Institute Applied Materials
Fundamentals of X-ray Diffraction X-Rays are electromagnetic radiation of exactly the same nature as light, but of very much shorter wavelength. The unit of measurement in the X-ray region is the angstrom (Å), equal to 10-10 m and X-rays used in diffraction have wavelengths lying in the range of 0.5-2.5 Å where as visible light is of the order of 6000 Å. X-rays are produced when any electrically charged particle of sufficient kinetic energy decelerates. Electrons are usually used for this purpose, the radiation being produced in an X-ray tube which contains a source of electrons and two metal electrodes. The high voltage maintained across these electrodes draws the electrons to the anode, or target, which they strike with high velocity. X-rays are produced at the point of impact and radiate in all directions. Most of the kinetic energy of the electrons striking the target is converted into heat, less than 1% being transformed into X-rays.
When the rays coming from the target are analyzed, they are found to contain a mixture of different wavelengths, and the variation of intensity with wavelength is found to depend on the tube voltage. The intensity is zero up to a certain wavelength, called the short wavelength limit (λSWL), increases rapidly to a maximum then decreases, with no sharp limit on the long wavelength side.
The smooth curves corresponding to voltages of 20kV or less represent polychromatic, continuous, or white radiation, since it is made up of rays of many wavelengths (similar to white light). White radiation is also called Bremsstrahlung, German for "braking radiation" because it is caused by electron deceleration.
Figure 1 illustrates interference (constructive) between waves scattering from two adjacent rows of atoms in a crystal. The net effect of scattering from a single row is equivalent to partial reflection from a mirror imagined to be aligned with the row. Thus, the angle of "reflection" equals the angle of incidence for each row. Interference then occurs between the beams reflecting off different rows of atoms in the crystal.
When X-rays are scattered from a crystal lattice, peaks of scattered intensity are observed which correspond to the following conditions:
1. The angle of incidence = angle of scattering.
2. The path length difference is equal to an integer number of wavelengths.
By finding the wavelengths () that allow the X-rays diffracting from the lower crystal plane to cover the distance from CB to BD and still be in constructive interference with the upper X-rays, Bragg's Law can be upheld. In equation form this is:
The condition for maximum intensity contained in Bragg's law above allow us to calculate details about the crystal structure, or if the crystal structure is known, to determine the wavelength of the X-rays incident upon the crystal.
During X-ray diffraction, we measure the angle (θ) by the angle between the arms of the diffractometer as they move. The "n" appears because of the periodicity of the wave and can be any positive integer. We are trying to find the spacing between the lattice planes, d, so the only variable left that we need to control in order to use the Bragg equation is the wavelength (λ) of the X-rays. That variable is controlled by the target that we use to create the beam. The following table lists some of the common X-ray wavelengths produced using different targets.
In our laboratory we commonly use Copper or Molybdenum targets, but other available targets include: Chromium, Iron, Cobalt, and Tungsten.
The different wavelengths produced by each target given in the table are present because each represents a different interest to a person analyzing X-ray diffraction data. (Eg. Someone who is focusing on the copper Kα1 peak will need to know where that peak should be. It will also be necessary to be able to eliminate the irrelevant peaks. In this case, the Kα2 and the Kβ peaks.)
With our Brucker-AXS D8 Advance 2 axes diffractometer (fig. 2) we can provide θ,2θ-scans (fig. 3), rocking curves, etc. From the measured scans peak positions and intensities can be obtained as well as widths and FWHMs, phases can be identified, indexed and compared with the ICDD PDF data base, etc. With help of Rietveld analysis crystal structures can be identified. These way phase analysis, crystallography analysis, thin film characterization, powder characterisation, texture and residual stress investigations, etc. can be performed.