 Solution

ID: past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2021/section-b/4/c/solution

A site in a primitive cubic lattice has six nearest neighbours, at

\pm a \hat{x}

\pm a \hat{y}

and

\pm a \hat{z}

t

along all six bonds gives

E (k) = E_{0} + 2 t [cos (k_{x} a) + cos (k_{y} a) + cos (k_{z} a)] .

The cosine sum ranges from

- 3

3

within the first Brillouin zone. Therefore

E_{m i n} = E_{0} - 6∣ t ∣, E_{m a x} = E_{0} + 6∣ t ∣, W = E_{m a x} - E_{m i n} = 12∣ t ∣.

For

t < 0

the minimum is at the zone centre; for

t > 0

the maximum is there.

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