scholarly journals Analysis of the structural continuity in twinned crystals in terms of pseudo-eigensymmetry of crystallographic orbits

IUCrJ ◽  
2013 ◽  
Vol 1 (1) ◽  
pp. 39-48 ◽  
Author(s):  
Mohamed Amine Marzouki ◽  
Bernd Souvignier ◽  
Massimo Nespolo
Keyword(s):  
Necessary Conditions ◽  
Space Group ◽  
Space Groups ◽  
The Individual ◽  
Structural Continuity

The reticular theory of twinning gives the necessary conditions on the lattice level for the formation of twins. The latter are based on the continuation, more or less approximate, of a substructure through the composition surface. The analysis of this structural continuity can be performed in terms of the eigensymmetry of the crystallographic orbits corresponding to occupied Wyckoff positions in the structure. If {\cal G} is the space group of the individual and {\cal H} a space group which fixes the twin lattice obtained as an intersection of the space groups of the individuals in their respective orientations, then a structural continuity is obtained if (1) the eigensymmetry of an orbit under {\cal G} contains the twin operation; (2) the eigensymmetry of a union of orbits under {\cal G} contains the twin operation; (3) the eigensymmetry of a split orbit under {\cal H} contains the twin operation; or (4) the eigensymmetry of a union of split orbits under {\cal H} contains the twin operation. The case of the twins in melilite is analysed: the (approximate) restoration of some of the orbits explains the formation of these twins.

scholarly journals Distribution of molecular pairs in Z′ = 2 structures

2006 ◽  
Vol 62 (5) ◽  
pp. 897-911 ◽  
Author(s):  
Anna Collins
Keyword(s):  
Single Molecules ◽  
Higher Symmetry ◽  
Space Group ◽  
Space Groups ◽  
Symmetry Operators ◽  
Independent Molecules ◽  
The Individual

The positions of pairs of independent molecules in Z′ = 2 structures have been surveyed for six of the most populated space groups for that class of structure. These results have been compared with the Z′ = 1 situation to reveal whether there are any fundamental differences in the construction of the asymmetric units in these cases. The results indicate that, broadly speaking, the packing of the molecular pairs in Z′ = 2 structures resembles that of single molecules in structures with Z′ = 1; this similarity may be chiefly attributed to the constraints imposed by the symmetry operators of the space group. However, there are key differences, which are particularly marked in the space groups with higher symmetry, that indicate that the asymmetric units in Z′ = 1 and Z′ = 2 structures are not directly comparable. In those cases where the positions of the pair centroids in Z′ = 2 structures are similar to the positions of molecular centroids for Z′ = 1 structures, it follows that the molecular centroids in Z′ = 2 structures must follow a different distribution. A different pattern is produced if the independent molecules in Z′ = 2 structures behave like the individual molecules in Z′ = 1 structures. These two scenarios combine to form the observed distributions of pair centroid positions.

Packing Effect on the Transfer Integrals and Mobility in α,α′-bis(dithieno[3,2-b:2′,3′-d]thiophene) (BDT) and its Heteroatom-Substituted Analogues

2011 ◽  
Vol 64 (12) ◽  
pp. 1587 ◽  
Author(s):  
Ahmad Irfan ◽  
Abdullah G. Al-Sehemi ◽  
Shabbir Muhammad ◽  
Jingping Zhang
Keyword(s):  
Electron Transfer ◽  
Electron Mobility ◽  
Hole Mobility ◽  
Experimental Investigations ◽  
Hole Transfer ◽  
Space Group ◽  
Space Groups ◽  
Transfer Integrals ◽  
Packing Effect ◽  
Good Agreement

Theoretically calculated mobility has revealed that BDT is a hole transfer material, which is in good agreement with experimental investigations. The BDT, NHBDT, and OBDT are predicted to be hole transfer materials in the C2/c space group. Comparatively, hole mobility of BHBDT is 7 times while electron mobility is 20 times higher than the BDT. The packing effect for BDT and designed crystals was investigated by various space groups. Generally, mobility increases in BDT and its analogues by changing the packing from space group C2/c to space groups P1 or . In the designed ambipolar material, BHBDT hole mobility has been predicted 0.774 and 3.460 cm2 Vs–1 in space groups P1 and , which is 10 times and 48 times higher than BDT (0.075 and 0.072 cm2 Vs–1 in space groups P1 and ), respectively. Moreover, the BDT behaves as an electron transfer material by changing the packing from the C2/c space group to P1 and .

