A tenfold twin of the CrB structure type

NiZr crystallized from an amorphous matrix or solidified from an undercooled melt exhibits a tenfold twinned microstructure, which is explained by an ideal twin model utilizing special geometric properties of the CrB structure type. The model is unique in several ways: (i) it contains no adjustable parameters other than a scaling factor accounting for the smallest interatomic distance; (ii) it features an irrational shift in the translational part of the twin operation; and (iii) it has many traits commonly observed for quasicrystals, connected to the occurrence of decagonal long-range orientational order, making NiZr the first experimental example of the recently introduced concept of Z-module twinning. It is shown how these remarkable properties of the tenfold twin's structure model are related to one another and founded in number theory as well as in the mathematical theory of aperiodic order.