Bose-Einstein condensation and thermodynamic functions of 4He film

Physica 96A (1979) 413-434 @ North-Holland

BOSE-EINSTEIN

Publishing Co.

CONDENSATION AND THERMODYNAMIC FUNCTIONS OF 4He FILM W.N. MEI and Y.C. LEE

Department of Physics

andAstronomy, State Unioersity of New York at Buffalo, Amherst, New York 14260, USA

Received 5 September

1978

A monolayer of 4He atoms is treated as a system of hard-sphere bosons in a thin film geometry, with a finite thickness. The method of pseudopotential is used to calculate first the energy spectrum, and then the Helmholtz free energy and other thermodynamic functions of the system. It is found that Bose-Einstein condensation exists below a definite temperature. Much like a liquid-gas transition, the boson system displays a high temperature normal phase, a low temperature condensed superfluid phase and a coexistence region. In the present treatment, the minimum momentum associated with the finite thickness of monolayer is used as a parameter. We find that the transition temperature is linearly proportional to the density of the “He film. After performing double-tangent construction of the Helmholtz free energy curve we find for the specific heat a rounded peak at the transition temperature, in agreement with recent experiments. The ratio of the superfluid density at the transition point to the transition temperature is found to be essentially a constant.

1. Introduction

Monolayer of 4He has been extensively studied in recent years both theoretically’“) and experimentally6). The experimental data corresponding to an area1 density p = 0.0279 k’ on grafoil showed that, as the temperature decreased the specific heat increased and passed through a rounded maximum at about 1.2 K6). This fact was interpreted as the occurrence of superfluidity in the two-dimensional boson system. In 1956, Penrose and Onsager’) suggested that Bose-Einstein condensation can occur in an infinite system of interacting bosons, stressing that the superfluidity of liquid helium was characterized by Bose-Einstein condensation, associated with the specific type of order called by London’) the long-range order of average momentum. In 1962, Yang? proposed that the phenomena of superfluidity and superconductivity were phases characterized by the existence of the off-diagonal long-range order (ODLRO) of the reduced density matrix in the coordinate space representation. He also demonstrated that the existence of Bose-Einstein condensation was associated with the behavior of the one particle reduced density matrix fiI in the coordinate representation such that (xlp,lx’) # 0 as IX- ~‘1+ ~0.Later, Hohenberg”), Chester, Fisher and Mermin”) proved that true 413

414

W.N. MEI AND Y.C. LEE

long range order in the boson operator +(x) cannot exist at any finite temperature in two dimensional systems. In 1973, Doniach”) argued that a model field theory with an n-component order parameter has a nonzero critical temperature for large n in two dimensions. This result was then extrapolated to the case n = 2 as an explanation of the superfluid phase transition in thin 4He films. More recently, the possibility of a transition at finite temperature into a low-temperature phase without long-range order was suggested and a jump in the superfluid density of 4He films as T, is approached from below was shown to exist, based on the classical two-dimensional XY mode?). In this paper we perform an explicit calculation on the thermodynamic properties of a 4He film of finite thickness at temperatures near the transition point by the method of pseudopotential I*-‘3. Based on the excitation spectrum obtained by lowest order perturbation treatment of the pseudopotential, a discontinuous jump in the occupation number of the zero-momentum level was found at a finite temperature T,. If we interpret the particles condensed into the zero-momentum level as the superfluid component of the liquid, this result seems to be consistent with those in references 5. Correspondingly, the specific heat would exhibit a discontinuity at the same temperature. These values of the discontinuities and Tf depended on the ratio a/L of the hard core radius to the thickness of the film. To first order in a/L, the ratio of the superfluid density at Ti to TL itself is found to be a constant which is, however, larger than the value in the theory of Kosterlitz and Thouless’). On the other hand, TL is shown to be linearly proportional to the density and generally decreases as the thickness of the film increases. In the limit of noninteracting bosons, the condensation into the zero-momentum states as well as the discontinuities were found to disappear completely, even if we kept the thickness L# 0. The thickness L entered our calculation through the matrix elements of the pseudopotential with respect to the unperturbed many-particle states of the thin film, although we include only the lowest eigenmode of the particle motion in the direction normal to the film. On closer examination of the above results based entirely on the first-order excitation spectrum by plotting the pressure versus volume at various temperatures, however, a thermodynamic instability such as that associated with the van der Waals gas-liquid transition was discovered. A Maxwell construction was necessary to remove the thermodynamic instability. After such a correction, the discontinuous jump in the zero-momentum state occupation number was found to spread over a small density range at a given T, much like the ratio of the amount of liquid to the total amount of fluid in the coexistence region during an ordinary liquid-gas transition. The specific heat then displayed not an abrupt jump but just a round peak at T& in agreement with experimen6’).

