Gittervektorfelt

Gittervektorfelt is a term used in mathematics and physics to describe a vector field defined on a lattice. It assigns to each lattice point a vector in a fixed vector space, representing quantities such as atomic displacement, velocity, or orientation of an order parameter. Formally, if Λ is a lattice (a discrete set of points generated by integer combinations of basis vectors) and V is a real vector space, a gittervektorfelt is a function F: Λ → V. In crystallography and solid-state physics such fields model how a quantity attached to lattice sites varies across the structure.

Common special cases include uniform fields, where F(p) is the same for all p, and more general displacement fields, where the vector at each site describes the local shift of the atom from its equilibrium position. The spatial variation can be studied through discrete differences, giving rise to notions analogous to gradient, divergence, and curl in the discrete setting. Through interpolation or coarse-graining, a lattice vector field can approximate a continuum vector field defined on the underlying crystal or manifold.

Applications span modeling of phonon displacements, elastic strains, and order-parameter textures in crystals. In numerical simulations, gittervektorfelt concepts appear in finite-difference or finite-element formulations on lattices. The concept also connects to discrete differential geometry, where edges carry vector data consistent with the lattice's symmetry.