Octagonal symmetry in low-discrepancy β-manganese

A low-discrepancy cubic variant of beta-Mn is presented exhibiting local octagonal symmetry upon projection along any of the three mutually perpendicular < 100 > axes. Ideal structural parameters are derived to be x(8c) = (2-root 2 / 16 and y(12d) = 1 / (4 root 2) for the P4(1)32 enantiomorph. A comparison of the actual and ideal structure models of beta-Mn is made in terms of the newly devised concept of geometrical discrepancy maps. Two-dimensional maps of both the geometrical star discrepancy D* and the minimal interatomic distance d(min) are calculated over the combined structural parameter range 0 <= x(8c) < 1/8 and 1/8 <= y(12) < 1/4 of generalized beta-Mn type structures, showing that the 'octagonal' variant of beta-Mn is almost optimal in terms of globally minimizing D* while at the same time globally maximizing dmin. Geometrical discrepancy maps combine predictive and discriminatory powers to appear useful within a wide range of structural chemistry studies. (C) 2014 International Union of Crystallography