Laticetranslatie

Laticetranslatie is a concept used in crystallography and solid-state physics to denote the translation operation of a lattice by one or more lattice vectors. It describes the symmetry that maps a periodic arrangement of atoms, spins, or other degrees of freedom onto itself.

In mathematical terms, for each primitive lattice vector a, there is a translation operator T_a that acts on the state or field such that T_a ψ(r) = ψ(r + a). The set of all translations forms an abelian group isomorphic to the lattice; they commute with the lattice Hamiltonian in the absence of external fields.

In quantum mechanics, eigenstates of a translationally invariant Hamiltonian can be chosen as Bloch states, which satisfy T_a ψ_k = e^{i k·a} ψ_k. The crystal momentum k is defined modulo reciprocal lattice vectors, leading to the Brillouin zone as the fundamental domain.

Translational symmetry under laticetranslatie constrains energy bands, determines selection rules for transitions, and underpins conservation of crystal momentum. It is essential in modeling electronic structure, phonons, and magnons in periodic solids.

Limitations and extensions: defects, surfaces, or external fields break or modify translational symmetry; in a magnetic field, translations acquire phase factors (projective representations). In finite or disordered systems, the symmetry is approximate rather than exact.