X-Ray Diffraction Analysis & Crystallite

X-ray Generation & Properties

X-rays are electromagnetic radiation with typical photon energies in the range of 100 eV - 100 keV. For diffraction applications, only short wavelength x-rays (hard x-rays) in the range of a few angstroms to 0.1 angstrom (1 keV - 120 keV) are used. Because the wavelength of x-rays is comparable to the size of atoms, they are ideally suited for probing the structural arrangement of atoms and molecules in a wide range of materials. The energetic x-rays can penetrate deeper into the materials and provide information about the bulk structure.

X-rays are produced generally by either x-ray tubes or synchrotron radiation. In a x-ray tube, which is the primary x-ray source used in laboratory x-ray instruments, x-rays are generated when a focused electron beam accelerated across a high voltage field bombards a stationary or rotating solid target. As electrons collide with atoms in the target and slow down, a continuous spectrum of x-rays are emitted, which are termed Bremsstrahlung radiation. The high energy electrons also eject inner shell electrons in atoms through the ionization process. When a free electron fills the shell, a x-ray photon with energy characteristic of the target material is emitted. Common targets used in x-ray tubes include Cu and Mo, which emit 8 keV and 14 keV x-rays with corresponding wavelengths of 1.54 Å and 0.8 Å, respectively. (The energy E of a x-ray photon and it's wavelength is related by the equation E = hc/λ, where h is Planck's constant and c the speed of light).

Lattice Planes and Bragg's Law

X-rays primarily interact with electrons in atoms. When x-ray photons collide with electrons, some photons from the incident beam will be deflected away from the direction where they original travel, much like billiard balls bouncing off one anther. If the wavelength of these scattered x-rays did not change (meaning that x-ray photons did not lose any energy), the process is called elastic scattering (Thompson Scattering) in that only momentum has been transferred in the scattering process. These are the x-rays that we measure in diffraction experiments, as the scattered x-rays carry information about the electron distribution in materials. On the other hand, In the inelastic scattering process (Compton Scattering), x-rays transfer some of their energy to the electrons and the scattered x-rays will have different wavelength than the incident x-rays. Diffracted waves from different atoms can interfere with each other and the resultant intensity distribution is strongly modulated by this interaction. If the atoms are arranged in a periodic fashion, as in crystals, the diffracted waves will consist of sharp interference maxima (peaks) with the same symmetry as in the distribution of atoms. Measuring the diffraction pattern therefore allows us to deduce the distribution of atoms in a material.

X-rays scattered from the periodic repeating electron density of a perfectly crystalline material give sharp diffraction peaks at angles that satisfy the Bragg relation, whether the crystal consists of atoms, ions, small molecules, or large molecules. Amorphous materials will also diffract X-rays and electron, but the diffraction is a much more diffuse, low frequency halo (the so called “amorphous halo”). Analysis of the diffraction peaks from amorphous materials leads to information about the statistical arrangement of atoms in the neighborhood of another atom.

Figure : X-rays interact with the atoms in a crystal [1].

In polymers, which are never perfectly crystalline, a superposition of both diffuse and sharp scattering occurs. First, the crystals present in polymers are very small in size. This leads to considerable broadening of the peaks as compared with fully crystalline materials. Second, there exists some fraction of noncrystalline region in even the most highly crystalline polymer [2, 3, 4].

Bragg’s law

When X-rays are scattered from a crystal lattice, observed peaks of scattered intensity, which correspond to the angle of incidence, should be equal to angle of scattering while the path length difference is equal to an integer number of wavelengths (see figure 1). Bragg derived Bragg’s law for the distance d between consecutive identical planes of atoms in the crystal:

nλ = 2d sin Ө

Where λ is the x-ray wavelength, Ө is the angle between the x-ray beam and these atomic planes and n corresponds to the order of diffraction.  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 [3, 5].

Crystallite size

It is important to realize that crystallite size can be obtained via a simple approach from peak’s full width at half maximum (FWHM) measurement by using the Debye-Scherrer equation [6].

Where :

• β is FWHM (in radians),
• λ is the x-ray wavelength, and
• Ө is the peak position in degree.

References :

1. http://en.wikipedia.org/wiki/Bragg's_law
2. Sperling, L. H., “An Introduction to Physical Polymer Science”, John Wiley and Sons, Inc., 2001, 3rd Ed.
3. Bradly, R. F., Jr., “Comprehensive Desk Reference of Polymer Characterization and Analysis”, American Chemical Society, Washington, D.C., 2003.   
4. Surman, D., “Percentage Crystallinity Determination by X-Ray Diffraction”,   http://www.kratos.com/XRD/Apps/pcent.html.
5. Alexander, “X-ray Diffraction Methods in Polymer Science”, Wiley, Interscience, New York, 1969.
6. Cullity, B. D., “Elements of X-ray Diffraction.” Prentice Hall, 2001, 3rd Ed.