Bose-Einstein condensation for general dispersion relations-V C Aguilera-Navarro, M de Llano and M A Solís

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Bose-Einstein condensation for general dispersion relations

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1999 Eur. J. Phys. 20 177

(http://iopscience.iop.org/0143-0807/20/3/307)

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Eur. J. Phys.20 (1999) 177182. Printed in the UK PII: S0143-0807(99)97312-9

BoseEinstein condensation forgeneral dispersion relations

V C Aguilera-Navarro, M de Llanoand M A Sols UNICENTRO, 85015-430 Guarapuava, PR, BrazilandInstituto de Fsica Teorica, Universidad Estadual Paulista, Rua Pamplona 145, 01405-900 - SaoPaulo, SP, Brazil Instituto de Investigaciones en Materiales, UNAM, Apdo. Postal 70-360, 04510 M exico, DF,Mexico Instituto de Fsica, UNAM, Apdo. Postal 20-364, 01000 Mexico, DF, Mexico

Received 8 September 1998, in final form 9 January 1999

Abstract. BoseEinstein condensation in an ideal (i.e. interactionless) boson gas can be studiedanalytically, at university-level statistical and solid state physics, in any positive dimensionality(d >0) for identical bosonswith anypositive-exponent (s >0) energymomentum (i.e.dispersion)relation. Explicit formulae with arbitrary d /s are discussed for: the critical temperature (non-zeroonly ifd/s > 1); the condensate fraction; the internal energy; and the constant-volume specificheat (found to possess a jump discontinuity only if d/s > 2). Classical results are recovered atsufficiently high temperatures. Applications to ordinary BoseEinstein condensation, as well asto photons, phonons, ferro- and antiferromagnetic magnons, and (very specially) to Cooper pairsin superconductivity, are mentioned.

1. Introduction

Experimental observations [1] reported in 1995 of BoseEinstein condensation (BEC) in ultra-cold alkali-atom gas clouds, as well as the 1996 Nobel Prize [2] for the discovery of superfluidphases in liquid helium-three, have spurred even greater interest [3] in this standard textbookexample of a phase transition.

In this paper we consider an ideal quantum gas of bosons which are either permanent(i.e. number-conserving as, e.g., 4 He atoms) orephemeral[4] (i.e.non-number-conserving as,e.g., photons), each possessing an excitation energy as a function of the wavenumber k , i.e.the bosonic dispersion relation

k = csks with s >0 (1)

in d >0 dimensions. For ordinary (non-relativistic) bosons of mass m in vacuo with quadraticdispersion relation,s = 2 andcs = h

2/2m. There exists a non-zero absolute temperatureTcbelow which a macroscopic occupation emerges for one quantum state (of infinitely many),only ifd > 2 [5]. (Thed = 2 case, in fact, displays the same [6] smooth, singularity-free,temperature-dependent specific heat for either bosons or fermions). The BoseEinstein (BE)distribution functionnk [e

[k (T)] 1]1 is by definition the average number of bosons ina given state k. When summed over all states it yields the total number of bosons N, each of

0143-0807/99/030177+06$19.50 1999 IOP Publishing Ltd 177


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178 V C Aguilera-Navarro et al

massm, of which, sayN0(T )are in the lowest state k ( 0 as one takes the thermodynamiclimit). Explicitly, at any given absolute temperature T,

N = N0(T )+ k=0

nk N0(T )+ k=0

1

e[k (T )] 1(2)

where 1/ kBT and (T ) 0 is the chemical potential. The latter inequality ensuresa non-negative summand, as by definition it must be. Einstein surmised that for T > Tc,N0(T ) is negligible compared with N; while for T < Tc, N0(T ) is a sizeable fraction ofN. At precisely T = Tc, N0(Tc) 0 and 0, while at T = 0 the last term in (2)vanishes so that N = N0(0) (namely, the absence of any exclusion principle). Note that = 0 forall T such that 0 < T < T c since from (2)N0(T ) = (e

1)1 implies thate = N0(T)/[N0(T )+ 1] < 1, and this quantity approaches 1

over thisentiretemperaturerange sinceN0(T )on cooling grows to a sizeable fraction ofNwhich in turn is macroscopic.

2. Transition temperature

SinceN0(T ), the number of bosons in the lowest energy state (with k = 0), just ceases to be

negligible (compared with N) at and below Tc, then N0(Tc)/N0(0) = 0 leads straightforwardly,after some algebra, from (2) to thegeneral Tc-formula

Tc =cs

kB

s (d/2) (2 )d n

2 d/2 (d/s)gd/s (1)

s/d ns/d (3)

where gd/s (1) are certain dimensionless numbers (see the appendix), () is the familiar

gamma function andn N/Ld thed-dimensional boson number-density. Also using (3), thecondensate fraction then simplifies to

N0(T )

N0(0)= 1 (T/Tc)

d/s T Tc (4)

which is 1 at T = 0 and 0 for T Tc. The reason for these simple, closed-expressionresults are the so-calledBose functions g(z) (see the appendix) to which the summation in (2)reduces. A somewhat different derivation of a similar, though not identical, result is found inthe PC-oriented textbook [8].

