Past exam of the physics A course of the University of Cambridge / 2021 / Paper 2 / Section A / 4

Table of contents
- Solution 4

Solution

The three primitive lattice translation vectors join a cube corner to the three adjacent face centres. Thus the Bravais lattice is a face-centered cubic lattice, with conventional cubic side

a

.

Write the given primitive lattice translation vectors as

a_{1}, a_{2}, a_{3}

. Their primitive unit cell volume is

Ω = a_{1} \cdot (a_{2} \times a_{3}) = \frac{a ^{3}}{4}, a_{2} \times a_{3} = \frac{a ^{2}}{4} (- 1, 1, 1) .

The reciprocal lattice convention

a_{i} \cdot b_{j} = 2 π δ_{ij}

then gives

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

Each reciprocal lattice vector has units

m^{- 1}

when

a

is in metres. Integer combinations generate a body-centered cubic lattice of conventional cubic side

4 π / a

; the point

(2 π / a) (1, 1, 1)

is its body centre. Directly taking the nine dot products checks the signs and the

2 π

normalization.

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