Binding of small number of He4 atoms

UE 5

Physica 108B (1981) 1381-1382 North-Holland Publishing Company

BINDING OF SMALL NUMBER OF He 4 ATOMS Stefan SunarI( and Sre6ko Kill(

Mechanical Faculty, University of Mostar, Yugoslavia *Department of Civil Engineering, University of S p l i t , Yugoslavia Abstract° The wave function used in the analysis includes the correlations of short-range exp(-(~/r) B) and of long-range exp(-sr)~ Using Monte Carlo calculation for the binding energy of the 4 atoms of He4 we obtained the value E=-O,Bo10"23jo A comparison of the results obtained for the trimer (He4)3 and for the molecule (He4)4 shows an increase in binding energy per atom° INTRODUCTION

E 2/31 +pl + -N = P 1F 2F °'"

At present t h e r e is no s a t i s f a c t o r y

theories of

quantum f l u i d s which considers the i n f i n i t e tem as a c o r r e l a t i o n particles,

sys-

in a group o f neighbouring

and regards each group as an i n t e r a c -

t i n g axstem c o n t a i n i n g a f i n i t e

As 11 <0 and 11F>6

in the limit p40 the ener-

gy remains negative for He 4 and it becomes positive for He3o This paper examines the problem of 4 atoms He4o

number o f p a r t i -

cleso One a t t e m p t in t h a t d i r e c t i o n based on the g e n e r a l i z e d c e l l model o f p a r t i c l e s was made in a previous paper 1)'. In g e n e r a l , the development

GROUND STATE ENERGY The v a r i a t i o n a l

E=

n s i d e r a t i o n o f a group c o n t a i n i n g a small number o f atoms° At the o u t s e t , when c o n s i d e r i n g the helium, i t was necessary to

ansatz is d e f i n e d by the expre-

ssion

o f such a theory would o b v i o u s l y lead to the co-

problem o f l i q u i d

f o r He3,

<~IHI~> <~I~>

(I)

'

where ~ is a symmetrical wave functiOn~ and H has the form

~2

take i n t o c o n s i d e r a t i o n the e x i s t e n c e o f helium

H = - 2--m ~i ~i + ½ i~ v ( r i J )

dimers~ the energy b i n d i n g 3 , 4 . o . 1 3 atomsp the

(2)

s t r u c t u r e o f such systems e t c . w i t h m = 6,624o10"27kg{ The Lennard-Jones p o t e The problem o f the e x i s t e n c e o f helium dlmers

n t i a l was chosen f o r the i n t e r a c t i o n p o t e n t i a l :

and t r i m e r s has been d e a l t w i t h in several papers2"5/°" Comparing the r e s u l t s f o r a f i n i t e

V(r) = 4oE{(o/r) 12 - (o/r) 6} ,

(3)

numb-

er o f p a r t i c l e s presented in the above mention-

where ~ = 14,1o10"23j and

o=2,556.1D-10mo The

ed papers w i t h r e s u l t s o b t a i n e d using the model

Bijl-Jastrow function was taken as the variate-

o f s e m i - f r e e gas of l i q u i d helium 6-81"~ i t can

onal function in the form suggested by K. Ljol j e 9)

be noted t h a t the r e s u l t s o f the many-boby theo r y lead to v a l i d conclusions about the e x i s t e nce o f a s t a b l e molecule He4 and the n o n - e x i s t e nce o f a s t a b l e molecule He3° Consequentlyp the energy per p a r t i c l e

in the model o f the semi-

- f r e e gas has the f o l l o w i n g form 6-8) E 2 ~ = Pl 1 +P 12 + °o°

for He 4

~(I,2 .... n) = ~ exp{-(~/rij)B-srij}

(4)

,B

and s are the variational parameters° As

the variational parameters a and B were defined in the previous papers I0'11'5), only s is variedo The expected value of energy (I) with the wave function (4) can be written in the form: E(s) =

0378.4363/81/0000-0000/$02.50 © North-HollandPublishingCompany

;

i
IB/I N

(5)

1381

1382 with

The binding energy of trimer He 4 is defined in

I52 ' IB =Y~ H*dT = y 2{ - Tm { ~ ~ ~ ijk

paper 13) and is approximately -0,138o10 -23 J,

,~.

(a 6 13 rij

-g-1 s)(cf3 B r i k - B ' l - s )

i.e.-0,046o10-23j/atomo

.+

Table that the binding energy per one atom in

riJ rik

-

It is evident from the

rij rik

a four-atom molecule of He 4 is not greater than -0,16o10-23j/atom. Consequently, the increase

+;~ }:( 13 ~8(1-13)r..-13"2-2sr.'1)} +

ij

ij

ij

+ 2 e .~ ~ (o/rij) 6 ((o/rij)6 -I)} dT

in binding energy per one atom is -0,I14"I0 -23 J/amom. Since the experimental binding energy (5')

I J

per atom in liquid helium amounts to -9,88o10 -23 J/amom, we can conclude that 87 particles are

and

sufficient to describe a real system° This resIN=f~2 dT=fexp{- Z .~ ((~/rij)!B+srij)}dr (5") i j

ult is only a qualitative orientation. All 87 particles cannot behave in the same way with

RESULTS AND DISCUSSION

regard to a single particale, since there are some first neighbour particles immediately su-

The computation of energy was performed using the iterative Monte Carlo procedure "Veges"12)o For particular values of parameter s the computation of the nominator took about 40 minutes, and for the denominator it took 7 minutes on the UNIVAC 1110. The results of the calculations are presented in Figure° For the value of

rrouding it, the second neighbour palticles etCo It should be emphasized that there is good agreement between the number we obtained and the Monte Carlo calculations of the infinite systems, and it is evident that it is sufficient to observe about a hundred particles in order to obtain results which approach experimental ones.

parameter s of about 0,14.1010 m -I the energy is at minimum and it is E = - 0,8oi0"23jo The

REFERENCES

respective values of parameter s obtained for two particles in two-dimensional motion 5) s = 0,075"1010 m -I and in the theree-dimenslo-

I) S°Kili~ and M.Ristig, 11 nuovo cimento 39, (1977) 248;

nal case of 13 p a r t i c l e s 10) s=0,17"1010 m"1

2) A.Bagchi, Physo Revo A3 (1971) I133;

demonstrate that our results confirm the expec-

3) RoL°Siddon and MoSchlek, Physo Rev. A9 (1974) 907;

ted behaviour of the wave function in the four-

4) FoCabral and LoW.Bruch, preprint 1979;

particle system. The wave function binds parti-

5) S.Kili~ and SoSunarl~, Fizika II(1979) 225;

cles more strongly than in the case of two ato-

6) KoLjolje, Fizika I (1968) 11~

ms a~d less strongly than in the system of 13

7) SoKili6, Fizika 2 (1970) I05;

a toms °

8) S.Sunari6, MoSoAo thesis, Sarajevo 1969 (unpublished)i

0

E(10-23j)

9) K°Ljolje, Reports of the University of Illinois, Urbana 1962; -0,2 -0,4

~

" ~

s

10) W°L.McMillan, Phys. Revo 1938 (1965) 442;

-0,6

-0,8 --1

• ,

0,1

i

0,12

(1010m "1)

I

0,14

i

0,16

Figure° Energy of 4 atoms He 4 as the function of parameter So

11) DoSchiff and L°Verlet, Physo Rev: 160 (1967) 208; 12) GoPo Lepage, SLAC-PUB-1839, 1977; 13) L.WoBruch and I.J.McGee, JoChem. Phys.59 (1973) 409.