Understanding X-Ray Diffraction for Crystal Structure Analysis

1. X-ray diffraction (XRD) X-ray is a form of electromagnetic radiation with a short wave length ranging from 0.01 to 10 nanometers and high energies in range of 100 ev to 100 Kev. Light is also electromagnetic radiation with a wavelength range from 400 to 700 nm. Figure -1- electromagnetic spectrum

2. X-ray diffraction is a technique used for determining the atomic and molecular structure of a crystal , in which the crystalline atomscause a beam of incident X-raysto diffract  into many specific directions. Bragg’s law Consider the two parallel planes of atoms and in Figure 2, which have the same h, k, and l Miller indices and are separated by the inter planar d spacing . Now assume that a parallel, monochromatic, and coherent (in-phase) beamof x-rays of wavelength is incident on these two planes at an angle . Two rays in this beam, labeled 1 and 2, are scattered by atoms P and Q. Constructive interference of the scattered rays and occurs also at an angle to the planes, nλ = dhklsinθ + dhklsinθ nλ= 2 dhklsinθ where, n is the order of reflection (1,2,…) , θ is angle of scattering, d = atom spacing , hkl = miller indices

3. Figure -2- interact of x-ray beam with atoms XRD is used to determine: • 1-Details of phases present in the sample • 2-Crystal Structure • 3-Lattice parameter • 4-Dislocation Density • 5-Residual Stress • 6-Crystal Size (using Scherrer Equation) • 7- hkl parameters ( miller indices )

4. crystal structure A crystalline material is one in which the atoms are situated in a repeating or periodic array over large atomic distances. Crystal consists of planes of atoms separated by d-spacing. Unit cell is a smallest group of atoms arranged in 3-dimensional which have lattice parameters ( lattice constant) represents the length of cell edges ( a,b,c).

5. Miller indices represent the family of planes and crystal direction For cubic structure, d-spacing = a / (h2 + k2 + l2)2 Modulus of elasticity for anisotropic materials at different orientations Metal [100] [110] [111] Aluminum 63.7 72.6 76.1 Copper 66.7 130.3 191. Iron 125.0 210.5 272

6. Specimen preparation : • powder the sample must be distributed uniformly on the sample holder • Bulk materials must be smooth after polishing. Figure -3- Powder sample holder of XRD

7. Phase identification Identification of crystalline substance and crystalline phases in a specimen is achieved by comparing the specimen diffraction spectrum with spectra of known crystalline substances. X-ray diffraction data from a known substance are recorded as a powder diffraction file (PDF). Most PDFs are obtained with Cu (Kα) radiation (wave length = 1.54 A0 )

8. Particles size = 100 nm , crystalline size is 20 nm.

10. Crystalline size determination D=Kλ / (β cos θ). K= 0.9 ( shape factor), D= crystal size, β = fWHM ( full width half max of the peak) in Rad , θ = bragg's angle Rad = (22 *FWHM) / (7 * 180) = FWHM* 0.01746 Residual stress In real life XRD peaks can be affected by stresses inside the material which are the effect of thermal or mechanical processing, or chemical contamination. Such effects can give rise to peak broadening (especially with grain size) and if residual stresses are present, peak shifting.