These formulae are valid for all d > 0 and s > 0. For s = 2, cs = h2/2mand d = 3

dimensions, equations (3) and (4) become

Tc =2h2n2/3

mkB[ (3/2)]2/3

3.31h2n2/3

mkBand

N0(T )

N0(0)= 1 (T/Tc)

3/2 (5)

since (3/2) 2.612. These are the familiar results for the ordinary BEC in 3D observedrecently [1]. Note also that for 0 < d s, Tc = 0, since (A2) diverges ford /s 1. Thisbehaviour of (A2) implies that BEC doesnotoccur fors-dispersion-relation bosons ford sdimensions, which is consistent with the well known fact that BEC does notoccur for freespace, quadratic-dispersion-relation (i.e. non-relativistic) bosons for dimensions equal to orsmaller than two. However, fors = 1 BECcanoccur for alld >1. In fact a lineardispersionrelation holds ([9, p 33], [10]) for a Cooper pair of electrons moving not in a vacuum but in the

Fermi sea. Such pairs can thus BoseEinstein condense [11], fortuitously, in all dimensionswhere actual superconductors have been found to exist, down to the quasi-one-dimensionalorganics [12] consisting of parallel chains of molecules. Although the creation/annihilationoperators of Cooper pairs do notobey the usual Bose commutation rules [9, p 38], theydosatisfy BE statistics[11] since an indefinitely large number of pairs, each with fixed momentak1and k2, correspond to different relative momenta k (k1 k2)/2 but add vectoriallyto thesametotal (centre-of-mass) momentumK (k1+ k2). Antiferromagneticmagnonsalso haves = 1, while ferromagnetic ones correspond to s = 2 [13, pp 458, 468], but neithercan BE condense as their number at any givenT is indefinite.


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BoseEinstein condensation 179

3. Internal energy

The internal energy U (Ld, T ) of an ideal many-boson system, where each boson has excitationenergyk, can be written as

U (Ld, T ) =k

k nk k

k

e(k ) 1(6)

and eventually leads to a general expression valid for all temperatures

U (Ld, T )

NkBT=

d

s

gd/s +1(z)

gd/s (1)td/s (7)

where t T /Tc. For T > Tc, N0(T ) = 0, so using (3) we obtain the remarkably simplerelation

gd/s (z) =gd/s (1)

td/s t 1. (8)

Thanks to (8), equation (7) simplifies for T Tc to

U (Ld, T )

NkBT=

d

s

gd/s +1(z)

gd/s (z)

T d/s t 1 (9)

where the limiting result follows from the fact (see equation (A2) below) that g(z) T z,

if >1. Equation (9) is a generalization of the classical partition theorem, more commonlyrecalled ford = 3 ands = 2.

ForT Tc, or t 1,z = 1, so equation (7) becomes

U (Ld, T )

NkBT=

d

s

gd/s +1(1)

gd/s (1)td/s Td/s . (10)

Ifd = 3and s = 1 as in a photon gas, this is just the StefanBoltzmann law U(T )/Ld = T4

of black-body radiation, witha constant. In this case, however,Tc = since =0 for

allT, as a consequence of the indefinitenessin the total number of particles.Figure 1 shows the internal energy (in units ofN kBT) as a function of temperature (in

units ofTc as given by (3)) for (d,s) = (3, 2),(2, 1)and (3, 1). Only the last case possessesa slope discontinuity precisely atTc, while the first two cases merely change in curvature asT increases, namely, from concave up to concave down. The asymptotes atU/NkBT =

32

,2 and 3 are just the respective classical (high-temperature) limits (9).

Figure 1. Internal energy U as a function oft = T /Tc for

d/s = 3/2, 2/1 and 3/1.Only the latter case exhibits a slope

discontinuity (dashed circle).


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The pressure can also be determined for any dand s, though we omit details, and leads toa generalization of the familiar relation P V = 2

3Ufor an ideal gas of either bosons or fermions

in the non-relativistic limit and confined within a volume V, namely

P Ld

=

s

dU (11)

which is cited in [7, p 190].

4. Specific heat

The constant-volume specific heat CVfor allTis defined by CV = (U/T )N,V. For T < Tc, = 0 andz = 1, and equation (10) leads to

CV(T ) = NkB(d/s)(d/s+ 1)gd/s +1(1)

gd/s (1)td/s Td/s . (12)

In three dimensions, Debye acoustic phonons (s = 1) correspond to d /s = 3 and hence thefamiliarT3 behaviour; Bloch (ferromagnetic) magnons (s = 2) tod/s = 3

2, [13, pp 124, 482],

and thus the well known T3/2 law. However, forT > Tc a result very different from (12)emerges. After some simple algebra, the specific heat jump (if any) atTc will be

[CV/NkB]Tc [CV(T

c ) CV(T+

c )]/NkB =(d/s)2 gd/s (1)

gd/s 1(1). (13)

