18. X-Ray Diffraction Techniques

Session Overview

Modules	Crystalline Materials
Concepts	Braggs’ law, x-ray diffraction of crystals: diffractometry, Laue, and Debye-Scherrer, crystal symmetry and selection rules
Keywords	x-ray diffraction, Braggs’ law, angle of incidence, angle of reflection, constructive interference, destructive interference, crest, trough, amplitude, wavelength, phase, monochromatic, coherent light, incoherent light, order of reflection, index of refraction, collimator, diffraction peak, rotational symmetry, Laue diffraction, quasicrystal, translational symmetry, long-range order, x-ray crystallography, Penrose tiles, William Henry Bragg, William Lawrence Bragg, Max von Laue, Roger Penrose, Peter Debye, Peter Scherrer, Dan Shechtman
Chemical Substances	copper (Cu), nickel (Ni), silicon (Si), aluminum-manganese alloy (Al-Mn)
Applications	growth of single-crystal Si, identification of planes and symmetry in crystals, Penrose tiles

Prerequisites

Before starting this session, you should be familiar with the prior topics in this module (Session 15 through Session 17), especially:

Miller indices for crystal directions and planes
SC, FCC, and BCC crystal structures
X-ray production methods and characteristic emission lines (Cu K_α, etc.)

Looking Ahead

X-ray diffraction is a popular technique to discover the structures of organic molecules such as proteins (Session 31) and, most famously, DNA (Session 32), as well as inorganic crystals. It is also used to determine the degree of long-range order and symmetry present in a crystal, or lacking in a glass, which is the topic of the next module (Session 21: Introduction to Glasses).

Learning Objectives

After completing this session, you should be able to:

Sketch the reflection of incident radiation off atomic planes, and derive Braggs’ law for this geometry.
Identify which planes produce x-ray diffraction peaks in FCC and BCC crystals.
Given a graph of x-ray intensity vs. angle, or the 2θ values of the diffraction peaks, determine the crystal structure and lattice constant of the sample.
Explain the difference between x-ray diffractometry and Laue diffraction.
Determine the types of symmetry present in a given tiling pattern.

Reading

Archived Lecture Notes #5 (PDF), Sections 4-6

Book Chapters	Topics
[Saylor] 12.3, “Structures of Simple Binary Compounds.”	Common structures of binary compounds, x-ray diffraction
[JS] 3.7, “X-Ray Diffraction.”	Diffraction, Braggs’ law and reflection rules; single-crystal, polycrystal, and powder diffraction techniques

Lecture Video

View video page

Resources

Lecture Slides (PDF - 5.3MB)

Lecture Summary

X-rays reflect off each atomic plane in a crystal, producing patterns of destructive and constructive interference according to Braggs’ law. One popular method of determining crystal structure, x-ray diffractometry, involves monochromatic x-rays bouncing off a rotating target; the resulting peaks indicate the identity and spacing of the close-packed planes, which are different for FCC and BCC. Another method, Laue diffraction, uses x-rays of multiple wavelengths and a fixed target, producing a pattern reflective of the symmetry present in the crystal structure. The cubic lattices include planes that have 1, 2, 3, or 4-fold rotational symmetry, but quasicrystals displaying 5-fold structures have been observed in experiments on Al-Mn alloys and generated mathematically as Penrose tiles.