Bose system with repulsive core and attractive tail

Physica 58 (1972) 263-276 o North-Holland

BOSE SYSTEM

WITH

Publishing

REPULSIVE

CO.

CORE AND ATTRACTIVE

TAIL

K. 111. KHANNAt International

Centre for Theoretical

Physics,

Trieste,

Italy

and A. N. PHUKAN Department

of Physics,

Dibrugarh

University,

India

Received 10 February 1971

Synopsis The properties of a dilute Bose system with strong repulsive core followed by a weak attractive tail are studied using Green-function methods. From the results it is concluded that such a system possesses properties which much resemble those of liquid helium (below the lambda-transition temperature). It is also concluded that such a system is a stable assembly and its transition temperature in the liquid state is comparable to the lambda transition temperature of liquid 4He.

1. Introduction. This paper is devoted to the study of a dilute Bose system of hard spheres interacting with an attractive potential, and the resemblance of the properties of such a system to those pertaining to liquid 4He. In an earlier paperr), the thermodynamics of a dilute Bose system with a purely repulsive interaction potential was studied using the well-known methods of statistical mechanics. The various thermodynamical functions resembled closely those of liquid 4He below the L-transition temperature. The specificheat variations, in particular, were found to conform closely to the experimentally observed variations of the specific heat with temperature for liquid 4He below T = TA. The dilute Bose system has been a good theoretical model for probing the properties of liquid helium 4He from a molecular point of view. Most of the earlier work on this line was developed by assuming a purely repulsive interaction between the bosons. Lee, Huang and Yangs) studied the lowtemperature equilibrium properties of a dilute Bose system of hard spheres using the pseudopotential methods) and calculated the energy levels near the ground state. Lee and Yanga) in another paper calculated the energy excitation spectrum which was used to study the thermodynamics of the system for various ranges of temperature below the critical point under the * On leave of absence from Department of Physics, Dibrugarh University, India. 263

264

conditions pa3 < phase transition

K. M. KHANNA

AND A. N. PHUKAN

1. But in none of these papers was the normal superfluidtemperature discussed. A realistic calculation, however,

was given by Khanna5) using the grand partition function of the assembly. The calculations made on the critical temperature (T,) for the dilute Bose system of hard spheres suggested 195) that a dilute Bose system of hard spheres with a density

of roughly

one tenth

that

of liquid

4He has a transition

temperature of T, = 2.116 K, which is very close to the value of T,J = 2.176 K for liquid 4He. The value of the hard-core diameter ‘a’ corresponding to the value of T, = 2.116 K is ‘a’ = 2.56 A; as can be calculated from ref. 5 that the value of ‘a’ corresponding to T, = 2.176 K is 2.84 A. This value of ‘a’ agrees very well with the value of ‘a’ calculated by Byckling6) who obtained the excitation spectrum for both a hard-sphere interaction and for the Lennard- Jones potential by the Green-function technique. The excitation spectra obtained by Byckling6) are in good agreement with the experimental results given by Henshaw and Woods?). This suggests that a dilute Bose system of hard spheres with a density of &p~ may have physical properties very similar to those of liquid 4He below the lambda point. Our earlier workr) was based on this indication as obtained from ref. 5. Although all these works, based on the assumption of a purely repulsive interaction between the Bose particles, have provided a good deal of insight into the problem of liquid helium, yet the possibility of introducing a weak attractive tail to the repulsive interaction has not been studied rigorously. Huangs), however, studied the Bose system of particles by bringing in an attractive tail to the repulsive potential by using the pseudopotential method. But Huang’ss) work was confined only to the study of the energy per particle in the ground state of the assembly as a function of the reciprocal of the density and it lacked the detailed discussion on the possibility of such a system predicting the properties of liquid 4He. An indication, however, is to be found in that papers) that a Bose system with repulsive core and an attractive tail may possess properties that are qualitatively similar to some of the properties of liquid 4He. In this paper we have studied the properties of a dilute Bose system of particles interacting through a repulsive core followed by an attractive tail using the Green-function technique due to Brueckner and Sawadag) and we have compared our results with those obtained by Huang8). Huang had obtained an expression for the energy per particle in the ground state as a function of the density of the system. We note here that there is a difference between the results of ref. 8 and ours. Whereas, in the case of Huang, the energy per particle in the ground state is positive or negative depending upon the value of p, in our case it is always positive. Our results, showing Ee/N as positive, might at first sight lead to the idea that our system is unbound and that it would expand infinitely in space. This, however, is not true. Since our system, by definition, is enclosed in a box, it is already a

