Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures

Abstract

Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting restrictions can be expressed in terms of linear relations of the a 2 , b 2 and a · b scalar products of the lattice basis vectors with rational coefficients. To characterize OD families and to check their validity, these lattice restrictions are expressed in the bases of different layers and combined. For a more familiar notation, they can be expressed in terms of the lattice parameters a, b and γ. Alternatively, the description of the lattice restrictions may be simplified by using centered lattices. The representation of the lattice restrictions in terms of scalar products is dependent on the chosen basis. A basis-independent classification of the lattice restrictions is outlined.

Symmetry Analysis of the Complex Polytypism of Layered Rare-Earth Tellurites and Related Selenites: The Case of Introducing Transition Metals

“…The slight shifts can form different polytypes and OD ("order-disorder") structures [19][20][21]. The symmetry analysis of such disordered compounds is based on the concepts of OD theory [22,23] and the symmetry of the OD structures is described by groupoids (instead of crystallographic space groups), which can contain the non-crystallographic symmetry operations [24][25][26]. This approach allows to determine the symmetry relations for the whole family of known polytypes and predict the hypothetical ones [27], which is important for the searching of new materials [28][29][30][31][32][33][34].…”