Interacting bosons and the properties of liquid 4He

Physica 76 (1974) 137-152 ©North-HollandPublishingCo.

INTERACTING BOSONS A N D THE PROPERTIES OF LIQUID 4 He K. M. KHANNA, B. K. DAS and O. P. SINHA Department of Applied Physics, Birla Institute of Technology, Mesra, Ranchi, Bihar, India

Received 10 January 1974

Synopsis The properties of a system of bosons interacting through a repulsive core followed by the combination of repulsive and attractive gaussian potentials have been studied using the reaction matrix and Green-function methods. The properties studied are the ground-state energy per particle (Eo/N), the transition temperature (Tc), the excitation energy (Ek) and the structure factor (Sk). These properties are quite similar to those observed for liquid 4 He below the lambda-transition temperature. It is emphatically brought out that the net 4 He_4 He interaction potential is very weak but repulsive.

1. Introduction. Recently I ) we have studied the properties of a system of bosons interacting through a repulsive core followed by the combination of repulsive and attractive gaussian potentials. For the first time, we have been able to predict the energy excitation spectrum of a system o f interacting bosons, which agrees almost exactly with the experimental energy excitation spectrum of liquid 4 He as obtained by Henshaw and Woods 2). To establish the suitability of the potential, we feel that it is necessary to perform some more calculations in order to see if it can also predict other properties o f liquid 4 He. Many theoretical physicists have studied the properties o f dilute Bose systems with special reference to liquid 4 He. In most o f the earlier work along this line a purely repulsive interaction between the bosons a- 13) was assumed. Lee et al. 4) studied the low-temperature equilibrium properties of a dilute Bose system of hard spheres by the pseudopotential m e t h o d 3). Luban and Grobman 9) studied the Bose-Einstein phase transition of an interacting system assuming that the particles interact through a central repulsive two-body potential. Goble and Trainor 11 ) studied the Bose system using a hard-core type interaction and the methods o f Brueckner and Sawada 14) and Parry and Ter Haar is ). Following L o n d o n 16 ), and using the basic assumptions made by Lee et al. 4), Khanna 12 ) calculated 137

138

K.M. KHANNA,B. K. DAS AND O. P. SINHA

the normal-superfluid phase-transition temperature. Khanna and Phukan in another paper ~3) studied the thermodynamic properties of a dilute hard-sphere Bose gas. Although all these calculations with a hard-core potential only, gave a good deal of insight into the properties of liquid 4 He, the introduction of an attractive tail to study the properties of liquid 4 He was also quite tempting. Huang ~ ) studied the properties of bosons by adding an attractive tail to the repulsive potential by using the pseudopotential method. To account for the liquid state the existence of an attractive tail was considered necessary. No rigorous work had been done so far with this motivation, till Khanna e t al. la) did so. They studied the properties of bosons having a hard core followed by a square-well interaction of the type18): I U(r) =

oo - Iio

0

for for for

r
a <~ r ~ b, r~>b

The methods used there ~8) were the Green function and the reaction matrix due to Brueckner and Sawada 14). The critical temperature (T c) and the ground-state energy per particle ( E o / N ) were calculated very satisfactorily. The commonly accepted interaction for the study of interacting bosons and hence the liquid 4 He is the Lennard-Jones 12-6 potential of the form U(r) = 4 e [(a/r) 12 - (a/r) 6 ]

(2)

with e = 14.11 × 10-16 erg and a = 2.556 A. When the hard-core followed by a square-well potential is compared with the Lennard-Jones form the former seems to be a very poor approximation. The potential of eq. (1) used earlier 18) has no repulsive element after the hard-core and there is no reason to believe that this is the most natural form of interaction. The most reliable potential, a combination of repulsive and attractive gaussians following the hard-core isa):

U(r)=

[4e

{exp

[-(r-al2 \/aR ] ]

{r-a~2 --exp [ - - \ / a A ] ]

r
for r >~a,

. . . . (3)

