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8.5 The free-electron gas 227
. These points all lie in the
and n z corresponds to a single-particle state ø n x ,n y ,n z
positive octant of this space. If we divide the octant into unit cubic cells, every
point representing a single-particle state lies at the corner of one of these unit
cells. Accordingly, we may associate a volume of unit size with each single-
particle state. Equation (8.56) may be rewritten in the form
2
8m e a E
2 2 2
n n n
x y z 2
h
which we recognize as the equation in n-space of a sphere with radius R equal
p
2
2
to 8m e a E=h . The number N (E) of single-particle states with energy less
than or equal to E is then the volume of the octant of a sphere of radius R
3=2
2
1 4ð ð 8m e a E 4ðv
3
N (E) R (2m e E) 3=2 (8:57)
8 3 6 h 2 3h 3
The number of single-particle states with energies between E and E dE is
ù(E)dE, where ù(E) is the density of single-particle states and is related to
N (E)by
dN (E) 2ðv 3=2 1=2
ù(E) (2m e ) E (8:58)
dE h 3
According to the Pauli exclusion principle, no more than two electrons, one
spin up, the other spin down, can have the same set of quantum numbers n x , n y ,
n z . At a temperature of absolute zero, two electrons can be in the ground state
2
2
2
2
with energy 3h =8m e a , two in each of the three states with energy 6h =8m e a ,
2
2
two in each of the three states with energy 9h =8m e a , etc. The states with the
lowest energies are ®lled, each with two electrons, until the spherical octant in
n-space is ®lled up to a value E F , which is called the Fermi energy. If there are
N electrons in the free-electron gas, then we have
8ðv 3=2
N 2N (E F ) (2m e E F ) (8:59)
3h 3
or
2=3
h 2 3N
E F (8:60)
8m e ðv
where equation (8.57) has been used. The Fermi energy is dependent on the
density N=v of the free-electron gas, but not on the size of the metallic crystal.
The total energy E tot of the N particles is given by
E F
E tot 2 Eù(E)dE (8:61)
0
where the factor 2 in front of the integral arises because each single-particle
state is doubly occupied. Substitution of equation (8.58) into (8.61) gives
