Structural realization of the polytope approach for the geometrical description ofthe transition of a quasicrystal into a crystalline phase

Abstract

A geometrical model is proposed for the recently observed transformation under theaction of highly localized stresses during the surface scratch test of the icosahedralphase into a body-centred cubic (BCC) phase, the disordered version of the B2phase with the CsCl structure, which occupies a large portion of the Al–Cu–Fephase diagram around the Al50(Cu, Fe)50 concentration. The model is founded ona polytope concept and the concept of the eight-dimensional root latticeE8. Inaccordance with these concepts many crystalline or quasicrystalline structures canbe derived from the polytope (the four-dimensional polyhedron) by fulfillingoperations of lowering its local curvature with subsequent mapping of decurvedpolytope fragments onto Euclidian three-dimensional space. The properties of theE8lattices give foundation to the possibility of mapping a quasicrystallinestructure on a crystalline structure. The structural transformation iseffected through intermediate atomic configurations coinciding with bothstructures, which are determined by a four-dimensional icosahedron (the{3, 3, 5}polytope). For the transformation of the icosahedron of the icosahedralphase into a rhombic dodecahedron of the cubic B2 phase, the cubicA15 structure plays the role of an intermediate configuration since itcan be represented as a three-dimensional packing of linearly interlacedchains of Frank–Kasper polyhedra with coordination numbers Z = 12 (icosahedron)and Z = 14. Thetransition between the rhombic dodecahedron of a B2 structure and the Frank–Kasper polyhedronwith Z = 14requires insertion of disclination quadruplets into some facets of the rhombicdodecahedron. The proposed geometrical model can be applied also to the polymorphicBCC–FCC transformation since the Miller indices of the Frank–Kasper polyhedronwith Z = 14coincide with the observed indices of habit planes of iron martensites.