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 Mn80Si15Al5.
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 45o). The layer sequence is ...AB'AB''... .
This fourfold stack can be viewed as a tiling composed of squares
and 45o 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.
Gähler F, 1997, Quasicrystals 6, Tokyo (in press).
Gähler F and Jeong H-C, 1995,
J. Phys. A: Math. Gen. 28, 1807-1817.
Gummelt P, 1996, Geometriae Dedicata 62, 1-17.
Huang Z and Hovmöller S, 1991, Phil. Mag. Letters 64, 83-88.
Jeong H-C and Steinhardt P J, 1994,
Phys. Rev. Letters 73, 1943-1946.
Jiang J-C, Hovmöller S and Zou X-D, 1995,
Phil. Mag. Letters 71, 123-129.