reciprocal lattice

(redirected from Dual lattice)

reciprocal lattice

[ri′sip·rə·kəl ′lad·əs]
(crystallography)
A lattice array of points formed by drawing perpendiculars to each plane (hkl) in a crystal lattice through a common point as origin; the distance from each point to the origin is inversely proportional to spacing of the specific lattice planes; the axes of the reciprocal lattice are perpendicular to those of the crystal lattice.
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Geometrically, they extend in the time direction (on the dual lattice) and can thus link with spatial Wilson loops.
The dual lattice of L is [L.sup.*] := {y [member of] span(L) | <x, y> [member of] Z, [for all]x [member of] L}.
the barycentric dual lattice and the barycentric decomposition lattice, as illustrated in Fig.
However, because these forms have different types of orientations: F and A are 'ordinary' differential forms with inner orientation whereas G (as well as *J) is a 'twisted' differential form with outer-orientation [5,14], they are associated with either the primal (F and A) or the dual lattice (G and *J) discussed above.
We denote that dual lattice of [LAMBDA] by [GAMMA] := {v [member of] V: v([lambda]) [member of] Z for all [lambda] [member of] [LAMBDA]}.
We define the dual lattice [[LAMBDA].sup.*] of [LAMBDA] as follows:
Since Sa is a rational elliptic surface with a singular fibre of Kodaira type [I.sub.3] at t = [infinity], [M.sub.[lambda]] is isomorphic to [E.sup.*.sub.6], the dual lattice of the root lattice [E.sub.6] under the assumption (*) that [S.sub.[lambda]] has no other reducible fibres (cf.
The dual lattice [L.sup.[perpendicular to].sub.W] is defined to be [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [W.sup.[perpendicular to]] = 2[pi][([W.sup.-1]).sup.T].
For each n [is greater than] 1, the dual lattice [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (also a subset of [R.sup.n+1] with minimal distance [square root of n/(n + 1)]; see Gruber and Lekkerkerker [1987] and Conway and Sloane [1993]), of [A.sub.n] is regular and is [nearly equals to] the unique (up to [nearly equals to]) n-dimensional regular lattice (with minimal distance 1) with regular inner product equal to -1/n.
Fixing the standard inner product <x, x> on [R.sup.n], we define the dual lattice [Laplace]f L by
SA is introduced to simultaneously reduce the lattice [DELTA](h)and its dual lattice [DELTA]([H.sup.#]), where [H.sup.*#] = [G.sup.H], and G = [([H.sup.*H] [H.sup.*]).sup.-1] [H.sup.*H].
A d-dimensional lattice polytope P [subset] [R.sup.d] is called reflexive if it contains the origin 0 as an interior point and its polar polytope is a lattice polytope in the dual lattice M := Hom(N, Z) [equivalent to] [Z.sup.d].