o n i t i s known t h a t exchan- g e e f f e c t s i n h a r d c o r e s y s t c m s a r e s m a l l 1 6 1 .

C o r r e s p o n d i n g l y e x a c t methods f o r t h e e x a c t i n t e g r a t i o n of f e r m i o n s y s t e m s h a v e n o t y e t b e e n de- v e l o p e d . We have shown

151

t h a t v a r i a t i o n a l c a l c u l a - t i o n s w i t h a n a n t i s y m m e t r i c c r y s t a l t r i a l f u n c t i o n c a n be c a r r i e d t h r o u g h w i t h no a d d i t i o n a l approxima- t i o n s ( s u c h a s t r u n c a t e d p e r m u t a t i o n e x p a n s i o n s ) by a n a p p r o p r i a t e g e n e r a l i z a t i o n o f t h e w e l l known sam- p l i n g method o f M e t r o p o l i s e t a l . / 7 / . We w i l l d e s - c r i b e t h e r e s u l t s o f t h e a p p l i c a t i o n o f t h i s method,

"Fermion Monte C a r l o " , i n s e c t i o n 4 below.

2. PROPERTIES OF CRYSTAL STATES OF %e.- We have mode- l e d h e l i u m a s a s y s t e m o f p a r t i c l e s i n t e r a c t i n g by Lennard-Jones p o t e n t i a l s u s i n g t h e d e Boer-Michels

/8/ p a r a m e t e r s :

We u s e t h i s p o t e n t i a l b e c a u s e i t h a s been s t u d i e d most e x t e n s i v e l y i n b o t h f l u i d and c r y s t a l p h a s e s and b e c a u s e e a r l i e r work / I / h a s s u g g e s t e d t h a t i t gave b e t t e r agreement w i t h e x p e r i m e n t a l e q u a t i o n s o f s t a t e t h a n had been i n d i c a t e d by v a r i a t i o n a l s t u - d i e s . I n a d d i t i o n we have e s t i m a t e d and i n c l u d e d i n o u r e q u a t i o n of s t a t e t h e p e r t u r b a t i o n a l e f f e c t o f t h e A x i l r o d - T e l l e r t r i p l e - d i p o l e p o t e n t i a l / 9 /

(3) The c a l c u l a t i o n s w e r e made on f . c . c . l a t t i c e s p a r t l y f o r t e c h n i c a l c o n v e n i e n c e and p a r t l y b e c a u s e most e a r l i e r v a r i a t i o n a l work and some GFMC c a l c u l a -

t i o n s / 2 / (on h a r d s p h e r e s y s t e m s ) was done w i t h t h i s s t r u c t u r e . We a r e i n t h e p r o c e s s o f r e p e a t i n g t h e v a r i a t i o n a l c a l c u l a t i o n s of Hansen and Levesque / l o /

( f . c . c . ) and o f Hansen and P o l l o c k / I I / (h.c.p.) t o b e t t e r p r e c i s i o n . P r e l i m i n a r y r e s u l t s i n d i c a t e t h a t t h e e q u a t i o n s of s t a t e a g r e e t o l e s s t h a n 0.1 K. We p l a n a l s o c o n f i r m a t o r y GFMC c a l c u l a t i o n s i n t h e

h , c . p . p h a s e .

We w i l l p r e s e n t , i n s u c c e e d i n g s e c t i o n s , r e - s u l t s o n t h e e q u a t i o n of s t a t e , i n c l u d i n g p r e s s u r e and c o m p r e s s i b i l i t y d e r i v e d t h e r e f r o m . I n a d d i t i o n we w i l l show s e l e c t e d d a t a on t h e s t r u c t u r e f a c t o r ,

t h e momentum d i s t r i b u t i o n , and o n t h e s i n g l e p a r t i - c l e d e n s i t y . Using s i m i l a r d a t a t o b e r e p o r t e d e l s e - where on t h e f l u i d p h a s e 1 1 2 1 , we have d e r i v e d t h e d a t a on t h e m e l t i n g - f r e e z i n g t r a n s i t i o n ( t o an

f . c . c . p h a s e ) .

