Crystal Structure and Lattices
Crystalline solids are built from a periodic repetition of atoms, and describing that periodicity with lattices, bases, and symmetry is the geometric foundation on which the rest of condensed matter physics is constructed.
Definition
A crystal structure is a periodic arrangement of atoms in space described by a Bravais lattice of translation vectors together with a basis of atoms attached to each lattice point; its symmetry is classified by point and space groups and probed in reciprocal space by diffraction.
Scope
This area covers the geometric description of crystalline order: the Bravais lattice and basis, the seven crystal systems and fourteen Bravais lattices, point and space group symmetry, the reciprocal lattice and Brillouin zones, and the experimental determination of structure through X-ray and neutron diffraction. It establishes the translational symmetry that underlies Bloch's theorem and band theory, and excludes the dynamical (phonon) and electronic responses treated in neighboring areas.
Sub-topics
Core questions
- How do a Bravais lattice and a basis together specify a crystal structure?
- What symmetry operations are allowed in periodic solids, and how do they organize crystals into systems and space groups?
- Why is the reciprocal lattice the natural setting for diffraction and for the electronic and vibrational spectra of solids?
- How do X-ray and neutron diffraction reveal atomic positions through the Bragg and Laue conditions?
Key concepts
- Bravais lattice, basis, and unit cell
- Seven crystal systems and fourteen Bravais lattices
- Point groups, space groups, and crystal symmetry
- Reciprocal lattice and Brillouin zones
- Bragg and Laue diffraction conditions
Clinical relevance
Crystallography underpins materials science, mineralogy, and structural biology; the lattice and reciprocal-lattice formalism developed here is the prerequisite for electronic band theory, phonon dynamics, and the interpretation of nearly every scattering experiment on ordered matter.
History
Bravais classified the fourteen space lattices in 1850; the diffraction of X-rays by crystals discovered by von Laue in 1912 and the simple reflection law formulated by W. L. Bragg in 1913 turned crystallography into a quantitative experimental science and confirmed the atomic lattice picture of solids.
Key figures
- Auguste Bravais
- Max von Laue
- William Lawrence Bragg
Related topics
Seminal works
- ashcroft1976
- kittel2005
- bragg1913
Frequently asked questions
- What is the difference between a lattice and a crystal structure?
- A lattice is the abstract periodic array of points generated by translation vectors; a crystal structure is obtained by attaching a basis of one or more atoms to every lattice point, so the same lattice can host many different structures.
- Why do physicists work in the reciprocal lattice?
- Periodicity in real space becomes a discrete set of points in reciprocal space, where the diffraction conditions, Brillouin zones, and the crystal momentum of electrons and phonons all take their simplest form.