Fermi and Boltzmann distributions - Example

1. At what energy (in units of kT) is the Fermi function within 1 % of the Maxwell-Boltzmann distribution function? What is the corresponding probability of occupancy?
   Answer: Replacing the Maxwell-Boltzmann distribution function by 1/x, the Fermi function equals 1/(1+x) and the two differ by 1 % if:
   1/(1+x) - 1/x = 0.01/x or 1/(1+x) - 1/x = -0.01/x
   Since the Fermi function is always smaller than the Boltzmann distriution (since x is always a positive number) one finds that x = 99 and
   E - E_F = kT ln 99 = 4.6 kT
   The corresponding values for the Fermi and Boltzmann functions are 0.01 and 0.0101.