Net topologies, space groups, and crystal phases

2006 ◽  
Vol 221 (12) ◽  
Author(s):  
Georg Thimm ◽  
Björn Winkler
Keyword(s):  
Crystal Structures ◽  
Space Group ◽  
Space Groups ◽  
Quotient Graphs ◽  
Crystal Phases ◽  
Symmetry Operations

Quotient graphs and nets — the graph theore tical correspondences of cells and crystal structures — are reintroduced independent from crystal structures. Based on this, the issue of iso- and automorphism of nets, the graph theoretical equivalent of symmetry operations, is closely examined. A result, it is shown that the topology of a net (that is the bonds in a crystal) constrains severely the symmetry of the embedding (that is the crystal), and in the case of connected nets the space group except for the setting. Several examples are studied and conclusions on phases are drawn (pseudo-cubic FeS

A Priori Prediction of Crystal Structures

1996 ◽  
Vol 453 ◽  
Author(s):  
I. D. Brown
Keyword(s):  
Crystal Structures ◽  
Short Range ◽  
A Priori ◽  
Range Order ◽  
Space Group ◽  
Space Groups ◽  
Unit Space ◽  
Two Factors ◽  
Symmetry Operations ◽  
Principle Of Maximum

AbstractThe arrangement of atoms in a crystal is determined by two factors, the bonding preferences of individual atoms (giving rise to short range order), and the translational symmetry operations of the space group (giving rise to long range order). Chemical rules can be used to determine which atoms are bonded, and hence the maximum possible symmetry of the formula unit. Space group theory is then used to find the space groups that are compatible with this symmetry. In favourable cases, using the principle of maximum symmetry, the structure can be completely determined, but in all cases the analysis yields insights into the restrictions that determine what crystal structures might be possible.

On the treatment of settings of space-groups and crystal structures by spezialized short Hermann-Mauguin space-group symbols

2002 ◽  
Vol 217 (4) ◽  
Author(s):  
H. Burzlaff ◽  
H. Zimmermann
Keyword(s):  
Crystal Structures ◽  
Shift Vector ◽  
Space Group ◽  
Space Groups ◽  
Matrix Description ◽  
Simple Rules ◽  
The Matrix ◽  
Free Parameters ◽  
Generating Operators

AbstractFrom the short Hermann-Mauguin space-group symbol a set of generating operators can be derived. The matrix description of the operators depends on three free parameters related to the origin of the setting. Simple rules allow the specification of an origin, the origin of the symbol. The use of any other origin is notated by appending a shift vector from the symbol origin to the new one selected.

scholarly journals Partial rotational lattice order–disorder in stefin B crystals

2014 ◽  
Vol 70 (4) ◽  
pp. 1015-1025 ◽  
Author(s):  
Miha Renko ◽  
Ajda Taler-Verčič ◽  
Marko Mihelič ◽  
Eva Žerovnik ◽  
Dušan Turk
Keyword(s):  
Crystal Lattice ◽  
Crystal Structures ◽  
Electron Density ◽  
Rejection Rate ◽  
Lattice Order ◽  
Space Group ◽  
Space Groups ◽  
Additional Layer ◽  
Diffraction Patterns ◽  
Ordered Layers

At present, the determination of crystal structures from data that have been acquired from twinned crystals is routine; however, with the increasing number of crystal structures additional crystal lattice disorders are being discovered. Here, a previously undescribed partial rotational order–disorder that has been observed in crystals of stefin B is described. The diffraction images revealed normal diffraction patterns that result from a regular crystal lattice. The data could be processed in space groupsI4 andI422, yet one crystal exhibited a notable rejection rate in the higher symmetry space group. An explanation for this behaviour was found once the crystal structures had been solved and refined and the electron-density maps had been inspected. The lattice of stefin B crystals is composed of five tetramer layers: four well ordered layers which are followed by an additional layer of alternatively placed tetramers. The presence of alternative positions was revealed by the inspection of electron-density score maps. The well ordered layers correspond to the crystal symmetry of space groupI422. In addition, the positions of the molecules in the additional layer are related by twofold rotational axes which correspond to space groupI422; however, these molecules lie on the twofold axis and can only be related in a statistical manner. When the occupancies of alternate positions and overlapping are equal, the crystal lattice indeed fulfills the criteria of space groupI422; when these occupancies are not equal, the lattice only fulfills the criteria of space groupI4.

