Diffraction

Short facts

• Elastic scattering (no energy loss).
• The wavelength of the radiation should be comparable to the interatomic spacing of the material.

Background

Crystalline, polycrystalline and amorphous materials.

The crystal lattice: unit cell and point lattices. The seven crystal systems. Space groups. Miller indices and Unit vectors. Reciprocal space.

Bragg's law

• Condition: The planes of atoms responsible for a diffraction peak have to behave as a specula mirror, so that the angle of incidence <math>\theta</math> equals the angle of reflection.
• Bragg's law

<math> n\lambda=2dsin\theta </math>

where n is an integer, <math>\lambda</math> is the wavelength of the radiation, d is the spacing of the crystal lattice planes and <math>\theta</math> is is the reflection angle.

• Allowed and forbidden reflections

In the FCC (face-centered cubic) lattice the Miller indices have to be all odd or all even.

In the BCC (body-centeres cubic) lattice the sum of h+k+l must be odd for an allowed reflection.

Reciprocal space

Bragg's law imposes that the angles of diffraction are inversely proportional to the spacing of the lattice planes. It is therefore helpful to introduce a coordinate system where the axes have the dimensions of inverse length (nm^-1). This system of coordinates is referred to as the reciprocal space.

• The limiting sphere and the Ewald's sphere:

The limiting sphere: The values of sin<math>\theta</math> will lie between -1 and 1 and therfore 1/d has to lie between 0 and <math>2/\lambda</math> for a diffraction pattern to be generated. (Bragg's law has to be fulfilled). A sphere of radius 2/<math>\lambda</math> is then drawn, where the diffraction crystal is located at the center of the sphere. This sphere, containing all the possible values of 1/d for a given <math>\lambda</math> is called the limiting sphere.

The Ewald's sphere: A smaller sphere of radius 1/<math>\lambda</math> (1/2 of the limiting sphere), that lies in the limiting spere, which is placed so that it just touches the limiting sphere and also the the position of the crystal (the center). This is the Ewals's sphere, or reflecting sphere.

• The reciprocal lattice

X-ray diffraction

Set-up

The X-ray diffractometer is composed of: a X-ray generator, a diffractometer assembly (controls the alignment of the beam as well as the position and orientation of the specimen and the X-ray detector), a detector assembly, X-ray data collection and processing systems.

• Generation of X-rays: X-rays are generated by acceleration of electrons on a pure metal target in ultra-high vacuum. Så må jeg si jeg ikke helt skjønner hva som skjer: hvilke elektroner dytter vekk hvilke? ++
• Characteristic radiation

Powder diffraction

Polycrystalline samples. The individual grains (single crystals) are randomly orientated in all directions)