Body-centered cubic lattice

A body-centered cubic lattice consists of the points of a cubic lattice together with their body centers. Its reciprocal lattice is face-centered cubic.

For

a_{1} = \frac{a}{2} (1, 1, 1), a_{2} = \frac{a}{2} (1, - 1, 1), a_{3} = a (0, 0, 1),

(1)

a reciprocal basis is

b_{1} = \frac{2 π}{a} (1, 1, 0), b_{2} = \frac{2 π}{a} (1, - 1, 0), b_{3} = \frac{2 π}{a} (- 1, 0, 1) .

(2)

Thus

Λ^{*} = (2 π / a) {(h, k, l) \in Z^{3} : h + k + l even}

and its shortest nonzero vectors have length

22 π / a

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Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 3 / 33B / Solution
Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 34D / Solution

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Body-centered cubic lattice

Reciprocal lattice of the 2023 Cambridge body-centered cubic basis

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