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In section 2 a brief description of the pseudopotential method is presented, and the analysis of the Helmholtz free energy of a thin film of hard-sphere bosons is also carried out. Section 3 is devoted to the calculation of all the thermodynamic quantities of the system together with the discussion of superfluid of the thin film. In section 4, we present the modified results after the Maxwell construction on the specific heat curve. A brief conclusion is given in the final section.

2. Theory We consider a dilute system of N identical spinless bosons of mass m, contained in a box of volume V, at very low temperature and density. The bosons interact with one another through binary collision characterized by the scattering length a, which is assumed to be positive. To simulate a monolayer, we treat a film of bosons with small but finite thickness L. This thickness L represents essentially the amplitude of the atomic excursion in the z-direction. In terms of a pseudopotential, the effective Hamiltonian for an imperfect gas of N identical particles of mass m may be taken to be”)

B=-g(v:+v;+.. .+v2,)+

y

z S(rtIIi-

rd

a

f ahI,

hJ,

(1)

where rIII is the three-dimensional position vector. The first term is the kinetic energy term, and the second term is the pseudopotential. Let the unperturbed wave functions be free-particle wave functions IGn) labelled by the occupation numbers {n,,,} where np,,, is the occupation number of bosons with three-dimensional momentum pIII. The energy levels to the first order in the hard-core radius a are E:, = (CD”,Z&B,)=

z2

nPlr, +

T (%,z NrIII, i
f-11+%).

(2)

The single-particle state are plane waves in x-y plane with 2-dimensional momentum p and standing wave in the z-direction of wave number k, values corresponding to higher energies in the z directional motion. It is this approximation which distinguishes our calculations. Actually L may be larger than a and we might have to take a few more standing wave modes in the z direction into statistical consideration. However, our results should not be affected qualitatively. Thus ‘pp,kz=

J-5

7 exp(i(p - rVh) cos klz,

(3)

416

W.N. MEI AND Y.C. LEE

where k, = AL,

(4)

V = NLv.

(5)

Here we have used the box boundary condition and required the wave function to vanish at z = *L/2. p, r are the two dimensional vectors and l/v is the area1 density. Let WI2

=

-

47rah' Sh, m

- rd.

(6)

Then

x [exp[-i(p, x

* rl + pz - r2}/h] cos k,z, cos krz2] - z2)}[exp[i{pI - rl + p; - r2}lh] cos k,z, cos kZz2]

{S(rl - r#(zI

- 67d2 mV

6

(7)

P1+Pz.Pi+Pi*

It follows that

(8)

z

nPnr = 2

n, 2

P

Pf9

n4 - 2

9

(n,)’ = N2 - 2 (nJ2.

P

(9)

P

Hence, (DE12 Oijl@~)= icj

s

[NZ- $N - i2 (np)‘].

The energies of the very low excited ni,o 4 N are approximately given by

(10)

P

states

of the system

with no 6 N,

(11) where no is the occupation number of the zero momentum ground state energy no = N, nP = 0 for pf 0 and we have Eo=N&-+Ng.

state. For the

(12)

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Note that the thickness L enters explicitly the expressions for the energy spectrum E, of eq. (11) via the matrix element eq. (7), and via the wave function (2/L)“’ cos(dL) in the z-direction. Thus, although no new statistical degree of freedom is added by placing all particles in the lowest standing wave mode in the z-direction, quantum mechanically this zero-point motion does affect the energy spectrum. Physically, as long as the interaction range is appreciably smaller than L, the interaction matrix element will depend on the extent L of the zero-point motion as given by eq. (7). In the absence of interaction, a approaches zero and the energy spectrum (11) reduces to that of a two-dimensional free boson system, even if we keep L# 0. Thus, the effect of the finite thickness L on the physical behavior of the system can be felt only through the interaction in our approximation. The partition function of the system is

Q(N) =

z exp(-PE,) n

=gexp[--P(F = exp(-N

n~&+~(N2-in3+~]] g)

z,!exp(-B

z0 [exp(nig) 3. np &)]

exp(-P

2)

