Consider a system of N bosons of spin zero, with orbitals a the single particle energies 0 and ε. The chemical potential is μ, and the temperature is τ. Find r such that the thermal average population of the lowest orbital is twice the population of the orbital at ε. Assume N >> 1 and make what approximations are reasonable.
If the atoms in a gas have integral spin (counting the sum of electronic and nuclear spins), they can form a boson condensate when the gas is cooled below the Einstein condensation temperature τE given by (72):

τE = (2πh2/M) (N/2.612V)2/3.

For atoms in the vapor phase the Einstein condensation temperature is very low because the number densities are very low: In (1995) early successful experiments were carried out at Boulder, MIT, and elsewhere. Such experiments, which are extraordinarily complex, mark the exciting forefront of the quantum gas field. A large literature on BEC experiments and theory is on the Web.
One set of experiments (MIT) started with a beam of sodium atoms exiting an oven at 600K at a concentration N/V of 1014 cm–3. What happens next is the result of a number of clever tricks with laser beams directed on one part or another of the beam of atoms. First the atoms are slowed by one laser beam from an exit velocity of 800 m s–1 to about 30 in s–1. This is stow enough for 1010 atoms in be happed within a magneto-optical trap. Further tricks, including evaporation, reduced the temperature of the gas to 2μK, the ultralow temperature TE at which the condensate was formed. The concentration at τE was again 1014 atoms/cm3,
The atoms in be condensed phase ace in the ground orbital aid expand only slowly once released from the trap. The atoms in excited states move relatively rapidly out of their steady-state positions. The positions of the atoms can be recorded as a function of time after release, using a laser beam. The number of atoms in excited orbitals is in good agreement with the τ3 law, (73). With this technique the signature of Bose-Einstein condensation is the sudden appearance of a sharp peak of atoms as the temperature is decreased through τE. The peak comes from light scattered by atoms in the condensate; the wings of the line from light scattered by atoms in excited orbitals,