Reciprocal lattice
, that are wavevectors k of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice
Reciprocal space (also called k-space) provides a way to visualize the results of the Fourier transform of a spatial function.
It is similar in role to the frequency domain arising from the Fourier transform of a time dependent function; reciprocal space is a space over which the Fourier transform of a spatial function is represented at spatial frequencies or wavevectors of plane waves of the Fourier transform.
In physical applications, such as crystallography, both real and reciprocal space will often each be two or three dimensional.
Reciprocal space comes into play regarding waves, both classical and quantum mechanical.
Because a sinusoidal plane wave with unit amplitude can be written as an oscillatory term
represents any integer, comprise a set of parallel planes, equally spaced by the wavelength
Some lattices may be skew, which means that their primary lines may not necessarily be at right angles.
whose periodicity is compatible with that of an initial direct lattice in real space.
Equivalently, a wavevector is a vertex of the reciprocal lattice if it corresponds to a plane wave in real space whose phase at any given time is the same (actually differs by
There is then a unique plane wave (up to a factor of negative one), whose wavefront through the origin
on the reciprocal lattice does always take this form, this derivation is motivational, rather than rigorous, because it has omitted the proof that no other possibilities exist.)
The Brillouin zone is a primitive cell (more specifically a Wigner–Seitz cell) of the reciprocal lattice, which plays an important role in solid state physics due to Bloch's theorem.
In pure mathematics, the dual space of linear forms and the dual lattice provide more abstract generalizations of reciprocal space and the reciprocal lattice.
follows the periodicity of this lattice, e.g. the function describing the electronic density in an atomic crystal, it is useful to write
Each plane wave in the Fourier series has the same phase (actually can be differed by a multiple of
As shown in the section multi-dimensional Fourier series[broken anchor],
, is itself a Bravais lattice as it is formed by integer combinations of its own primitive translation vectors
of primitive translation vectors for the reciprocal lattice derived in the heuristic approach above and the section multi-dimensional Fourier series[broken anchor].
The cross product formula dominates introductory materials on crystallography.
This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency.
The direction of the reciprocal lattice vector corresponds to the normal to the real space planes.
(cubic, tetragonal, orthorhombic) have primitive translation vectors for the reciprocal lattice,
[3] One path to the reciprocal lattice of an arbitrary collection of atoms comes from the idea of scattered waves in the Fraunhofer (long-distance or lens back-focal-plane) limit as a Huygens-style sum of amplitudes from all points of scattering (in this case from each individual atom).
, where when there are j = 1,m atoms inside the unit cell whose fractional lattice indices are respectively {uj, vj, wj}.
Whether the array of atoms is finite or infinite, one can also imagine an "intensity reciprocal lattice" I[g], which relates to the amplitude lattice F via the usual relation I = F*F where F* is the complex conjugate of F. Since Fourier transformation is reversible, of course, this act of conversion to intensity tosses out "all except 2nd moment" (i.e. the phase) information.
On the down side, scattering calculations using the reciprocal lattice basically consider an incident plane wave.
The first, which generalises directly the reciprocal lattice construction, uses Fourier analysis.
Therefore, L^ is the natural candidate for dual lattice, in a different vector space (of the same dimension).
The other aspect is seen in the presence of a quadratic form Q on V; if it is non-degenerate it allows an identification of the dual space V* of V with V. The relation of V* to V is not intrinsic; it depends on a choice of Haar measure (volume element) on V. But given an identification of the two, which is in any case well-defined up to a scalar, the presence of Q allows one to speak to the dual lattice to L while staying within V. In mathematics, the dual lattice of a given lattice L in an abelian locally compact topological group G is the subgroup L∗ of the dual group of G consisting of all continuous characters that are equal to one at each point of L. In discrete mathematics, a lattice is a locally discrete set of points described by all integral linear combinations of dim = n linearly independent vectors in Rn.