(13)

where (14) is the thermal wavelength of the particle. For convenience, we have separated out the state p = 0 from the states p# 0. We first carry out the prime summation over all np (p# 0) with a fixed value (N - no) for the partial sum z ,,zO and then carry out the summation over all possible value of no from no = 0 to no = N. It is convenient to introduce a new variable 5, 5 = no/N = 0, l/N, 2/N,. . . , 1,

(1%

which denotes the fraction of the particles lying in the-state p = 0. In terms of 5, the expression for the partition function can be written as: Q(N) = exp(-N

$)

g [exp(N6’g)Qo{N(l

- t)}] exp(-Np

g),

(16) where Qo{n} stands for the partition function of a “fictitious” system of n noninteracting bosons, confined to area NV of the actual film (from which the

418

W.N. MEI AND Y.C. LEE

state p = 0 has been “artificially removed” and hence all the n particles of the system are distributed over the states pf 0). We then obtain: +log

Q(N)= =

-+$++log’$I[exp(N

g&*)Qo{N(l

3ah* -z+ilog’$

g

.6=0

-g)}]--p

h2k2 --.? 2m

5* - PA,{N( 1- 011

h*k*

-P,,,

(17)

where A(s) stands for the free energy of the “fictitious” system. The logarithm of the sum Z5 can be replaced as usual by the logarithm of the largest term in the sum, the error committed in doing so being statistically negligible. The value ,$ of the variable 5 which corresponds to the largest term of the sum can be obtained by setting the &derivative of the general term in the sum equal to zero. N

[

3ah2 ,5-B$AoW(l-5)1] =O. t=F

Consequently,

the free energy of the actual system will be given by:

1 3ah2 A(N, 5) = - p log C?(N) = N --pLV

1 p N g C

p - @A,{N( 1 - c)}] + 2

N (19)

On the right-hand side of eq. (19), the second term represents the contribution of the repulsive hard-core interaction to the free energy. That this term decreases as the fraction 5 of the ground-level occupation increases from 0 to 1 is a manifestation of the well-known enhancement of momentum space attraction due to spatial repulsion’2). Now, the free energy A,{N(l - 5)) of the fictitious system is that of the free boson gas contained in the two-dimensional area NV, A,{N(l-

[)}= N(1 -[)kTlogz

- NV $g2(z).

(20)

where g*(z)

=

“$] 5.

(21)

Here z is the fugacity of the “fictitious” by the equation. NV N( 1 - 5) = g,(z) = -

A2

$

two-dimensional

log( 1 - z)

system, determined

(22)

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for OCZ
(23)

where

g,(z) =

“$, 5.

(24)

It is obvious from eq. (20) that A,{N(l - 5)) is a negative quantity. Since we expect that the free energy of a free many-particle system confined in a fixed area to be a monotonic function of the number of particles, therefore it should increase as the number of particles N(1 - 4) decreases towards zero (or as 5 increases toward l), with 0 s z < 1. From eq. (19) we see that in the limit of a 40 the free energy would be minimized by a vanishing 5 for this fictitious two-dimensional system, implying that there could be no Bose-Einstein condensation? when we turn off the interaction. In fact, it is just the opposite directions of variations of the first two terms on the right-hand side of eq. (19) with respect to 5 that give rise to an optimum value between 0 and 1, leading to condensation. In order to determine .$, we have to differentiate A(N, 5) with respect to 5: N$+3[-NkTlogz+(N(l-[)kT-yg,(z)} Substitution

91

=O. c=g I=’ (24)

of eq. (22) into (24) leads to

$+logf=o. Combining eq. (25) with eq. (22) we obtain (l-c)=-clog(l-exp(-Fct])

(26)

we can solve eq. (26) for c and then eq. (25) for ,Zfor given o and T provided l>fsOand 1zlaO. Let us plot eq. (26) and discuss the properties of the solution. There are three curves in fig. 1, for fixed u. (i) is for T smaller than T,; there are two solutions. (ii) is for T equal to T,; there is only one solution, and (iii) is for T greater than T,; there is no solution. Let y=(l-l)=-slog(l-exp

( -x ‘“Zi}).

(27)

The existence of solution f at small T implies the occurrence of condensation. The value of T, can be calculated as follows. At T = T, there is only t For three-dimensional system, 6 takes on the smallest possible value consistent with the condition on 6 given by an equation similar to eq. (22) with N fixed and 0 6 z < 1.

