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]

References

  1. ↑ Neil W. Ashcroft and N. David Mermin, Solid State Physics (Harcourt: New York, 1976).
  2. ↑ 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)
  3. ↑ J. B. Suck, M. Schreiber, and P. Häussler, eds., Quasicrystals: An Introduction to Structure, Physical Properties, and Applications (Springer: Berlin, 2004).
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