2 . ] . T h e e q u a t i o n of S t a t e of t h e C r y s t a l Phase.- T a b l e 1 c o n t a i n s r e s u l t s f o r t h e e q u a t i o n o f s t a t e and r e l a t e d q u a n t i t i e s .

T a b l e I

E n e r g i e s o f t h e f . c . c . h e l i u m c r y s t a l

The f i r s t column g i v e s t h e d e n s i t y i n r e d u c e d u n i t s . E2 i s t h e e x a c t e i g e n v a l u e u s i n g t h e Lennard-Jones p o t e n t i a l , <V3> i s t h e p e r t u r b a t i o n e s t i m a t e of V3, and E i s t h e sum o f E2 and <V >. The l a s t coaumn g i v e s t h e p e r t u r b a t i o n e s t i m a 2 e of t h e e n e r g y con- t a i n e d i n r e f e r e n c e 1 2 1 . It s h o u l d b e compared w i t h E

.

A l l e n e r g i e s a r e i n d e g r e e s K e l v i n p e r p a r t i c l e .

2

We f i t t e d t h e r e s u l t s t o a polynomial

F i g u r e I shows t h e computed d a t a , t h e f i t o f E q . ( 4 ) , and t h e e x p e r i m e n t a l r e s u l t s o f Edwards and Pandorf 1131. We o b s e r v e t h a t t h e polynomial g i v e s a f a i r t h f u l r e p r e s e n t a t i o n o f t h e d a t a .

At t h e l o w e s t s t a b l e c r y s t a l d e n s i t y , t h e com- p u t e d r e s u l t i s a b o u t 0 . 7 K p e r p a r t i c l e h i g h e r t h a n e x p e r i m e n t . The d i s c r e p a n c y d e c r e a s e s t o a b o u t 0.2 K a t t h e h i g h e s t d e n s i t y we h a v e t r e a t e d . The d i f f e r e n c e i n m a g n i t u d e and s h a p e i s u n l i k e l y t o a r i s e from t h e d i f f e r e n c e between f . c . c . and h . c . p . ; i t i s s u r e l y a consequence o f t h e i n a d e q u a c y o f t h e Lennard-Jones p o t h n t i a l , Eq. ( I ) . The somewhat wrong s h a p e of E(p) i s r e f l e c t e d i n t h e d i s a g r e e - ment between c a l c u l a t e d and e x p e r j m e n t a l p r e s s u r e s

and c o m p r e s s i b i l i t i e s , shown i n f i g u r e s 2 and 3.

From t a b l e I we s e e t h a t t h e n e t e n e r g y of t h e s y s t e m comes a s a r a t h e r d e l i c a t e c a n c e l l a t i o n

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between the kinetic and potential energies.

crystal energies vs p

T /

Fig. 1 : Energy as a function of reduced density.

The two lower curves give the results of the GFMC calculations. The lowest solid curve is without LVg>;

the upper dashed curve includes <V3>. The two upper curves give the results of our variational calculations. Again the lower curve is without <V3> andthe upper curve includes this correction. The experimen-

tal values, reference 1131, are shown as small solid squares.

140

-

¹²⁰

^-

E

loo-

-

2 8 0 -

3

2

6 0 -

E

4 0 - 20

-

Rig. 2 : The pressure as a function of reduced density. The solid curve is the experimental data, reference 1131. The points with error bars show the results of the GFMC calculations.

It would be extremely interesting to have experimental data on <T> or <V> separately ; it appears pos- sible that the former can be measured 1141.

Kalos, Levesque, and Verlet /2/ studied fluid and f.c.c. crystal ground states of hard-sphere quantum systems. They suggested a perturbation theory

connecting such results with those for hard-core po- tentials. The relation is similar to but simpler than the corresponding one for classical fluids.and crystals 1151.

Fig. 3 : The compressibility as a function.of reduced density. The solid curve is the experimental data, reference 1131, the points with error bars show the results from the GFMC calculations.

