0198506961.pdf

318 Appendix F: The statistical mechanics of Bose–Einstein condensation

Note that the strength of the interaction between the atoms does not

appear in this treatment—the value ofTCdoes not depend on the scat-

tering length. This shows thatBEC arises from quantum statistics.In

real experiments there must be interactions so that atoms have a finite

collision cross-section, otherwise there would not be any mechanism for

establishing thermal equilibrium and evaporative cooling would not be

possible. (A non-interacting Bose gas has some curious properties.)

F.2.1 Bose–Einstein condensation in a harmonic trap

The volume of the trapped atomic cloud depends on temperature as

V∝T^3 /^2 (from eqn 10.16); hence we find that for a trapped atom the

equation equivalent to eqn F.11 is

N−N 0 ∝T^3. (F.15)

This dependence on the cube ofTarises because the density of states for

particles in a harmonic trap is different to that given in eqn F.10 for a

gas in a box of fixed volume (i.e. an infinite square-well potential). This

affects the way that the states fill up and hence the conditions for BEC.

An argument analogous to that leading to eqn F.14 gives the fraction in

thegroundstateas

N 0

N

=1−

(

T

TC

) 3

. (F.16)

This is a stronger dependence onT/TCthan in a homogeneous gas.

AtT=0. 99 TCthis equation predicts a condensate fraction ofN 0 /N=

0 .03, so that even just belowTCacloudofN∼ 106 trapped atoms gives

1 /N 0 1, and this partly justifies the assumptions made after eqn F.7.

Typically, experiments are carried out at aroundT/TC∼ 0 .5, or below,

where only a fraction (0.5)^3 =0.125 of the atoms remain in the thermal

cloud. This gives a sufficiently pure condensate for most purposes and

further evaporative cooling would cut deeply into the condensate and

(^7) A large condensate has a chemical po- reduceN 0. 7
tential that is considerably greater than
the energy of the harmonic oscillator
ground state; however, this turns out
not to seriously affect results such as
eqn F.16.