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Atomic displacement parameters and anisotropic thermal ellipsoid lengths and angles

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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.

Keywords: Euler angles; anisotropic temperature factor matrix; anisotropic thermal ellipsoid; atomic displacement matrix; metric matrix; mixed matrix

Document Type: Research Article


Publication date: May 1, 2004


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