They determined the energy of crystal states for the Lennard-Jones potential. Table I shows that within their somewhat large errors, the perturbation theory gives results in agreement with ours (for the energy without <V3>). Further work on helium crystals based upon a hard-sphere reference system would seem to be very fruitful.

Table I1 shows a comparison of our GFMC results with Hansen's variational calculations /16/. The present energies are lower by a nearly constant dif- ference of about 1 K. This discrepancy is extremely important in comparisons with experimental equations of state, but it is in a sense, misleading. There is, as we have noted, a significant cancellation between

.

I n fact the variational results for the latter are in error by about 2-6 %. When <T>

and <V> are of the order of 35 K , the error in the net binding is considerably amplified. Nevertheless the variational method with trial functions of the form of Eq. (1) are not adequate for quantitative assessments of alternative force laws, and it is clear that the Lennard-Jones potential is inadequate.

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Tnblc _{1 1}

Comparison of E x a c t and V a r i a t i o n a l E n e r g i e s

The f i r s t column g i v e s t h e d e n s i t y i n reduced u n i t s . E2, <V> and <T> a r e t h e t o t a l , p o t e n t i a l and k i n e t i c e n e r g i e s computed by t h e GFMC method u s i n g t h e Lennard-Jones poten- t i a l . (E2)

,

^<V> and <T> a r e t h e same q u a n t i t i e s c a l c u l a t e d u s i n g t h e v a r i a t i o n a l method.

The l a s t cglumn g i v e s AE

I E ~ -

a s a p e r c e n t a g e o f <T>

.

A l l e n e r g i e s a r e i n

d e g r e e s K e l v i n p e r a r t i c l e . v

We a r e c u r r e n t l y engaged i n a s t u d y of t h e e f f e c t s of o t h e r p o t e n t i a l s t h a t have b e e n proposed

F i g . 4 : The s i n g l e p a r t i c l e d i s t r i b u t i o n a t t h r e e reduced d e n s i t i e s . D i s t a n c e s a r e measured i n u n i t s of ^0.

It seems l i k e l y t h a t t h e d i s a g r e e m e n t between v a r i a t i o n a l and GFMC c a l c u l a t i o n s h a s i t s o r i g i n i n t h e n e g l e c t of e x p l i c i t three-body c o r r e l a t i o n s i n t h e t r i a l f u n c t i o n s used i n t h e f o r m e r . S i m i l a r d i s - c r e p a n c i e s o c c u r i n t h e f l u i d p h a s e and t h e r e i s e v i d e n c e /17,18/ t h a t t h e y can b e removed f o r t h e most p a r t b y u s e of b e t t e r t r i a l f u n c t i o n s .

2 . 2 . T h e M e l t i n g - F t e e z i n g T r a n s i t i o n . - GFMC c a l c u l a - - t i o n s have a l s o been performed f o r t h e f l u i d p h a s e . Using t h e e q u a t i o n of s t a t e obtained thereby and a nume- r i c a l c o n s t r u c t i o n of t h e Maxwell d o u b l e t a n g e n t , we

have o b t a i n e d f r e e z i n g and m e l t i n g d e n s i t i e s and r e l a t e d d a t a f o r a n f . c . c .

-

f l u i d t r a n s i t i o n . These a r e shown i n t a b l e 111 a l o n g w i t h e x p e r i m e n t a l v a l u e s and t h e v a r i a t i o n a l v a l u e s of Hansen 1161.

T a b l e 111

potential pL03 p S d p(atmos) AV(cc/mola)

The M e l t i n g - F r e e z i n g T r a n s i t i o n

T h e f i r s t column g i v e s t h e p o t e n t i a l and method used.

The n e x t two columns g i v e t h e f r e e z i n g and m e l t i n g d e n s i t i e s i n reduced u n i t s . Column f o u r g i v e s t h e t r a n s i t i o n p r e s s u r e i n atmospheres. The l a s t column g i v e s t h e volume d i f f e r e n c e between t h e s o l i d and l i q u i d i n c u b i c c e n t i m e t e r s p e r mole. We f i n d no d i f f e r e n c e i n t h e m e l t i n g and f r e e z i n g d e n s i t i e s , t r a n s i t i o n p r e s s u r e and volume d i f f e r e n c e i f we omit t h e c o r r e c t i o n s d u e t o <V >.

