Editing Reciprocal lattice (section)

== Wave-based description ==
[[File:Superstructures in low-energy electron diffraction (LEED).svg|thumb|Adsorbed species on the surface with 1×2 superstructure give rise to additional spots in low-energy electron diffraction (LEED).]]

=== Reciprocal space ===
Reciprocal space (also called {{mvar|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. The domain of the spatial function itself is often referred to as [[spatial domain]] or real space. In physical applications, such as crystallography, both real and reciprocal space will often each be two or three dimensional. Whereas the number of spatial dimensions of these two associated spaces will be the same, the spaces will differ in their [[quantity dimension]], so that when the real space has the dimension length ('''L'''), its reciprocal space will have [[inverse length]], so '''L'''<sup>−1</sup> (the reciprocal of length).

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 {{nowrap begin}}<math>\cos(kx - \omega t + \varphi_0)</math>,{{nowrap end}} with initial [[Phase (waves)|phase]] {{nowrap begin}}<math>\varphi_0</math>,{{nowrap end}} [[angular wavenumber]] <math>k</math> and [[angular frequency]] {{nowrap begin}}<math>\omega</math>,{{nowrap end}} it can be regarded as a function of both <math>k</math> and <math>x</math> (and the time-varying part as a function of both <math>\omega</math> and {{nowrap begin}}<math>t</math>).{{nowrap end}} This complementary role of <math>k</math> and <math>x</math> leads to their visualization within complementary spaces (the real space and the reciprocal space). The spatial periodicity of this wave is defined by its wavelength {{nowrap begin}}<math>\lambda</math>,{{nowrap end}} where {{nowrap begin}}<math>k \lambda = 2\pi</math>;{{nowrap end}} hence the corresponding wavenumber in reciprocal space will be {{nowrap begin}}<math>k = 2\pi / \lambda</math>.{{nowrap end}}

In three dimensions, the corresponding plane wave term becomes {{nowrap begin}}<math>\cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_0)</math>,{{nowrap end}} which simplifies to <math>\cos(\mathbf{k} \cdot \mathbf{r} + \varphi)</math> at a fixed time {{nowrap begin}}<math>t</math>,{{nowrap end}} where <math>\mathbf{r}</math> is the position vector of a point in real space and now <math>\mathbf{k}=2\pi \mathbf{e} / \lambda</math> is the [[wave vector|wavevector]] in the three dimensional reciprocal space. (The magnitude of a wavevector is called wavenumber.) The constant <math>\varphi</math> is the phase of the [[wavefront]] (a plane of a constant phase) through the origin <math>\mathbf{r}=0</math> at time {{nowrap begin}}<math>t</math>,{{nowrap end}} and <math>\mathbf{e}</math> is a unit [[normal vector]] to this wavefront. The wavefronts with phases <math>\varphi + (2\pi)n</math>, where <math>n</math> represents any [[integer]], comprise a set of parallel planes, equally spaced by the wavelength {{nowrap begin}}<math>\lambda</math>.{{nowrap end}}

=== Reciprocal lattice ===
In general, a geometric lattice is an infinite, regular array of vertices (points) in space, which can be modelled vectorially as a [[Bravais lattice]]. Some lattices may be skew, which means that their primary lines may not necessarily be at right angles. In reciprocal space, a reciprocal lattice is defined as the set of [[wave vector|wavevector]]s <math>\mathbf{k}</math> of plane waves in the [[Fourier series]] of any function <math>f(\mathbf{r})</math> 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 <math>(2\pi)n</math> with an integer <math>n</math>) at every direct lattice vertex.

One heuristic approach to constructing the reciprocal lattice in three dimensions is to write the position vector of a vertex of the direct lattice as {{nowrap begin}}<math>\mathbf{R} = n_1\mathbf{a}_1 + n_2\mathbf{a}_2 + n_3\mathbf{a}_3</math>,{{nowrap end}} where the <math>n_i</math> are integers defining the vertex and the <math>\mathbf{a}_i</math> are [[Linear independence|linearly independent]] primitive translation vectors (or shortly called primitive vectors) that are characteristic of the lattice. There is then a unique plane wave (up to a factor of negative one), whose wavefront through the origin {{nowrap begin}}<math>\mathbf{R} = 0</math>{{nowrap end}} contains the direct lattice points at <math>\mathbf{a}_2</math> and {{nowrap begin}}<math>\mathbf{a}_3</math>,{{nowrap end}} and with its adjacent wavefront (whose phase differs by <math>2\pi</math> or <math>-2\pi</math> from the former wavefront passing the origin) passing through {{nowrap begin}}<math>\mathbf{a}_1</math>{{nowrap end}}. Its angular wavevector takes the form {{nowrap begin}}<math>\mathbf{b}_1 = 2\pi \mathbf{e}_1 / \lambda_{1}</math>,{{nowrap end}} where <math>\mathbf{e}_1</math> is the unit vector perpendicular to these two adjacent wavefronts and the wavelength <math>\lambda_1</math> must satisfy {{nowrap begin}}<math>\lambda_1 = \mathbf{a}_1 \cdot \mathbf{e}_1</math>{{nowrap end}}, means that <math>\lambda_1</math> is equal to the distance between the two wavefronts. Hence by construction <math>\mathbf{a}_1 \cdot \mathbf{b}_1 = 2\pi</math> and {{nowrap begin}}<math>\mathbf{a}_2 \cdot \mathbf{b}_1 = \mathbf{a}_3 \cdot \mathbf{b}_1 = 0</math>.{{nowrap end}}

Cycling through the indices in turn, the same method yields three wavevectors <math>\mathbf{b}_j</math> with {{nowrap begin}}<math>\mathbf{a}_i \cdot \mathbf{b}_j = 2\pi \, \delta_{ij}</math>{{nowrap end}}, where the [[Kronecker delta]] <math>\delta_{ij}</math> equals one when <math>i=j</math> and is zero otherwise. The <math>\mathbf{b}_j</math> comprise a set of three primitive wavevectors or three primitive translation vectors for the reciprocal lattice, each of whose vertices takes the form {{nowrap begin}}<math>\mathbf{G} = m_1\mathbf{b}_1 + m_2\mathbf{b}_2 + m_3\mathbf{b}_3</math>,{{nowrap end}} where the <math>m_j</math> are integers. The reciprocal lattice is also a [[Bravais lattice]] as it is formed by integer combinations of the primitive vectors, that are <math>\mathbf{b}_1</math>, <math>\mathbf{b}_2</math>, and <math>\mathbf{b}_3</math> in this case. Simple algebra then shows that, for any plane wave with a wavevector <math>\mathbf{G}</math> on the reciprocal lattice, the total phase shift <math>\mathbf{G} \cdot \mathbf{R}</math> between the origin and any point <math>\mathbf{R}</math> on the direct lattice is a multiple of {{nowrap begin}}<math>2\pi</math>{{nowrap end}} (that can be possibly zero if the multiplier is zero), so the phase of the plane wave with <math>\mathbf{G}</math> will essentially be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevector <math>\mathbf{G}</math> 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 form]]s and the [[dual lattice]] provide more abstract generalizations of reciprocal space and the reciprocal lattice.