BOSE

SYSTEM

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CORE

AND

ATTRACTIVE

TAIL

265

bound system and hence the question of the system being unbound does not arise. We also note that the attractive

tail introduced

by us is weaker than

the one introduced by Huang*). Huangs) has remarked and shown that the occupation of the low-lying energy levels of an interacting Bose system between the ground state and a nonzero momentum state (K # 0) is essentially uniform when the number of levels between 0 < j/z] < Ke is small compared to N. We feel that under certain conditions this may lead to the absence of particles in the lowest (k = 0) energy level. Obviously such a result will be physically uninteresting since the lowest level of a condensed Bose system is always macroscopically occupied. In our opinion a non-uniform distribution should be more reasonable. 2. Theoretical calculations. Let us consider a system of N helium particles enclosed in a box of volume Q. This assembly of helium particles is assumed to be interacting through a potential composed of a hard core followed by a potential well of the form +=J V(r) =

-Va (0

for

Y I a,

for

a
for

r < b.

(1)

The ground-state energy can be calculated by using the theory developed it is given by by Brueckner and Sawadas) and, to a good approximation, Eo = +Nptoo,oo,

(2)

where p is the number density of helium particles. Using the same notation as given in ref. 9, the t matrix is given by the integral equationa)

To solve the integral eq. (3) for the t matrix, it is assumed that the motion of the centre of mass can be separated. This approximation is accurate in the ground state. Writing Q = a(~~1- qs), 4’ = $(q; - q$), eq. (3) becomes t cl’,9 = Vcv, q + C V,,, ,IG$‘) rl” where the Green function

t,“, q,

(4

is

G(q”) = -[~~“~/rn + 2N(t,-,,a

+ t,~,_,~ -

to, o)]-l.

(5)

In getting to eq. (5), the centre-of-mass momentum is dropped with respect to 4”. This does not affect the propagator in too,oo. To solve the integral equation fort, we transform the equation to coordinate space where the infinite repulsion can be handled easily. The matrix element

266

K. M. KHANNA

of t can now be written t,t,,

AND

A. N. PHUKAN

as

= (#J*,>@a) = j dr d+;,(r)

t(r, y’) $a(~‘)>

(6)

where the 4’s are plane waves for relative momentum q and q’. In analogy with the scattering problem, the wave function can be defined as9)

J t(r, Y’) &(Y’) dr’.

V(r) &(Y) : The elements

of the t matrix

(7)

are thus obtainable

from

t a’,4 = f d4,*~(~) J’(r) Ah). The wave function @)

satisfies

(8)

the integral

equation

= $;(v) + J dr’Gz(r, r’) V(r’) &(Y’).

(9)

Obviously, +i( r ) is the free-particle wave function. To know Z&(Y) we have to know the wave distortion due to the interaction potential. Knowing S&(Y), we can calculate the t matrix from eq. (8). The method for knowing &(Y) is also given by Khanna and Bhattacharyaia) where it turns out that &Y) = Jl(qr) is the free-particle wave function. Here Jl(qr) is the spherical Bessel function. Proceeding in this way one can find that in the zeroth approximation, M

a

and from eq. (2) we get

Eo

_=-I

’

”

N

-4rr~drr”V(r)(l

Go@,a)

-

;;;;;I;i

)3,

(11)

cl

where the Green functions

are given by M

x[$+%{4xn)

