## Thermal Ellipsoids and Chi-Squared

In solids, atoms vibrate about their equilibrium positions. At higher temperatures atomic motion increases, but not alyways in the way we expect. For instance, the figure below shows iso-probability contours for atoms in the unit cell of a perovskite crystal with chemical formula CsSnBr$_3$ at 300 degrees Kelvin (about 80 degrees Farenheit). The thermal ellipsoids pictured below enclose 95% of the probability density of atomic motion.

In this post we'll explore atomic displacement distributions in the harmonic approximation, and figure out how to calculate the iso-contours of the distribution.