INTERACTING BOSONS AND THE PROPERTIES OF LIQUID 4He

139

where/.tR and/~A are, respectively, the repulsive and attractive ranges after the hard core. In our earlier communication 1 ) we studied the excitation spectrum (Ek-k) for a high-density (2.06 × 1022 particles cm -3) Bose gas using the potential o f eq. (3). For completeness we realized that it is necessary to do the calculations for a low-density 18) (2 × 1021 particles cm -3) system o f bosons interacting.through the potential constructed by us ~) and given in eq. (3). A number of quantities like Eo/N, T c and the structure factor (Sk) had not been calculated in ref. 1. We have first calculated the values o f Eo/N, Tc, Ek and Sk for a low-density system interacting through the potential o f eq. (3) using the formalism developed earlier 18) and then for a high-density one using the formalism of refs. 1, 19, 20 and 21. The Ek-k and the Sk-k graphs drawn for p = 2 × 1021 particles cm -3 , a = 2.1 )~ and m*/m = 1.2 show no similarity with what is known about these quantities for liquid 4He. But the ( E 0 / N ) - p graph and the Tc~ p graph start behaving differently when p crosses the value of 7 × 10 particles cm -3 . Hence we obtained the E k - k and the S k - k graphs for this density using the formalism developed for a high-density system I ). The E k - k graph does not show any peculiar behaviour. The S k - k graph has a peak at k = 2.0 A, which is the experimental value, but the corresponding Sk (= 1.1 2) value is lower t h a n the experimentally observed Sk for the peak (= 1.34) in the S k - k graph 22 ) for liquid 4 He. The valley part does not show much similarity with the experiment. We then drew the S k - k graph for p = 2.06 X 1022 particles cm- 3 which is the actual density of liquid 4 He at the k-point. The S k - k graph for this density gives a nice qualitative agreement with the experimental graph showing the peak and the valley at the same values o f k. But the peak in the S k - k graph is a little higher than the experimental value. This shows that for a value o f p between 7 X 1021 particles cm -3 and 2.06 × 1022 particles cm -3 , we may get the peak in the S k - k graph which may agree with the experimental value. With this in mind we did a large number o f calculations for Sk. We find that for p = 1.2 X 1022 particles cm -3 we get the E k - k graph which distinctly shows the p h o n o n - r o t o n characteristics and the S k - k graph gives a peak at k = 2.0 A, Sk = 1.34 which is also the experimental value for the peak. For the high-density (2.06 × 1022 particles crn-3) system Eo/N = 0.8843 X 10-16 erg and Tc = 2.49. The depletion effects are small and we have neglected them in our calculations and assumed N ~ No throughout the calculations.

2. General derivations. We consider a system o f N bosons enclosed in a box o f volume V. It is understood that the density p is held constant even

140

K.M. KHANNA, B. K. DAS AND O. P. SINHA

when N and V are allowed to approach infinity. This system of bosons is supposed to be interacting through a potential composed of a hard core followed by a combination of both repulsive and attractive gaussians of the form given by eq. (3). The hamiltonian of the system in the second-quantization formalism is, h k2 ) H = ~ a~ ak + ~ k \ 2m*

+ + ~

klk2kak4

Uklkz,k3k 4 akl ak 2 ax3 a k 4 ,

(4)

where h k is the m o m e n t u m and m is the mass of 4He (bosons) whose destruction and creation operators are, respectively, ak and a~. In eq. (4) Uklk2,k3k4 is the matrix element for the two-body potential. Following Brueckner and Sawada ~4) the effective hamiltonian for our system of bosons becomes, Heft = ½ N o ( N o - l ) too,oo + ~ [(h 2 k 2 / 2 m *) + No (tok, ok + tok, ko -- too,oo) a~ ak k4~O +

+

+ ½ No ( t k - k , o o a k a_ k + too,k - k ak a - k ] ,

(5)

where the symbols have the meanings expressed in ref. 14. Liu, Liu and Wong 23) solved eq. (5) to get the expression for excitation energy Ek as, Ek = [(Notk + h 2 k 2 / 2 m * ) 2 -N2o t~] ~-,

(6)

where t k = tok,k o + tok, ko -- too,o o.

