Three-dimensional Space Groups Research Articles

Spectroscopy (MIR), chromatography (RP-HPLC) and chemometrics applied to soluble peptides to discriminate the geographic origin of coalho cheese

In the present study, the discrimination of the origin of the traditional brazilian Coalho cheese was investigated by means of peptide profiles using a quick method Mid-Infrared Spectroscopy (MIR) compared to Reversed-Phase High Performance Liquid Chromatography (RP-HPLC) and associated with different chemometric analysis. Soluble peptides were extracted from Coalho cheeses obtained from four different mesoregions of Bahia, Brazil, from seven different producing factories. 70 samples were collected within 10 days of manufacture and analyzed using RP-HPLC and MIR. The RP-HPLC indicated five markers peptides for cheese differentiation and the MIR revealed spectroscopic profiles of amides with distinct absorptions, varying according to geographic origin. Using the Principal Component Analysis, the MIR correctly discriminated the peptide profiles of the cheeses by origin, presenting distinct groups in three-dimensional space and the Artificial Neural Networks models were efficient in correctly classifying all cheeses in their respective groups of validation, associated with low error rates (RMSEVRP-HPLC = 0.0165, RMSEVMIR = 0.0488). The results showed that RP-HPLC and MIR discriminated the origin of Coalho cheese and the MIR presented itself as an alternative of lower operational cost, fast, efficient and robust for this investigation.

MSGCorep: A package for corepresentations of magnetic space groups

Motivated by easy access to complete corepresentation (corep) data of all the 1651 magnetic space groups (MSGs) in three-dimensional space, we have developed a Mathematica package MSGCorep to provide an offline database of coreps and various functions to manipulate them, based on our previous package SpaceGroupIrep. One can use the package MSGCorep to obtain the elements of any MSG and magnetic little group, to calculate the multiplication of group elements, to obtain the small coreps at any k-point and full coreps of any magnetic k-star for any MSG and show them in a user-friendly table form, to calculate and show the decomposition of direct products of full coreps between any two specified magnetic k-stars, and to determine the small coreps of energy bands. Both single-valued and double-valued coreps are supported. In addition, the 122 magnetic point groups (MPGs) and their coreps are also supported by this package. To the best of our knowledge, MSGCorep is the first package that is able to calculate the direct product of full coreps for any MSG and able to determine small coreps of energy bands for general purpose. In a word, the MSGCorep package is an offline database and tool set for MSGs, MPGs, and their coreps, and it is very useful to study the symmetries in magnetic (type-I, -III, and -IV MSGs) and nonmagnetic (type-II MSGs) materials. Program summaryProgram title:MSGCorepCPC Library link to program files:https://doi.org/10.17632/zt9r8pp8kr.1Developer's repository link:https://github.com/goodluck1982/MSGCorepLicensing provisions: GNU General Public Licence 3.0Programming language: WolframExternal routines/libraries used:SpaceGroupIrep (https://github.com/goodluck1982/SpaceGroupIrep)Nature of problem: The package MSGCorep provides offline database and tools for easy access to 1651 magnetic space groups, 122 magnetic point groups, and their corepresentations (coreps). MSGCorep is the first package that is able to calculate the direct product of full coreps for any magnetic space group and able to determine small coreps of energy bands for general purpose.Solution method: Coreps of a magnetic group are constructed from the representations of the maximal unitary subgroup of the magnetic group. Based on the complete representation data of 230 space groups provided by SpaceGroupIrep package, complete corep data of 1651 magnetic space groups are derived in MSGCorep package. A more convenient equation of induced corep, i.e. Eq. (2), is introduced to calculate the full coreps of all MSGs.

Seeing mesoatomic distortions in soft-matter crystals of a double-gyroid block copolymer.

Supramolecular soft crystals are periodic structures that are formed by the hierarchical assembly of complex constituents, and occur in a broad variety of 'soft-matter' systems1. Such soft crystals exhibit many of the basic features (such as three-dimensional lattices and space groups) and properties (such as band structure and wave propagation) of their 'hard-matter' atomic solid counterparts, owing to the generic symmetry-based principles that underlie both2,3. 'Mesoatomic' building blocks of soft-matter crystals consist of groups of molecules, whose sub-unit-cell configurations couple strongly to supra-unit-scale symmetry. As yet, high-fidelity experimental techniques for characterizing the detailed local structure of soft matter and, in particular, for quantifying the effects of multiscale reconfigurability are quite limited. Here, by applying slice-and-view microscopy to reconstruct the micrometre-scale domain morphology of a solution-cast block copolymer double gyroid over large specimen volumes, we unambiguously characterize its supra-unit and sub-unit cell morphology. Our multiscale analysis reveals a qualitative and underappreciated distinction between this double-gyroid soft crystal and hard crystals in terms of their structural relaxations in response to forces-namely a non-affine mode of sub-unit-cell symmetry breaking that is coherently maintained over large multicell dimensions. Subject to inevitable stresses during crystal growth, the relatively soft strut lengths and diameters of the double-gyroid network can easily accommodate deformation, while the angular geometry is stiff, maintaining local correlations even under strong symmetry-breaking distortions. These features contrast sharply with the rigid lengths and bendable angles of hard crystals.

