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A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
Unit cell definition using parallelepiped with lengths a, b, c and angles between the sides given by α, β, γ [1]. A lattice constant or lattice parameter is one of the physical dimensions and angles that determine the geometry of the unit cells in a crystal lattice, and is proportional to the distance between atoms in the crystal.
A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in ...
Vectors and planes in a crystal lattice are described by the three-value Miller index notation. This syntax uses the indices h, k, and ℓ as directional parameters. [4] By definition, the syntax (hkℓ) denotes a plane that intercepts the three points a 1 /h, a 2 /k, and a 3 /ℓ, or some multiple thereof. That is, the Miller indices are ...
In the mathematical study of order, a metric lattice L is a lattice that admits a positive valuation: a function v ∈ L → ℝ satisfying, for any a, b ∈ L, [1] + = + and > > (). Relation to other notions
In general terms, ideal lattices are lattices corresponding to ideals in rings of the form [] / for some irreducible polynomial of degree . [1] All of the definitions of ideal lattices from prior work are instances of the following general notion: let be a ring whose additive group is isomorphic to (i.e., it is a free -module of rank), and let be an additive isomorphism mapping to some lattice ...
A modular lattice of order dimension 2. As with all finite 2-dimensional lattices, its Hasse diagram is an st-planar graph.. In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition,
A lattice is positive definite if the norm of all nonzero elements is positive. The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (a i, a j), where the elements a i form a basis for the lattice. An integral lattice is unimodular if its determinant is 1 or −1.