[

(2J+ 1) (Jz(Tw)” Gd@4

c.2

1 +

Go@, a)

(12) a

BOSE

SYSTEM

WITH

REPULSIVE

CORE

AND

ATTRACTIVE

TAIL

267

and dq Jo@)

Job9

0

a Co

1 Go(a, a) -

+

4~

(13) a

The energy calculations are done by taking into account the wave distortion due to the potential given by eq. (l), as long as too, 00 and the energy denominators are positive. To solve eq. (2) for Eo/N, we have to determine the values of Go(a, a) and Go(a, r) from eqs. (12) and (13), respectively. In order to facilitate calculations, let us make the following simplifying approximations :

$;(y) = J&r) - Jz(qa) &(a, r)/Gz(a>4, Go@,4 Go@,a)

Gz(a,4 Gz(a,a) -

(14)

-m/47&+

a

-m/47&%

Y

(15)

With these approximations, we calculate the value of Go(u, a) for the S state of the system from eq. (12). The appropriate substitutions on the right-hand side of eq. (12) give Co Go(u, a) = -

+ 2~

+

-

(-&--~~dq~{$

2 Go(a, u) +

Go(;,a)

-4$dr:2cl

1 +2f

-

G,,(u, a)

+-

-+)V(+

+ TcVou(b-u)(5a

-+)-jr’

+ b) - F

(16)

(b -

.,3)]-t

(17)

The integral in eq. (17) can be solved by the iteration method given below.

K. M. KHANNA

268

A. N. PHUKAN

In the first approximation

3. Iteration method. the denominator gives

AND

except

q%@/m and determine

we leave all the terms of the value of Go(a, a). This

co

G,,(a, a) = - -

m 2ns s 0

%!?!?dq_-$. q2a2

(18)

Substituting this value of Ge(a, a) in the demominator of eq. (17) and solving for Ga(a, a), we get, after dropping terms which are very small compared to unity,

s 00

G&z, a) = -

sin2qa ~ 2x2 o (a2q2 + 8xpa3) m

dqJ

or we can write -

[G,,(a, a)]-1 = 2

[1

+ (8xpa3)~].

(19)

Substituting the value of Ge(a, a) from eq. (19) into the integral and solving the integral, we get -[Go@,

a)]-l

=

2;[

1 +

(anpa3)1

+

“;“].

in eq. (17)

(20)

This procedure can be carried out to any desired order, although the calculations become very tedious. Every subsequent term in the bracket in eq. (20) will be larger and in order to have a rough measure of the radius of convergence, let us require that the third term be smaller than the first, i.e. pa3 <

1.

(21)

Eq. (21) gives the value of p or the limit of dilution of the system to be studied by the method of iteration given above. Systems not satisfying the above conditions have to be studied by the self-consistent iteration process. Substituting eqs. (l), (15) and (20) in eq. (1 1) we get

Eo _~_ N

1 + (8npa3)~

+

3

16xpa3

This is the basic expression that we shall use in our present study and all our calculations are confined within the limits of validity of the above theory of Brueckner and Sawadas), namely, that Eo/N is non-negative. Now the energy of a non-interacting assembly is all kinetic. If, however, a perturbation is switched on adiabatically, some of the particles from the ground state will go over to excited states. But if the perturbation is weak, so that no violent disturbances appear in the system, there will be only very few

