Laue equations
Encyclopedia
In crystallography
Crystallography
Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

, the Laue equations give three conditions for incident waves to be diffracted
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...

 by a crystal lattice. They are named after physicist Max von Laue (1879 — 1960). They reduce to the Bragg law.

Equations

Take to be the wavevector for the incoming (incident) beam and to be the wavevector for the outgoing (diffracted) beam. is the scattering vector and measures the change between the two wavevectors.

Take to be the primitive vectors
Primitive cell
Used predominantly in geometry, solid state physics, and mineralogy, particularly in describing crystal structure, a primitive cell is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions...

 of the crystal lattice. The three Laue conditions for the scattering vector, or the Laue equations, for integer values of a reflection's reciprocal lattice
Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...

 indices
Miller index
Miller indices form a notation system in crystallography for planes and directions in crystal lattices.In particular, a family of lattice planes is determined by three integers h, k, and ℓ, the Miller indices. They are written , and each index denotes a plane orthogonal to a direction in the...

 (h,k,l) are as follows:


These conditions say that the scattering vector must be oriented in a specific direction in relation to the primitive vectors of the crystal lattice.

Relation to Bragg Law

If    is the reciprocal lattice vector
Reciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...

, we know  . The Laue equations specify  . Whence we have    or  .
From this we get the diffraction condition:

Since : (considering elastic scattering):
.

The diffraction condition    reduces to the Bragg law  .
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