Bose Condensation

Consider Bose condensation for an arbitrary dispersion law in D dimensions (see Figure 1.1). Assume a relation between energy and momentum of the form Find a relation between D and σ for Bose condensation to occur.

Figure 1.1

SOLUTION

For Bose particles,

(1)

where τ is the temperature in energy units. The total number of particles in a Bose distribution is

(2)

Substituting into the integral gives

(3)

The condition for Bose condensation to occur is that, at some particular temperature, the chemical potential goes to zero. Then the number of particles outside the Bose condensate will be determined by the integral

(4)

This integral should converge since N is a given number. Expanding around x = 0 in order to determine conditions for convergence of the integral yields

(5)

So, this integral diverges at D/σ ≤ 1, and there is no Bose condensation for this region. (For instance, in two dimensions, particles with ordinary dispersion law E = p²/2m would not Bose-condense.) In three dimensions, D/σ = 3/2 > 1, so that Bose condensation does occur.