BOSE

SYSTEM

WITH

REPULSIVE

CORE

AND

ATTRACTIVE

TAIL

269

particles in the states above the ground state. This means that the excited states appearing in the system will be microscopically occupied singleparticle states. Owing to the very weak nature of the perturbation switched on, the ground state of the system existing before the perturbation continues to remain macroscopically occupied even after the perturbation. This macroscopically occupied single-particle state will therefore be defined as the ground state of the interacting system while the next microscopically occupied single-particle state will be the first excited state, ~1 say. We impose the restriction that, since the interaction is weak, ~1 will be approximately equal to the energy per particle in the ground state. Thus the assembly of bosons, below its transition point, can be assumed to be distributed among two levels, one called the ground state and the other the first excited state. When we switch on the interaction there will be some depletion of the ground state, but it is shown to be very small (see sec. 6) and hence, No m N. After the interaction is switched on, the system may have a number of internal levels, but in our calculation we have considered the relevance of only one of them just above the ground state. It is generally feltli) that the ground state should be a macro-state and the constituent single-particle states should be microscopically occupied. However, we have assumed that the ground state is a macroscopically occupied single-particle state. 4. Two-level afiproximation. The Bose-Einstein condensation temperature of this gas can be expressed as a function of ~1. If N is the total number of particles in the assembly, then

N=

C

1

i(scFeo)es(Ei--/c)_

1

where et = $42~ is the kinetic energy of the particle. At temperatures above the Bose-Einstein condensation temperature, the sum over ~1 can be replaced by the integral to give

where

m

I@, 4) =

x4 dx l (eafX - 1) ' r(& + 1) s

(25)

0

At ,u (k = tion. N’ =

= 0, the particles begin to condense 0). The particles in the excited states The condensation temperature T, N, where N is the total number of

to the state of zero kinetic energy do not take part in the condensais determined by the condition particles in the system and N’ is

270

K. M. KHANNA

AND

A. N. PHUKAN

the number of particles in the higher states with k # 0. At T = Tc, N’ becomes smaller than N and N - N’ particles condense to the zero state. Thus N=Q(

2~~~Tc):[((0,:)+I(~,j)].

where

(27) Now, it ~1 --f co, 1(/3&i) < 1, and the system reduces to an assembly of noninteracting bosons for which the condensation temperature may be denoted by To. Since I(0, +) = 2.612, we get

(28) Combining T, s

eqs. (26) and (28) we can finally To( 1 -

write

0.255 e--E1’kY’o).

(29)

Thus we have arrived at a means to calculate the transition temperature T, of our interacting assembly of bosons in terms of the condensation temperature To of an ideal Bose gas. Eq. (29) shows that T, is less than To. This result is obviously easily understandable, since the transition temperature of an interacting assembly of bosons, say liquid 4He, (2.176 K) is known to be less than the transition temperature of a non-interacting assembly of bosons (3.14 K). 5. Cahlations. Using eq. (22) we calculate the value of Eo/N and hence of ~1 to be used in eq. (29). Taking p = 2 x 1021 particles cm-s, Vo = 5 x 1016 erg, ‘a’ = 2.84 A, we have calculated T, from eq. (29) for different of ‘b’. The results are shown in table I. TABLE

values

I

5.00

0.4834

x

lo-16

2.424

5.50

0.3003

x

10-16

2.393

5.84 6.00 6.34

0.1292 x 0.0464 x negative

lo-16 lo-16

2.361 2.342

The value of p used in our present calculation is obtained from ref. 5 and is about one tenth of the value of p for liquid 4He at the il point. The value of ‘a’ (= 2.84 A) is the one used in our earlier investigationsi). We record

BOSE

SYSTEM

WITH

REPULSIVE

CORE

AND

ATTRACTIVE

TAIL

271

here that this value of ‘a’ is supported by the works of Khannas) and Bycklings). Va is obtained from the 4He-4He interatomic potential as given by Slater and Kirkwoodis).

We note that the value of ‘b’ = 6.00 fi gives

T, = 2.342 K which is quite close to Tn = 2.176 K. Because of the limitations of our basic theorys), we cannot use negative Eo/N values in our present calculations and therefore we choose from table I the value of ‘b’ = 6.00 A as the one to be used in our next calculation of Eo/N from eq. (29). The basic reason we do so is that this value of Tc, corresponding to ‘b’ = 6.00 8, is close to the il transition temperature of liquid 4He. We have carried out the numerical calculations for p values ranging from p = 2 X 102i particles cm-s to p = 10 X 1021 particles cm-a and the results are given in table II. Similarly, table III is obtained from eq. (22). TABLE II EoIN

