15. The Fermi-Dirac occupation probability is given by P
FD
= 1/
e
∆E/kT
+ 1
, and the Boltzmann occu-
pation probability is given by P
B
= e
−∆E/kT
. L et f be the fractional difference. Then
f =
P
B
− P
FD
P
B
=
e
−∆E/kT
−
1
e
∆E/kT
+ 1
e
−∆E/kT
.
Using a common denominator and a little algebra yields
f =
e
−∆E/kT
e
−∆E/kT
+ 1
.
The solution for e
−∆E/kT
is
e
−∆E/kT
=
f
1
− f
.
We take the natural logarithm of both sides and solve for T . The result is
T =
∆E
k ln
f
1
− f
.
(a) Letting f equal 0.01, we evaluate the expression for T :
T =
(1.00 eV)(1.60
× 10
−19
J/eV)
(1.38
× 10
−23
J/K) ln
0.010
1
− 0.010
= 2.5 × 10
3
K .
(b) We set f equal to 0.10 and evaluate the expression for T :
T =
(1.00 eV)(1.60
× 10
−19
J/eV)
(1.38
× 10
−23
J/K) ln
0.10
1
− 0.10
= 5.3 × 10
3
K .