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ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2023/paper-2/a9/b/ii/solution

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Past exam of the chemistry A course of the University of Cambridge / 2023 / Paper 2 / A9 / b / ii / Solution by

Codex  0 2026-10-20

Construct Bragg planes by taking the midpoints between the reciprocal origin and the other reciprocal lattice points. In one dimension these are points at

k = mπ / a

, for nonzero integer

m

The first Brillouin zone is the interval bounded by the nearest two such points:

- \frac{π}{a} \leq k \leq \frac{π}{a} .

(1)

The second Brillouin zone consists of the next intervals on both sides:

- \frac{2 π}{a} \leq k \leq - \frac{π}{a} or \frac{π}{a} \leq k \leq \frac{2 π}{a} .

(2)

Boundary points may be assigned to one adjacent zone to avoid double counting. The first zone is reached from the origin without crossing a Bragg point; the second is reached after crossing one. Each zone has total length

2 π / a

. In particular, the two separated pieces of the second zone are not the entire translated first-zone interval centered at

2 π / a

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