Intergrowth polytypoids as modulated structures: a superspace description of the Sr n (Nb,Ti) n O3n + 2 compound series

2001 ◽  
Vol 57 (4) ◽  
pp. 471-484 ◽  
Author(s):  
L. Elcoro ◽  
J. M. Perez-Mato ◽  
R. L. Withers
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Separation Distance ◽  
Space Group ◽  
Space Groups ◽  
Cell Parameters ◽  
Average Structure ◽  
Layer Stacking ◽  
Compound Series ◽  
Three Dimensional Space

A new, unified superspace approach to the structural characterization of the perovskite-related Sr n (Nb,Ti) n O3n + 2 compound series, strontium niobium/titanium oxide, is presented. To a first approximation, the structure of any member of this compound series can be described in terms of the stacking of (110)-bounded perovskite slabs, the number of atomic layers in a single perovskite slab varying systematically with composition. The various composition-dependent layer-stacking sequences can be interpreted in terms of the structural modulation of a common underlying average structure. The average interlayer separation distance is directly related to the average structure periodicity along the layer stacking direction, while an inherent modulation thereof is produced by the presence of different types of layers (particularly vacant layers) along this stacking direction. The fundamental atomic modulation is therefore occupational and can be described by means of crenel (step-like) functions which define occupational atomic domains in the superspace, similarly to what occurs for quasicrystals. While in a standard crystallographic approach, one must describe each structure (in particular the space group and cell parameters) separately for each composition, the proposed superspace model is essentially common to the whole compound series. The superspace symmetry group is unique, while the primary modulation wavevector and the width of some occupation domains vary linearly with composition. For each rational composition, the corresponding conventional three-dimensional space group can be derived from the common superspace group. The resultant possible three-dimensional space groups are in agreement with all the symmetries reported for members of the series. The symmetry-breaking phase transitions with temperature observed in many compounds can be explained in terms of a change in superspace group, again in common for the whole compound series. Inclusion of the incommensurate phases, present in many compounds of the series, lifts the analysis into a five-dimensional superspace. The various four-dimensional superspace groups reported for this incommensurate phase at different compositions are shown to be predictable from a proposed five-dimensional superspace group apparently common to the whole compound series. A comparison with the scarce number of refined structures in this system and the homologous (Nb,Ca)6Ti6O20 compound demonstrates the suitability of the proposed formalism.

Hydrate von Li3VO4: Synthese und Strukturchemie / Hydrates of Li3VO4: Synthesis and Structural Chemistry

2007 ◽  
Vol 62 (10) ◽  
pp. 1235-1245 ◽  
Author(s):  
Simone Schnabel ◽  
Caroline Röhr
Keyword(s):  
Hydrogen Bonds ◽  
Building Blocks ◽  
Water Molecules ◽  
Space Group ◽  
Space Groups ◽  
Monoclinic Form ◽  
Lithium Sulfide ◽  
3D Network ◽  
Coordination Spheres ◽  
A Chain

