Discussion
The dispersion relation for a simple cubic crystal is given by :
\[ \begin{aligned} E(\vec{k}) = &E_{0} - 2\beta(\cos k_{x}a \\ &+ \cos k_{y}a + \cos k_{z}a), \end{aligned} \]
where β is constant having dimension of energy and α is lattice constant. The width of the energy band is :
Answer: D
The band width is 12β.
Each cosine ranges from -1 to +1, so their sum ranges from -3 to +3:
\[ \begin{aligned} E_{min} &= E_{0} - 2\beta(3) = E_{0} - 6\beta \quad (\vec{k} = 0) \\ E_{max} &= E_{0} + 6\beta \quad (k_{x}, k_{y}, k_{z} = \pm\pi/a) \\ \Delta E &= 12\beta \end{aligned} \]
(The paper names the lattice constant α in the text but writes a in the formula.) This is the tight-binding band of a simple cubic lattice, whose width is 2zβ with z = 6 nearest neighbours. 4β is the width of a one-dimensional chain, 8β that of a square lattice, and 2β is a single hopping energy.
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