A Peptoid with Extended Shape in Water.

The term "peptoids" was introduced decades ago to describe peptide analogues that exhibit better physicochemical and pharmacokinetic properties than peptides. Oligo(N-substituted glycine) (oligo-NSG) was previously proposed as a peptoid due to its high proteolytic resistance and membrane permeability. However, oligo-NSG is conformationally flexible, and ensuring a defined shape in water is difficult. This conformational flexibility severely limits the biological application of oligo-NSG. Here, we propose oligo(N-substituted alanine) (oligo-NSA) as a peptoid that forms a defined shape in water. The synthetic method established in this study enabled the first isolation and conformational study of optically pure oligo-NSA. Computational simulations, crystallographic studies, and spectroscopic analysis demonstrated the well-defined extended shape of oligo-NSA realized by backbone steric effects. This new class of peptoid achieves the constrained conformation without any assistance of N-substituents and serves as a scaffold for displaying functional groups in well-defined three-dimensional space in water, which leads to effective biomolecular recognition.

Topological Classification of Crystalline Insulators through Band Structure Combinatorics

We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure. The algorithm applies to crystals with broken time-reversal, particle-hole, and chiral symmetries in any dimension. The presented results match the mathematical structure underlying the topological classification of these crystals in terms of K-theory, and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify the allowed topological phases in any possible two-dimensional crystal in class A. We also show how the same procedure can be used to classify the allowed phases for any three-dimensional space group. Employing these classifications, we study transitions between topological phases within class A that are driven by band inversions at high symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure, and identify whether or not they give rise to intermediate Weyl semimetallic phases.

Development of Spiroligomer-Peptoid Hybrids.

Creating functional macromolecules that possess the diversity and functionality of proteins poses an enormous challenge, as this requires large, preorganized macromolecules to facilitate interactions. Peptoids have been shown to interact with proteins, and combinatorial libraries of peptoids have been useful in discovering new ligands for protein binding. We have created spiroligomer-peptoid hybrids that have a spirocyclic core that preorganizes functional groups in three-dimensional space. By utilizing spiroligomers, we can reduce the number of rotatable bonds between functional groups while increasing the stereochemical diversity of the molecules. We have synthesized 15 new spiroligomer monomer amines that contain two stereocenters and three functional groups (67-84% yields from a common hydantoin starting material) as well as a spiroligomer trimer 25 with six stereocenters and five functional groups. These 16 amines were used to synthesize five first-generation spiroligomer-peptoids hybrids.

ОБ АЛГОРИТМИЧЕСКИХ ПРОБЛЕМАХ В ГРУППАХ КОКСТЕРА

The main algorithmic problems of group theory posed by M. Dehn are the problem of words, the problem of the conjugation of words for finitely presented groups, and the group’s isomorphism problem. Among the works related to the study of the M. Dehn’s problems, the most outstanding ones are the work of P. S. Novikov who proved the undecidability of the problem of words and the conjugacy problem for finitely presented groups as well as the undecidability of the problem of isomorphism of groups. In this regard, the main algorithmic problems and their various generalizations are studied in certain classes of groups. Coxeter groups were introduced by H. S. M. Coxeter: every reflection group is a Coxeter group if its generating elements are reflections with respect to hyperplanes limiting its fundamental polyhedron. H. S. M. Coxeter listed all the reflection groups in three-dimensional Euclidean space and proved that they are all Coxeter groups and every finite Coxeter group is isomorphic to some reflection group in the three-dimensional Euclidean space which elements have a common fixed point. In an algebraic aspect Coxeter groups are studied starting with works by J. Tits who solved the problem of words in certain Coxeter groups. The article describes the known results obtained in solving algorithmic problems in Coxeter groups; the main purpose of the paper is to analyze of the results of solving algorithmic problems in Coxeter groups that were obtained by members of the Tula algebraic school ’Algorithmic problems of theory of the groups and semigroups ’ under the supervision of V. N. Bezverkhnii. It reviews assertions and theorems proved by the authors of the article for the various classes of Coxeter groups: Coxeter groups of large and extra-large types, Coxeter groups with a tree-structure, and Coxeter groups with n-angled structure. The basic approaches and methods of evidence among which the method of diagrams worked out by van Kampen, reopened by R. Lindon and refined by V. N. Bezverkhnii concerning the introduction of R-cancellations, special R-cancellations, special ring cancellations as well as method of graphs, method of types worked out by V. N. Bezverkhnii, method of special set of words designed by V. N. Bezverkhnii on the basis of the generalization of Nielsen method for free construction of groups. Classes of group considered in the article include all Coxeter groups which may be represented as generalized tree structures of Coxeter groups formed from Coxeter groups with tree structure with replacing some vertices of the corresponding tree-graph by Coxeter groups of large or extra-large types as well as Coxeter groups with n-angled structure.

