Octagonal symmetry in low-discrepancy β-manganese

A low-discrepancy cubic variant of β-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 and for the P4₁32 enantiomorph. A comparison of the actual and ideal structure models of β-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 and of generalized β-Mn type structures, showing that the `octagonal' variant of β-Mn is almost optimal in terms of globally minimizing D^* while at the same time globally maximizing d_min. Geometrical discrepancy maps combine predictive and discriminatory powers to appear useful within a wide range of structural chemistry studies.