The crystallographic chameleon: when space groups change skin

2016 ◽  
Vol 72 (5) ◽  
pp. 523-538 ◽  
Author(s):  
Massimo Nespolo ◽  
Mois I. Aroyo
Keyword(s):  
Space Group ◽  
Practical Applications ◽  
Space Groups ◽  
Common Substructure ◽  
Group Information ◽  
Alternative Setting ◽  
The Common ◽  
Alternative Settings ◽  
Unnecessary Complication ◽  
Symmetry Operations

VolumeAofInternational Tables for Crystallographyis the reference for space-group information. However, the content is not exhaustive because for many space groups a variety of settings may be chosen but not all of them are described in detail or even fully listed. The use of alternative settings may seem an unnecessary complication when the purpose is just to describe a crystal structure; however, these are of the utmost importance for a number of tasks, such as the investigation of structure relations between polymorphs or derivative structures, the study of pseudo-symmetry and its potential consequences, and the analysis of the common substructure of twins. The aim of the article is twofold: (i) to present a guide to expressing the symmetry operations, the Hermann–Mauguin symbols and the Wyckoff positions of a space group in an alternative setting, and (ii) to point to alternative settings of space groups of possible practical applications and not listed in VolumeAofInternational Tables for Crystallography.

scholarly journals VI. Tabulated data for the examination of the 230 space-groups by homogeneous X-rays

1924 ◽  
Vol 224 (616-625) ◽  
pp. 221-257 ◽  
Keyword(s):  
Crystal Structures ◽  
Chemical Society ◽  
Unit Cell ◽  
X Rays ◽  
Screw Axis ◽  
Monoclinic System ◽  
Space Group ◽  
X Ray ◽  
Space Groups ◽  
Relative Arrangement

The object of the present paper is to express the conclusions of mathematical crystallography in a form which shall be immediately useful to workers using homogeneous X-rays for the analysis of crystal structures. The results are directly applicable to such methods as the Bragg ionisation method, the powder method, the rotating crystal method, etc., and summarise in as compact a form as possible what inferences may be made from the experimental observations, whichever one of the 230 possible space-groups may happen to be under examination. It is only in certain cases that the spacings of crystal planes as determined by the aid of homogeneous X-rays agree with the values of those spacings which would be expected from ordinary crystallographic calculations. In the majority of cases the relative arrangement of the molecules in the unit cell leads to apparent anomalies in the experimental results, the observed spacings of certain planes or sets of planes being sub-multiples of the calculated spacings. The simplest case (fig. 8) of such an apparent anomaly is found in the space-group C 2 2 of the monoclinic system, where the presence of a two-fold screw-axis, because it interleaves halfway the (010) planes by molecules which are exactly like those lying in the (010) planes, except that they have been rotated through 180°, leads to an observed periodicity which is half the periodicity to be inferred from the dimensions of the unit cell, that is, leads to an observed spacing for (010) which is half the calculated. All screw-axes produce similar results, and, in general, a p -fold screw-axis leads to an observed spacing for the plane perpendicular to it which is 1/ p th that to be inferred from the dimensions of the cell. Besides those produced by the screw-axes, other abnormalities arise out of the presence of glide-planes. The simplest case of this is shown by the space-group C s 2 (fig. 4) of the monoclinic system, in which the second molecule is obtained from the first by a reflection in a plane parallel to (010) and half a primitive translation parallel to that plane. If we look along a direction perpendicular to this glide-plane, the projections of the two molecules on the (010) plane are indistinguishable except in position, which is equivalent to saying that, for the purposes of X-ray interference, certain planes perpendicular to this plane of projection are interleaved by an identical molecular distribution. Furthermore, since the translation associated with the glide-plane must always be half a primitive translation parallel to the glide-plane, we know that the interleaving is always a submultiple of the full spacing and the periodicity is again reduced in a corresponding manner. The use of this method for discriminating between the various space-groups of the monoclinic system was described by Sir Wm. Bragg in a lecture to the Chemical Society. In the present paper the method has been extended to the whole of the 230 space-groups possible to crystalline structures. In general, it may be said that if a crystal possesses a certain glide-plane, a certain set of planes lying in the zone whose axis is perpendicular to that glide-plane will have their periodicity reduced by one-half.

scholarly journals Some more space-group corrections

2014 ◽  
Vol 70 (9) ◽  
pp. 834-836 ◽  
Author(s):  
Lawrence M. Henling ◽  
Richard E. Marsh
Keyword(s):  
Crystal Structures ◽  
Cambridge Structural Database ◽  
Space Group ◽  
Space Groups ◽  
Structural Database

A survey of approximately 100 000 entries in recent releases of the Cambridge Structural Database (CSD) has uncovered 156 crystal structures that were apparently described in inappropriate space groups. We have revised these space groups and prepared CIFs containing the new coordinates and brief comments describing the revisions.