Stoichiometric hydrates of Li3VO4, the hexahydrate and two polymorphs of the octahydrate, were prepared by evaporation of alkaline aqueous solutions 1 molar in LiOH and 0.5 molar in the metavanadate LiVO3 at r. t. with or without the addition of Lithium sulfide, i. e. at different pH values. Their crystal structures have been determined and refined using single crystal X-ray data; all lithium and hydrogen atom positions were localised and refined without contraints. All three title compounds crystallise in non-centrosymmetric space groups. The water molecules belong to the tetrahedral coordination spheres of the Li cations, i. e. they are embedded as water of coordination exclusively. The tetrahedral orthovanadate(V) anions VO3−4 and the LiO4 tetrahedra are connected via common O corners to form building units which are further held together by strong, nearly linear hydrogen bonds. The hexahydrate Li3VO4 ・ 6H2O (space group R3, a = 962.9(2), c = 869.2(2) pm, Z = 3, R1 = 0.0260) contains isolated orthovanadate(V) anions VO3−4 surrounded by a 3D network of cornersharing Li(H2O)4 tetrahedra forming rings of three, seven and eight units. The water molecules are ‘isolated’ in the sense that no hydrogen bonds are formed between water molecules. The octahydrate is dimorphous: The triclinic polymorph of Li3VO4 ・ 8H2O (space group P1, a = 592.6(2), b = 651.3(2), c = 730.2(4) pm, α = 89.09(2), β = 89.43(2), γ = 88.968(12)°, Z = 1, R1 = 0.0325) contains two types of chains of tetrahedra: One consists of corner-sharing Li(H2O)4 tetrahedra only, the second one is formed by alternating LiO4 and VO4 tetrahedra, also sharing oxygen corners. Only one water molecule is ‘isolated’, the other seven form a branched fragment of a chain with hydrogen bonds between them. In the monoclinic form of Li3VO4・8H2O (space group Pc, a = 732.6(1), b = 653.7(1), c = 1292.9(3) pm, β = 112.21(1)°, Z = 2, R1 = 0.0289) a fragment of a chain of three LiO4 tetrahedra, two of which share a common edge, and one VO4 tetrahedron represent the formular unit. These building blocks are connected via hydrogen bonds formed by three ‘isolated’ water molecules and a chain fragment of five connected water molecules.

Absolute configurations of Emycin D, E and F; mimicry of centrosymmetric space groups by mixtures of chiral stereoisomers

1999 ◽  
Vol 55 (4) ◽  
pp. 607-616 ◽  
Author(s):  
Martina Walker ◽  
Ehmke Pohl ◽  
Regine Herbst-Irmer ◽  
Martin Gerlitz ◽  
Jürgen Rohr ◽  
...  
Keyword(s):  
Correct Choice ◽  
Unit Cell ◽  
Monoclinic Space Group ◽  
Space Group ◽  
Space Groups ◽  
Centrosymmetric Space Group ◽  
Distance Restraints ◽  
Similar Distance ◽  
Triclinic Unit Cell ◽  
R Factors

The crystal structures of Emycin E (1), di-o-bromobenzoyl-Emycin F (2) and o-bromobenzoyl-Emycin D (3) have been determined by X-ray analysis at low temperature. Emycin E and o-bromobenzoyl-Emycin D both crystallize with two molecules in a triclinic unit cell. These two structures can be solved and refined either in the centrosymmetric space group P\bar 1, with apparent disorder localized at or around the expected chiral centre, or in the non-centrosymmetric space group P1 as mixtures of two diastereomers without disorder. Only the latter interpretation is consistent with the chemical and spectroscopic evidence. Refinements in the centrosymmetric and non-centrosymmetric space groups are compared in this paper and are shown to favour the chemically correct interpretation, more decisively so in the case of the bromo derivative as a result of the anomalous dispersion of bromine. Structures (1) and (3) provide a dramatic warning of the dangers inherent in the conventional wisdom that if a structure can be refined satisfactorarily in both centrosymmetric and non-centrosymmetric space groups, the former should always be chosen. In these two cases, despite apparently acceptable intensity statistics and R factors (5.87 and 3.55%), the choice of the centrosymmetric space group leads to the serious chemical error that the triclinic unit cell contains a racemate rather than two chiral diastereomers! The weakest reflections are shown to be most sensitive to the correct choice of space group, underlining the importance of refining against all data rather than against intensities greater than a specified threshold. The use of similar-distance restraints is shown to be beneficial in both P1 refinements. Di-o-bromobenzoyl-Emycin F crystallizes in the monoclinic space group P21 with one molecule in the asymmetric unit and so does not give rise to these problems of interpretation. The absolute configuration of the two bromo derivatives, and hence the Emycins in general, was determined unambiguously as S at the chiral centre C3.

