Crystallography of homophase twisted bilayers: coincidence, union lattices and space groups.

This paper presents the basic tools used to describe the global symmetry of so-called bilayer structures obtained when two differently oriented crystalline monoatomic layers of the same structure are superimposed and displaced with respect to each other. The 2D nature of the layers leads to the use of complex numbers that allows for simple explicit analytical expressions of the symmetry properties involved in standard bicrystallography [Gratias & Portier (1982). J.

Atomic scale analyses of {\bb Z}-module defects in an NiZr alloy.

Some specific structures of intermetallic alloys, like approximants of quasicrystals, have their unit cells and most of their atoms located on a periodic fraction of the nodes of a unique {\bb Z}-module [a set of the irrational projections of the nodes of a (N > 3-dimensional) lattice]. Those hidden internal symmetries generate possible new kinds of defects like coherent twins, translation defects and so-called module dislocations that have already been discussed elsewhere [Quiquandon et al. (2016).

{\bb Z}-module defects in crystals.

An analysis is presented of the new types of defects that can appear in crystalline structures where the positions of the atoms and the unit cell belong to the same {\bb Z}-module, i.e. are irrational projections of an N > 3-dimensional (N-D) lattice Λ as in the case of quasicrystals.

Merohedral twins revisited: quinary twins and beyond.

A twin is defined as being an external operation between two identical crystals that share a fraction of the atomic structure with no discontinuity from one crystal to the other. This includes merohedral twins, twins by reticular merohedry as well as coherent twins by contact where only the habit plane is shared by the two adjacent crystals (epitaxy). Interesting and original cases appear when the invariant substructure is built with positions belonging to the same {\bb Z}-module as, for example, the quinary twin structure first drawn by Albrecht Dürer [(1525).

Atomic clusters and atomic surfaces in icosahedral quasicrystals.

This paper presents the basic tools commonly used to describe the atomic structures of quasicrystals with a specific focus on the icosahedral phases. After a brief recall of the main properties of quasiperiodic objects, two simple physical rules are discussed that lead one to eventually obtain a surprisingly small number of atomic structures as ideal quasiperiodic models for real quasicrystals. This is due to the fact that the atomic surfaces (ASs) used to describe all known icosahedral phases are located on high-symmetry special points in six-dimensional space.

Atomic structure of the (Al,Si)CuFe cubic approximant phase.

The structure of the alpha-(Al,Si)CuFe approximant phase is determined by a single-crystal X-ray diffraction study and compared to the ideal structure obtained by the perpendicular shear method of the parent icosahedral phase. It is shown that the local environments (typical atomic clusters) of the two phases are similar and expand significantly farther than the size of the unit cell of the approximant. The orbit Al(2) issuing from the theoretical icosahedral model corresponding to the inner dodecahedron of the Mackay-type cluster is not found in the approximant and is replaced by a partially occupied inner icosahedron with an unusually large Debye-Waller factor.