 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,

$$f(\vec{r}) = \sum \limits_{\vec{G}}f_{\vec{G}}e^{i\vec{G}\cdot\vec{r}}.$$

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,

$$\vec{q} = \vec{k}'-\vec{k}=\vec{G}.$$

• 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 know the diffraction condition $\Delta\vec{k}=\vec{G}$.