Gitterbasis

Gitterbasis is a term used in mathematics and crystallography to refer to the basis of a lattice. In mathematics, a lattice in R^n is a discrete subgroup generated by integer linear combinations of a set of linearly independent vectors. The vectors forming this generating set are called the lattice basis. Any lattice L can be written as L = { z1 b1 + z2 b2 + ... + zn bn : z_i ∈ Z }. The number of basis vectors equals the rank of the lattice; when the vectors span the ambient space, the lattice is full rank.

Typically the basis is represented as the columns of a matrix B, with det(B) giving the volume of the fundamental parallelepiped spanned by the basis. The choice of basis is not unique; different bases related by integer unimodular transformations (n×n matrices with determinant ±1 and integer entries) describe the same lattice.

In crystallography, the term has a related but distinct meaning. The lattice refers to the regular array of lattice points in space, while the basis (or motif) is the arrangement of atoms associated with each lattice point. The crystal structure results from the combination of the lattice and the basis. Thus, the Gitterbasis helps determine symmetry properties and physical behavior by defining the repeating unit in space.

Examples include the two-dimensional square lattice with basis vectors (1,0) and (0,1), and the hexagonal lattice with a different admissible pair of vectors. In computational contexts, the lattice basis enables coordinate representations, transformations, and volume calculations essential for simulations and crystallographic analysis.