Physics Letters A 375 (2011) 1637–1639

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Physics Letters A

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The internal energy and thermodynamic behaviour of a boson gas below theBose–Einstein temperature

F.A. Deeney ∗, J.P. O’Leary

Physics Department, National University of Ireland, Cork, Ireland

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 February 2011Accepted 28 February 2011Available online 5 March 2011Communicated by V.M. Agranovich

We have examined the issue of the kinetic energy of particles in the ground state of an ideal boson gas.By assuming that the particles, on dropping into the ground state, retain the kinetic energy they possessat the Bose–Einstein temperature T B , we obtain new expressions for the pressure and internal energyof the gas below T B , that are free of the difficulties associated with the corresponding expressions incurrent theory. Furthermore, these new equations yield a value for the maximum density temperature inliquid 4He that is very close to the measured value.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

In the current treatment of the statistical mechanics of anideal gas of identical bosons, it is assumed that when particlesdrop into the ground state of the system, they lose their ki-netic energy (e.g. [1,2]). Presumably this assumption is made be-cause the ground state is usually designated the ‘zero momentumstate’ of the system, and not because there is any experimen-tal evidence that this is the case. On this basis expressions arederived for the internal energy and pressure of the gas belowthe Bose–Einstein temperature, and the equation of state is es-tablished. The expressions so obtained, however, are fraught withdifficulties; the values obtained for the isothermal compressibilityand the heat capacity, for instance, are infinite over the temper-ature range from T B to zero. Furthermore, the internal energyvanishes at absolute zero, in contravention of the Heisenberg un-certainty principle. Here we take the alternative approach of as-suming the kinetic energy of the particles at the temperatureT B to be retained by the system, as the particles drop into theground state. In this way we find new expressions for the inter-nal energy and pressure below T B that have none of the problemsassociated with the existing theory. In addition, the new theorypredicts that a density maximum should exist in any ideal bo-son gas at some temperature below T B . In the case of a gas of4He, this should occur at a temperature ∼ 1.88 K. We have ex-tended the theory to examine the case of liquid 4He, and showthat the predicted result agrees very well with the observed den-sity maximum that is observed in this liquid at a temperature of2.18 K.

* Corresponding author.E-mail address: f.a.deeney@ucc.ie (F.A. Deeney).

0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2011.02.066

2. Existing theory

The expressions for the internal energy and pressure of an idealboson gas of N identical particles contained in a volume V , at tem-peratures above and below T B , as derived in the standard existingtheory, may be written as follows [1,2]. For T > T B one has

U (V , T ) = 3

2NkV T g5/2(z)〈λth〉−3 T > T B (1)

and

P V = NkT g5/2(z)/g3/2(z) T > T B (2)

where g3/2(z) and g5/2(z) are Bose–Einstein functions and z =eμ/kT . λth is the mean thermal de Broglie wavelength for the par-ticles, defined by

〈λth〉 =√

2π h̄2

mkT

To obtain the corresponding expressions for temperatures be-low T B , the assumption is made that the gas particles, on droppinginto the ground state at temperatures below T B , lose all of theirkinetic energy. These particles thus stop moving and no longercontribute to the internal energy or pressure of the gas. Hence oneobtains the expressions

U = 3

2cV T 5/2 T < T B (3)

and

P = cT 5/2 T < T B (4)

where c = g5/2(1)k5/2(m/2π h̄2)3/2 is a constant.

1638 F.A. Deeney, J.P. O’Leary / Physics Letters A 375 (2011) 1637–1639

As we mentioned in our introduction, it has always been evi-dent that there are serious problems with the physical implicationsof expressions (3) and (4). These include the following:

(i) The disappearance of the kinetic energy of the particles asthey drop into the ground state, leads to the difficult conceptof these particles remaining fixed in position in space.

(ii) According to Eq. (3), the internal energy of the system tendsto zero as the temperature approaches 0 K, implying that nowall of the particles will be static at that temperature, contra-dicting the Heisenberg uncertainty principle.

(iii) From Eq. (4), P no longer depends on the volume, so thatthe isothermal compressibility, κT = − 1

V ( ∂V∂ P )T , is infinite over

the full temperature range T = 0 → T B . Furthermore, sinceC P = V TκT (∂ P/∂T )2

V + CV , the heat capacity C P is also infi-nite at these temperatures. These unphysical results are some-times dismissed as arising due to the omission of interactionsbetween the particles when dealing with an ideal gas [3]. Ifone considers the equivalent classical gas, however, the quan-tities take the form κT = 1

P and C P = 5/2Nk, respectively i.e.they remain finite and well behaved at all temperatures. Yetthe only difference between the two gases is that, in the quan-tum gas, the particles are indistinguishable.

