Raumgitter

Raumgitter, in crystallography and solid-state physics, denotes the idealized three-dimensional lattice formed by all points of space that are related by a discrete set of translations. It provides the periodic scaffold on which atoms or motifs are arranged in a crystal.

Mathematically, a Raumgitter in three-dimensional Euclidean space is the set of all integer linear combinations of three non-coplanar primitive vectors a1, a2, a3: L = {n1 a1 + n2 a2 + n3 a3 | n1, n2, n3 ∈ Z}. The lattice describes the positions of lattice points; a basis (or motif) of atoms placed at each lattice point generates the full crystal structure. The choice of primitive vectors is not unique, and different vector sets can describe the same lattice.

In three dimensions there are 14 distinct Bravais lattices, categorized by translation symmetry and lattice centering (primitive, body-centered, face-centered, etc.). The Raumgitter embodies the translational symmetry of a crystal, while the basis accounts for the specific atomic arrangement within each unit cell. The combination determines many physical properties and diffraction behavior.

The concept extends to reciprocal space, where the reciprocal lattice, derived from the Raumgitter, is used to interpret X-ray, neutron, and electron diffraction patterns. Real crystals realize a Raumgitter with a particular basis, yielding the observed crystal structure and its symmetry.

Beyond crystallography, Raumgitter can be viewed as a discrete subgroup of Euclidean space and has analogues in two or higher dimensions. It remains a foundational concept for describing spatial periodicity in physics, chemistry and materials science.