(7)

Following our earlier work I ) and making the substitution, too,k - k = t k - k,oo, we get the following expression for Ek, Ek = [(h2k2 /2m*) 2 + (h2k2 /m*) No too,oo sin ka/ka]~,

(8)

where we have used the relation tk = too,oo sin ka/ka.

(9)

Again the ground-state energy per particle to a good approximation is given by 14).

141

INTERACTING BOSONS AND THE PROPERTIES OF LIQUID 4 He

Eo/iV = ½ p too,oo.

( 1O)

Eqs. (8) and eq. (10) show that to calculate the excitation energy and the ground-state energy per particle we have to calculate too,oo.

3. Evaluation of too,oo. Following our earlier works1,1s,2°,21 ) we get, too,oo = ( - [ G o ( a , a ) ]

-1 + 47r ~ drr 2 U(r) [1 -Go(a,r)/Go(a,a)]2). (11)

The iteration method which is applicable to low-density systems gives 19) -

[Go(a,a)]-I = (47ra/m*) [1 + (87rpa3) ~- + ~rrpa 3 ].

(12)

The second term of the r.h.s, of eq. (1 1) for the potential of eq. (3) becomes: 47ra/2 e (#~ - #~). Here Go (a, r)/Go (a, a) is approximated as a/r. Finally the expression for too,oo becomes

too,oo = (4rra/m*) [1 + (8~rpa3) ~ + / ~ lrpa 3 ] (13)

+ 47r3/2e ( # ~ - - / . t ~ ) .

Substituting the value of too,oo from eq. (13) in eqs. (8) and (10) we get the expression for Ek and Eo/N, respectively, as follows:

Ek =

I[h2k2) 2

+

4~h4 P[1 + (81rpa3) ~ + ~ m,-----T-

+ 4~r3/2 h 2 pe (#~ _ # ~ ) k sin

am*

?rpa3]ksinka

ka

(14)

and Eo_

N

27rpah2[ + ~ T1+(87rpa3)~ r p a 3 ] m * + 2*r ~/~ pe (u~,

-~*~).

The expression for the structure factor

Sk = h2 k 2/2m*E k.

(15)

(Sk) is given by ~ ) (16)

142

K.M. KHANNA, B. K. DAS AND O. P. SINHA

Substituting the values of Ek from eq. (14) we can calculate Sg. Using eq. (15) the Tc can be calculated la) (see table II).

4. Numerical calculations and tables. Using eq. (15) we have recorded in table I the values of Eo/N, for the hard-core, the gaussian potentials and the total potential separately for different values of 1//9. Table II gives the values of Eo/N and Tc for increasing values o f p calculated from eqs. (15) and (17), respectively. The values of Ek using eq. (14) for p = 2 × 1 021 particles cm -3 are calculated for different m o m e n t u m transfers k. Using eq. (19) the values of Ek for p = 7 X 1 021 particles cm -3 , 2.06 × 1 022 particles cm -3 and 1.2 × 1022 particles cm -3 have been calculated for different k. All these Ek values are recorded in table III. Table IV gives the Sk values calculated from eq. (16) for P = 2 × 1021 particles cm -3 and from eq. (20) for p = 7 × 1 021 particles cm -3 for different values of k. The other parameters used throughout the calculations are e = 14.1 1 × 10 -16 erg, a = 2.1 3,,/a~t= 0.1103 A~2 ; ~t~, = 0.2206 )~2 and m*/m = 1.2 where m = 6.646 × 10 -~4 g. The required values for different pa 3 have been obtained from the X2 --pa 3 graph of Brueckner 19). Results of table I are plotted in fig.1. Results of table II are plotted in figs. 2 and 3. Results of table III are plotted in fig. 4. Results o f table IV TABLE I

