Discussion
Consider a 2-D square lattice of lattice constant a. The ratio of the kinetic energy of a free electron at a corner of the first Brillouin zone (\(E_{c}\)) to that of an electron at the midpoint of a side face of the same zone (\(E_{m}\)) is \(E_{c}/E_{m}\) =
Answer: B
The ratio is 2.
The first Brillouin zone of a square lattice is the square \(|k_{x}|, |k_{y}| \le \pi/a\). A free electron has \(E = \hbar^{2}k^{2}/2m\), so the ratio of energies is the ratio of \(k^{2}\):
\[ \begin{aligned} \text{corner: } k^{2} &= \left(\frac{\pi}{a}\right)^{2} + \left(\frac{\pi}{a}\right)^{2} \\ &= \frac{2\pi^{2}}{a^{2}} \\ \text{mid-side: } k^{2} &= \frac{\pi^{2}}{a^{2}} \\ E_{c}/E_{m} &= 2 \end{aligned} \]
\(\sqrt{2}\) is the ratio of the wave vectors, not of the energies, which go as \(k^{2}\). 1/2 inverts the ratio, and 1 would mean the corner is as close to the centre as the mid-side point.
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