Editing Reciprocal lattice (section)

=== 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.