Laue equations

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 upon scattering by a crystal lattice. They are named after physicist Max von Laue (1879–1960).

The Laue equations can be written as ${\displaystyle \mathbf {\Delta k} =\mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {G} }$ as the condition of elastic wave scattering by a crystal lattice, where ${\displaystyle \mathbf {\Delta k} }$ is the scattering vector, ${\displaystyle \mathbf {k} _{\mathrm {in} }}$, ${\displaystyle \mathbf {k} _{\mathrm {out} }}$ are incoming and outgoing wave vectors (to the crystal and from the crystal, by scattering), and ${\displaystyle \mathbf {G} }$ is a crystal reciprocal lattice vector. Due to elastic scattering ${\displaystyle |\mathbf {k} _{\mathrm {out} }|^{2}=|\mathbf {k} _{\mathrm {in} }|^{2}}$, three vectors. ${\displaystyle \mathbf {G} }$, ${\displaystyle \mathbf {k} _{\mathrm {out} }}$, and ${\displaystyle -\mathbf {k} _{\mathrm {in} }}$ , 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 (the direction along ${\displaystyle \mathbf {k} _{\mathrm {out} }}$). If the equation is not satisfied, then for any scattering direction, only some lattice points scatter the incoming wave. (This physical interpretation of the equation is based on the assumption that scattering at a lattice point is made in a way that the scattering wave and the incoming wave have the same phase at the point.) It also can be seen as the conservation of momentum as ${\displaystyle \hbar \mathbf {k} _{\mathrm {out} }=\hbar \mathbf {k} _{\mathrm {in} }+\hbar \mathbf {G} }$ since ${\displaystyle \mathbf {G} }$ is the wave vector for a plane wave associated with parallel crystal lattice planes. (Wavefronts of the plane wave are coincident with these 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.