Discussion
For an ideal Fermi gas in three-dimensions, the electron velocity \(v_{F}\) at the Fermi surface is related to electron concentration n as :
Answer: D
The Fermi velocity varies as \(n^{1/3}\).
For a free electron gas in three dimensions the Fermi wave vector is fixed by the electron density:
\[ \begin{aligned} k_{F} &= (3\pi^{2}n)^{1/3} \\ v_{F} &= \frac{\hbar k_{F}}{m} \propto n^{1/3} \end{aligned} \]
The Fermi energy, \(E_{F} = \hbar^{2}k_{F}^{2}/2m\), goes as \(n^{2/3}\), which is the trap in option A. The velocity is the square root of an energy, so it goes as \(n^{1/3}\).
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