Arithmetic Crystal Classes Research Articles

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.

Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6.

This paper gives classification results for crystallographic groups in dimensions up to 6 which refine earlier enumeration results. Based on the classification data, the asymptotic growth of the number of space-group types is discussed. The classification scheme for crystallographic groups is revisited and a new classification level in between that of geometric and arithmetic crystal classes is introduced and denoted as harmonic crystal classes. Enantiomorphic pairs are determined on all classification levels from space-group types to crystal families and the enantiomorphic pairs of fixed-point-free space groups are given. A general algorithm to compute enantiomorphic pairs is described.

Report of a subcommittee on the nomenclature of n-dimensional crystallography. II. Symbols for arithmetic crystal classes, Bravais classes and space groups.

The Second Report of the Subcommittee on the Nomenclature of n-Dimensional Crystallography recommends specific symbols for R-irreducible groups in 4 and higher dimensions (nD), for centrings, for Bravais classes, for arithmetic crystal classes and for space groups (space-group types). The relation with higher-dimensional crystallographic groups used for the description of aperiodic crystals is briefly discussed. The Introduction discusses the general definitions used in the Report.

On the classification and enumeration of the irreducible co-representations of magnetic space groups

A generalization of Wintgen's method for classifying space-group representations is made to enable the infinite set of co-representations of magnetic space groups to be classified and enumerated. Using this method, all special points, lines and planes in the Brillouin zone model for magnetic crystals are identified. Arithmetic crystal-classes of magnetic space groups are introduced, fully listed and used to classify magnetic space groups and their co-representations. A version of Burnside's theorem adapted to co-representations is constructed and used to solve completely the enumeration problem of co-representations of magnetic space groups.

Arithmetic equivalence of point groups for quasiperiodic structures

Necessary and sufficient conditions are formulated for an n-dimensional arithmetic point group such that it may be the symmetry group of a d-dimensional quasiperiodic but not periodic, i.e. incommensurate, structure with Fourier modulus of rank n. Only point groups leaving invariant a d-dimensional subspace (the physical space) are considered. For an arithmetic point group describing an incommensurate structure, all equivalent choices for the internal space are related by the normalizer in Gl (n, \bb Z) of the point group. Also, the conditions on arithmetic equivalence of two point groups allowing an incommensurate structure are discussed. These conditions yield a further partition of the arithmetic crystal classes.

Space groups rare for organic structures. II. Analysis by arithmetic crystal class

Twelve arithmetic crystal classes contain only one space-group type; others contain various numbers up to sixteen (mmmP and 4/mmmP). In the multimember classes the number Nsg of substances having a particular space-group type can be modelled quantitatively (unweighted R2^{\sim}_{\lt} 0.05) by Nsg= Acc exp {Σ(Xi)cc[xi]sg}, where Acc is a normalizing constant for the arithmetic crystal class, [xi]sg is the number of symmetry elements [2, 21, m, a, b, c, d, n, 3, {\bar 3}, 31, 32, 4,...] of type xi in the unit cell of the space-group type, (Xi)cc is a parameter characteristic of the arithmetic crystal class and the symmetry element, and the summation is over all the elements xi that are to be considered. In many cases the parameters Xi are equal, within their estimated standard deviations, throughout a cohort larger than the arithmetic crystal class (such as a geometric crystal class or an entire crystal system). The above equation can then be applied to the larger cohort, with arithmetic crystal class included as an additional 'symmetry element' in the sum in the exponent. There is at present no theory to account for the different popularities of different arithmetic crystal classes.

On crystallography in higher dimensions. III. Results inR4

An explicit classification of lattices and crystallographic groups of 4-dimensional space R4 is given. There are (in R4): 710 arithmetic crystal classes; 227 geometric crystal classes belonging to 118 isomorphism types of groups; 64 Bravais classes corresponding to 64 Bravais types of lattices; 33 crystal systems; 23 crystal families.

Determination of n-dimensional space groups by means of an electronic computer

A method is presented to determine the n-dimensional space groups if the arithmetic crystal classes are known. It considers space groups as group extensions and gives a relation between the nonequivalent extensions and a set of vectors in a certain linear vector space. In this way we transform the group extension problem into a problem of linear algebra. In fact the method can be used to determine the nonequivalent extensions of a free Abelian group by an arbitrary finitely generated group. The algorithm has been used to determine four-dimensional space groups by means of an electronic computer.

Relativistic crystallographic point groups in two dimensions

Abstract Two-dimensional relativistic point groups are investigated and various equivalence classes of them are explicitly given. It is shown that every two-dimensional relativistic point group can always be generated by at most three elements (the total inversion, a proper Lorentz transformation of infinite order, and a mirror, i.e an improper Lorentz transformation). There are 7 abstract point groups (C1, C2, C∞, D2, D∞, C∞ × C2 and D∞ × C2) divided into an infinite number of geometric (or R- equivalent) and arithmetic (or Z-equivalent) crystal classes, which are indicated. The relativistic geometric crystal classes (conjugated crystallographic subgroups of the Lorentz group) are also given. All elements of GL(2,Z) that can be interpreted as crystallographic transformations of the Minkowskian metric space are discussed. The role of point group symmetry in physical systems having a crystallographic structure in space and time is considered and the importance of point groups for recognizing such systems in nature is underlined.

Crystallographic groups in space and time: III. Four-dimensional Euclidean crystal classes corresponding to generalized magnetic point groups

To determine the non-isomorphic space groups one needs the arithmetic crystal classes. We limit ourselves to four-dimensional point groups which are 3 + 1-reducible in GL(4, R). These are called GM point groups. It is shown that there exists an one-to-one correspondence between n-dimensional GM point groups and ( n - 1)- dimensional magnetic point groups. So in four dimensions the GM point groups are known. For these groups corresponding arithmetic point groups are determined by constructing all lattices left invariant by such a group. The arithmetic crystal classes are found by eliminating equivalent arithmetic point groups. The arithmetic crystal classes and Bravais classes are explicitly given.