A topological proof of the modified Euler characteristic based on the orbifold concept

The notion of the Euler characteristic of a polyhedron or tessellation has been the subject of in-depth investigations by many authors. Two previous papers worked to explain the phenomenon of the vanishing (or zeroing) of the modified Euler characteristic of a polyhedron that underlies a periodic tessellation of a space under a crystallographic space group. The present paper formally expresses this phenomenon as a theorem about the vanishing of the Euler characteristic of certain topological spaces called topological orbifolds. In this new approach, it is explained that the theorem in question follows from the fundamental properties of the orbifold Euler characteristic. As a side effect of these considerations, a theorem due to Coxeter about the vanishing Euler characteristic of a honeycomb tessellation is re-proved in a context which frees the calculations from the assumptions made by Coxeter in his proof. The abstract mathematical concepts are visualized with down-to-earth examples motivated by concrete situations illustrating wallpaper and 3D crystallographic space groups. In a way analogous to the application of the classic Euler equation to completely bounded solids, the formula proven in this paper is applicable to such important crystallographic objects as asymmetric units and Dirichlet domains.

Extension of Hall symbols of crystallographic space groups to magnetic space groups

The Hall symbols for describing unambiguously the generators of space groups have been extended to describe any setting of the 1651 types of magnetic space groups (Shubnikov groups). A computer program called MHall has been developed for parsing the Hall symbols, generating the full list of symmetry operators and identifying the transformation to the standard setting.

Normally supportive sublattices of crystallographic space groups

The tabulation of normal subgroups of 3D crystallographic space groups that are themselves 3D crystallographic space groups (csg's) is an ambitious goal, but would have a variety of applications. For convenience, such subgroups are referred to as `csg-normal' while normal subgroups of the crystallographic point group (cpg) of a crystallographic space group are referred to as `cpg-normal'. The point group of a csg-normal subgroup must be a cpg-normal subgroup. The present work takes a significant step towards that goal by tabulating the translational subgroups (a.k.a. sublattices) that are capable of supporting csg-normal subgroups. Two necessary conditions are identified on the relative sublattice basis that must be met in order for the sublattice to support csg-normal subgroups: one depends on the operations of the point group of the space group, while the other depends on the operations of the cpg-normal subgroup. Sublattices that meet these conditions are referred to as `normally supportive'. For each cpg-normal subgroup (excluding the identity subgroup 1) of each of the arithmetic crystal classes of 3D space groups, all of the normally supportive sublattices have been tabulated in symbolic form, such that most of the entries in the table contain one or more integer variables of infinite range; thus it could be more accurately described as a table of the infinite families of normally supportive sublattices. For a given pair of cpg-normal subgroup and normally supportive sublattice, csg-normal subgroups of the space groups of the parent arithmetic crystal class can be constructed via group extension, though in general such a pair does not guarantee the existence of a corresponding csg-normal subgroup.

A general algorithm for generating isotropy subgroups in superspace

This paper presents a general algorithm for generating the isotropy subgroups of superspace extensions of crystallographic space groups involving arbitrary superpositions of multi-korder parameters from incommensurate and commensuratekvectors. Several examples are presented in detail in order to illuminate each step of the algorithm. The practical outcome is that one can now start with any commensurate parent crystal structure and generate a structure model for any conceivable incommensurate modulation of that parent, fully parameterized in terms of order parameters of irreducible representations at the relevant wavevectors. The resulting modulated structures have (3 +d)-dimensional superspace-group symmetry. Because incommensurate structures are now commonly encountered in the context of many scientifically and technologically important functional materials, the opportunity to apply the powerful methods of group representation theory to this broader class of structural distortions is very timely.

ISOSUBGROUP: an internet tool for generating isotropy subgroups of crystallographic space groups

ISOSUBGROUP, the newest member of the ISOTROPY Software Suite (http://iso.byu.edu), generates isotropy subgroups of crystallographic space groups based on superpositions of multiple irreducible representations, along with a wealth of information about each one. Like the original ISOTROPY program, its scope is general rather than being restricted to common types of order parameters of a user-specified parent structure. But like the newer ISODISTORT program, its user-friendly interface has menu-driven selections. This combination of features has been oft requested but unavailable until now. Program output includes information about the subgroup symmetry, ferroic species, phase-transition continuity, active k vectors, domains and secondary order parameters.

Generation of Basis Vectors for Magnetic Structures and Displacement Modes

Increasing attention is being focused on the use of symmetry-adapted functions to describe magnetic structures, structural distortions, and incommensurate crystallography. Though the calculation of such functions is well developed, significant difficulties can arise such as the generation of too many or too few basis functions to minimally span the linear vector space. We present an elegant solution to these difficulties using the concept of basis sets and discuss previous work in this area using this concept. Further, we highlight the significance of unitary irreducible representations in this method and provide the first validation that the irreducible representations of the crystallographic space groups tabulated by Kovalev are unitary.

The 17,803 types of double antisymmetry space groups

"Double antisymmetry, as a type of color symmetry, refers to the consideration of two independent color-reversing operations. We found that there are 17,803 types of double antisymmetry space groups. The ""types"" of double antisymmetry space groups being the proper affine equivalence classes thereof, i.e. if two groups differ only by changes in origin, lattice constants, and angles then they are considered the same type of group, just as with the 230 types of crystallographic space groups. The significance of listing the types of double antisymmetry space groups is that it gives all possible symmetries of a crystal when two independent antisymmetries are considered in conjunction with the conventional 3D Euclidean space symmetry. Examples from our listing of their properties and symmetry diagrams (available online) will be given."