3

The p r e s e n t r e s u l t s f o r t h e t r a n s i t i o n d e n s i - r i e s a r e about 10 % h i g h e r t h a n experiment ; a s a consequence t h e c a l c u l a t e d m e l t i n g p r e s s u r e i s much t o o l a r g e . The volume change of t h e two p h a s e s i n 1.6 cc/mole, f a i r l y c l o s e t o 1.9 cc/mole, t h e expe- r i m e n t a l v a l u e . The d i s c r e p a n c i e s may b e due t o t h e u s e of f . c . c . r a t h e r t h a n h . c . p . Although, a s we have n o t e d , t h e e n e r g y d i f f e r e n c e s a r e s m a l l , t h e d o u b l e t a n g e n t c o n s t r u c t i o n i s v e r y s e n s i t i v e t o s m a l l d i f f e r e n c e s . I t i s l i k e l y t h a t t h e u s e of t h e Lennard-Jones p o t e n t i a l i s a c a u s e f o r a l e a s t p a r t of t h e d i s a g r e e m e n t . U n c e r t a i n t i e s i n t h e three-body p o t e n t i a l a r e u n l i k e l y t o be a problem a s i n d i c a - ted i n t h e l a s t column of t a b l e 111.

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JOURNAL DE PHYSIQUE

It i s i l l u m i n a t i n g t o compare t h e f r e e z i n g and m e l t i n g o f t h e Lennard-Jones and of t h e h a r d - s p h e r e systtem. K a l o s , Levesque, and V e r l e t 121 sug- g e s t e d t h a t one s h o u l d s c a l e t h e h a r d - s p h e r e system i n s u c h a way t h a t t h e e q u i l i b r i u m d e n s i t y o f l i q u i d h e l i u m c o r r e s p o n d s t o a reduced h a r d - s p h e r e d e n s i t y of n a 3 = 0.2138, where a i s t h e d i a m e t e r . S c a l i n g o u r Lennard-Jones t r a n s i t i o n d e n s i t i e s i n t h e same way, we f i n d r e d u c e d t r a n s i t i o n d e n s i t i e s o f 0.276 and 0.30 f o r t h e hard-sphere system. T h e s e a r e 20 % l a r g e r t h a n were g i v e n i n d i r e c t c a l c u l a t i o n s f o r t h e l a t t e r 121. The i n c l u s i o n of t h e e f f e c t s o f t h e a t t r a c t i v e p a r t of t h e p o t e n t i a l c a r r i e d o u t i n r e - f e r e n c e / 2 / b r i n g s t h e t r a n s i t i o n d e n s i t i e s i n t o agreement w i t h o u r s a l t h o u g h t h e r e s u l t s a r e n o t p r e c i s e .

We c o n c l u d e t h a t , a s i n t h e c l a s s i c a l s y s t e m , t h e t r a n s i t i o n i s a p a c k i n g phenomenon m e d i a t e d p r i - m a r i l y by t h e e q u i v a l e n t h a r d - s p h e r e f o r t h e two- body p o t e n t i a l w i t h s i g n i f i c a n t enhancement by t h e a t t r a c t i v e p a r t of t h e p o t e n t i a l .

2 . 3 . The S i n g l e P a r t i c l e D e n s i t y . - Because we expect exchange t o b e e x t r e m e l y r a r e and b e c a u s e we have used a l o c a l i z e d i m p o r t a n c e f u n c t i o n , v i z . Eq. ( I ) ,

i n t h e p r e s e n t c a l c u l a t i o n s , we p r e s e n t d a t a o n l y f o r p ( r ) , t h e s p h e r i c a l a v e r a g e o f ~ ( f l ) , a t t h r e e d i f f e r e n t d e n s i t i e s . T a b l e I V g i v e s t h e second moments o f t h e s e v e r a l p ( r ) and a l i n e a r c o m b i n a t i o n ,

-

0 , o f t h e f o u r t h and s q u a r e o f t h e second moments

which v a n i s h e s f o r a p u r e Gaussian.

