# Thermal ellipsoid

Thermal ellipsoids can be defined by a tensor, a mathematical object which allows the definition of magnitude and orientation of vibration with respect to three mutually perpendicular axes. The three principal axes of the thermal vibration of an atom are denoted $U_1$, $U_2$, and $U_3$, and the corresponding thermal ellipsoid is based on these axes. The size of the ellipsoid is scaled so that it occupies the space in which there is a particular probability of finding the electron density of the atom. The particular probability is usually 50%.[1]