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

A geometrical model is proposed for the recently observed transformation under the action of highly localized stresses during the surface scratch test of the icosahedral phase into a body-centred cubic (BCC) phase, the disordered version of the B2 phase with the CsCl structure, which occupies a large portion of the Al-Cu-Fe phase diagram around the Al-50(Cu, Fe)(50) concentration. The model is founded on a polytope concept and the concept of the eight-dimensional root lattice E-8. In accordance with these concepts many crystalline or quasicrystalline structures can be derived from the polytope (the four-dimensional polyhedron) by fulfilling operations of lowering its local curvature with subsequent mapping of decurved polytope fragments onto Euclidian three-dimensional space. The properties of the Es lattices crive foundation to the possibility of mapping a quasicrystalline structure on a crystalline structure. The structural transformation is effected through intermediate atomic configurations coinciding with both structures, which are determined by a four-dimensional icosahedron the (3, 3, 51 polytope). For the transformation of the icosahedron of the icosahedral phase into a rhombic dodecahedron of the cubic B2 phase, the cubic A15 structure plays the role of an intermediate configuration since it can be represented as a three-dimensional packing of linearly interlaced chains of Frank-Kasper polyhedra with coordination numbers Z = 12 (icosahedron) and Z = 14. The transition between the rhombic dodecahedron of a B2 structure and the Frank-Kasper polyhedron with Z = 14 requires insertion of disclination quadruplets into some facets of the rhombic dodecahedron. The proposed geometrical model can be applied also to the polymorphic BCC-FCC transformation since the Miller indices of the Frank-Kasper polyhedron with Z = 14 coincide with the observed indices of habit planes of iron martensites.