T a b l e I V

Moments o f t h e S i n g l e P a r t i c l e D i s t r i b u t i o n F u n c t i o n The f i r s t column g i v e s t h e d e n s i t y i n r e d u c e d u n i t s . The n e x t t h r e e columns g i v e t h e s e c o n d , f o u r t h and s i x t h moments o f t h e s i n g l e p a r t i c l e d i s t r i b u t i o n .

= < r 4 >

-

513 < r 2 > 2 . A l l l e n g h t s a r e i n u n i t s o f u.

A m e a s u r e o f t h e d e p a r t u r e o f t h e s i n g l e a r t i c l e d i s t r i b u t i o n from G a u s s i a n form i s $ / < r 2 >

! .

The s h a r p e r l o c a l i z a t i o n a s d e n s i t y i n c r e a s e s i s c l e a r l y shown. The e v i d e n c e f o r non-Gaussian beha- vier i s ambiguous :

-

0 l i e s i n t h e r a n g e from z e r o

t o 1.8 x lo-?. A p o s i t i v e d e v i a t i o n from G a u s s i a n b e h a v i o r i s p r o b a b l e a t low d e n s i t i e s . I n f i g u r e 5 s u c h d e v i a t i o n s c a n be o b s e r v e d i n t h e t a i l of t h e d i s t r i b u t i o n .

log p(r

I I I I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 r 2 ( i n

2)

F i g . 5 : The l o g a r i t h m o f t h e s i n g l e p a r t i c l e d e n s i - t y f o r pu3 = 0.526 p l o t t e d a s a f u n c t i o n of t h e s q u a r e o f t h e d i s t a n c e . The s t r a i g h t l i n e shows per- f e c t G a u s s i a n b e h a v i o r . Our d a t a shows a s l i g h t t e n - dency t o f a l l o f f l e s s s l o w l y a t l a r g e d i s t a n c e s .

To d e t e r m i n e whether p ( z ) had any s i g n i f i c a n t d e p a r t u r e from s p h e r i c a l symmetry, we c o n s t r u c t e d one d i m e n s i o n a l p r o j e c t i o n s i n 9 d i r e c t i o n s . These d i s - t r i b u t i o n s and t h e i r second moments e x h i b i t e d no si- g n i f i c a n t asymmetry. T h i s i s t o b e c o n t r a s t e d w i t h t h e s t r o n g a n i s o t r o p y found i n comparable s t u d i e s f o r c l a s s i c a l c r y s t a l s by Young and A l d e r 1191.

The second moment of r , < r 2 > , may be used t o compute Lindemann's r a t i o , y = <r2>'/2 /d where d i s t h e n e a r e s t n e i g h b o r s e p a r a t i o n i n t h e l a t t i c e . A t o u r m e l t i n g d e n s i t y , we i n t e r p o l a t e a v a l u e o f 0.267

+

0.002. Computations on o t h e r quantum s y s t e m s have g i v e n v a l u e s i n t h e r a n g e from 0.27 t o 0.30 / 2 , 4 , 5 / . I t a p p e a r s t h a t t h e narrow i n t e r v a l f o r y d e f i n e s a quantum a n a l o g o f t h e c l a s s i c a l ~ i n d e m a n n ' s m e l t i n g r u l e .

