X-Ray Diffraction
X-ray diffraction (XRD) is an effective method for determining the crystal structure of materials. It detects crystalline materials having crystal domains greater than 3-5 nm. It is used to characterize bulk crystal structure and chemical phase composition.
Crystalline & Amorphous materials
Materials can be classified as
- Crystalline material: Crystalline material can be single crystal or polycrystalline
- Amorphous material
Crystalline material
Crystalline materials are composed of atoms arranged in a regular ordered pattern in three dimensions. This periodic arrangement is known as crystal structure. It extends over distance much larger than the inter atomic separations. In single crystal this order extends through out the entire volume of the material. There are seven crystal system: cubic, tetragonal, orthorhombic, rhombohedral, hexagonal, monoclinic and triclinic. Different crystal structures are based on framework of one of the 14 Bravais lattice.
Parallel planes of atoms intersecting the unit cell are used to define directions and distances in the crystal. The ‘d spacing' is defined as the distance between adjacent planes. The orientation and interplaner spacing (d) of these lattice planes are defined by three integers h,k,l called Miller Indices. The (hkl) designate a crystal face or family planes throughout a crystal lattice.
Polycrystalline materials consist of many small single crystal regions called grains. Grains are separated by grain boundaries. The grains can have different shape and size and are disoriented with respect to each other.
Amorphous materials: When the atoms are not arranged in a regular periodic manner the material is called amorphous. Such material posses only short range order, distance less than a nanometer.
X-Ray Diffraction:
X-ray is a form of electromagnetic radiation having range of wavelength from 0.01-0.7 nm which is comparable with the spacings between lattice planes in the crystal. Spacing between atoms in metals ranges from 0.2-0.3 nm. When an incident beam of X-rays interacts with the target atom, X-ray photons are scattered in different directions. Scattering is elastic when there is no change in energy between the incident photon and the scattered photon. In inelastic scattering the scattered photon loses energy. These scattered waves may super impose and when the waves are in phase then the interference is constructive and if out of phase then destructive interference occurs. Atoms in crystal planes form a periodic array of coherent scatterers. Diffraction from different planes of atoms produces a diffraction pattern, which contains information about the atomic arrangement within the crystal.
Bragg's law
The X-ray beams incident on a crystalline solid will be diffracted by the crystallographic planes. Bragg's law is a simple model explaining the conditions required for diffraction. It is given as n 
, where 
,
n is an integer and 
For parallel planes of atoms, with a spacing dhkl between the planes, constructive interference occurs only when Bragg's law is satisfied. In diffractometers, the X-ray wavelength is fixed. Consequently, a family of planes produces a diffraction peak only at a specific angle θ. The spacing between diffracting planes of the atoms determinses the peak positions. The peak intensity is determined by the atoms are in the diffracting plane. The Fig. 1 explains the Bragg's law. Two in-phase incident waves, beam 1 and beam 2, are deflected by two crystal planes (Z and Z1). The diffracted waves will be in phase when the Bragg's Law, nλ = 2d sin θ , is satisfied. In order to keep these beams in phase, their path difference (SQ + QT) has to equal one or multiple X-ray wavelengths (nλ) i.e SQ + QT = nλ or SQ + QT = 2PQ sin θ = 2d sin θ = nλ. Hence the path difference depends on the incident angle (θ) and spacing between the parallel crystal planes (d).
Fig. 1. Braggs analysis for X-ray diffraction by crystal planes