Solid State Physics

The Bravais lattice, or space lattice, is an infinite array of points, determined by the

lattice vectors , where such that every is an integer. The 's are the three primitive vectors of the Bravais lattice; in three dimensions they must have a

nonzero product.       There are an infinite number of different choices for

the primitive vectors of a given lattice. For example, ; ; and

will describe the same lattice.   The lattice spacings are the lengths of the shortest possible set of primitive vectors.

All three crystals in Figure 1.1 have the same Bravais lattice. Note that not all symmetric arrays of points are Bravais lattices! For example, Figure 1.2 shows a honeycomb lattice and a choice for its Bravais lattice and basis.  

   Figure 1.2: Example of a regular array of points that is not a Bravais lattice (honeycomb lattice).

Wigner–Seitz cell

A Wigner–Seitz cell is an example of another kind of primitive cell. The primitive unit cell (or simply primitive cell) is a special case of unit cells which has only one lattice point combined and shared by eight other primitive cells. It is the most "primitive" cell one can construct, and it is a parrallelepiped. The general unit cell has an integral number of lattice points. The simple cubic lattice is the only primitive unit cell conventionally. The body centered cubic (BCC) and face centered cubic (FCC) lattices are simply unit cells, not primitive.

The general mathematical concept behind the primitive cell is termed the fundamental domain or the Voronoi cell. The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone.

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Primitive lattice CellAn ideal crystal is constructed by the infinite repetition of identical structural units in space. In the simplest crystals the structural unit is a single atom, as in copper, silver, gold, iron, aluminium, and the alkali metals.

The structure of all crystals can be described in terms of a lattice, with a group of atoms attached to every lattice point. The group of atoms is called the basis; when repeated in space it forms the crystal structure. The basis consists of a primitive cell, containing one single lattice point. Arranging one cell at each lattice point will fill up the entire crystal.  

The Lattice and the Basis

A crystal is a regular, ordered arrangement of atoms over a large scale. The atoms may be of a single type or the repetition of a complex arrangement of many different types of atoms. The crystal can be thought of as consisting of two separate parts: the lattice and the basis. The lattice is an ordered arrangement of points in space, while the basis consists of the simplest arrangement of atoms which is repeated at every point in the lattice to build up the crystal structure. A good analogy, is patterned wallpaper. The basis is like a motif on the wallpaper and the lattice would be periodic pattern of points on which of the motif is repeated. In Figure 1, the lattice points correspond with the centre of the basis, but this does not have to be the case.

The Fourteen Bravais Lattices

The ways in which we can specify the lattice points in space and keep translational symmetry is limited. In 1848, Auguste Bravais demonstrated that there are in fact only fourteen possible point lattices and no more. For his efforts, the term Bravais lattice is often used in place of point lattice.

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SystemNumber of

LatticesLattice Symbol

Restriction on crystal cell angle

Cubic 3P or sc, I or bcc,

F or fcca=b=c

α =β =γ=90°

Tetragonal 2 P, Ia=b≠c

α=β =γ=90°

Orthorhombic 4 P, C, I, Fa≠b≠ c

α=β =γ=90°

Monoclinic 2 F, Ca≠b≠ c

α=β=90 °≠β

Triclinic 1 Pa≠b≠ cα≠β≠γ

Trigonal 1 Ra=b=cα=β =γ

<120° ,≠90°

Hexagonal 1 Pa=b≠c

α =β =90°γ=120°

Table 1. Seven crystal systems make up fourteen Bravais lattice types in three dimensions.

P - Primitive: simple unit cellF - Face-centred: additional point in the centre of each faceI - Body-centred: additional point in the centre of the cellC - Centred: additional point in the centre of each endR - Rhombohedral: Hexagonal class only

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Properties of cubic system:

Corner points =

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Body centred = N

Face centered =

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