2 . 4 . P a i r c o r r e l a t i o n and S t r u c t u r e F u n c t i o n . - We have computed t h e s p h e r i c a l i z e d 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 ) and a c o r r e s p o n d i n g s t r u c t u r e f u n c t i o n S ( k ) . F i g u r e 6 shows t h e l a t t e r a t d e n s i t i e s p a 3 = 0.526 and 0.589. The r i s e o f t h e f i r s t peak i n t h e l a t t e r c a s e i s a n o t h e r m a n i f e s t a t i o n of t h e s h a r p e r l o c a l i z a t i o n o f t h e p a r t i c l e s a t h i g h e r d e n s i t y . But, compared t o a n a l o g o u s c l a s s i c a l s y s t e m s , t h e c r y s t a l i s r a t h e r s o f t . F o r t y p i c a l c l a s s i c a l systems S ( k ) a t f r e e z i n g i s a b o u t 2.85 a t t h e f i r s t peak. The h i g h e s t v a l u e o b s e r v e d i n o u r c a l c u l a t i o n s a t 1.25 t i m e s f r e e z i n g d e n s i t y , i s 1.97. We s u g g e s t t h a t measurements on S ( k ) i n t h e c r y s t a l phase would be w e l l w o r t h w h i l e a s a s e n s i t i v e t e s t o f t h e s t r u c - t u r e s i n c e a c c u r a t e c a l c u l a t i o n s a r e now p o s s i b l e .

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k (onverse sigmas)

F i 6 : The s t r u c t u r e f a c t o r a t two d e n s i t i e s ; paa'= 0.526 ( l e f t f i g u r e ) and po3 = 0.589 ( r i g h t f i g u r e )

.

2.5 The Momentum D i s t r i b u t i o n . - There appear t o b e no measured d a t a which r e l a t e t o t h e o f f - d i a g o n a l one- body d e n s i t y , p l ( r ) o r i t s F o u r i e r t r a n s f o r m , n ( k ) , which g i v e s t h e moaentum d i s t r i b u t i o n . We p r e s e n t our c o m p u t a t i o n a l r e s u l t s i n t h e hope of s t i m u l a t i n g such measurements.

F i g u r e 7 d i s p l a y s a s p h e r i c a l i z e d n ( k ) f o r t h e c r y s t a l a t pa3 = 0.526 a l o n g w i t h r e s u l t s f o r t h e f l u i d phase a t pa3 = 0.438. The b r o a d e r d i s t r i - b u t i o n which c h a r a c t e r i z e s t h e c r y s t a l r e f l e c t s t h e s h a r p e r l o c a l i z a t i o n i n t h a t phase and t h e a s s o c i a - t e d 5 0 % i n c r e a s e i n k i n e t i c energy. Within a p r e c i - s i o n of a b o u t 1 % we o b s e r v e no c o n d e n s a t e f r a c t i o n i n t h e c r y s t a l .

L5.00

LO. 03

35.00

33.00

25.03

F i g . 7 : The momentum d i s t r i b u t i o n . The s o l i d c u r v e i s f o r t h e c r y s t a l phase a t pa3 = 0.526 ; t h e dashed c u r v e i s f o r t h e h i g h e s t s t a b l e d e n s i t y i n t h e f l u i d phase ; pa3 = 0.438.

3 . VARIATIONAL CALCULATIONS ON 3 ~ e CRYSTALS. -Using the fermion Monte C a r l o s a m p l i n g C e p e r l e y , C h e s t e r , and

Kalos / 5 / performed v a r i a t i o n a l c a l c u l a t i o n s of b.c.c. p h a s e s o f s o l i d 3 ~ e u s i n g a t r i a l f u n c t i o n of t h e form

That i s , an a n t i s y n m e t r i c c o m b i n a t i o n of G a u s s i a n o r b i t a l s r e p l a c e s t h e p r o d u c t i n Eq. ( 1 ) . They u s e d t h e same d e n s i t y and v a r i a t i o n a l p a r a m e t e r s which Hansen and Levesque 1161 used i n a c a l c u l a t i o n of a Boltzmann c r y s t a l w i t h t h e 3 ~ e mass f o r each atom.