Eo/N for various potentials and densities (× 10-21

1/19 cm3 particle-1 ) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9' 1.0 1.25 1.5 1.6 1.8 1.9 2.0 2.5

Eo/N

Eo/N

repulsive

gaussians

(1 0-16 erg) 2.477 1.445 0.985 0.723 0.571 0.471 0.396 0.3~1 0.299 0.227 0.182 0.170 0.147 0.137 0.129 0.099

6 0 4 1 8 5 6 7 4 1 0 7 6 93

Eo/N total

(1 0 -16 erg)

( 10 -16 erg)

0.053 0.035 0.026 0.021 0.017 0.015 0.013 0.011 0.010 0.008 0.007 0.006 0.005 0.005 0.005 0.004

2.423 1.409 0.958 0.701 0.553 0.455 0.383 0.329 0.288 0.219 0.175 0.163 0.141 0.132 0.124 0.095

7 47 87 5 84 37 437 93 75 06 16 87 97 66 37 3

3 53 73 5 56 73 363 57 85 64 24 23 03 04 23 63

INTERACTING BOSONS AND THE PROPERTIES OF LIQUID 4He

143

TABLE II

Eo/Nand Tc from eqs. (15) and (17) p (X 10 21 particles cm -3)

E/N (1 0-~° erg)

Tc (K)

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

0.2889 0.6870 1.2015 1.81 2.5145 3.296 4.1655 5.241 6.1495 7.238

2.390 2.465 2.532 2.613 2.698 2.766 2.833 2.901 2.956 2.989

TABLE III E k using eq. (19) k

(A -l )

0.1 0.3 0.5 0.7 0.9 1.1 1.3

Ek(K) p = 2 X 10 21 particles cm 3

particles cm -3

p = 1.2X 10 22 particles cm -3

0.3305 1.043 1.543 3.102 4.587 6.468 8.690

0.9432 2.932 4.657 6.197 7.506 8.664 9.842

1.447 4.235 6.713 8.716 10.12 10.90 11.37

p=7X

10 21

1.5

11.37

11.34

11.31

1.7 1.9 2.1 2.3 2.5

14.47 18.05 22.05 26.52 31.43

13.42 16.33 20.08 24.67 29.97

11.93 13.66 17.19 22.04 27.97

are p l o t t e d in fig. 5. T h e e x p e r i m e n t a l S k - k g r a p h 22 ) is also d r a w n t o scale in fig. 5 f o r c o m p a r i s o n .

5. Discussion and conclusions. We have p l o t t e d the values o f E o / N f o r d i f f e r e n t values o f 1/p in fig. 1. T h e values o f E o / N f o r the h a r d - c o r e and the c o m b i n e d gaussians are also d r a w n in fig. 1. T h e s h a p e o f the h a r d - c o r e

144

K. M. KHANNA, B. K. DAS AND O. P. SINHA TABLE IV Sk from eqs. (16) and (20)

k (A -1 )

0.5 1.0 1.5 2.0 2.5 3.0 4.0 5.0 6.0 7.0 8.0 9.0

Sk P = 2 X 10 21 particles cm -a

t0 = 7 X 1021 particles cm-3

0.6494 0.9259 1.002 1.0108 1.005 1.000 1.000 1.000 1.000 1.000 1.000 1.000

0.2725 0.625 1.002 1.12 1.054 1.000 0.999 1.007 1.000 0.9984 1.002 1.000

p = 2.06 X 10 22 particles cm -3 0.1415 0.3752 1.009 2.086 1.28 0.9926 0.9853 1.026 0.9816 0.977 1.006 0.9816

p= 1.2X 10 22 particles crn-3 0.1881 0.4760 1.005 1.339 1.129 1.000 1.000 1.014 1.000 1.000 1.000 1.000

