Physics Letters A 375 (2011) 1637–1639
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Physics Letters A
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1638 F.A. Deeney, J.P. O’Leary / Physics Letters A 375 (2011) 1637–1639
As we mentioned in our introduction, it has always...
F.A. Deeney, J.P. O’Leary / Physics Letters A 375 (2011) 1637–1639 1639
We may compare this prediction with experiment by ...
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The internal energy and thermodynamic behaviour of a boson gas below the Bose–Einstein temperature

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

  1. 1. Physics Letters A 375 (2011) 1637–1639 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla The internal energy and thermodynamic behaviour of a boson gas below the Bose–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 2011 Accepted 28 February 2011 Available online 5 March 2011 Communicated 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 possess at the Bose–Einstein temperature TB , we obtain new expressions for the pressure and internal energy of the gas below TB , that are free of the difficulties associated with the corresponding expressions in current theory. Furthermore, these new equations yield a value for the maximum density temperature in liquid 4 He 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 an ideal gas of identical bosons, it is assumed that when particles drop 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 momentum state’ of the system, and not because there is any experimen- tal evidence that this is the case. On this basis expressions are derived for the internal energy and pressure of the gas below the Bose–Einstein temperature, and the equation of state is es- tablished. The expressions so obtained, however, are fraught with difficulties; the values obtained for the isothermal compressibility and the heat capacity, for instance, are infinite over the temper- ature range from TB to zero. Furthermore, the internal energy vanishes 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 temperature TB to be retained by the system, as the particles drop into the ground state. In this way we find new expressions for the inter- nal energy and pressure below TB that have none of the problems associated with the existing theory. In addition, the new theory predicts that a density maximum should exist in any ideal bo- son gas at some temperature below TB . In the case of a gas of 4 He, this should occur at a temperature ∼ 1.88 K. We have ex- tended the theory to examine the case of liquid 4 He, and show that the predicted result agrees very well with the observed den- sity maximum that is observed in this liquid at a temperature of 2.18 K. * Corresponding author. E-mail address: f.a.deeney@ucc.ie (F.A. Deeney). 2. Existing theory The expressions for the internal energy and pressure of an ideal boson gas of N identical particles contained in a volume V , at tem- peratures above and below TB , as derived in the standard existing theory, may be written as follows [1,2]. For T > TB one has U(V , T ) = 3 2 NkV T g5/2(z) λth −3 T > TB (1) and P V = NkT g5/2(z)/g3/2(z) T > TB (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¯h2 mkT To obtain the corresponding expressions for temperatures be- low TB , the assumption is made that the gas particles, on dropping into the ground state at temperatures below TB , lose all of their kinetic energy. These particles thus stop moving and no longer contribute to the internal energy or pressure of the gas. Hence one obtains the expressions U = 3 2 cV T 5/2 T < TB (3) and P = cT 5/2 T < TB (4) where c = g5/2(1)k5/2 (m/2π¯h2 )3/2 is a constant. 0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2011.02.066
  2. 2. 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 implications of expressions (3) and (4). These include the following: (i) The disappearance of the kinetic energy of the particles as they drop into the ground state, leads to the difficult concept of these particles remaining fixed in position in space. (ii) According to Eq. (3), the internal energy of the system tends to zero as the temperature approaches 0 K, implying that now all 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 that the isothermal compressibility, κT = − 1 V (∂V ∂ P )T , is infinite over the full temperature range T = 0 → TB . Furthermore, since CP = V T κT (∂ P/∂T )2 V + CV , the heat capacity CP is also infi- nite at these temperatures. These unphysical results are some- times dismissed as arising due to the omission of interactions between the particles when dealing with an ideal gas [3]. If one considers the equivalent classical gas, however, the quan- tities take the form κT = 1 P and CP = 5/2Nk, respectively i.e. they remain finite and well behaved at all temperatures. Yet the 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) ≈ 0.513NkTB T = TB (5) when the temperature of the gas is TB . Using the relationship P V = 2U/3 [2], we then obtain U ≈ 0.77NkTB so that the average kinetic energy of a gas particle, when T = TB , is ≈ 0.77kTB . Since there is no experimental evidence that this energy is lost to the surroundings during Bose–Einstein condensation, we here make the assumption that it is retained in the system instead. This means that at any temperature below TB , the particles in the con- densate have an internal energy U0(T ) = N0(T ) × 0.77kTB (6) where N0(T ) = N(1 − (T /TB )3/2 ) is the number of particles in the ground state at that temperature. The total internal energy of the gas below the temperature TB is then U = 3 2 cV T 5/2 + 0.77N 1 − (T /TB )3/2 kTB T < TB (7) From this we see that, at absolute zero, the gas has the residual internal energy U0(0) = 0.77NkTB (8) which is equal to the kinetic energy of the gas at T = TB . This re- sult is in excellent agreement with the estimate of the zero point energy made by Fetter and Walecka [4], using a simple argument