CLUSTER APPROACH TO OCTAGONAL QUASICRYSTALS - AN EXAMPLE: MN-SI-AL

S.I. Ben-Abraham

Department of Physics, Ben-Gurion University of the Negev, POB 653, IL-84105 Beer-Sheba, Israel
e-mail: benabr@bgumail.bgu.ac.il

Octagonal quasicrystalline phases closely related to the b-Mn structure have been observed in the Mn-Al-Si, Cr-Ni-Si, V­Ni­Si, Mo-Cr-Ni systems (cf. Huang and Hovmöller 1991, Jiang, Hovmöller and Zou 1995) and interpreted in terms of the octagonal Ammann-Beenker tiling (ABT). It is interesting to apply the cluster approach to octagonal quasicrystals (Jeong and Steinhardt 1994, Gähler and Jeong 1995, Gummelt 1996). Every tile in ABT is covered by at least one and at most three decorated elementary octagonal patches (OP). Maximizing the density of OPís (to minimize energy) and respecting the vertex atlas one can certainly construct ABT, but it is not known whether this is the unique result. Gähler (1997) convincingly argued that the negative is extremely improbable. Ignoring the vertex atlas one arrives at a wrong vertex which trivially forces a periodic tiling. Some experimentally observed clusters can be adequately modeled by plain arrowed OPís, while others are better represented by oriented OP's with additional inner vertex decoration. Here I apply the cluster approach to a structure model put forward by Jiang, Hovmöller and Zou (1995) for an octagonal phase observed in an alloy of nominal composition Mn₈₀Si₁₅A_l5. The structure was determined by high resolution electron microscopy. Its symmetry is 8₄/mmc. It is a layer structure composed of octagonal layers A alternating with tetragonal layers B' and B'' (mutually rotated by 45^o). The layer sequence is ...AB'AB''... . This fourfold stack can be viewed as a tiling composed of squares and 45^o rhombi. Projecting the stack onto the basal plane cancels the asymmetry B'B'' but produces the equivalent of decorating the square edges with arrows as in an ABT. Arrowed squares impose arrowing on the rhombi which is, in fact, redundant. Finally, covering by OP's will produce an ABT. It remains to be shown that the construction is unique. It is interesting to note that the b-Mn structure is a periodic square-and-rhomb tiling with systematically changed arrowing n the tiles' boundaries become oriented: all squares one way, all rhombi the other.

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