7. (a) Eq. 42-5 gives
"
8 2Ä„m3/2
N(E) = E1/2
h3
for the density of states associated with the conduction electrons of a metal. This can be written
n(E) =CE1/2
where
" "
8 2Ä„m3/2 8 2Ä„(9.109 × 10-31 kg)3/2
C = = =1.062 × 1056 kg3/2/J3 · s3 .
h3 (6.626 × 10-34 J·s)3
1
Now, 1 J = 1 kg·m2/s2 (think of the equation for kinetic energy K = mv2), so 1kg = 1J·s2·m-2.
2
Thus, the units of C can be written (J·s2)3/2·(m-2)3/2·J-3·s-3 =J-3/2·m-3. This means
C =(1.062 × 1056 J-3/2·m-3)(1.602 × 10-19 J/eV)3/2 =6.81 × 1027 m-3·eV-3/2 .
(b) If E =5.00 eV, then
-3/2
n(E) =(6.81 × 1027 m-3·eV )(5.00 eV)1/2 =1.52 × 1028 eV-1·m-3 .