Thermal motion of the sulfate group in sodium alum NaAl(SO4)2.12H2O

Abstract

In the structure refinement of Na alum]by,Cromer, Kay & Larson'([Acta Cryst. (1967). 22, 182])the unusual anisotropic thermal parameters of the SO4 group were explained by a coupled rotation about and translation along the threefold axis. S and O(1) are on the threefold axis and 0(2) is in a general position. 0(2) thus follows a spiral path on the surface of a cylinder. In the present work the scattering over the spiral path was calculated by integrating numerically and taking numerical derivatives. It was then evident that a torsional oscillation also exists. Based on the formation developed by Kay & Behrendt, a model that adds a torsional oscillation to the screw motion was calculated. In addition, thermal parameters parallel and perpendicular to the threefold axis were applied to the whole group. The thermal motion is then described by only five parameters instead of the ten used in the earlier anisotropic refinement. A result of this treatment is that the bond lengths can be expressed in terms of equilibrium positions and are as accurate as the data and potentials allow. The resulting bond lengths are S-O(1)= 1.476 (6) and S-O(2)= 1.491 (4) .~. The uncorrected lengths from the anisotropic refinement are 1.461 and 1.459 .~. The Cruickshank rigid body correction increases these bond lengths to 1.475 and 1-483/~ and the Schomaker-Trueblood analysis increases them to 1.476 and 1.500/~,.