part is quite similar to those o b t a i n e d b y H u a n g 17 ) and K h a n n a et al. 18). T h e shape o f the gaussian p a r t is o f course, t o some e x t e n t d i f f e r e n t and unlike the earlier works it runs parallel to the abscissa with a slight curvature n e a r the o r d i n a t e . H u a n g 17 ) and K h a n n a et al. Is) got b o t h positive and negative values for E o / N d e p e n d i n g u p o n the values o f 1/p. T h e potential used b y H u a n g 17) was a p s e u d o p o t e n t i a l and b y K h a n n a et al. la) it was a hard core f o l l o w e d by a square well. But in o u r case the total p o t e n t i a l always gives a positive Eo/N, n o m a t t e r w h a t the value o f 1/p is. This fact shows t h a t the repulsive p a r t o f the i n t e r a c t i o n p o t e n t i a l d o m i n a t e s , and predicts t h a t the net c o n t r i b u t i o n o f the p o t e n t i a l should always be positive to get the e x c i t a t i o n s p e c t r u m which fits best 1 ) to that o f liquid 4 He. T h e result is in c o n f o r m i t y with the t h e o r y o f B r u e c k n e r and Sawada 19) w h i c h requires that the r e a c t i o n - m a t r i x e l e m e n t t0o,oo should always be positive. F r o m table II and figs. 2 and 3, respectively, we find t h a t the e n e r g y per particle in the g r o u n d state (Eo/N) and the critical t e m p e r a t u r e o f the transition (Tc) increase with an in6rease o f p. This result is quite in line with the earlier w o r k s o f K h a n n a 12 ) and K h a n n a et al. is). T h e E o / N versus p graph for low density is curved up to p = 7 X l021 particles cm -a and after this value o f p the graph is a straight line (fig. 2). This means Eo/N increases slowly up to p = 7 X l02a particles cm- a, and after this it increases faster. Thus the i n t e r a c t i o n p o t e n t i a l b e c o m e s effective o r starts c o n t r i b u t i n g to the e n e r g y as the density o f interacting

145

INTERACTING BOSONS AND THE PROPERTIES OF LIQUID 4 He

1"(

1-4

1"2

'0

1

0"8

0-6

0-4

02

I/P

o

0.4-

0.8

tN 1 ..q.

/0 -2t P A R T / C I . . E ~ t.6 2.0

2;4.

2. ~

~

i

-0.1 -0"2 - 0"3

Fig. 1. a: Eo/N-I/p graph for the hard-core potential only (table I); b: Eo/N-1/p graph for the combined gaussian potential (table I); c: Eo/N-1/p graph for the total potential of eq. (3) (table I).

146

K.M. KHANNA, B. K. DAS AND O. P. SINHA

5

I

4

c6 tu

0

1

2

3

4-

.5

e

7

8

P I I V 10 8 PAKT"1CLES CPI -~

Fig. 2.

Eo/N-p graph for different p values (table II).

9

I0

147

INTERACTING BOSONS AND THE PROPERTIES OF LIQUID 4He

2-9

~8

2?

2.6

I

i

i

I

i

9 P IN

I0 m PARTICLE3

CH "~

Fig. 3. Tc- p graph for different p values (table II).

I

1o

148

K.M. KHANNA, B. K. DAS AND O. P. S1NHA

bosons approaches that of liquid 4 He. Hence a dilute gas of bosons interacting through the potential of eq. (3) behaves roughly like a non-interacting gas and the effects of interaction start showing up only after the density approaches that o f liquid 4 He near the X-point. The T c - p graph is a straight line up to p -~ 7 X 10 21 particles cm -s and becomes curved for P > 7 × 1 011 particles cm -a . This behaviour once again confirms that the interaction potential becomes effective as the density approaches that of liquid 4 He at the X-point, or for O > 7 X 10 21 particles cm -3 . The E k - k graph for low-density (O a 3 "~ 1) systems interacting through the potential of eq. (3), shows no similarity to what is observed experimentally for liquid 4 He. So is the case with the structure factor. Thus a dilute gas of bosons interacting through a potential of eq. (3) has practically no property similar to that o f liquid 4 He. But the present study of the dilute gas has clearly brought out that for p > 7 X 1021 particles cm -3 , various properties of the interacting Bose gas start showing some peculiar behaviour. It indicates that if we study various properties of the highdensity gas, and if the agreement with the corresponding properties of liquid 4 He could be obtained, the authenticity of the interaction potential of eq. (3) could be better established. Since the properties of the low-density system of bosons interacting through the potential given by eq. (3) do not agree with those of liquid 4He, we present in the next section calculations ofEo/N, Tc and Sk for a high-density system of bosons interacting through the same potential.

