Determine the energy at which the Boltzmann approximation may be considered valid.
Calculate the energy, in terms of k T and E_{F}, at which the difference between the Boltzmann approximation and the Fermi-Dirac function is 5 percent of the Fermi function.
We can write
\frac{\exp \left[\frac{-\left(E-E_{F}\right)}{k T}\right]-\frac{1}{1+\exp \left(\frac{E-E_{F}}{k T}\right)}}{\frac{1}{1+\exp \left(\frac{E-E_{F}}{k T}\right)}}=0.05
If we multiply both numerator and denominator by the 1+\exp () function, we have
\exp \left[\frac{-\left(E-E_{F}\right)}{k T}\right] \cdot\left\{1+\exp \left[\frac{E-E_{F}}{k T}\right]\right\}-1=0.05
which becomes
\exp \left[\frac{-\left(E-E_{F}\right)}{k T}\right]=0.05
or
\left(E-E_{F}\right)=k T \ln \left(\frac{1}{0.05}\right) \approx 3 k T
Comment
As seen in this example and in Figure 3.35, the E-E_{F} \gg k T notation is somewhat misleading. The Maxwell-Boltzmann and Fermi-Dirac functions are within 5 percent of each other when E-E_{F} \approx 3 k T.