P (particles cm-s)

(erg)

2 x

1021

0.0464

x

lo-16

3 x

1021

0.2331

x

lo-16

4 x

1021

0.5930

x

10-16

6 x

1021

1.6749

x

lo-16

8 x 10 x

1021 1021

3.2324

x

lo-16

5.986

x

lo-16

TABLE III

Eo/N repulsive

‘If

EoIN attractive

(erg)

Eo/N combined

(erg)

(erg)

0.250

x

lo-21

2.045

x

lo-16

-1.3312

x

lo-16

0.714

x

0.285

x

lo-21

1.639

x

lo-16

-1.1560

x

lo-16

0.983

x lo-16 x lo-16

10-16

0.330

x

10-21

1.293

x

lo-16

-0.9909

x

10-16

0.302

0.400

x

10-21

0.980

x

lo-l6

-0.8257

x

lo-16

0.154

x

IO-16

0.500

x

10-21

0.707

x

lo-16

-0.6606

x

10-16

0.046

x

lo-16

0.660

x

10-21

0.468

x

lo-16

-0.4954

x

10-16

-0.027

x

lo-16

1.000

x

10-21

0.264

x

lo-l6

-0.3303

x

lo-16

-0.066

x 10-16

1.250 1.50

x x

1O-21 10-21

0.199 0.155

x x

lo-16 lo-16

-0.2640 -0.2180

x x

lo-16 lo-16

-0.065 -0.063

x lo-l6 x lo-16

1.75

x

10-21

0.127

x

lo-16

-0.1883

x

lo-16

-0.061

x lo-16

2.00

x

10-21

0.107

x

lo-16

-0.165

x

lo-16

-0.058

x lo-l6

6. Depletion of the ground state. Written of particles, eq. (26) gives

in terms of the number density

(30)

272

K. M. KHANNA AND A. N. PHUKAN

Using T, = 2.342K, equation gives p -

po = 0.2186

~1 = Es/N = 0.046x lo-16 erg from X 1O23 particles

cm-a

table

I, this

(31)

and hence pa/p = 0.964.

(32)

This shows that the number density of particles in the ground state changes by hardly more than 5 potential composed of a repulsive core followed switched between the bosons in the ground state. of the ground state in our case is much less than

of an assembly of bosons per cent when a two-body by an attractive well is Therefore, the depletion that calculated by Byck-

Fig. 1. Variation of Eo/N with p.

BOSE SYSTEM WITH

linge),

where,

REPULSIVE

for pure hard-sphere

CORE AND ATTRACTIVE

interactions,

n&z = 0.71

TAIL

273

and for the

Lennard-Jones potential lze/n = 0.76. This means that the depletion effect is less in the case of a potential which is partly repulsive and partly attractive and the existence of attractive interactions reduces the depletion. This is precisely what is contained in our eq. (32). Since the depletion of the ground state has been shown to be small, we have neglected its effect in our calculations. 7. Graflhs. Results of table II are plotted in fig. 1. Fig. 2 gives the plot of the experimental values obtained by Keesom et al. 13~14) for the internal energy for liquid helium at different temperatures. This plot is given to interpret the results of table II (see sec. 8). In fig. 3 we have plotted Ee/N against 1/p with the help of eq. (22). (1) is ’ f or p urely repulsive potentials, (2) for

I

I

!

I

I

I

I

T(K)

Fig. 2. Variation

of internal energy with temperature. for liquid 4He.

