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Everything moves like a wave and exchanges energy and momentum like a particle. When waves move through a crystal they diffract. Light, sound, neutrons, atoms, and electrons are all diffracted by crystals.

**Reading**

Kittel chapter 2: Crystal diffraction or R. Gross und A. Marx: Strukturanalyse mit
Beugungsmethoden

- You should know that every periodic function can be expressed as a Fourier series,
*f*() = Σ*r**f*exp(_{G}*i*), where the**G·r**'s are the reciprocal lattice vectors.*G* - You should be able to determine the reciprocal lattice vectors of any Bravais lattice.
- You should know that the reciprocal lattice of an orthorhombic lattice (
*a,b,c*) is also an orhtorhombic lattice (2π/*a*,2π/*b*,2π/*c*); the reciprocal lattice of fcc is bcc and the reciprocal lattice of bcc is fcc. - You should be able to construct the first Brillouin zone of any reciprocal lattice.
- You should know the different forms of the diffraction condition given in the table below.
- You should be able to explain how the Bravais lattice and the size of the unit cell can be determined in a diffraction experiment.
- You should know how to calculate structure factors. These are the complex coefficients of the Fourier series.
- You should know how the square of the structure factor can be measured in a diffraction experiment and how this information can be used to determine the basis of the crystal structure. (The basis is the pattern of atoms that are repeated at every Bravais lattice site to create the crystal.)
- You should be able to define: powder diffraction, neutron diffraction, LEED, and Ewald shphere.

Equivalent statements of the diffraction condition | ||

Bragg's law | Bragg-Gleichung | |

Laue condition | Laue-Bedingung | |

Diffraction condition 1 | Diffraktion Bedingung 1 | |

Diffraction condition 2 | Diffraktion Bedingung 2 |

The shape and the dimensions of the unit cell can be deduced from the position of the Bragg reflections; the content of the unit cell, on the other hand, must be determined from the intensities of the reflections.

*Solid State Physics*, Ibach and Lüth

- Beugung am Kristallgitter - Günter Krois und Markus Kurz, 2008
- Cut-out patterns to make your own models of the Brillouin zones
- Zusammenfassung für die Prüfungsvorbereitung
- Punkte hoher Symmetrie des fcc-Gitters
- Body centered cubic real space and reciprocal space lattices.
- Video about x-ray diffraction. You should know more than Bob. Specifically you should know how to determine the Bravais lattice from an x-ray diffraction measurement and for a very good grade you should be able to explain how to determine the locations of the atoms in the basis.

**Resources**

CSIC Crystallography website

PowderCell - a program to visualize crystal structures, calculate the corresponding powder patterns and refine experimental curves

The International Centre for Diffraction Data

Guide for the exercise:
X-ray diffraction within the course 511.121 Praktikum für Fortgeschrittene

International Tables for Crystallography: Structure Factor

Advanced Certificate in Powder Diffraction on the Web

Brillouin zones at the University of Cambridge

- Teaching pamphlets from the International Union of Crystallography