420

W.N. MEI AND Y.C. LEE Ii)

T=O.g”K

iii) T=l.Z99”K=

Tc

(iii) T= 1.5 OK

\\

Y,TI \\

p =2.79x1d4cm2 (1 =I.sA L.758

(iii1

O.I0.1

0.2

0.3

0.4 &,0.5

0.6

0.7

0.8

0.9

Fig. 1. The graphic solution of eq. (26) for fixed temperature T and volume V.

one intersection. Therefore, the straight line y = 1 - 4 is also the tangent of y = -(v/h*) log(1 - exp{-(3ah*/Lu)&} at the intersection f= &, i.e. v (3ah*/Lv) exp{- (3ah*/Lv)~~ I F=k 1 - exp{- (3aA’/L~)z}

(28)

(1+%>.

(29)

or

&f&log c

To solve for T, we substitute

& int eq. (26), obtaining

or A:_ L ,-5;;*og(l+~)+log(l+~).

(30)

Thus, Tc = 2mk[(L/3a)

h2P log( 1 + 3a/L) + log( I+ L/3a)]

(31)

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and (L/3a) log( 1+ 3alL) 5c = (L/3a) log( 1 + 3alL) + log( 1 + L/3a) S l.

(32)

It is interesting to observe that, to the first order of a/L for which our perturbative excitation spectrum given by eq. (11) is valid, the ratio [&T’,/p&= (27rfi2/m), a constant depending only on the atomic mass of the helium atoms. We also observe that T, varies linearly with the particle area1 density and, at a fixed p, both T, and & generally decrease as the thickness L of the film increases. In the limit that a/L approaches zero, both & and T, approach zero as l/log(L/3a), meaning that there is no condensation as we switch off the hard-sphere interaction. This result agrees with Fisher and Barber’s conclusion’) that there is no condensation in the noninteracting 4He slab. For comparison we may consider the equation corresponding to eq. (26) in the three-dimensional system, 1-4=qg3,Z

A

(

exp 1 $&

(26’)

4).

While the right-hand side of eq. (26),

(-f2>gdew{-(3aA2/Lu)FH diverges logarithmically as a + 0, rendering a solution ,$ in the range from 0 to 1 impossible, the right-hand side of eq. (26’) approaches a finite constant in the same a +O limit so that c exists in the range 0 to 1 whenever (u/N) s g3,2(l)h3. The appearance of g3/2in eq. (26’) rather than g, in eq. (26) is, of course, due to the additional degrees of freedom associated with the third dimension. In the presence of interaction, a# 0 and the argument of the function g,(exp{-(3ah2/Lu)l}) becomes smaller than 1, rendering the right-hand side of eq. (26) finite and hence a solution I< 1 possible. We remark that we are not allowed to take the limit L approaching zero since our perturbation procedure is valid only when II is the smallest length in the system. If we were to let L-0, the delta-function pseudopotential would cause divergence, spoiling the perturbation calculation. Actually we are dealing with a quasi-two-dimensional system. The possibility of condensation for such a system is not ruled out by Hohenberg’s work’? which applies only to true two-dimensional system. Now, let us examine the behavior of the free energy. According to eq. (19) and ignoring the last constant term we have: v

= (1 - f)kT log f - 3 kTg,(z) + $$

kT(2 - p),

(33)

422

W.N. MEI AND

Y.C.

LEE

subject to the condition eq. (22) and eq. (25) A(N, s> -=kT[(l-f)log(l-exp[-:(1-c)}) N (34) at T=O, (35) Form eq. (35) we see that as long as a f 0 the_Helmholtz free energy is at its minimum when $ = 1, that is to say all the particles are condensed into its ground level at zero temperature. In order to investigate the finite temperature behavior, we plot in fig. 2 the Helmholtz free energy against & the zero momentum state occupation number. From the preceding results we see that, starting with temperatures below T,, there are two solutions that correspond to two extrema in the plot of A(N)/N versus 5 One corresponds to a local maximum, the other a local minimum. The local minimum represents the state of thermodynamic equilibrium to the merging of the local minimum and the p=2.79xK3%ti2) o=I.8H L=7.5%

0.76

(i) T=l.141°K (ii) T=l.2Z°K (iii1 T=l.299OK

0.72 ; \ 0 &

I

0.68 0.64

(i)

cf $ 0.60 LL ;

0.56

z

3

(ii1

! 0.48 -

Fz I _I 044w

0.36 QOO

0.10

0.20 0.30 0.40 ZERO MOMENTUM

0.50 0.60 0.70 Q80 OCCUPATION NUMBER

Fig. 2. The Helmholtz free energy A versus zero momentum temperature T and volume o based on eq. (34).

