ABSTRACT
To obtain an expression for the Fermi energy εF, deˆned in Section 12.3, consider a system of N fermions of spin 12 and mass m in a container of volume V at T = 0 K. In zero-applied magnetic ˆeld, the mS = ± 12 spin states
are degenerate, and it is convenient to introduce a degeneracy factor in expressions that involve the density of single particle states. e fermions ˆll the lowest energy states up to the energy εF = µ. By equating the number of particles to the number of states in the range 0 to εF, an expression for εF is obtained. In general, for fermions, N n er r r= ∑ = ∑ +−1 1/( ).( )b e m Conversion of the sum to an integral gives
N f=
∞∫2 0 r e e e( ) ( ) ,d (13.1) where ρ(ε) is the density of states and the factor 2 is the spin degeneracy factor. f ee b e m( ) = +−1 1/( )( ) is the Fermi function, which at T = 0 K has the form f(ε) = 1 for e eʺ F and f(ε) = 0 for e e> F. is may be seen in Figure 12.1. At T = 0 K, Equation 13.1 may therefore be written as
N = ∫2 0 r e ee ( ) .dF (13.2)
From Equation 4.14, the density of single particle states for particles in a box is r e p e( ) ( / )( / ) / /= V m4 22 2 3 2 1 2 , and substitution in Equation 13.2 gives
N V m V m
= ⎛
⎝
⎞
⎠
= ⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
e pF =
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
2 3
m N
V
.