Dealing with Aperiodic Protein Crystal Structures

Protein crystals can be aperiodic. They will diffract X-rays, and are therefore a crystal, but their diffraction is not periodic. This means that their diffraction pattern cannot be simply be indexed by a typical three-dimensional unit cell and space group. Aperiodic crystals include "quasi-crystals" as well as "modulated" crystals. In the latter case, the modulation can be positional or occupational and these modulations can be incommensurate with the normal periodic lattice [1]. An overview of aperiodic protein crystal diffraction will be presented with examples. The discussion will then focus on the characteristics of incommensurately modulated crystals followed by a more detailed discussion of how to solve these crystals. The following details of structure solution will be presented: (1) data collection perils; (2) specialized diffraction data processing in (3+1)D space using a q-vector [2]; (3) how to get an approximation of the structure in 3D space; (4) the assignment of the (3+1)D space group; and the ultimate (5) crystallographic refinement in superspace[3]. Future directions and needs will be discussed.

Tabulation of irreducible representations of the crystallographic space groups and their superspace extensions

New tables of irreducible representations (IRs) are introduced for the 230 crystallographic space groups (SGs) in three-dimensional space, at both special and non-special k vectors, and for their extensions to (3 + d)-dimensional superspace (`superspace-extended SGs' or SSESGs). Neither a tabulation of SG IR matrices for non-special k vectors nor a tabulation of SSESG IR matrices for d > 1 have been previously published. These tabulations are made possible by a new form in which the IR matrices of SGs are separated as a product of a translation part T and a point-operation part P, and where the IR matrices of SSESGs are separated as a product of a phase-shift part Q and a point-operation part P(s). Both T and Q have a simple prescribed form that does not need to be tabulated. Also, the new IR matrices are in a convenient block form which allows one to see by inspection which parts of the matrices and the associated order parameters belong to which arm of the star of k. In addition to complex IR matrices, real physically irreducible representation (PIR) matrices are tabulated. The new IR and PIR tables are available on the ISO-IR website (http://stokes.byu.edu/iso/irtables.php) in both convenient human-readable and computer-readable forms.

Three-dimensional space groups two-dimensional hyperbolic orbifolds and 'sponge' groups

Three-dimensional space groups two-dimensional hyperbolic orbifolds and 'sponge' groups

Interactive 3D Space Group Visualization with CLUCalc and the Clifford Geometric Algebra Description of Space Groups

A new interactive software tool is described, that visualizes 3D space group symmetries. The software computes with Clifford (geometric) algebra. The space group visualizer (SGV) originated as a script for the open source visual CLUCalc, which fully supports geometric algebra computation. Selected generators (Hestenes and Holt, JMP, 2007) form a multivector generator basis of each space group. The approach corresponds to an algebraic implementation of groups generated by reflections (Coxeter and Moser, 4th ed., 1980). The basic operation is the reflection. Two reflections at non-parallel planes yield a rotation, two reflections at parallel planes a translation, etc. Combination of reflections corresponds to the geometric product of vectors describing the individual reflection planes. We first give some insights into the Clifford geometric algebra description of space groups. We relate the choice of symmetry vectors and the origin of cells in the geometric algebra description and its implementation in the SGV to the conventional crystal cell choices in the International Tables of Crystallography (T. Hahn, Springer, 2005). Finally we briefly explain how to use the SGV beginning with space group selection. The interactive computer graphics can be used to fully understand how reflections combine to generate all 230 three-dimensional space groups.

Supramolecular Patterns Controlled by Electron Interference and Direct Intermolecular Interactions

Whereas all 230 three-dimensional space groups occur in organic crystals, out of only 17 plane groups some highly symmetric ones such as p31m have not yet been observed in two-dimensional (2D) crystals of organic molecules. Here a kagome network with p31m symmetry is reported for cobalt phthalocyanine on Cu(111). This unusual structure results from substrate-induced reduction of molecular symmetry and substrate-mediated interaction via quantum interference of surface electrons. These interactions provide additional control over the symmetry of 2D crystals of phthalocyanines and lead to a variety of other symmetries in self-assembled arrays.

