Lattice VectorsEdit
I can’t present this from a political perspective, but here is a neutral, encyclopedia-style article on Lattice Vectors.
Lattice vectors are the building blocks used to describe the regular, repeating patterns that underpin crystalline materials and many mathematical constructions. In three-dimensional space, they encode the translational symmetry of a lattice, allowing every point in the lattice to be reached by integer linear combinations of a small set of generating vectors. These vectors, often denoted a1, a2, and a3, span what is called a Bravais lattice in the context of crystallography, and they play a central role in areas as varied as solid-state physics, materials science, and computational modeling lattice Bravais lattice.
A lattice is not just a collection of points; it is a precise mathematical object that couples geometry to symmetry. The lattice vectors capture the periodicity of the structure, while a separate motif, or basis, attached to each lattice point describes the arrangement of atoms or molecules within the unit cell. In a crystal, the position of any lattice point can be written as R = n1 a1 + n2 a2 + n3 a3, where n1, n2, n3 are integers. This distinction between lattice vectors and the motif is fundamental for understanding diffraction, electronic structure, and vibrational properties. See basis for related concepts and crystal structure for how lattice repetition combines with motifs to form real materials.
Definition and notation
In an n-dimensional Euclidean space, a lattice Λ is the set of all integer linear combinations of a set of primitive vectors {a1, a2, ..., an}: Λ = { n1 a1 + n2 a2 + ... + nn an : ni ∈ Z }. In three-dimensional space, a1, a2, and a3 are called the primitive lattice vectors. The parallelepiped spanned by these vectors is the primitive cell, whose volume V is V = |det[a1 a2 a3]|, computed as the absolute value of the determinant of the matrix whose columns are the vectors a1, a2, a3. The lattice is independent of the choice of origin, although the specific coordinates of lattice points depend on where you place the origin.
A convenient way to work with lattice vectors is to collect them into a matrix A, whose columns are a1, a2, a3. Any lattice point has a coordinate vector n = (n1, n2, n3)^T in the lattice basis, and its Cartesian position is R = A n. Conversely, to express a Cartesian vector in lattice coordinates one uses n = A^{-1} R. This matrix formalism is especially useful when converting between different coordinate systems, such as from a crystal basis to a Cartesian basis or vice versa.
Lattice vectors are distinct from the vectors that describe displacements within the motif (the basis). The lattice describes where translation by a lattice vector leaves the system unchanged, while the basis tells you what is associated with each lattice point. See reciprocal lattice for how lattice vectors pair with reciprocal vectors to describe wave phenomena, and see Miller indices for a notation system used to specify lattice planes and directions.
Reciprocal lattice and plane waves
Associated with every lattice Λ is a reciprocal lattice Λ*, consisting of all vectors kb such that exp(i kb · R) = 1 for every R ∈ Λ. If a1, a2, a3 are the primitive lattice vectors, the reciprocal lattice vectors can be taken as b1 = 2π (a2 × a3) / (a1 · (a2 × a3)), b2 = 2π (a3 × a1) / (a1 · (a2 × a3)), b3 = 2π (a1 × a2) / (a1 · (a2 × a3)). Equivalently, if A is the matrix with columns a1, a2, a3, then B = 2π (A^{-1})^T has columns b1, b2, b3. The reciprocal lattice is central to describing diffraction patterns, electronic band structures, and phonon dispersions, since waves with wavevector k that satisfy k · R ∈ 2πZ for all R ∈ Λ are consistent with the periodic structure. Miller indices, which label crystallographic planes, are naturally interpreted in terms of reciprocal-lattice vectors: a plane with normal vector h b1 + k b2 + l b3 corresponds to the plane indices (hkl) in the conventional notation. See diffraction and Miller indices for related topics.
The volume of the primitive cell in reciprocal space is V* = (2π)^3 / V, reflecting the reciprocal relationship between real-space and reciprocal-space descriptions. This duality underpins much of solid-state physics, from X-ray diffraction to the construction of Bloch states in band theory.
Coordinate systems and transformations
Lattice vectors are often expressed in a Cartesian basis or in a basis tied to the crystal’s symmetry. If the Cartesian basis is {ex, ey, ez}, then each primitive vector can be written as ai = Σj A_{ji} ej, where A is the matrix of Cartesian components. Changing the lattice basis corresponds to a linear transformation of coordinates in which the new basis vectors are linear combinations of the old ones. The determinant of the transformation matrix gives the ratio of the volumes of the old and new primitive cells, a quantity that must remain positive for a physical lattice.
In practice, one frequently encounters two common representations: - The primitive cell representation, where {a1, a2, a3} are linearly independent and generate the lattice with the smallest possible volume. - The conventional cell representation, chosen to make the lattice symmetry explicit. The conventional cell may contain more than one lattice point, but its edges are aligned with a high-symmetry orientation of the crystal. Transformations between these representations are standard fare in crystallography and solid-state physics, and they preserve the underlying lattice structure. See crystal structure and unit cell for related concepts.
Common Bravais lattices and two-dimensional examples
A lattice in three dimensions that cannot be reduced to a smaller, equivalent lattice by a change of basis is a Bravais lattice. There are 14 distinct Bravais lattices in 3D, arising from the different ways to arrange primitive vectors to reflect symmetry. In two dimensions, there are five Bravais lattices. Representative examples include: - Simple cubic (3D): a1 = (a, 0, 0), a2 = (0, a, 0), a3 = (0, 0, a). - Body-centered cubic (3D): a2 and a3 chosen to reflect additional lattice points at the cell center. - Face-centered cubic (3D): vectors arranged to reflect face-centered packing while preserving a high degree of symmetry. - Hexagonal (3D) and tetragonal, orthorhombic, and other systems, each with their own primitive or conventional choices. - In 2D, square, rectangular, hexagonal (often called honeycomb in specific realizations), and oblique lattices illustrate the variety of symmetry classes.
These lattices underlie a great diversity of materials, from metals to semiconductors to mineral structures. See Bravais lattice for a detailed taxonomy and crystal structure for how these lattices combine with motifs to form real materials.
Lattice vectors in practice: basis, cells, and motifs
A lattice by itself encodes translational symmetry; the full crystal structure requires attaching a motif to each lattice point. The combination of a lattice Λ with a basis B (the set of atoms or substructures attached to a lattice point) yields the full crystal structure. Altering either the lattice (the choice of a1, a2, a3) or the basis can produce markedly different physical properties, such as electronic band structure, phonon spectra, and mechanical behavior. See basis (crystal) for more on how motifs attach to lattice points, and electronic band structure for a concrete application.
In computational settings, lattice vectors enable the construction of supercells by repeating the primitive cell. Supercells are essential in simulations that impose periodic boundary conditions, allowing the study of defects, phonons, and finite-temperature effects within a finite model. Lattice reduction techniques, such as Niggli reduction or Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction, help find compact, well-conditioned bases for numerical work. See supercell and Niggli reduction for related topics.
Applications and perspectives
Lattice vectors are used to: - Describe crystal directions and planes, enabling interpretation of diffraction patterns and habitus in materials science. See Miller indices. - Build and solve models of electronic structure, such as tight-binding and nearly free electron models, where Bloch’s theorem relies on the periodicity encoded by the lattice vectors. See band structure. - Analyze lattice dynamics and phonons, where the dynamical matrix is constructed from the lattice and the interactions between unit cells. See phonons. - Facilitate computational materials science, where lattices define simulation cells and boundary conditions. See crystallography.