Bravais Lattices (CONTINUACION)

Bravais lattices are the underlying structure of a crystal. A 3D Bravais lattice is defined by the set of vectors R

where the vectors a1 are the basis vectors of the Bravais lattice. (Do not confuse with a lattice with a basis.) Every point of a Bravais lattice is equivalent to every other point. In an elemental crystal, it is possible that the elemental ions are located at the vertices of a Bravais lattice. In general, a crystal structure will be a Bravais lattice with a basis. The symmetry group of a Bravais lattice is the group of translations through thelattice vectors together with some discrete rotation group about (any) one of thelattice points. In the problem set (Ashcroft and Mermin, problem 7.6) you will showthat this rotation group can only have 2-fold, 3-fold, 4-fold, and 6-fold rotation axes. There are 5 different types of Bravais lattice in 2D: square, rectangular, hexago-nal, oblique, and body-centered rectangular. There are 14 di erent types of Bravaislattices in 3D. The 3D Bravais lattices are discussed in are described in Ashcroft andMermin, chapter 7 (pp. 115-119). We will content ourselves with listing the Bravaislattices and discussing some important examples.

Bravais lattices can be grouped according to their symmetries. All but one canbe obtained by deforming the cubic lattices to lower the symmetry.

*Cubic symmetry: cubic, FCC, BCC.

*Tetragonal: stretched in one direction, a x a x c; tetragonal, centered tetragonal.

*Orthorhombic: sides of 3 di erent lengths a x b x c, at right angles to eachother; orthorhombic, base-centered, face-centered, body-centered.

*Monoclinic: One face is a parallelogram, the other two are rectangular; mono-clinic, centered monoclinic.

*Triclinic: All faces are parallelograms.

*Trigonal: Each face is an a x a rhombus.

*Hexagonal: 2D hexagonal lattices of side a, stacked directly above one another,with spacing c.

Examples:

*Simple cubic lattice: ai = a xi.

*Body-centered cubic (BCC) lattice: points of a cubic lattice, together with thecenters of the cubes = interpenetrating cubic lattices o set by 1/2 the body-diagonal.

Examples: Ba, Li, Na, Fe, K, Tl

*Face-centered cubic (FCC) lattice: points of a cubic lattice, together with the centers of the sides of the cubes, = interpenetrating cubic lattices o set by 1/2 a face-diagonal.

Examples: Al, Au, Cu, Pb, Pt, Ca, Ce, Ar.

*Hexagonal Lattice: Parallel planes of triangular lattices

Bravais lattices can be broken up into unit cells such that all of space can be recovered by translating a unit cell through all possible lattice vectors. A primitive unit cell is a unit cell of minimal volume. There are many possible choices of primitive unit cells. Given a basis, a1;a2;a3, a simple choice of unit cell is the region:

The volume of this primitive unit cell and, thus, any primitive unit cell is:

An alternate, symmetrical choice is the Wigner-Seitz cell: the set of all pointswhich are closer to the origin than to any other point of the lattice. Examples: Wigner-Seitz for square=square, hexagonal=hexagon (not parallelogram), oblique=distortedhexagon, BCC=octohedron with each vertex cut o to give an extra square face (A+Mp.74).