Lecture 20 X-Ray Diffraction (XRD) Theory and Analytical Technique

X-Ray Analysis X-rays discovered in 1895 Fundamental to understanding of crystal structure and symmetry Powder diffraction analyses are a simple and inexpensive method for identifying minerals, especially fine-grained minerals

X-ray Crystallography X-ray wavelengths are on the same order of magnitude as atomic spacings. Crystals thus make excellent diffraction gratings Can use the geometry of the x-ray spots to determine geometry of grating (i.e. the crystal) nλ = 2d sin θ

Bragg Diffraction Diffraction from a three dimensional periodic structure such as atoms in a crystal is called Bragg Diffraction. Similar to diffraction though grating. Consequence of interference between waves reflecting from different crystal planes. Constructive interference is given by Bragg's law: Where λ is the wavelength, d is the distance between crystal planes, θ is the angle of the diffracted wave. and n is an integer known as the order of the diffracted beam. nλ = 2d sin θ Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure. Following Bragg's law, each dot (or reflection), in this diffraction pattern forms from the constructive interference of X-rays passing through a crystal. The data can be used to determine the crystal's atomic structure.

X-ray Generation X-rays – High energy*, highly penetrative electromagnetic radiation *E = hc/λ λ(X-rays) = 0.02-100Å (avg. ~1 Å) λ(visible light) = 4000-7200Å X-ray Vacuum Tube Cathode (W)– electron generator Anode (Mo, Cu, Fe, Co, Cr) – electron target, X-ray generator Our instrument uses a copper target

When light hits an electron, the electron jumps to a higher energy level, then drops back to its original, shell, emitting light X-ray Spectra Continuous spectra (white radiation)– range of X-ray wavelengths generated by the absorption (stopping) of electrons by the target Characteristic X-rays – particular wavelengths created by dislodgement of inner shell electrons of the target metal; x-rays generated when outer shell electrons collapse into vacant inner shells K peaks created by collapse from L to K shell; K peaks created by collapse from M to K shell K K X

X-ray Crystallography Methods Single-Crystal: Laue Method Several directions simultaneously fulfill Bragg equations Good for symmetry, but poor for analysis because distorted Fig 7.39 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

X-ray Diffraction (Bragg’s Law) nλ = 2d sinθ Defines the spacing (d) of atomic planes and incident angle (θ) at which X-rays of a particular wavelength will reflect in phase (i.e., diffract) GE+EH = nλ θ’ ≠ nλ GE + EH is the path difference, waves add if equal to nλ

X-ray Crystallography Methods: Single-Crystal: Precession Use motors to move crystal & sensor to satisfy Bragg equations for different planes without distortions Fig 7.40 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

X-ray Crystallography Methods Powder Easiest Infinite orientations at once, so only need to vary q , the angle of the incident beam of x-ray light.

Powder Diffraction Method Requires random orientation of very fine crystals Incident beam of a certain X-ray wavelength will diffract from atomic planes oriented at the appropriate θ angles for the characteristic d spacing Random orientation of crystals will produce more intense diffraction peaks for particular angles that correspond to characteristic atomic planes

Powder Diffraction Plots θ = arcsin (nλ / 2d) λ(Cu) = 1.54Å d - Qtz [101] = 3.342 θ = 13.32° ; 2θ = 26.64° Quartz

Interpreting X-ray data We will use the data obtained to identify the mineral determine the dimensions of the unit cell.