3. New theory

From expression (2) above we obtain

P V = NkT g5/2(1)/g3/2(1) ≈ 0.513NkT B T = T B (5)

when the temperature of the gas is T B . Using the relationshipP V = 2U/3 [2], we then obtain

U ≈ 0.77NkT B

so that the average kinetic energy of a gas particle, when T = T B ,is ≈ 0.77kT B .

Since there is no experimental evidence that this energy is lostto the surroundings during Bose–Einstein condensation, we heremake the assumption that it is retained in the system instead. Thismeans that at any temperature below T B , the particles in the con-densate have an internal energy

U0(T ) = N0(T ) × 0.77kT B (6)

where N0(T ) = N(1 − (T /T B )3/2) is the number of particles in theground state at that temperature. The total internal energy of thegas below the temperature T B is then

U = 3

2cV T 5/2 + 0.77N

(1 − (T /T B)3/2)kT B T < T B (7)

From this we see that, at absolute zero, the gas has the residualinternal energy

U0(0) = 0.77NkT B (8)

which is equal to the kinetic energy of the gas at T = T B . This re-sult is in excellent agreement with the estimate of the zero pointenergy made by Fetter and Walecka [4], using a simple argumentbased on the Heisenberg uncertainty principle. They show that themean zero point energy of a particle in an ideal boson gas, consist-ing of N particles in a volume V , and hence confined to a volumeof ∼ V /N , is approximately the same as the mean kinetic energyof a particle at the Bose–Einstein temperature, in the same gas, un-der the same conditions. We can thus identify U0(T ) with the zeropoint energy of the gas, defined in the usual way as that compo-nent of the internal energy of a system that does not vanish asT → 0.

Thus the first two difficulties that are present in the existingtheory, no longer exist in our new model. Looking next at expres-sion (4) for the pressure, this is now modified to become

P = cT 5/2 + 0.513(1 − (T /T B)3/2)NkT B/V (9)

where, again, we have used the relationship PV = 2U/3. Thus,when T = 0 K, the gas exerts a finite pressure P0(0) =0.513NkT B/V , which is the same as the pressure exerted by thegas at T = T B . The isothermal compressibility may then be writtenas

κT = 1

V

(∂V

∂ P

)T

= 1

P − cT 5/2(10)

This is a much more satisfactory expression for κT than hitherto,since its form is similar to κT = 1/P that applies in the idealclassical gas, with the additional term cT 5/2 arising due to the in-troduction of the principle of indistinguishability.

Furthermore, the compressibility is now finite at all tempera-tures except at T = T B . To understand the meaning of the latter,we note that infinite values of compressibility and heat capacities,at a particular temperature, are indicative of a phase change takingplace at that point (e.g. Ref. [2] chapter 11). Hence the divergencesat T = T B can be interpreted as arising due to the onset of Bose–Einstein condensation. We thus find that the final difficulty listedabove for the existing theory, is also absent in our analysis.

We then obtain a new equation of state for an ideal boson gasat temperatures below T B , to replace Eq. (4). This may be writtenas

PV = cV T 5/2 + 0.513(1 − (T /T B)3/2)NkT B T < T B (11)

Taken together with Eq. (2) for temperatures above T B , it expressesthe thermodynamic behaviour of an ideal boson gas at all temper-atures.

4. Density maximum in liquid 4He

A further feature of our new theory is the prediction that ex-trema in the pressure and density of an ideal boson gas, will occurat some critical temperature between 0 K and T B . To see this, con-sider expression (9). Keeping the volume of the system constant,we find the expression

Tc = 0.308

c

(N

V

)k

T 1/2B

(12)

for the critical temperature Tc at which the pressure will be a min-imum. Writing c = g5/2(1)k5/2(2πm/h2)3/2 as before, and usingthe expression for the number density

N

V= ζ(3/2)

(2πmk

h2

)3/2

T 3/2B (13)

Eq. (12) simplifies to

Tc = 0.308ζ(3/2)

g5/2(1)T B = 0.308 × 2.612

1.341≈ 0.60T B (14)

In the case of an ideal gas of 4He, for example, T B ≈ 3.13 K, and apressure minimum is predicted to occur at ≈ 1.88 K. Furthermore,if we allow the volume of the gas to vary but keep the pressureconstant, the density of the gas will pass through an extremum,in this case a maximum value. At that point d(N/V )/dT = 0 andusing Eq. (9) again, but now keeping P fixed, we find that a max-imum occurs in N/V at exactly the same temperature Tc givenby (11). Hence, for an ideal helium gas, the density should passthrough a maximum at a temperature ≈ 1.88 K.

