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In crystallography and solid state physics, the Laue equations relate incoming waves to outgoing waves in the process of elastic scattering, where the photon energy or light temporal frequency does not change by scattering, by a crystal lattice. They are named after physicist Max von Laue.
The Laue equations can be written as Δ k = k o u t − k i n = G {\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} } as the condition of elastic wave scattering by a crystal lattice, where k i n {\displaystyle \mathbf {k} _{\mathrm {in} }} , k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} , and G {\displaystyle \mathbf {G} } are an incoming wavevector, an outgoing wavevector, and a reciprocal lattice vector for the crystal respectively. Due to elastic scattering | k o u t | 2 = | k i n | 2 {\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}} , three vectors. G {\displaystyle \mathbf {G} } , k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} , and − k i n {\displaystyle -\mathbf {k} _{\mathrm {in} }} , form a rhombus if the equation is satisfied. If the scattering satisfies this equation, all the crystal lattice points scatter the incoming wave toward the scattering direction. If the equation is not satisfied, then for any scattering direction, only some lattice points scatter the incoming wave. It also can be seen as the conservation of momentums as ℏ k o u t = ℏ k i n + ℏ G {\displaystyle \hbar \mathbf {k} _{\mathrm {out} }=\hbar \mathbf {k} _{\mathrm {in} }+\hbar \mathbf {G} } since G {\displaystyle \mathbf {G} } is the wavevector for a plane wave associated with parallel crystal lattice planes.
The equations are equivalent to Bragg's law; the Laue equations are vector equations while Bragg's law is in a form that is easier to solve, but these tell the same content.