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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-3/34b/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 34B / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

Take

a_{1} = (- 1, 1, 1) /2, a_{2} = (1, - 1, 1) /2, a_{3} = (1, 1, - 1) /2.

(1)

Their integer span consists exactly of the cubic points together with the body-centre points, so the BCC set is a Bravais lattice. Using the inverse matrix supplied in the question, reciprocal primitive vectors are

b_{1} = 2 π (0, 1, 1), b_{2} = 2 π (1, 0, 1), b_{3} = 2 π (1, 1, 0) .

(2)

These generate a face-centred cubic lattice; thus the reciprocal lattice of BCC is FCC, with the displayed scale.

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