Laue equations

The Laue condition, according to Max von Laue, is an equivalent to the Bragg condition description of diffraction effects in crystals. It provides information on the occurrence of diffraction peaks at elastic scattering of X-rays, electrons or neutrons in crystals.

To explain the X-ray diffraction, there are two equivalent approaches. In both crystals are viewed as rigid periodic structures of microscopic objects. At the Bragg theory, the atoms are arranged in crystal lattice planes in parallel with a constant distance. At these levels it comes to the specular reflection of radiation. When von Laue theory starts from different assumptions:

• Describe the crystal as a Bravais lattice
• On the lattice sites sit identical microscopic objects that scatter the incident radiation
• Reflections interferes constructively only in the directions for the radiation scattered from the grid points

The Laue condition is: is obtained if and only constructive interference when the change of the wave vector in the scattering process corresponds to a reciprocal lattice vector.

Laue condition goes from pure crystal lattice ( lattice point on a point-like scatterer ) and indicates the direction in which diffraction peaks are observed. The relative intensity of the reflections depends on the structure of the base, the stray capacity of the base and of atoms of the thermal motion of atoms, it is described by the structural factor.

Derivation of the Laue condition

The distance between two scattering centers ( grid points ) is a lattice vector. The wave vector of the incident radiation is he who was of the scattered. This results in the following path difference ( path difference ):

For constructive interference, the path difference must be an integer multiple of the wavelength:

Equating provides:

Assuming elastic scattering of the wave number of the incident and reflected beam is equal to: . For all lattice vectors must apply:

This is exactly the equation for reciprocal lattice vectors:

The Laue condition is thus: is obtained if and only constructive interference when the change of the wave vector in the scattering process corresponds to a reciprocal lattice vector.

To illustrate the Laue condition see Ewald sphere.

Laue equations

Reciprocal lattice vectors can be expressed as a linear combination of the primitive lattice vectors of the reciprocal lattice, here are the Miller indices (usually, but here confusion with wavenumber because of it):

Similarly, the lattice vectors as a linear combination of the primitive lattice vectors can be represented by:

The scalar product of primitive lattice vectors of the local space and the reciprocal space is:

By forming the scalar product of the above Laue condition with the primitive position vectors, we get the three Laue equations:

Alternative formulation of the Laue condition

You can still write in an alternative form, the Laue condition. You squaring the Laue condition and use:

Parts by:

For a given, this is a plane equation in the Hessian normal form. The projection of the direction is constant. A wave vector of the incident radiation satisfies the Laue condition when its tip lies in a Bragg plane. A Bragg plane, the center plane perpendicular to the line connecting the origin and a point in the reciprocal space. This plane equation corresponding to adjacent points in the reciprocal space of the design specification of the Wigner- Seitz cell of the reciprocal lattice (first Brillouin zone).

It follows from the alternative formulation of the Laue condition: you get exactly then constructive interference when the tip of the incident wave vector lies on the edge of a Brillouin zone.

Equivalence of Laue and Bragg condition

Judging from in and out, so we have:

The angle between and is:

Square root provides:

The scalar product between a reciprocal lattice vector and a lattice vector gives:

For a given this is a plane equation for a plane grating, which is perpendicular to this plane. Written as the following linear combination, so the vector is perpendicular to the lattice plane. The interplanar spacing is

With and obtained from the Bragg condition ( n is the order of the reflection ):

Diffraction peak

• Laue: change of the wave vector of the reciprocal lattice vector
• According to Bragg: reflection at lattice planes of the crystal lattice, which is perpendicular to and is their spacing.