To represent anisotropic thermal motion, a total of six parameters per atom are necessary. Three of these parameters provide the orientations of the principal axes of the ellipsoid produced by anisotropic thermal motion; the other three parameters represent the magnitudes of displacement along the ellipsoid axes. One of the common representations of these anisotropic thermal parameters is:

where h, k and l are Miller indices; a*, b* and c* are reciprocal cell lengths; and, the U_ij are the six anisotropic thermal parameters expressed in terms of mean-square amplitudes of vibration.

With thermal effects included, the complete expression for the calculated structure factor is



The equation above includes just an isotropic thermal parameter; if the treatment is anisotropic, there would be six thermal parameters in that second exponential. In order to refine the structure, the coordinates and temperature factors must be adjusted so these calculated F_hkl's match, as closely as possible, the observed F_hkl's derived from the experimentally measured intensities. Therefore, there are n equations (where n is the number of reflections measured) in (9N+4M) unknowns (where N is the number of anisotropic atoms and M is the number of isotropic atoms). There is an additional adjustable parameter - a scale factor - that removes the sample size effect from the experimental intensities (assuming that a large crystal would produce greater intensities than a small crystal while the calculated intensities depend only on the number of electrons in the unit cell). Thus, there are (9N+4M)+1 adjustable parameters.