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

2. 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

3. 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

4. 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.

5. X-ray Generation
X-rays – High energy*, highly penetrative electromagnetic radiation
*E = hc/λ λ(X-rays) = Å (avg. ~1 Å)
λ(visible light) = Å
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

6. 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

7. 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

8. 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

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

10. 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.

11. 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

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

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