5.1. H i g h - d e n s i t y c a l c u l a t i o n s . For a high-density system of bosons we obtain the following expression for Eo IN,

Eo/N = (1/2V)[(xZhZ/2aZm *) + 4zr 3/2 eo (U~ - U.~)] •

(18)

For the values of the parameters given in ref. 1, i.e., p = 2.06 × 10 22 particles cm -a , a = 2.1 A and m*/m = 1.2, we get Eo/N ~-- 0.8843 X 10 -16 erg. Substituting this value of Eo/N in eq. (17) we get Tc - 2.49 K. The values of Ek and Sk for a high-density gas of bosons interacting through the potential given by eq. (3) are 1 )

EkHigh

=

\ 2m*

+ 2aam.2 k sin ka

+ 4rr 3/2 pe"h 2 (/1~ - # ~ , ) k sin ka

m*a

,

(19)

INTERACTING BOSONS AND THE PROPERTIES OF LIQUID 4He

S k =-h 2 k 2/2m*

EkHig h,

149 (20)

For P = 7 X 1 0 21 particles cm -3 the peak value of Sk in the S k - k graph (fig. 5) lies at Sk = 1.12 and k = 2.0 A. This value of Sk is lower than the experimentally observed peak. F o r p = 2.06 × 1022 particles cm -a the peak in the S k - k graph falls at S k = 2.07 and k = 2.0 A. In both cases the peaks in the S k - k graphs fall at the same k value, which is also the experimental value o f k (fig. 5). The position of the peak in the former case is lower and in the latter case it is higher than the experimental result. The value for/9 = 2.06 X 1022 particles cm -a gives good qualitative agreement with the experimental result. These calculations indicated that to get a peak which agrees quantitatively with the experimental result the value of p may be between 7 X 1021 particles cm -3 and 2.06 × 1022 particles cm -a . We then did a number of calculations for different p values (not all of them are reported here for brevity). Finally, for p = 1.2 X 10 22 particles cm -3 we get an S k - k graph which shows a peak at Sk = 1.34 and k = 2.0 .&, which are exactly the experimental values for the peak in the S k - k graph for liquid 4 He. The E k - k graphs are drawn in fig. 4. The graph for p = 7 X 1021 p a r t i c l e s c m - 3 shows no peculiar behaviour, but for p = 1.2 × 10 22 p a r t i c l e s c m - 3 it seems to show what is known as phonon-roton characteristics for liquid 4 He. Now for p = 2.06 X 1022 particles c m - 3 and an interaction potential given by eq. (3) we get an E k - k graph I ) which agrees exactly with the experimental one. But for this value of p the S k - k graph does not give exact agreement with experiment. It is only for p = 1.2 X 10 22 particles c m - 3 and interaction potential given by eq. (3) that the S k - k graph agrees almost exactly with experiment. This value o f p is certainly not the density of liquid 4 He at the ),-point. Therefore, we must decide which of the two physical properties, Ek or Sk should be treated as more fundamental for the description of the behaviour of liquid 4 He. However, if we consider Ek and Sk to be equally important and if we accept the value of p "-- 2.06 X 1022 particles c m - 3 at the )t-point, we come to the conclusion that the interaction potential given by eq. (3) is slightly stronger than the actual 4 H e - 4 He potential. In fact, the potential given by eq. (3) is already a comparatively weak potential. Thus, our calculations confirm that the 4 He_4 He potential is very weak, which is the fundamental reason that liquid 4 He does not become solid even down to absolute zero of temperature unless an external pressure o f 25 atm is applied. It may be necessary to look for an interaction potential for the 4 He particles which is still weaker than that given by eq. (3).

