crystallography Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

, a lattice plane of a given

Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 + n_ ...

is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a lattice plane is a plane whose intersections with the lattice (or any crystalline structure of that lattice) are periodic (i.e. are described by 2d Bravais lattices).Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: New York, 1976). A family of lattice planes is a collection of equally spaced parallel lattice planes that, taken together, intersect all lattice points. Every family of lattice planes can be described by a set of integer

Miller indices Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''� ...

that have no common divisors (i.e. are

relative prime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...

). Conversely, every set of Miller indices without common divisors defines a family of lattice planes. If, on the other hand, the Miller indices are not relative prime, the family of planes defined by them is not a family of lattice planes, because not every plane of the family then intersects lattice points. Conversely, planes that are ''not'' lattice planes have ''aperiodic'' intersections with the lattice called

quasicrystal A quasiperiodic crystal, or quasicrystal, is a structure that is ordered but not periodic. A quasicrystalline pattern can continuously fill all available space, but it lacks translational symmetry. While crystals, according to the classical cr ...

s; this is known as a "cut-and-project" construction of a quasicrystal (and is typically also generalized to higher dimensions).J. B. Suck, M. Schreiber, and P. Häussler, eds., ''Quasicrystals: An Introduction to Structure, Physical Properties, and Applications'' (Springer: Berlin, 2004).

References

Crystallography Geometry {{geometry-stub