based on the Heisenberg uncertainty principle. They show that the mean 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 volume of ∼ V /N, is approximately the same as the mean kinetic energy of a particle at the Bose–Einstein temperature, in the same gas, un- der the same conditions. We can thus identify U0(T ) with the zero point energy of the gas, defined in the usual way as that compo- nent of the internal energy of a system that does not vanish as T → 0. Thus the first two difficulties that are present in the existing theory, 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 /TB )3/2 NkTB /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) = 0.513NkTB /V , which is the same as the pressure exerted by the gas at T = TB . The isothermal compressibility may then be written as 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 ideal classical 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 = TB . 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 taking place at that point (e.g. Ref. [2] chapter 11). Hence the divergences at T = TB can be interpreted as arising due to the onset of Bose– Einstein condensation. We thus find that the final difficulty listed above for the existing theory, is also absent in our analysis. We then obtain a new equation of state for an ideal boson gas at temperatures below TB , to replace Eq. (4). This may be written as PV = cV T 5/2 + 0.513 1 − (T /TB )3/2 NkTB T < TB (11) Taken together with Eq. (2) for temperatures above TB , it expresses the thermodynamic behaviour of an ideal boson gas at all temper- atures. 4. Density maximum in liquid 4 He A further feature of our new theory is the prediction that ex- trema in the pressure and density of an ideal boson gas, will occur at some critical temperature between 0 K and TB . 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 1/2 B (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 using the expression for the number density N V = ζ(3/2) 2πmk h2 3/2 T 3/2 B (13) Eq. (12) simplifies to Tc = 0.308 ζ(3/2) g5/2(1) TB = 0.308 × 2.612 1.341 ≈ 0.60TB (14) In the case of an ideal gas of 4 He, for example, TB ≈ 3.13 K, and a pressure minimum is predicted to occur at ≈ 1.88 K. Furthermore, if we allow the volume of the gas to vary but keep the pressure constant, the density of the gas will pass through an extremum, in this case a maximum value. At that point d(N/V )/dT = 0 and using Eq. (9) again, but now keeping P fixed, we find that a max- imum occurs in N/V at exactly the same temperature Tc given by (11). Hence, for an ideal helium gas, the density should pass through a maximum at a temperature ≈ 1.88 K.
  3. 3. 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 4 He, which is a real boson system with atoms interacting via van der Waals’ forces. Here one finds that, despite the presence of these forces, the same arguments as were used in the ideal gas should broadly apply. Fol- lowing the standard van der Waals’ approach [2], the equation of state above TB can be modified by changing expression (2) to ob- tain the following approximate form for the equation of state of liquid 4 He, Pe = {NkT /V }g5/2(z)/g3/2(z) + PV deW T > TB (15) where the extra term PV deW represents the effect of the van der Waals forces on the gas pressure. At temperatures below TB , the modified Eq. (9) becomes P = cT 5/2 + 1 − (T /TB )3/2 NkTB /3V + PV deW T < TB (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 now has the presence of the term ∂ PV deW /∂T . The van der Waals’ force is not expected to have a large temperature dependence, so the extra term will be small. Neglecting this term to first approxima- tion, the theory predicts that the density maximum in liquid 4 He should occur at a temperature ≈ 1.88 K, which is in very good agreement with the measured value of 2.18 K [5]. The difference between the two may be attributed to the action of the van der Waals’ force. The latter has the effect of increasing the overlap be- tween the wave functions of neighbouring particles, compared to the ideal gas equivalent, thereby raising the value of TB and with it the value of Tc. By including the additional term U0 in the internal energy of the condensate at temperatures below TB , our new theory suc- ceeds in predicting the occurrence of a density maximum in an ideal boson gas, and in giving a very good estimate of the temper- ature at which this phenomenon should occur in liquid 4 He. We have already commented upon this in a general way elsewhere [6]. There we noted that liquid 4 He is a particularly simple liquid, in that its atoms are spherically symmetrical and the interatomic forces are purely van der Waals’ in nature. It is difficult to envisage how a density maximum could occur in such a system other than in the way described above, since such a phenomenon requires the presence of two independent sources of kinetic energy, one of which decreases and one of which increases with temperature change. The point at which the effects of these two mechanisms intersect will then give rise to a maximum in the density of liquid 4 He. 5. Conclusion In conclusion, we have examined the issue of the kinetic energy of particles in the ground state of a boson gas. By assuming that the kinetic energy of these particles, as they drop into the con- densate, is retained in the system, we obtain expressions for the internal energy and pressure of the gas below the temperature TB , that lack the difficulties associated with the corresponding expres- sions in the current theory. In addition, the new analysis predicts that a density maximum will occur in the gas at some critical tem- perature below TB . On applying the theory to the case of liquid 4 He, a value is predicted for this temperature that is in very good agreement 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, New York, 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.

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