150

K . M . K H A N N A , B. K. D A S A N D O. P. S I N H A

22

20

ta

16

a,)

(>3

o~

0.51

J.~

1,5"

1-0

2.1

2-41,

~.7

Fig. 4. a: E k - k g r a p h for p = 2 X 10 21 particles c m -a ; b: E k - k g r a p h for p = 7 X 10 21 particles cm -a ; c: Ek-k g r a p h for p = 1.2 X 10~ particles cm -a.

I N T E R A C T I N G B O S O N S A N D T H E P R O P E R T I E S O F L I Q U I D 4 He

151

1.9

I-7

co)

l_

"~ I

o.9~

(> 7~

0.5

(b)

i/\~,i

I W

-~e)

~

i~ i i i

coL)

~ -II11. f I~r

~)

03

o.J 0

.1

3

4-

5

~

'7 /<

8

9

]:?.

IN 4 °-j

Fig. 5. a: S k - k g r a p h for p = 2 X 1 021 particles cm -3 ; b: S k - k g r a p h for p : 7 X 1021 particles c m - 3 ; c: S k - k g r a p h for p = 2.06 X 1022 particles c m -3 ; d: S k - k g r a p h for p -- 1.2 X 10 22 particles c m , 3 ; e: e x p e r i m e n t a l g r a p h d u e t o A c h t e r et al. 22).

152

K.M. KHANNA, B. K. DAS AND O. P. SINHA

A c k n o w l e d g e m e n t. Financial assistance f r o m the Co uncil o f Scientific and Industrial Research, New Delhi, India is gratefully a c k n o w l e d g e d .

REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22)

Khanna, K. M. and Das, B. K., Physica 69 (1973) 611. Henshaw, D. G. and Woods, A. D. B., Phys. Rev. 121 (1961) 1266. Huang, K. and Yang, C. N., Phys. Rev. 105 (1957) 767. Lee, T. D., Huang, K. and Yang, C. N., Phys. Rev. 106 (1957) 1135. Lee, T. D. and Yang, C. N., Phys. Rev. 112(1958) 1419. Lee, T. D. and Yang, C. N., Phys. Rev. 113 (1959) 1405. Luban, M., Phys. Rev. 128 (1962) 965. Isihara, A. and Yee, D. D. H., Phys. Rev. 136A (1966) 618. Luban, M. and Grobman, W. D., Phys. Rev. Letters 17 (1966) 182. Nisteruk, C. J. and Isihara, A., Phys. Rev. 154 (1967) 150. Goble, D. F. and Trainor, L. E. H., Canad. J. Phys. 46 (1968) 839. Khanna, K. M., Journ. Phys. Soc. Japan 27 (1969) 1093. Khanna, K. M. and Phukan, A. N., Ind. J. pure and appl. Phys. 9 (1971) 156. Brueckner, K. A. and Sawada, K., Phys. Rev. (1957) 1117, 1128. Parry, W. E. and Ter Haar, D., Ann. Phys. 19 (1962)496. London, F., Superfluids, vol. II, Wiley (New York). Huang, K., Phys. Rev. 115 (1959) 765. Khanna, K. M. and Phukan, A. N., Physica 58 (1972) 263. Brueckner, K. A., The Many Body Problem, John Wiley (New York, 1959). Khanna, K. M. and Phukan, A N., Physica 60 (1972) 488. Khanna, K. M. and Das, B. K., Czech. J. Phys. B23 (1973). Achter, E. K. and Meyer, L., Phys. Rev. (1961) 188, 291 ; Phys. Rev. A4 811(E) (1971). 23) Liu, L., Liu, L. S. and Wong, K. W., Phys. Rev. 135A (1964) 1166. 24) Wilks, J., The properties of Liquid and Solid Helium, Clarendon Press (Oxford, 1967) p. 277.