0.90 E

occupation

1.00

number

-$ for fixed

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local maximum into a point of inflection. For T greater than T,the curve rises monotonically as f increases, in which case f= 0 yields the minimum of A(N)/N.For a fixed p, as the temperature increases all points on each curve will be lowered since the derivative (aA/aT), = --s~0. For T < T,,A(f= 0) is greater than the local minimum Ah,,, and the system prefers to stay in the condensed phase with i# 0, But, as we keep increasing the temperature the point at f= 0, i.e. A(f= 0),is depressed at a faster rate than the point Ati,,. Therefore, at a certain TL slightly less than T, A(f = 0)will become equal to Ati” at 55 for T a TL,the system will undergo a transition from the condensed phase with a finite discontinuity into the normal phase (f= 0) abruptly since A(,$=0)will then be smaller than Amine That is to say, the local minimum Amin corresponding to a g # 0 solution of eq. (27) is no longer the absolute minimum of A(<). Therefore, we will take this temperature TL as the critical temperature of the phase transition. At this point the Helmholtz free energy is still continuous. At the slightly higher T = T,,the curve has no local maximum or minimum but only a point of inflection (see fig. 2). For T > T,, the curve is monotonically rising and there is no more condensation. This set of curves shown in fig. 2 are qualitatively like those Helmholtz free energy curve obtained from the Landau-Ginzburg theory by including up to sixth order terms in the order l.OOp=2.79xl~‘4&~) a= I.8i3 k

Z =

L=7.5a

.80

o.gol “\, 0.70

5 p 0.60B z g 0.50-

0.40

0.60

‘0.80 1.00 1.20 1.40 TEMPERATURE T FK)

1.60

1.80

2.00

Fig. 3. The zero momentum occupation number 5 versus temperature T for fixed volume tr based on eq. (26).

424

W.N.

MEI AND

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LEE

parameter, with the coefficient of the fourth order term ~0. Numerically the two temperatures T, and Tf are found to be very close. In any case, we can determine numerically the critical temperatures and obtain the Helmholtz free energy in both phases. The behavior of 5 as a function of temperature has been obtained numerically and is shown in fig. 3. It is interesting again, to examine the ratio of the superfluid density at the critical point, pS = (Sp to the critical temperature TS. In fig. 4, pS is plotted against Ti for various values of a/l. It is seen that, for a given a/L, ps is essentially proportional to TS, and the proportionality constant increases slightly as the thickness L increases. In fig. 5 we see that the transition temperature TS is proportional to the area1 density p and, for a given p, the value of Ti decreases as L increases. Thus the behavior of TL and ,$1_ follows the same pattern as that of T, and &. We remark that a constant ratio for p,(Tf)lkTS is a 1so found in the theory of Kosterlitz and Thouless’) although the numerical value for the constant is smaller than our p,(TL)/kTL by a factor of 4. The linear dependence of TL on p is in agreement with experiment6). That TL decreases as L increases may provide an alternative interpretation of the recent experimental observation6) that the transition temperature for the second layer of 4He film on a substrate is lower than that for the first layer since the atoms in the second layer should be freer to move in the direction normal to the film and hence correspond to a large L value. P 5.0

T

.

(i)

x

L= L= L= L= L=

IO.58 9.58 5.58 7.58 6.58

I 0.8

, 0.9

/,

(ii) 4.5- (iii)

0

(iv) IV)

A

> 3.0Iv, zl o

2.5-

2.01

1.0

Fig. 4. Superlluid

I I I.1 1.2 TRANSITION

density

I 1.3 1.4 TEMPERATURE

p, = &

versus

I 1.5

1.6

Tr for different

I I.7

I I.8

a/L values.

, l.g T;YK

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425

h

CP 3.0-

E :

(i)

l

(iv) A W x

L=l0.5H

L.7.51 L.6.58.

Fig. 5. Transition temperature Tf versus particle areal density p for different a/L values.

3. Results and discussion For T < TL the system is in the condensed zero. We have 45, NJ ---==T[(1-Alogi-~g,(i)+~~(2-k~] N

phase where f is greater than

(36)

together with

and (l-F)=-;log(l-r,,

OSf
(38)

The pressure is given by (39) where the terms proportional (37) and eq. (38).

to (~E/lau)~ and (c%/&J)~ cancelled out due to eq.