F.A. Deeney, J.P. O’Leary / Physics Letters A 375 (2011) 1637–1639 1639

We may compare this prediction with experiment by examin-ing the variation with temperature of liquid 4He, which is a realboson system with atoms interacting via van der Waals’ forces.Here one finds that, despite the presence of these forces, the samearguments as were used in the ideal gas should broadly apply. Fol-lowing the standard van der Waals’ approach [2], the equation ofstate above T B can be modified by changing expression (2) to ob-tain the following approximate form for the equation of state ofliquid 4He,

P ′e = {NkT /V }g5/2(z)/g3/2(z) + P V deW T > T B (15)

where the extra term P V deW represents the effect of the van derWaals forces on the gas pressure. At temperatures below T B , themodified Eq. (9) becomes

P ′ = cT 5/2 + (1 − (T /T B)3/2)NkT B/3V + P V deW T < T B (16)

The density maximum, as before, will occur when d(N/V )dT= 0. The only difference from the ideal gas case is that one nowhas the presence of the term ∂ P V deW /∂T . The van der Waals’ forceis not expected to have a large temperature dependence, so theextra term will be small. Neglecting this term to first approxima-tion, the theory predicts that the density maximum in liquid 4Heshould occur at a temperature ≈ 1.88 K, which is in very goodagreement with the measured value of 2.18 K [5]. The differencebetween the two may be attributed to the action of the van derWaals’ force. The latter has the effect of increasing the overlap be-tween the wave functions of neighbouring particles, compared tothe ideal gas equivalent, thereby raising the value of T B and withit the value of Tc .

By including the additional term U0 in the internal energy ofthe condensate at temperatures below T B , our new theory suc-ceeds in predicting the occurrence of a density maximum in anideal boson gas, and in giving a very good estimate of the temper-ature at which this phenomenon should occur in liquid 4He. Wehave already commented upon this in a general way elsewhere [6].There we noted that liquid 4He is a particularly simple liquid,in that its atoms are spherically symmetrical and the interatomicforces are purely van der Waals’ in nature. It is difficult to envisage

how a density maximum could occur in such a system other thanin the way described above, since such a phenomenon requiresthe presence of two independent sources of kinetic energy, oneof which decreases and one of which increases with temperaturechange. The point at which the effects of these two mechanismsintersect will then give rise to a maximum in the density of liquid4He.

5. Conclusion

In conclusion, we have examined the issue of the kinetic energyof particles in the ground state of a boson gas. By assuming thatthe kinetic energy of these particles, as they drop into the con-densate, is retained in the system, we obtain expressions for theinternal energy and pressure of the gas below the temperature T B ,that lack the difficulties associated with the corresponding expres-sions in the current theory. In addition, the new analysis predictsthat a density maximum will occur in the gas at some critical tem-perature below T B . On applying the theory to the case of liquid4He, a value is predicted for this temperature that is in very goodagreement with the experimentally observed value.

Acknowledgement

The authors wish to thank Dr. J.J. Lennon for his valuable assis-tance throughout this work.

References

[1] K. Huang, Introduction to Statistical Physics, Taylor and Francis, London, NewYork, 2001, Chap. 11.

[2] R.K. Pathria, Statistical Mechanics, 2nd ed., Butterworth–Heinemann, 1996,Chap. 7.

[3] D.C. Mattis, R.H. Swendsen, Statistical Mechanics, 2nd ed., World Scientific Pub-lishing, 2008.

[4] A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw–Hill, New York, 1971.

[5] K. Mendelssohn, Liquid helium, in: S. Flugge (Ed.), Low Temperature Physics II,in: Handbuch der Physik, vol. XV, Springer-Verlag, Berlin, 1962, p. 373.

[6] F.A. Deeney, J.P. O’Leary, Phys. Lett. A 358 (2006) 53.