MAS.020UF Introduction to Solid State Physics

Crystal diffraction

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. In a diffraction experiment, parallel waves strike a periodic structure. For a crystal, the waves scatter off the atoms. Most of the time the waves scattered from the different atoms have different phases and this results in destructive interference. Under certain conditions, the waves scattered from all of the atoms add constructively and there is a diffraction peak. The shape of the unit cell can be determined from the angle at which this diffraction peak is observed and the arrangement of the atoms in the unit cell can be deduced from the intensities of the diffraction peaks.

To understand diffraction it is necessary to work with periodic functions in three dimensions. These are often expressed as a Fourier series of the form,

Here the $\vec{G}$'s are the reciprocal lattice vectors of the Bravais lattice. In a diffraction experiment, the incoming wave is described by wave vector $\vec{k}$ and the scattered wave is described by a wave vector $\vec{k}'$. A diffration peak will be observed if the scattering vector $\vec{q}$ equals one of the reciprocal lattice vectors,

Reading
First read the sections on periodic functions in the outline.
Kittel chapter 2: Crystal diffraction or R. Gross und A. Marx: Strukturanalyse mit Beugungsmethoden

For the exam
• You should know that every periodic function can be expressed as a Fourier series,$f(\vec{r}) = \sum_{\vec{G}}f_{\vec{G}}\exp ( i\vec{G}\cdot\vec{r} )$, where the $\vec{G}$'s are the reciprocal lattice vectors.
• 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 diffraction condition $\Delta\vec{k}=\vec{G}$.
• 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.

Resources
CSIC Crystallography website
International Tables for Crystallography: Structure Factor
Advanced Certificate in Powder Diffraction on the Web
Brillouin zones at the University of Cambridge