The reciprocal lattice of a full-rank point lattice point
lattice
where dot product . If the columns of an invertible
matrix basis of
where the superscript transpose . Indeed,
integer for every iff covolumes satisfy
In solid-state physics, the reciprocal lattice is usually normalized as basis matrix is
See also Brillouin Zone ,
Covolume ,
Point Lattice
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References Barvinok, A. Math 669: Combinatorics, Geometry and Complexity of Integer Points. Lecture notes. Ann Arbor, MI: University of Michigan, 2011.
https://sites.lsa.umich.edu/barvinok/wp-content/uploads/sites/1434/2025/05/latticenotes669.pdf . Lagarias,
J. C.; Lenstra, H. W. Jr.; and Schnorr, C.-P. "Korkin-Zolotarev Bases
and Successive Minima of a Lattice and Its Reciprocal Lattice." Combinatorica 10 ,
333-348, 1990. https://doi.org/10.1007/BF02128669 . Tong,
D. "Band Structure." In Lectures on Applications of Quantum Mechanics ,
pp. 52-56. https://www.damtp.cam.ac.uk/user/tong/aqm/solid2.pdf .
Cite this as:
Weisstein, Eric W. "Reciprocal Lattice."
From MathWorld https://mathworld.wolfram.com/ReciprocalLattice.html
Subject classifications
is the point
lattice
is the dot product. If the columns of an invertible
matrix 
form a basis of 
, so that 
, then
denotes the transpose. Indeed,
is an integer for every 
iff 
. Taking the reciprocal twice gives 
, and the covolumes satisfy
.
For example, the reciprocal of 
, for 
, is 
.
, so its defining condition is 
and its basis matrix is 
(Tong, p. 52).
