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).

as the condition of elastic wave scattering by a crystal lattice, where

(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.)

indicating each lattice point is an integer linear combination of the primitive vectors.)

must satisfy, called the Laue equations, are the following: where numbers

, called Miller indices, determines a scattering vector

Hence there are infinitely many scattering vectors that satisfy the Laue equations as there are infinitely many choices of Miller indices

(This is the meaning of the Laue equations as shown below.)

This condition allows a single incident beam to be diffracted in infinitely many directions.

However, the beams corresponding to high Miller indices are very weak and can't be observed.

These equations are enough to find a basis of the reciprocal lattice (since each observed

For an incident plane wave at a single frequency

For simplicity we take waves as scalars here, even though the main case of interest is an electromagnetic field, which is a vector.

We can think these scalar waves as components of vector waves along a certain axis (x, y, or z axis) of the Cartesian coordinate system.

The incident and diffracted waves propagate through space independently, except at points of the lattice

of the crystal, where they resonate with the oscillators, so the phases of these waves must coincide.

By rearranging terms, we get Now, it is enough to check that this condition is satisfied at the primitive vectors

This ensures that if the Laue equations are satisfied, then the incoming and outgoing (diffracted) wave have the same phase at each point of the crystal lattice, so the oscillations of atoms of the crystal, that follows the incoming wave, can at the same time generate the outgoing wave at the same phase of the incoming wave.

(We use the physical, not crystallographer's, definition for reciprocal lattice vectors which gives the factor of

for a crystal in diffraction, and this is the meaning of the Laue equations.

(In other words, the incoming and diffracted waves are at the same (temporal) frequency.

) is an equation for a plane (as the set of all points indicated by

This indicates the plane that is perpendicular to the straight line between the reciprocal lattice origin

is by definition the wavevector of a plane wave in the Fourier series of a spatial function which periodicity follows the crystal lattice (e.g., the function representing the electronic density of the crystal), wavefronts of each plane wave in the Fourier series is perpendicular to the plane wave's wavevector

, and these wavefronts are coincident with parallel crystal lattice planes.

This means that X-rays are seemingly "reflected" off parallel crystal lattice planes perpendicular

with respect to the lattice planes; in the elastic light (typically X-ray)-crystal scattering, parallel crystal lattice planes perpendicular to a reciprocal lattice vector

for the crystal lattice play as parallel mirrors for light which, together with

as the distance between adjacent parallel crystal lattice planes and