Atomic displacement parameters and anisotropic thermal ellipsoid lengths and angles

Relations between direct and reciprocal basis vectors and the atomic displacement matrix and anisotropic temperature-factor matrix are provided for generic dimension n, introducing a few new symbols. The moments of the probability distribution of the anisotropic thermal ellipsoids are provided as functions of dimension n. For specific dimensions n = 2 and n = 3, new explicit formulas for the semi-axis lengths and rotation angles of the thermal ellipsoids are derived from the atomic displacement matrix, taking into account certain special cases. For dimension n = 3, the Euler angles are defined. The resulting formulas can be used, for example, to display the thermal ellipsoids of atoms or for further analysis of molecular vibrations.