15. The Fermi-Dirac occupation probability is given by PFD =1/ e"E/kT +1 , and the Boltzmann occu-
pation probability is given by PB = e-"E/kT . L et f be the fractional difference. Then
1
-
PB - PFD e-"E/kT
e"E/kT +1
f = = .
PB e-"E/kT
Using a common denominator and a little algebra yields
e-"E/kT
f = .
e-"E/kT +1
The solution for e-"E/kT is
f
e-"E/kT = .
1 - f
We take the natural logarithm of both sides and solve for T . The result is
"E
T = .
f
k ln
1 - f
(a) Letting f equal 0.01, we evaluate the expression for T :
(1.00 eV)(1.60 × 10-19 J/eV)

T = =2.5 × 103 K .
0.010
(1.38 × 10-23 J/K) ln
1 - 0.010
(b) We set f equal to 0.10 and evaluate the expression for T :
(1.00 eV)(1.60 × 10-19 J/eV)

T = =5.3 × 103 K .
0.10
(1.38 × 10-23 J/K) ln
1 - 0.10

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