The f e r m i o n c a l c u l a t i o n gave a s l i g h t l y h i g h e r energy (1.38

+

0.1 K ) t h a n was found by Hansen and Levesque (1.07 ^.10 . 3 K) and a h i g h e r p r e s s u r e (7.9 i 0.5 a s opposed t o 4 . 6 ) . I t may b e t h a t t h e p r e s s u r e depends upon t h e symmetry of t h e w a v e f u n c t i o n , a l - though a d d i t i o n a l c o m p u t a t i o n s t o improve and a s s e s s a l l e r r o r s would be n e c e s s a r y t o b e s u r e . I n t h a t c a l c u l a t i o n "exchange" could b e o b s e r v e d i n t h e s e n s e t h a t a p a r t i c l e was found l o c a l i z e d a t a d i f - f e r e n t l a t t i c e p o s i t i o n from t h a t a t which i t was s t a r t e d . Such a jump o c c u r r e d o n c e i n 2 x 105 s t e p s of t h e random walk.

4. SUMMARY.- An a c c u r a t e n u w r i c a l method i s a v a i l a - b l e f o r t h e t r e a t m e n t o f Bose c r y s t a l s . C a l c u l a t i o n s

done w i t h i t f o r f . c . c . p h a s e s of 4 ~ e u s i n g t h e s t a n d a r d Lennard-Jones p o t e n t i a l show t h e l a t t e r t o be i n s u b s t a n t i a l l y b e t t e r agreement w i t h e q u a t i o n of s t a t e d a t a t h a n had been i n d i c a t e d by p r e v i o u s v a r i a t i o n a l c a l c u l a t i o n s . The d e f e c t i n t h e l a t t e r i s l i k e l y t o b e t h e n e g l e c t of e x p l i c i t three-body c o r r e l a t i o n s i n t h e ground s t a t e .

N e v e r t h e l e s s , t h e Lennard-Jones p o t e n t i a l i s i n a d e q u a t e t o p r e d i c t c o r r e c t l y t h e e q u a t i o n of s t a - t e and presumably o t h e r o b s e r v a b l e s . We a r e p r e s e n - t l y engaged i n a s s e s s i n g t h e u t i l i t y of a l t e r n a t i v e p o t e n t i a l s f o r p r e d i c t i n g t h e e q u a t i o n of s t a t e , u s i n g hard-sphere p e r t u r b a t i o n t h e o r y a s a s c r e e n i n g method. GFMC c a l c u l a ~ i o n w i l l b e c a r r i e d o u t f o r

t h o s e p o t e n t i a l s which a p p e a r p r o m i s i n g . But a s a d e f i n i t e s t e s t of t h e f o r c e law and of t h e t h e o r y i n g e n e r a l i t would b e w o r t h w h i l e t o have a d d i t i o n a l e x p e r i m e n t a l d a t a on c r y s t a l p h a s e s of 4 ~ e , p a r t i c u - l a r l y f o r t h e s t r u c t u r e f u n c t i o n . Measurements of momentum d i s t r i b u t i o n and of t h e k i n e t i c energy would be v a l u a b l e i f t h e y c a n b e performed.

As f o r s o l i d 3 ~ e , what i s most needed i s a r e f i n e m e n t o f t h e comparison of t h e c a l c u l a t i o n s u s i n g Boltzmann and Fermi s t a t i s t i c s t o s e e whether

(8)

t h e p r e l i m i n a r y d i f f e r e n c e s a r e r e a l . C l e a r l y r e l e - v a n t measurements would b e h i g h l y d e s i r a b l e a t t h e s ame t i m e

.

The a u t h o r s would l i k e t o t h a n k t h e Aspen C e n t e r f o r P h y s i c s f o r p r o v i d i n g t h e o p p o r t u n i t y f o r t h e c o m p l e t i o n of t h i s p a p e r .

R e f e r e n c e s

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Rev. A

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(1974) 2178.

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P h y s . Rev. B

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(1978) 1070.

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16

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/ 6 / McMahan, A.K. and Guyer, R.H., Phys. Rev. A

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Phys.

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5

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165

(1968) 293.

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140 (1965) A 8 1 6 .

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1 1 4 1 Woods, A.D.B. and S e a r s , V.F., J. P h y s .

10

(1977) L341.

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48 (1976) 587.

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1 1 6 1 Hansen, J . P . , J . P h y s i q u e C o l l o q .

2

(1970)C3-67.

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171 Chang, C.C. and Campbell, C.E., P h y s . Rev. B

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(1974) 1256.