426

W.N. MEI AND Y.C. LEE

The grand partition function _(J!is given by (40) The internal energy is (41) The entropy is (42) where terms proportional to (&?/‘laT), and (@/U), again cancelled out due to eq. (37) and eq. (38). The chemical potential or Gibbs’ free energy is given as (43) The specific heat at constant volume is given as

and the specific heat at constant pressure is given as (45) where (46) (47) (48)

+a($)“=-@B=

.F[log f + (3alL) log( 1 - Z)] l-(3a/L)f/(l_f) *

(49)

Therefore, all the thermodynamical quantities can be calculated and plotted after we have solved 5 and Z numerically. Although we cannot obtain these quantities explicitly as functions of v and T, the T + 0 limiting expressions can be obtained from eq. (25) and eq. (26):

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(51) and

3a

pz

h2

(52)

2L22rmv and

h2 N - 2L 2rmv’

tJ,3”

(53)

These are consistent with the previous expression eq. (11) of the excitation spectrum of the hard core system, E., by setting all n, = 0 for p # 0. Also, as T-SO, S-0, which is consistent with the third law of thermodynamics. Also, we have C, and C,, approaching zero as T approaches zero. In the normal phase, ,$= 0 and we have from eq. (36), eq. (38) - eq. (47).

A(N) _ kT logr-;g,(z)+F; N [

I

(54)

together with * = 1 _ ,-A2/c,

(55) (56) (57)

(58) (59) G A p=%=%+PvL=kT

6a A2 logz+Lu, C 1

~“-(~)=k[~g2~r)-~1ea~~*,“],

(60)

(61)

c = c,_ [(WWv12 P

(62)

[c?Plav > T

’

W.N.

428

MEI AND

Y.C.

LEE

(63) kT A2

=-7;

eeA2” +@ L I. C1 -e-“+

(64)

The expressions in the condensed phase can be reduced to those of the normal phase by setting s= 0. The expressions for the normal phase can be reduced to those for the free bosons case by setting a = 0. In fig. 6 the numerically computed C, is plotted against T. When plotting the C, curve, we chose L = 7.5 A to fit the location6) of the peak Tf = 1.2 K for p = 2.79 x lOI (cm-2). We observe that L is approximately twice the hard core diameter, which is consistent with the previous assumption that the hard-core radius a is the smallest length parameter. We see that the peak in C, is a consequence of the onset of the Bose-Einstein condensation. As noted previously the condensation and hence the peak in C, will disappear if we let the interaction parameter a go to zero, reflecting the known fact that the spatial repulsion (a > 0) enhances the momentum space attraction (condensation). The other prominent feature is the discontinuous jump in C, at the critical temperature. 0.40-

.y \ P

2 P ‘p

ox0.32-

p=2797d4km2, (1=I.Ba L=7.5a

5 0.28 w I ii 0.249 = 0.205 :: s 0.16t 2

0.12-

ii g 0.08: co 0.04-

0.001 0.40

Fig. 6. The specific (61).

0.60

heat at constant

1.20 0.80 1.00 TEMPERATURE

volume

C. versus

I.40 T (‘IO

1.60

temperature

1.80

2.00

T based on eq. (44) and eq.

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429

This jump was also found in 3-dimensional imperfect boson gas”) which behaves differently from the free boson gas. Similarly, when we plot the numerical result of pressure against temperature, we find an abrupt jump. The same is also found in numerically obtained isotherm curves. These are shown in figs. 7 and 8. The discontinuous jump of the isotherm at the critical point corresponds to a thermodynamic instability. The reason for such a thermodynamic instability, (P/au) > 0, is because we take the energy spectrum eq. (11) too literally. As pointed out in ref. 12, the instability can be removed by a Maxwell construction just as in the case of a van der Waals gasu). Note that the Helmholtz energy has already been minimized by choosing the smaller numerical value between the local minimum and the normal phase value (see fig. 2) when it is used to derive the isotherms in fig. 8. This minimization procedure, also mentioned in ref. 12, is still not sufficient, as we just saw, to eliminate the unphysical behavior arising from the use of the energy spectrum of eq. (11) too literally. An abrupt jump in the order parameter was also found by Nelson and Kosterlitz’) in the recent calculation of the XY model. However, after reconstructing a la Maxwell the isotherm curve by smoothening out the discontinuous jump with a flat portion, which represents the region of coexistence between the condensed phase and the normal phase, the kink originally present in the curve plotting AIN against 11for a fixed T (see fig. 7)

p=279x10+‘4m-2, a= l.SA L=3.6%

l.lO-

_

IDO-

Is E p 0.90e 2 70 0.00F a

_/I_

w O.?O‘L

0.50 I

LziLLl_ 0.40

0.60

0.00 1.00 1.20 1.40 TEMPERATURE T (‘K)

