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 x(8c) = (2-\sqrt{2})\big/16 and y(12d) = 1\big/(4 \sqrt{2}) 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 dmin are calculated over the combined structural parameter range 0 \leq x(8c) \,\lt\, 1/8 and 1/8 \leq y(12d)\, \lt\, 1/4 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 dmin. Geometrical discrepancy maps combine predictive and discriminatory powers to appear useful within a wide range of structural chemistry studies.

Octagonal symmetry in low-discrepancy β-manganese.

The layer-by-layer growth of Ammann-Beenker graph (a quasi-periodic graph with eightfold symmetry) has been experimentally and theoretically studied. The limiting form for the growth of Ammann-Beenker graph is established in the form of a regular octagon, whose vertices are found explicitly. The lower and upper bounds of this form, which coincide with the growth form, are proven rigorously.

Layer-by-Layer Growth of Ammann-Beenker Graph

Domain structures in rapidly solidified Mn76Fe4Si20 alloys

CCAST WORLD LAB,CTR CONDENSED MATTER & RADIAT PHYS,BEIJING,PEOPLES R CHINA. ACAD SINICA,BEIJING LAB ELECTRON MICROSCOPY,BEIJING 100080,PEOPLES R CHINA.;CAO, W (reprint author), ACAD SINICA,INST METAL RES,ATOM IMAGING SOLIDS LAB,WENHUA RD,SHENYANG 110015,PEOPLES R CHINA

On the microstructure of a rapidly quenched Mn4Si alloy

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On the microstructure of a rapidly quenched Mn4Si alloy