PX284 - N3 - bose-einstein condensation

N = \int_{0}^{\infty} (2 s + 1) \frac{V}{(2 π)^{2}} {(\frac{2 m}{ℏ})}^{3 / 2} \frac{E^{1 / 2}}{\exp (β (E - μ)) - 1} d E

k_{B} T_{C} = \frac{2 π ℏ^{2}}{m} {(\frac{n}{2.612})}^{2 / 3}

this is because at low temperatures, all the bosons want to be in the ground state
it can be shown that the fraction of particles in the ground state:

\frac{n_{0}}{n} = 1 - {(\frac{T}{T_{c}})}^{3 / 2}

below $T_{C}$ , a macroscopic number of particles occupy the same quantum state
this new state is called a bose-einstein condensate (BEC)
by analogy with liquid/gas phase change, but BEC doesn't require interparticle interactions
BEC observed in ultracold atoms