I .60

1.80

2.00

Fig. 7. Pressure P versus temperature T for fixed volume v based on eq. (39) and eq. (56).

430

W.N. MEI AND Y.C. LEE

a

i

2.32

=IB%

t

1.84-

Fig. 8. Pressure P versus volume u for fixed temperature T based on eq. (39) and eq. (56). The dash-dot line indicates the Maxwell construction.

will correspondingly be smoothened out by a double-tangent construction’2*“) This new A/N will imply, in turn, a modified energy spectrum different from eq. (11). The relation between the zero momentum state occupation number and the volume at a given T is also modified by the Maxwell construction, as shown in fig. 10. Therefore, the value of g range continuously from 0 to &,,,,, The discontinuous jump has disappeared. as u varies from t&,al t o L)fon&nscd. Instead, a straight line joins the nonvanishing value of &,nd at &,ndensedto the point 5 = 0 at vnormal.This straight line represents the coexistence region between the normal phase and the condensed phase. The above behavior of ,$ is analogous to the behavior of the ratio of the amount of liquid to the total amount of fluid in the coexistence region of an ordinary classical liquid-gas transition. At a given T < TL the system displays a superfluid phase for v u,,.,,,~ and a coexistence region for o,,-,, > v > &on& The quantity p&O,,dcnsedrepresents the density of the superfluid phase at the transition. More easily accessible to experiment is the specific heat C, as a function of T. The discontinuity in C, at the critical point in fig. 6 can be traced back to that in the order parameter r of our system. With the smoothening out of ,$ by the Maxwell construction, the discontinuous peak in C, will also be rounded off, corresponding more closely to

BOSE-EINSTEIN

CONDENSATION

“condensed VOLUME

~xIO-‘~

431

OF 4He FILM

(Cm’)

bormol

Fig. 9. The Helmholtz free energy A versus volume o for fixed temperature and eq. (54). The dash-dot line indicates the double-tangent construction.

50

T based on eq. (36)

the experimental observation6). The width of the peak depends on the size of the coexistence region. The improved C, curve can be obtained from the modified isotherms as follows. For each fixed temperature, we can obtain the numerical value of the modified Helmholtz free energy A/N as a function of u from fig. 9. Then, for a given u we can obtain values of A/N for various T from the same figure. The specific heat C, can then be obtained as C, = -T(a2($2”‘)n

(65)

We have plotted a set of pressure versus volume curves (fig. 8) and Gibbs free energy versus volume curves (fig. 11) around the neighborhood of T = 1.2 K. The Maxwell construction has been performed, and the numerical value of the Helmholtz free energy has been found for the particular density p = 2.79 x 10” (cm-‘). For each temperature in that neighborhood, numerical differentiation has been performed. The peak on the specific curve is smoothened into a rounded one, and the peak value is approximately 2.2 k which is smaller than the original value but still larger than the experimental one.

432

W.N. MEI AND Y.C. LEE

a=l.EsA

L=75a T=l.2OK

0

om-

5

2 0.50

-

‘\

g 0.40-

‘\

E 0.30k 0.20N O.lO-

‘\ ‘\

‘1. 0.00 ’ 0 :a 3 * a r .k. ’ 0.20 0.22 0.24 0.26 020 0.30 032 0.34 0.36 038 040 0.42 ‘Q44046 %ondenssd %ormal VOLUME v~ld~&~~

s ( 0.48 Cl50

Fig. 10. The zero momentum occupation number 6 versus volume u for fixed temperature based on eq. (26). The dash-dot line indicates the coexistence region.

T

a=lBH Lz7.58 To 0.88 jj

0.84-

u OBOLT 2 0.76 cn g 0.72 n 0.68 0.64 0.60

-

0.56

-

Fig. 11. The Gibbs free energy versus volume D for fixed temperature (60). The dash-dot line indicates the coexistence region.

