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.

Question

Determine the energy at which the Boltzmann approximation may be considered valid.

Calculate the energy, in terms of [latex]k T[/latex] and [latex]E_{F}[/latex], at which the difference between the Boltzmann approximation and the Fermi-Dirac function is 5 percent of the Fermi function.

Accepted Answer

We can write

[latex] \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 [/latex]

If we multiply both numerator and denominator by the [latex]1+\exp ()[/latex] function, we have

[latex] \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[/latex]

which becomes

[latex]\exp \left[\frac{-\left(E-E_{F}\right)}{k T}\right]=0.05[/latex]

or

[latex]\left(E-E_{F}\right)=k T \ln \left(\frac{1}{0.05}\right) \approx 3 k T[/latex]

Comment

As seen in this example and in Figure [latex]3.35[/latex], the [latex]E-E_{F} \gg k T[/latex] notation is somewhat misleading. The Maxwell-Boltzmann and Fermi-Dirac functions are within 5 percent of each other when [latex]E-E_{F} \approx 3 k T[/latex].

Question 3.8: Determine the energy at which the Boltzmann approximation ma...

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