Statistical Mechanics - Physics at Oregon State University

114 CHAPTER 5. FERMIONS AND BOSONS
this is impossible! For such a combination of density and temperature we need
the first term in equation (5.133). For large values of the volume , the value of
λ will be close to one, and the value of the second term in equation (5.133) can
be replaced by (2S + 1)G. In the thermodynamic limit we have
or
lim
V →∞
2S + 1
V
λ = 1 −
λ
1 − λ = n − (2S + 1)nQ(T )G (5.136)
2S + 1
V (n − (2S + 1)nQ(T )G)
(5.137)
Since in most experiments the temperature is varied and not the density, it
is customary to define a critical temperature TE, the Einstein temperature, by
n = nQ(TE)(2S + 1)G, or
2 2π n
TE =
MkB (2S + 1)G
2
3
(5.138)
The Einstein temperature for Helium is about 3K. The first term in the formula
for n, equation (5.133), gives the number of particles in the orbital with the
lowest energy. If the temperature is higher than the Einstein temperature,
there are essentially no particles in this orbital. If the temperature is below the
Einstein temperature, the number of particles in this orbital becomes very large.
More precisely, the Einstein temperature is the dividing point between having
or not having a macroscopically large number of particles in the ground state
orbital. In the thermodynamic limit this is the only orbital with a macroscopic
number of particles, but for finite volumes other orbitals can be important, too.
As we have shown before, for a volume of 1 m 3 we need a temperature less than
10 −18 K for all particles to be in the ground state.
The relative number of particles in the ground state orbital, (2S+1)fBE(ɛ111)/N,
for a temperature below the Einstein temperature follows from
(2S + 1) fBE(ɛ111)
V
= n − (2S + 1)λ −3
T G (5.139)
and since the thermal wavelength is proportional to T − 1
2 we can write this as
the fraction of particles in the ground state orbital:
(2S + 1) fBE(ɛ111)
N
3
2 T
= 1 −
TE
(5.140)
This fraction is zero for T > TE. At T = 0, all particles are in the ground state
orbital.
Note that there is some confusion in defining an orbital. If we define an
orbital by the quantum numbers nx, ny, nz only, the total number of particles
in that orbital is given by (2S + 1)fBE(ɛ(nx, ny, nz)). If we define an orbital by
the quantum numbers n plus a quantum number s for spin, the total number
of particles in that orbital is fBE(ɛ(nx, ny, nz, s)). But now we have 2S + 1