Editing Reciprocal lattice (section)

=== 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}}