Frequency distribution of space groups in soluble and membrane proteins and their complexes

2021 ◽  
Vol 77 (6) ◽  
pp. 187-191
Author(s):  
Rajneesh K. Gaur
Keyword(s):  
Membrane Proteins ◽  
Frequency Distribution ◽  
Data Bank ◽  
Frequency Distributions ◽  
Space Group ◽  
Space Groups ◽  
Group Frequency ◽  
Different Types ◽  
Analytical Assessment ◽  
Three Space

The space-group frequency distributions for two types of proteins and their complexes are explored. Based on the incremental availability of data in the Protein Data Bank, an analytical assessment shows a preferential distribution of three space groups, i.e. P212121 > P1211 > C121, in soluble and membrane proteins as well as in their complexes. In membrane proteins, the order of the three space groups is P212121 > C121 > P1211. The distribution of these space groups also shows the same pattern whether a protein crystallizes with a monomer or an oligomer in the asymmetric unit. The results also indicate that the sizes of the two entities in the structures of soluble proteins crystallized as complexes do not influence the frequency distribution of space groups. In general, it can be concluded that the space-group frequency distribution is homogenous across different types of proteins and their complexes.

Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces

2017 ◽  
Vol 73 (5) ◽  
pp. 387-402 ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Sajdeh Sajjadi ◽  
Bernard Shiffman ◽  
Steven M. Zucker
Keyword(s):  
General Theory ◽  
Rigid Body ◽  
Three Dimensional ◽  
Molecular Replacement ◽  
Translation Group ◽  
Common Occurrence ◽  
Space Group ◽  
Space Groups ◽  
Relevant Case ◽  
The Subject

In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

scholarly journals Application of the genetic algorithm to the task of compiling the curriculum

2018 ◽  
Author(s):  
V. A. Turchina ◽  
D. O. Tanasienko
Keyword(s):  
Higher Education ◽  
Genetic Algorithm ◽  
Genetic Algorithms ◽  
Necessary Conditions ◽  
Fitness Function ◽  
Educational Process ◽  
Heuristic Methods ◽  
Educational Groups ◽  
The Individual ◽  
Negative Property

One of the main tasks in organizing the educational process in higher education is the drawing up of a schedule of classes. It reflects the weekly student and faculty load. At the same time, when compiling, there are a number of necessary conditions and a number of desirable. The paper considers seven required and four desirable conditions. In this paper, one of the well-known approaches that can be used in drawing up a curriculum is consid-ered. The proposed scheme of the genetic algorithm, the result of which is to obtain an approximate solution to the problem of scheduling with the need to further improve it by other heuristic methods. To solve the problem, an island model of the genetic algorithm was selected and its advantages were considered. In the paper, the author's own structure of the individual, which includes chromosomes in the form of educational groups and genes as a lesson at a certain time, is presented and justified. The author presents his own implementations of the genetic algorithms. During the work, many variants of operators were tested, but they were rejected due to their inefficiency. The biggest problem was to maintain the consistency of information encoded in chromosomes. Also, two post-steps were added: to try to reduce the number of teacher conflict conflicts and to normalize the schedule - to remove windows from the schedule. The fitness function is calculated according to the following principles: if some desired or desired property is present in the individual, then a certain number is deducted from the individual's assessment, if there is a negative property, then a certain number is added to the assessment. Each criterion has its weight, so the size of the fine or rewards may be different. In this work, fines were charged for non-fulfillment of mandatory conditions, and rewards for fulfilling the desired

scholarly journals From space to superspace and back: Superspace Group Finder

2008 ◽  
Vol 41 (6) ◽  
pp. 1182-1186 ◽  
Author(s):  
Ivan Orlov ◽  
Lukas Palatinus ◽  
Gervais Chapuis
Keyword(s):  
Crystal Structure ◽  
Dimensional Space ◽  
Flexible Structures ◽  
Three Dimensional ◽  
Real Space ◽  
Space Group ◽  
Space Groups ◽  
Group A ◽  
Modular Composition ◽  
Three Dimensional Space

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.

2. Symmetry

2016 ◽  
pp. 24-38
Author(s):  
A. M. Glazer
Keyword(s):  
Crystal Structure ◽  
Reflection Point ◽  
Notation System ◽  
Space Group ◽  
Space Groups ◽  
Different Types ◽  
Point Groups

In order to explain what crystals are and how their structures are described, we need to understand the role of symmetry, for this lies at the heart of crystallography. ‘Symmetry’ explains the different types of symmetry: rotational, mirror or reflection, point, chiral, and translation. There are thirty-two point groups and seven crystal systems, according to which symmetries are present. These are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Miller indices, lattices, crystal structure, and space groups are described in more detail. Any normal crystal belongs to one of the 230 space group types. Crystallographers generally use the International Notation system to denote these space groups.

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