T based on eq. (43) and eq.

BOSEEINSTEIN

CONDENSATION

OF ‘He FILM

433

l.OO-

5

0.60-

2 a

0.40-

2 5

0.20-

i5 z Fif 2

o.oo-

-0.20-

2 ii2

-0.40-

I -O.bO-

-

I.OOl 040 0.60

0.80

1.00 1.20 TEMPERATURE

1.40 T(“K)

1.60

I.80

2.00

Fig. 12. The Helmholtz free energy A versus temperature T for fixed volume tl based on eq. (36) and eq. (54).

4. Conclusion

Based on the method of pseudopotential representing hard-core interaction or low-energy scattering length, we have calculated the thermodynamic properties of a thin film consisting of interaction bosons. The excitation spectrum of the system as a result of lowest order perturbation treatment of the pseudopotential leads to discontinuities in the population in the zero momentum level as well as in the specific heat at a finite temperature T6. However, thermodynamic instability results when the excitation spectrum is taken too literally. The result as modified by the Maxwell construction is similar in nature to an ordinary liquid-gas transition, showing a low temperature condensed superfluid phase, a high-temperature normal phase and a coexistence region. A rounded peak appears in the specific heat at the transition temperature. This indicates a phase transition from the normal phase to the low-temperature, superfluid phase, consistent with the experimen&) and theoretica14’5) results of other workers. Owing to the nature of the approximation involved our results should be valid qualitatively but are not expected to yield detailed, quantitative answer. It is straightforward to derive

434

W.N. MEI AND Y.C. LEE

the phonon spectrum as low temperature excitations of the system by the method of Bogoliubov, after assuring ourselves, as is done in this paper, that the thin film does go into Bose-Einstein condensation at low enough temperatures. Thus, the calculation of the thermodynamic properties for T 4 T, can be improved without too much difficulty. However, other schemes of approximation must be sought in order to improve the calculations for the region very close to the transition point.

Acknowledgment

We would like to thank Drs. David Bishop illuminating conversation concerning experimental the two-dimensional superfluid phase transition.

and F. Gasparini for an and theoretical features of

References 1) 2) 3) 4) 5)

6)

7) 8) 9) 10) 11) 12) 13) 14) 15)

R.L. Siddon and M. Schick, Phys. Rev. A9 (1974) 907, 1753. O’Hipo’lito and R. Lobo, Phys. Rev. B14 (1976) 3892. M.N. Barber and ME. Fisher, Phys. Rev. A8 (1973) 1124. S. Doniach, Phys. Rev. Lett. 31 (1973) 1450. J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181. J.M. Kosterlitz, J. Phys. C7 (1976) 1046. D.R. Nelson, Phys. Rev. B16 (1977) 1217. D.R. Nelson and J.M. Kosterlitz, Phys. Rev. Lett. 39 (1977) 1201. Various experimental techniques have been applied to the study of supertluidity in thin film. For recent works using different methods see J.A. Herb and J.G. Dash, Phys. Rev. Lett. 31 (1973) 1377, M. Bretz, Phys. Rev. Lett. (1973) 1447, J.M. Scholtz, E.O. Mclean and Rudnick, Phys. Rev. lett. 32 (1974) 147, S.E. Polanco and M. Bretz, Phys. Rev. B17 (1978) 151. 0. Penrose and L. Onsager, Phys. Rev. 104 (1956) 576. F. London, Superfluid (Dover, New York, 1960). C.N. Yang, Rev. of Mod. Phys. 34 (1%2) 694. P.C. Hohenberg, Phys. Rev. 158 (1%7) 383. G.V. Chester, M.E. Fisher and N.D. Mermin, Phys. Rev. 185 (1%9) 760. K. Huang and C.N. Yang, Phys. Rev. 105 (1957) 767, K. Huang, J.M. Luttinger and C.N. Yang, Phys. Rev. 105 (1957) 776. T.D. Lee and C.N. Yang, Phys. Rev. 112 (1958) 1419, T.D. Lee, K. Huang and C.N. Yang, Phys. Rev. 106 (1957) 1135. K. Huang, Imperfect Bose Gas, in Studies in Statistical Mechanics, Vol. 2, J. deBoer and G.E. Uhlenbeck, eds. (North-Holland Publ. Co., Amsterdam, 1%3). E. Stanley, Introduction to Phase Transition and Critical Phenomena, The International Series of Monographs on Physics.