Crystal LatticeEdit
A crystal lattice is the orderly, repeating arrangement of points in a crystalline solid. Each point represents a motif—an atom, a small group of atoms, or a molecule—that is carried through space according to a set of translation vectors. The geometry and symmetry of this arrangement govern a wide range of material properties, from how light diffracts through the solid to how electrons move and how the material responds to stress. In practice, scientists separate the lattice from the motif, describing the full crystal as a lattice with a basis attached to each lattice point.
The crystal structure of a material is described by two complementary notions: the Bravais lattice, which encodes the periodic array of lattice points, and the basis, which specifies what resides at each point. Together, they determine the overall arrangement of atoms in the crystal. This framework allows the abstract periodicity of space to be connected to real materials such as table salt, quartz, and many metals. For a general introduction to the repeating framework, see Bravais lattices and Crystal structure. X-ray diffraction and other scattering techniques exploit the lattice to infer the arrangement of atoms in a sample, linking observed patterns to the underlying lattice geometry X-ray crystallography.
Bravais lattices
There are 14 distinct three‑dimensional Bravais lattices, which can be grouped into seven crystal systems based on symmetry and lattice parameters. The lattice is defined by three translation vectors a1, a2, a3, and the angles between them. In many common materials, the simplest representations are the primitive lattices, such as the simple cubic, and the centered lattices, such as body-centered cubic Body-centered cubic and face-centered cubic Face-centered cubic structures. The seven crystal systems are named as cubic, tetragonal, orthorhombic, monoclinic, triclinic, hexagonal, and rhombohedral (also referred to as trigonal in some conventions). See also Crystal system for related concepts.
In practice, many materials crystallize with a small repeating unit called the unit cell, the smallest region that, when translated through the lattice vectors, reproduces the entire lattice. The choice of unit cell is not unique, but the lattice vectors and the symmetry of the lattice determine the conventional cell used to describe the structure. Examples include simple cubic Simple cubic, body-centered cubic Body-centered cubic, and face-centered cubic Face-centered cubic lattices, each with characteristic packing and coordination numbers that influence properties like density and mechanical strength. Topics such as the lattice parameter and the metric tensor are discussed in detail in sections on unit cells and lattice geometry Unit cell.
Lattice types and structures
Materials can be categorized by the nature of the motif attached to each lattice point.
Ionic crystals: The lattice includes ions arranged to maximize electrostatic stability, as seen in Sodium chloride and related salts. The alternating pattern of cations and anions leads to high melting points and characteristic diffraction signatures, while defects such as vacancies and antisites can influence diffusion and conductivity. See also Ionic crystal.
Covalent networks: In these lattices, atoms are bound by directional covalent bonds forming an extended network. Diamond is the classic example, with a robust, stiff structure and high hardness. Linkages to other covalent materials appear in discussions of Covalent network lattices and related materials Diamond (material).
Metallic lattices: These lattices feature a sea of delocalized electrons that bind a continuous array of positively charged ions. Metals such as copper and aluminum adopt close-packed arrangements in many cases, contributing to properties like electrical conductivity, malleability, and reflectivity. See Metallic lattice.
Molecular crystals: discrete molecules occupy lattice points and interact through weaker forces such as van der Waals interactions. Ice, solid CO2 (dry ice), and various organic crystals fall into this class, with properties that reflect both the molecular geometry and lattice packing.
The choice of lattice type strongly influences material properties, such as electronic band structure in semiconductors, diffusion rates in solids, and phonon spectra that govern heat capacity and thermal transport. See also Crystal structure and Crystal chemistry for broader connections.
Reciprocal lattice and diffraction
The reciprocal lattice is a conceptual construction dual to the real-space lattice, enabling compact descriptions of wave propagation and scattering. Diffraction phenomena arise when waves (such as X-rays, neutrons, or electrons) interact with the periodic lattice, producing constructive interference at angles determined by the geometry of the lattice and the wavelength of the incident radiation. Bragg’s law, nλ = 2d sin θ, ties the observed diffraction angles to the spacing between lattice planes, while the structure factor encodes how the motif at each lattice point scatters. These ideas underpin X-ray crystallography and related methods used to determine atomic positions with high precision Bragg's law Reciprocal lattice.
Defects, dynamics, and properties
Real crystals deviate from perfect periodicity, and such defects play outsized roles in material behavior.
Point defects: vacancies, interstitials, and impurities alter local chemistry and diffusion pathways. They affect electrical conductivity in semiconductors and catalytic activity in supported materials.
Line defects: dislocations enable plastic deformation and influence yield strength; their interaction with the lattice governs mechanical performance.
Grain boundaries: the junctions between crystalline grains introduce misorientation and can dominate diffusion and mechanical properties in polycrystalline materials.
Lattice dynamics, including vibrational modes called phonons, shape thermal properties and electron–phonon interactions. The Debye model and related theories connect lattice vibrations to heat capacity and thermal expansion, while advanced simulations using density functional theory Density functional theory and molecular dynamics help predict how real lattices behave under stress, temperature changes, or irradiation.
Methods of investigation
A suite of experimental and computational tools probes crystal lattices:
X-ray crystallography X-ray crystallography uses diffraction to reveal atomic positions within a crystal.
Electron diffraction and transmission electron microscopy provide structural information at very small scales, including local lattice distortions and defects.
Neutron diffraction is especially useful for locating light atoms (such as hydrogen) and for studying magnetic ordering in certain materials.
Scanning probe techniques, such as atomic force microscopy and scanning tunneling microscopy, visualize surface lattices and local electronic structure.
Computational methods, including density functional theory Density functional theory and ab initio calculations, model lattice energetics, phonons, and defect formation.
Applications and materials
Lattice structure is central to many technologies:
Semiconductors: the lattice in materials like silicon and gallium arsenide sets electronic band structure, carrier mobility, and device performance. See also Semiconductor and Silicon.
Optics and photonics: periodic dielectrics and photonic crystals manipulate light via lattice-induced band gaps and Bragg scattering, with applications in waveguides and sensors. See Photonic crystal.
Structural and functional materials: lattice quality, defects, and grain structure govern strength, toughness, and diffusion-limited processes in metals and ceramics.
Functional materials: perovskites, quartz, and related crystals exhibit useful properties for photovoltaics, nonlinear optics, and sensors; their performance is intimately tied to lattice geometry and stability. See Perovskite and Quartz (crystal).
Catalysis and energy storage: lattice sites and defects influence adsorption energies and ionic mobility, affecting catalytic activity and ion transport in batteries and fuel cells.