p42 015

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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 .


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