Statistical Mechanics - Physics at Oregon State University

90 CHAPTER 5. FERMIONS AND BOSONS
= (2S + 1)
orb
∞
∞
∞
nx=1 ny=1 nz=1
If h(ɛ) is an arbitrary function of the energy we have
h(ɛo) = (2S + 1)
orb
∞
∞
nx=1 ny=1 nz=1
(5.2)
∞
h(ɛ(nx, ny, nz)) (5.3)
Since the orbital energies do not depend on spin the summation over spins
simply gives us a factor 2S + 1.
Integrals are easier.
Next, we transform the summation variables to k and multiply by 1 =
∆kx∆ky∆kz( π
L )−3 . This gives:
orb
h(ɛo) = (2S + 1)( L
π
)3
k
h(ɛ( k))∆kx∆ky∆kz
(5.4)
If L is large enough the sum can be replaced by an integral and we have ( with
V = L 3 ):
1
V
h(ɛo) =
orb
2S + 1
π 3
pos
d 3 k h( 2k2 ) + error (5.5)
2M
where the k-integration runs over all k-vectors with positive components. If we
extend the integral over all of k-space, using the fact that ɛ( k) is symmetric, we
get
1
V
h(ɛo) =
orb
2S + 1
(2π) 3
d 3 k h( 2k2 ) + error (5.6)
2M
This gives, of course, the additional factor 23 . The error is typically of order 1
V
and hence is unimportant in the limit V → ∞. Practical calculations always use
a finite value for V and in that case the magnitude of the error is determined
by the smoothness of the function h on a scale π
L ! This is an important point
to keep in mind.
Grand partition function.
with
The grand energy for a gas of fermions is
Ω(T, µ, V ) = −(2S + 1)kBT
nx,ny,nz
log(Znx,ny,nz(T, µ, V )) (5.7)