29. The average energy of the conduction electrons is given by
"
1
Eavg = EN(E)P (E) dE
n
0
where n is the number of free electrons per unit volume, N(E) is the density of states, and P (E) is the
occupation probability. The density of states is proportional to E1/2, so we may write N(E) =CE1/2,
where C is a constant of proportionality. The occupation probability is one for energies below the Fermi
energy and zero for energies above. Thus,
EF
C 2C
5/2
Eavg = E3/2 dE = EF .
n 5n
0
Now
" EF
2C
3/2
n = N(E)P (E) dE = C E1/2 dE = EF .
3
0 0
We substitute this expression into the formula for the average energy and obtain
2C 3 3
5/2
Eavg = EF = EF .
3/2
5
2CEF 5
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