## 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, K_{α}, K_{β}, L_{α}, L_{β}, 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 |

### Prerequisites

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**(K_{α}, K_{β}, etc.) for a given element to the**electron shell structure**of that element.

## Reading

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 |

[JS] 3.2, "Metal Structures." | Body-centered cubic, face-centered cubic/cubic close-packed, and hexagonal close-packed structures; atomic packing factor; plane stacking |

[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

> Download from iTunes U (MP4 - 194MB)

> Download from Internet Archive (MP4 - 194MB)

### Resources

### 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.

## Homework

### Textbook Problems

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

## For Further Study

### Supplemental Readings

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

### People

Wilhelm Röntgen – 1901 Nobel Prize in Physics

### Other OCW and OER Content

Content | Provider | Level | Notes |
---|---|---|---|

Lattice Planes and Miller Indices | DoITPoMS | Undergraduate | |

Crystal Structure | Connexions | Undergraduate |