In crystallography, a lattice plane of a given Bravais lattice 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).[1] 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 that have no common divisors (i.e. are relative prime). 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.[2]
Conversely, planes that are not lattice planes have aperiodic intersections with the lattice called quasicrystals; this is known as a "cut-and-project" construction of a quasicrystal (and is typically also generalized to higher dimensions).[3]
^Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: New York, 1976).
^H., Simon, Steven (2020). The Oxford Solid State Basics. Oxford University Press. ISBN 978-0-19-968077-1. OCLC 1267459045.{{cite book}}: CS1 maint: multiple names: authors list (link)
^J. B. Suck, M. Schreiber, and P. Häussler, eds., Quasicrystals: An Introduction to Structure, Physical Properties, and Applications (Springer: Berlin, 2004).
a latticeplane of a given Bravais lattice is any plane containing at least three noncollinear Bravais lattice points. Equivalently, a latticeplane is...
(the latticeplanes), the distance d between adjacent latticeplanes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by...
In physics, the reciprocal lattice emerges from the Fourier transform of another lattice. The direct lattice or real lattice is a periodic function in...
crystallography for latticeplanes in crystal (Bravais) lattices. In particular, a family of latticeplanes of a given (direct) Bravais lattice is determined...
vector for a plane wave associated with parallel crystal latticeplanes. (Wavefronts of the plane wave are coincident with these latticeplanes.) The equations...
than one plane of atoms in the crystal. The translational invariance of a crystal lattice is described by a set of unit cell, direct lattice basis vectors...
origin without crossing any Bragg plane. Equivalently, this is the Voronoi cell around the origin of the reciprocal lattice. There are also second, third...
The surface along which the lattice points are shared in twinned crystals is called a composition surface or twin plane. Crystallographers classify twinned...
atomic bonds along a line in the lattice. A plane in the lattice is sheared, resulting in 2 oppositely faced half planes or dislocations. These dislocations...
circles on a plane. In one dimension it is packing line segments into a linear universe. In dimensions higher than three, the densest lattice packings of...
and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms...
trigonal) and two lattice systems (hexagonal and rhombohedral). While commonly confused, the trigonal crystal system and the rhombohedral lattice system are...
sink. Dislocation loops are formed if vacancies form clusters on a latticeplane. If these vacancy concentration expand in three dimensions, a void forms...
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of...
an ordered pair of complex numbers that defines a lattice in the complex plane. This type of lattice is the underlying object with which elliptic functions...
In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, K {\displaystyle \scriptstyle \mathbf {K} } , at right...
A lattice system is a set of Bravais lattices. Space groups are classified into crystal systems according to their point groups, and into lattice systems...
used to represent the orientation distribution of crystallographic latticeplanes in crystallography and texture analysis in materials science. Consider...
unit cell of the crystalline phase. Each peak represents a certain latticeplane and can therefore be characterized by a Miller index. If the symmetry...
It contains four sets of parallel planes of points and lines, each plane being a two dimensional kagome lattice. A second expression in three dimensions...
group by the Bravais lattice is a finite group which is one of the 32 possible point groups. A glide plane is a reflection in a plane, followed by a translation...
quasicrystal itself be aperiodic, this slice must avoid any latticeplane of the higher-dimensional lattice. De Bruijn showed that Penrose tilings can be viewed...
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