Experimental

resultsls. 14)

274

K. M. KHANNA

AND A. N. PHUKAN

2.0_ 2.41.6-

1

1.2-

0.8 -

0.4G kJ q

n 0.0

0

w”

n

0 ”

d

-

0

-0.4

1

-1.2

-0.8

I

2.0

l/P

Fig. 3. Variation of Eo/N with l/p.

purely attractive potentials and (3) for the combination and the attractive potentials.

of the repulsive

8. Disctmion. Our calculations for the critical temperature T, of an interacting assembly of bosons show that (see table I) T, decreaseswith increasing ‘b’ for a fixed value of I/O.Increase in ‘b’ means broadening of the region of the attractive well, and broader attractive well implies stronger attractive interactions. This means that the suppresssion of the critical temperature is due to the existence of the attractive interactions between the helium atoms. Moreover, as shown in table I, when ‘b’ exceeds the value of 6.00 8, Ee/N, and hence the reaction matrix of eq. (2), becomes negative invalidating the applicability of the basic theory of Brueckner and Sawada g). Thus we restrict the value of ‘b’ to 6.00 A. From table I, we arrive at the conclusion that an assembly of bosons interacting through a repulsive potential followed by an attractive tail possesses a critical temperature

BOSE SYSTEM WITH T, =

2.342 K which

REPULSIVE

CORE AND ATTRACTIV-E

is quite close to the lambda

transition

TAIL

275

temperature

Tn = 2.176 K for liquid 4He. From table II we find that the energy per particle in the ground state increases with increasing p. These results are plotted in fig. 1, while fig. 2 gives the plot of the experimental values of the internal energy for liquid 4He at different temperatures as observed by Keesom et al. 13114). We see from fig. 2 that the internal energy increases with temperature up to 2.2 K. Now, as found by Onnes and Bocksis), the density of 4He increases with temperature up to T = 2.2 K and then drops. On the basis of this we may interpret fig. 2 as showing an increase in the internal energy with density up to T = 2.2 K. Since the internal energy per particle and the energy per particle Eo/N calculated by us can be treated as equivalent, the results in fig. 1 showing an increase in Eo/N with p seem to conform roughly to the experimental observations on liquid 4He presented in fig. 2. In fig. 3 we have plotted the records of table III. The shapes of the attractive and repulsive curves are quite similar to the corresponding shapes obtained by Huangs). But the combined graph agrees with that obtained by Huangs) only up to the point where the curve touches the I/p axis. After this point our graph is more or less parallel to the 1/p axis and this does not agree with the result obtained by Huangs). The reason for this disagreement may be due to the fact that the attractive tail we introduced in the interaction potential is weaker than that considered by Huang. Finally, we conclude that a dilute Bose system with repulsive core and attractive tail, and having only one excited level besides the ground-state level at &a=O, may have properties similar to those of liquid 4He at very low temperatures. Acknowledgments. One of us (K.M.K.) is grateful to Professors A. Salam and P. Budini, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. Thanks are also due to Professors F. C. Auluck and J. Mohanty for helpful discussions, to the Indian National Science Academy, Kothari Scientific and Research Institute and to the Swedish International Development Authority for financial assistance.

REFERENCES 1) Khanna, K. M. and Phukan, A. N., Indian J. pure appl. Phys., to be published 1971. 2) Lee, T. D., Huang, K. and Yang, C. N., Phys. Rev. 106 (1957) 1135. 3) Huang, K. and Yang, C. N., Phys. Rev. 105 (1957) 767. 4) Lee, T. D. and Yang, C. N., Phys. Rev. 112 (1958) 1419. 5) Khanna, K. M., J. Phys. Sot. Japan 27 (1969) 1093. 6) Byckling, E., Phys. Rev. 145 (1966) 71. 7) Henshaw, D. G. and Wood, A. W. D., Phys. Rev. 121 (1961) 1266.

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SYSTEM

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CORE

AND

ATTRACTIVE

TAIL

Huang, K., Phys. Rev. 115 (1959) 765. Brueckner, K. A. and Sawada, K., Phys. Rev. 106 (1957) 1117, 1128. Khanna, K. M. and Bhattacharya, P., Indian J. pure appl. Phys. 5 (1967) 253. Girardeau,

M., J. math.

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W. H. and Clausius,

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Keesom,

W. H. and Keesom,

K., Proc. Roy. Acad. Amsterdam 35 (1932) 307. 35 (1932) 736. A. B., Proc. Roy. Acad. Amsterdam

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p. 2.