9 Bose–Einsteingases
Thischapterdiscusses the properties ofanidealBose–Einstein (BE)gas, which
without anyinteractions nevertheless shows a remarkable phase transition, the ‘Bose–
Einstein condensation’. This property is relevant to liquid^4 He and to the behaviour
ofgroups of‘coldatoms’. But anotherimportant application ofBE statisticsistothe
‘phoney’ bosongases, photons and phonons.
9 .1 Properties of an ideal Bose–Einstein gas
9 .1.1 The Bose–Einsteindistribution
In Chapter 5 ((5.11) and (5.13)) we derived the form of the distribution function, i.e.
the number of particles per state of energyεin thermal equilibrium. For agas of ideal
bosonsthedistributionis
fffBE(ε)= 1 /[Bexp(ε/kkkBT)− 1 ] (9.1)For clarity the subscript BE will be omitted in the rest of this chapter. The parameterB
istobedeterminedfromthenumbercondition
∑
ni=N,whichcauseditsappearance
in the first place (see again Chapter 5 ). We shall discuss this number condition in
terms ofBfor theboson gas,butitis entirely equivalent to the use ofαorμfrom the
identities
B=exp(−α)=exp(−μ/kkkBT)Before comingto thedetermination ofB,themain taskofthis section, we can observe
some of its properties just from inspection of the distribution (9.1). To be specific, let
us measure the one-particle energiesεusing the groundstate as the zero ofenergy,i.e.
ε 0 =0. Now to makephysicalsense, weknow thatfor allεthe distributionf(ε)must
bepositive. Becauseoftheboson−1inthedenominator, thisrequiresB>1. IfBwere
negative, then atleast the ground-state occupation would be negative! Furthermore,
ifone were to supposeB= 1 ,then theground-state occupation would beinfinite,
97