Extension of three-dimensional space-group generators to the (3+1) case

1999 ◽  
Vol 32 (3) ◽  
pp. 452-455
Author(s):  
Kazimierz Stróż
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Space Group ◽  
Space Groups ◽  
Symmetry Operations ◽  
Three Dimensional Space

A method of building up the generators of 775 (3+1)-dimensional superspace groups is proposed. The generators are based on the conventional space-group generators selected by Wondratschek and applied in theInternational Tables for Crystallography(1995, Vol. A). By the method, the generation of (3+1) space groups is found to be easier, the description of symmetry operations is closer to that used for the conventional space groups, and ambiguities in the (3+1) group notation are avoided.

Matching selenium-atom peak positions with a different hand or origin

2002 ◽  
Vol 35 (3) ◽  
pp. 368-370 ◽  
Author(s):  
G. David Smith
Keyword(s):  
Computer Program ◽  
Iterative Procedure ◽  
Polar Axis ◽  
Polar Space ◽  
Selenium Atom ◽  
Space Group ◽  
Space Groups ◽  
Fortran 90 ◽  
Symmetry Operations

An algorithm is described for matching and correlating two or more sets of peaks or atoms. The procedure is particularly useful for matching putative selenium atoms from a selenium-atom substructure as obtained fromEmaps from two or more random-atom trials. The algorithm will work for any space group exceptP1. For non-polar space groups, the procedure is relatively straightforward. For polar space groups, the calculation is performed in projection along the polar axis in order to identify potential matching peaks, and an iterative procedure is used to eliminate incorrect peaks and to calculate the displacement along the polar axis. The algorithm has been incorporated into a computer program,NANTMRF, written in Fortran 90. Less than 0.5 s are required to match 27 peaks in space groupP21, and the output lists the correct origin, enantiomorph, symmetry operations, and provides the relative displacements between pairs of matching peaks.

scholarly journals Mathematical Crystallography in the 21st century

2014 ◽  
Vol 70 (a1) ◽  
pp. C18-C18
Author(s):  
Marjorie Senechal
Keyword(s):  
Crystal Structures ◽  
21St Century ◽  
New Materials ◽  
Space Group ◽  
Space Groups ◽  
Group Assignment ◽  
New Directions

The solution of simple crystal structures in 1914 sent crystallographers to the library stacks to dig out, dust off, and learn to use the lattices and space groups that mathematicians had discovered in the century before. In 2014 space group assignment is a job for computers, but mathematical crystallography is anything but routine. New materials stretch the boundaries of "pattern," simulations stretch the boundaries of "possible," and mathematicians study structures on all scales, from nano to n-dimensional. In this talk I will outline challenges and new directions posed by 21st century crystallography.

A list of organic kryptoracemates

2010 ◽  
Vol 66 (1) ◽  
pp. 94-103 ◽  
Author(s):  
László Fábián ◽  
Carolyn Pratt Brock
Keyword(s):  
Crystal Structures ◽  
Inversion Center ◽  
Racemic Compound ◽  
Ordered Structures ◽  
Compound A ◽  
Space Groups ◽  
Structural Database ◽  
Relationship Of ◽  
Symmetry Operations ◽  
Centrosymmetric Molecule

A list of 181 organic kryptoracemates has been compiled. This class of crystallographic oddities is made up of racemic compounds (i.e. pairs of resolvable enantiomers) that happen to crystallize in Sohnke space groups (i.e. groups that include only proper symmetry operations). Most (151) of the 181 structures could have crystallized as ordered structures in non-Sohnke groups. The remaining 30 structures do not fully meet this criterion but would have been classified as kryptoracemates by previous authors. Examples were found and checked with the aid of available software for searching the Cambridge Structural Database, for generating and comparing InChI strings, and for validating crystal structures. The pairs of enantiomers in the true kryptoracemates usually have very similar conformations; often the match is near-perfect. There is a pseudosymmetric relationship of the enantiomers in about 60% of the kryptoracemate structures, but the deviations from inversion or glide symmetry are usually quite easy to spot. Kryptoracemates were found to account for 0.1% of all organic structures containing either a racemic compound, a meso molecule, or some other achiral molecule. The centroid of a pair of enantiomers is more likely (99.9% versus 99% probability) to be located on an inversion center than is the centroid of a potentially centrosymmetric molecule.

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