Bravais lattice

A three-dimensional Bravais lattice is

Λ = {n_{1} a_{1} + n_{2} a_{2} + n_{3} a_{3} : n_{i} \in Z},

(1)

where the primitive vectors

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

are linearly independent.

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Face-centered cubic lattice

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A face-centered cubic lattice has lattice points at every corner and face center of a conventional cube. One primitive cell has one quarter of the conventional cube's volume.

Lattice point

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A lattice point is a vector

l = n_{1} a_{1} + n_{2} a_{2} + n_{3} a_{3}

whose coefficients in a chosen primitive basis are integers.

Reciprocal lattice ( $Λ^{*}$ )

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The reciprocal lattice is

Λ^{*} = {q : q \cdot l \in 2 π Z for every l \in Λ} .

(1)

Ω = a_{1} \cdot (a_{2} \times a_{3})

, its primitive vectors are

b_{1} = 2 π \frac{a _{2} \times a _{3}}{Ω}, b_{2} = 2 π \frac{a _{3} \times a _{1}}{Ω}, b_{3} = 2 π \frac{a _{1} \times a _{2}}{Ω},

(2)

and satisfy

a_{i} \cdot b_{j} = 2 π δ_{ij}

Wigner-Seitz cell

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The Wigner--Seitz cell of a lattice point consists of points at least as close to it as to every other lattice point. Its faces lie on perpendicular bisectors to neighbouring lattice points.

Reciprocal lattice and Wigner-Seitz cell of the unit triangular lattice

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For direct primitive vectors

(1, 0)

and

(- 1/2, 3 /2)

, reciprocal primitive vectors are

(2 π, \frac{2 π}{3}), (0, \frac{4 π}{3}) .

(1)

The reciprocal Wigner--Seitz cell is a regular hexagon of area

8 π^{2} / 3

and circumradius

4 π /3

Body-centered cubic lattice

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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.

Reciprocal lattice of the 2023 Cambridge body-centered cubic basis

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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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A Bravais lattice is a concept in crystallography that describes a specific arrangement of points in space, which represents the periodic repetition of a motif in three-dimensional space. It is defined by a set of discrete points that are arranged in a pattern that repeats at regular intervals, effectively forming the basis for the structure of a crystalline solid.

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