16. Crystallographic Notation & X-Rays

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Session Overview

Modules Crystalline Materials
Concepts crystal coordinate systems, Miller indices, introduction to x-rays, generation of x-rays
Keywords Bravais lattice, crystal system, unit cell, face-centered cubic, simple cubic, body-centered cubic, Miller indices, crystallography, crystallographic notation, lattice constant, close-packing, packing density, lattice point, interplanar spacing, gas discharge tube, x-ray tube, target anode, discovery of x-rays, scintillation screen, characteristic emission lines, Kα, Kβ, Lα, Lβ, William H. Miller, Wilhelm Röntgen
Chemical Substances barium platinum cyanide (BaPt(CN)4), copper (Cu), brass (Cu-Zn), zinc (Zn), wood, steel
Applications x-ray spectroscopy, medical/dental x-rays, quality assurance of welds, airport baggage scans


Before starting this session, you should be familiar with:

  • Basic 3D coordinate geometry and trigonometry, including vectors and planes
  • Photon frequency, wavelength, and energy (Session 3)
  • Atomic absorption and emission of photons (Session 4)
  • Cubic crystal structures (Session 15)

Looking Ahead

Session 17 and Session 18 explain more about the use of x-rays for investigating the structure of crystals and molecules.

Learning Objectives

After completing this session, you should be able to:

  • Calculate key properties of the cubic lattices, such as atoms per unit cell, nearest and second-nearest neighbor distances, packing density, and the relationship between atomic radius r and lattice constant a.
  • Write the Miller indices for any direction, plane, or family of directions or planes, and calculate the distance and angle between any two directions and/or planes.
  • Given a material and a crystal direction or plane, sketch the appropriate crystal structure and indicate the correct direction or plane on the sketch.
  • Explain how x-rays were produced in 1895, and how Röntgen’s experimental observations lead him to conclude that they were a previously unknown form of electromagnetic radiation.
  • Explain how the properties of x-rays produce the observed results in the following applications: dental x-rays; quality assurance of welds; airport baggage scans.
  • Relate the energies of the characteristic emission lines (Kα, Kβ, etc.) for a given element to the electron shell structure of that element.


Archived Lecture Notes #4 (PDF), Section 4

Archived Lecture Notes #5 (PDF), Section 1

Book Chapters Topics
[Saylor] 12.2, "The Arrangement of Atoms in Crystalline Solids." The unit cell; packing of spheres
Buy at Amazon [JS] 3.2, "Metal Structures." Body-centered cubic, face-centered cubic/cubic close-packed, and hexagonal close-packed structures; atomic packing factor; plane stacking
Buy at Amazon [JS] 3.6, "Lattice Positions, Directions, and Planes." Lattice points and translations; lattice directions and planes; Miller indices; families of directions and planes; planar and linear atomic density

Lecture Video

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Lecture Slides (PDF)

Lecture Summary

Miller indices are a standard mathematical notation describing planes in crystals, derived from where the plane intercepts each coordinate axis. In a specific material with a known lattice constant and crystal structure, this allows the calculation of angles and distances between planes and directions of interest. For convenience, crystallographers sometimes refer to families of planes or directions, which all have the same indices but use different origins.

X-rays are well-suited for measuring atomic-level structure because their wavelengths are of the same order as typical lattice constants. Such short wavelengths require high energies, typically created by sending high-voltage electrons into an anode, where they ionize electrons from the lowest energy levels. Electrons from higher energy levels cascade down to replace them, emitting photons with a highly characteristic set of wavelengths, corresponding to the specific energy levels of the anode material. The discovery of x-rays by Wilhelm Röntgen in 1895 heralded the development of many important modern technologies, including medical radiography, security screening, and industrial inspection of metal parts.


Problems (PDF)

Solutions (PDF)

Textbook Problems

Buy at Amazon [JS] Chapter 3, Sample Problems 8-10, 13-19; Practice Problems 11-14, 16-21

For Further Study

Supplemental Readings

Buy at Amazon Thomas, A. M. K. The Invisible Light: 100 Years of Medical Radiology. Cambridge, MA: Wiley-Blackwell, 1995. ISBN: 9780865426276.


William Hallowes Miller

Wilhelm Röntgen1901 Nobel Prize in Physics

Other OCW and OER Content

Content Provider Level Notes
Lattice Planes and Miller Indices DoITPoMS Undergraduate  
Crystal Structure Connexions Undergraduate  


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