Because g(1) = for 1 (see the appendix) there is no jump discontinuity in the specificheat for all d /s 2. The commonest instance of this is the 3D ideal Bose gas (withs = 2)exhibiting merely a cusp in its temperature-dependent specific heat at Tc, i.e. a discontinuityonly in the slope but not in the value ofCV(T ). Also, since g(1) ()if > 1 (see theappendix), ford/s > 1 one has the quantity (to be discussed further)

CV(Tc)

CV(T

c )

=(d/s)2(d/s)

(d/s+ 1)(d/s 1)(d/s + 1)

. (14)

At high temperatures (i.e. the classical regime) occupation in any given state k is expectedto be minute, so from equations (1) and (2) with each summand very small

N T e

k

eCs ks

(15)

ultimately allows one to write

CV/NkBT d/s. (16)

This is the generalized DulongPetit law. Recalling that CV = (U/T )N,V this checkswith (9).

Figure 2 depicts the constant-volume specific heat CV(in units ofN kB) versust T /Tc.

Only for (d,s) = (3, 1) is there a jump discontinuity, while for (2, 1) and (3, 2) the singularityis merely a cusp, all in keeping with the properties of the internal energy already mentionedin connection with figure 1. Figure 3 illustrates how the specific-heat jump discontinuityvanishes for all d /s 2, as evident from (13), and how it rises for d/s > 2. Remarkably,the BCS value of 0.588 (marked on the figure) associated [15] with an ideal gas offermionicexcitations (called bogolons or bogoliubons) is only slightly smaller than the value 0.609marked by the dashed lines and which corresponds [11] to an ideal gas ofbosonicCooper pairswiths = 1 in three dimensions. A bogolon quasiparticle [16] is a linear combination of afermion particle and a fermion hole.


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BoseEinstein condensation 181

Figure 2.Constant-volume specific heat CVas a function

oft= T /Tcfor d/s = 3/2, 2/1, and3/1. Only thelatter

case exhibits a jump discontinuity (dashed vertical line).

Figure 3. Magnitudeof jump discontinuity in thespecific

heat as a function of d/s , the 3D bosonic value of

0.609 [11] being just above the BCS value of 0.588 [15]

corresponding tofermionicexcitations.

5. Conclusions

We have presented a didactic discussion appropriate for university-level physics of the closed,analytical forms assumed by several thermodynamic functions of an ideal gas ofNbosons ina volumeLd, in d >0 dimensions, for bosons each of energy k = cs k

s , withs >0, wherecs is a constant andk the boson wavenumber. BoseEinstein condensation occurs (i.e. with anon-zero transition temperature Tc)only if d/s >1. Moreover, ifd/s >1, Tc n

s/d, wheren N/Ld, and the condensate fraction N0/N = 1 (T/Tc)

d/s , where N0 is the numberof bosons in the lowest (k = 0) single-boson quantum state. The system internal energyU = (d/s)PLd, whereP is the thermodynamic pressure, and for T < Tc,U T

d/s +1. Theconstant-volume specific heat has a jump discontinuity at T

conly ifd/s >2. Finally, at high

temperatures the expected classical limits, such as equipartition and the DulongPetit law,are recovered.

Note added in proof. After this paper was completed, we learned that some of the results reported here previouslyappeared in [17].

Acknowledgments

MAS and ML acknowledge partial support from UNAM-DGAPA-PAPIIT (Mexico)#IN102198 and CONACyT (Mexico) #27828 E.

Appendix. A word about Bose integrals

The volume Vd(R) of a hypersphere of radius R in d 0 dimensions is found to beVd(R) =

d/2Rd/ (1 +d /2) [7, p 504]. For d = 3 this is just 4 R3/3; for d = 2 it isthe area R2 of a circle of radiusR; ford = 1 it is the diameter 2R of a line of radius R;and ford = 0 it is unity. Using this ford > 0 , and since the allowed states of a particle ina box with sides L correspond to the sites in k-space of a simple-cubic lattice with latticespacing 2/L, the summation in (2) over the d-dimensional vector k becomes an integral over


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positive k |k|, which in the thermodynamic limit (where 2/L becomes infinitesimallysmall) is simply

k=0

2 d/2

(d/2)

L

2

d

0+

dk kd1 (A1)

with the first prefactor reducing as it should to 2, 2 and 4 ford= 1, 2 and 3, respectively.The sum in (2) is then an elementary integral expressed in terms of the so-calledBose integrals[7, pp 159, 506]:

g(z) 1

()

0

dxx1

z1ex 1=

l=1

zl

l (A2)

wherez e/kBT is known as the fugacity. Forz = 1 and 1 equation (A2) coincideswith the Riemann zeta-function (), which converges for > 1 and diverges for = 1

when it becomes the celebrated harmonic series g1(1) (1) = 1 + 12

+ 13

+ . Forz = 1and 0<

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Bose-Einstein condensation for general dispersion relations-V C Aguilera-Navarro, M de Llano and M A Solís