From space to superspace and back: Superspace Group Finder

The symmetry of a commensurately modulated crystal structure can be described in two different ways: in terms of a conventional three-dimensional space group or using the superspace concept in (3 +d) dimensions. The three-dimensional space group is obtained as a real-space section of the (3 +d) superspace group. A complete network was constructed linking (3 + 1) superspace groups and the corresponding three-dimensional space groups derived from rational sections. A database has been established and is available at http://superspace.epfl.ch/finder/. It is particularly useful for finding common superspace groups for various series of modular (`composition-flexible') structures and phase transitions. The use of the database is illustrated with examples from various fields of crystal chemistry.

Magnetic group maximal subgroups of index ≤4

The one-, two- and three-dimensional magnetic space groups and the two- and three-dimensional magnetic subperiodic groups are considered. The maximal subgroups of index < or =4 of a representative group of each type in the reduced superfamilies of these magnetic groups are tabulated.

Symmetry, crystallography, and art

Crystallography, a scientific discipline about the properties and structure of crystals, and the arts seem to be two different areas of human endeavor. There are, indeed, many differences between them – however, there are also similarities. The concept of symmetry can be used as a bridge to recognize such similarities. Symmetry is everywhere, in the hard sciences, in the arts, in nature, and in our everyday life; therefore it seems to be a natural choice for connecting the seemingly disparate fields of crystallography and the arts. In the following, first we look at science and the arts through one of the simplest symmetries, reflection. This is followed by considerations of chirality, which is related to reflection in a special way. Further, space groups will be discussed, the one-, two-, and three-dimensional space groups with examples from both the sciences and the arts. The symmetries of crystal structures belong to the three-dimensional space groups.

Phase Transitions in Aperiodic Crystals

We review phase transitions involving aperiodic crystals. Because the symmetry of aperiodic crystals is not a three-dimensional space group, but an n-dimensional superspace group, the usual procedure in the Landau theory has to be reconsidered. There are transitions where the dimension of the (super)space changes, and transitions where that is not the case. Moreover, different classes of aperiodic crystals pose specific problems. Phase transitions with modulated phases, incommensurate composites, quasicrystals, and incommensurate magnetic structures are discussed.

Crystallographic space groups in geometric algebra

We present a complete formulation of the two-dimensional and three-dimensional crystallographic space groups in the conformal geometric algebra of Euclidean space. This enables a simple new representation of translational and orthogonal symmetries in a multiplicative group of versors. The generators of each group are constructed directly from a basis of lattice vectors that define its crystal class. A new system of space group symbols enables one to unambiguously write down all generators of a given space group directly from its symbol.

Twinsvs. modular crystal structures

AbstractThe crystallographic nature of modular structures is analysed and discussed in terms of their symmetry and structure building operations. The modular crystal structures we discuss in this paper are homogeneous edifices, built by one or more types of modules related by a point-space operation, and the whole structure is described by an ordinary three-dimensional triperiodic space group. Because the term “twin” is often used, with or without modifiers, to describe symmetry operations that act as building mechanisms in modular structures, a comparison is made with twins, that are instead heterogeneous edifices, described not by a space group but by a (polychromatic) point group, where the individuals are related by operations in the vector space. A unifying scheme is proposed where the passage from heterogeneous to homogeneous structures, only apparently justified by a change of dimensions of the individuals, implies a change of space where the operations are defined.

Internal rotation and space groups

Relationships between crystallographic space groups and the symmetry groups of molecular models with multiple internal rotations were explored. With position vectors defined as functions of internal rotation coordinates and after defining unit cells for internal rotation models, it was possible to transform these unit cells into unit cells of two- or three-dimensional space groups. Specific examples are given for models with two or three internal rotations. The vector of the symmetry numbers σ j of the internal rotation model is the equivalent of the wavevector k in crystals.

Th. Hahn (Ed.) Brief Teaching Edition of Space Group Symmetry, Vol A of International Tables of Crystallography 5th revised edition. Dordrecht (Kluwer), 2002. 160 pp. Price €27, US$25, £16. ISBN 0-7923-6591-7

The full edition of Volume A has for a long time been an indispensable tool in most crystallography laboratories. This Brief Teaching Edition consists of direct extracts from the full volume, chosen with classroom teaching in mind and published in paperback. It is also aimed at researchers who wish to become more familiar with the space-group tables or to have an inexpensive laboratory handbook. This book is considerably smaller than the full Volume A, being only 160 A4 pages rather than over 900. The greatest reduction is from the careful selection of just 24 of the 230 three-dimensional space groups, in order to include groups from all seven crystal systems, those that occur frequently in both organic and inorganic crystals and from which the general positions of other space groups may be derived. There will always be cases for including other groups (e.g. Pm -3 m (perovskite, CsCl) and Im -3 m (bcc), …