Lattices

Lattices in two dimensions: detailed discussion

There are five 2D lattice types as given by the crystallographic restriction theorem. Below, the wallpaper group of the lattice is given in parentheses; note that a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. If the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n.

• a rhombic lattice, also called centered rectangular lattice or isosceles triangular lattice (cmm), with evenly spaced rows of evenly spaced points, with the rows alternatingly shifted one half spacing (symmetrically staggered rows):

• a hexagonal lattice or equilateral triangular lattice (p6m)

• a square lattice (p4m):

• a rectangular lattice, also called primitive rectangular lattice (pmm):

• more generally, a parallelogrammic lattice, also called oblique lattice (p2)(with asymmetrically staggered rows):

For the classification of a given lattice, start with one point and take a nearest second point. For the third point, not on the same line, consider its distances to both points. Among the points for which the smaller of these two distances is least, choose a point for which the larger of the two is least. (Not logically equivalent but in the case of lattices giving the same result is just "Choose a point for which the larger of the two is least".)

The five cases correspond to the triangle being equilateral, right isosceles, right, isosceles, and scalene. In a rhombic lattice, the shortest distance may either be a diagonal or a side of the rhombus, i.e., the line segment connecting the first two points may or may not be one of the equal sides of the isosceles triangle. This depends on the smaller angle of the rhombus being less than 60° or between 60° and 90°.

The general case is known as a period lattice. If the vectors p and q generate the lattice, instead of p and q we can also take p and p-q, etc. In general in 2D, we can take a p + b q and c p + d q for integers a,b, c and d such that ad-bc is 1 or -1. This ensures that p and q themselves are integer linear combinations of the other two vectors. Each pair p, q defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental parallelogram.

The fundamental domain of the period lattice.

The vectors p and q can be represented by complex numbers. Up to size and orientation, a pair can be represented by their quotient. Expressed geometrically: if two lattice points are 0 and 1, we consider the position of a third lattice point. Equivalence in the sense of generating the same lattice is represented by the modular group: represents choosing a different third point in the same grid, represents choosing a different side of the triangle as reference side 0-1, which in general implies changing the scaling of the lattice, and rotating it. Each "curved triangle" in the image contains for each 2D lattice shape one complex number, the grey area is a canonical representation, corresponding to the classification above, with 0 and 1 two lattice points that are closest to each other; duplication is avoided by including only half of the boundary. The rhombic lattices are represented by the points on its boundary, with the hexagonal lattice as vertex, and i for the square lattice. The rectangular lattices are at the imaginary axis, and the remaining area represents the parallelogrammetic lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis.

Lattices in three dimensions

The 14 lattice types in 3D are called Bravais lattices. They are characterized by their space group. 3D patterns with translational symmetry of a particular type cannot have more, but may have less symmetry than the lattice itself.

Lattices in complex space

A lattice in Cn is a discrete subgroup of Cn which spans the 2n-dimensional real vector space Cn. For example, the Gaussian integers form a lattice in C.

Every lattice in Rn is a free abelian group of rank n; every lattice in Cn is a free abelian group of rank 2n.

Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikipedia.org/wiki/Lattice_(group)