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.

Algorithms for deriving crystallographic space-group information

1999 ◽  
Vol 55 (2) ◽  
pp. 383-395 ◽  
Author(s):  
R. W. Grosse-Kunstleve
Keyword(s):  
Three Dimensional ◽  
Basis Matrix ◽  
Space Group ◽  
Space Groups ◽  
Group Type ◽  
Group Information ◽  
Commercial Applications ◽  
Crystallographic Space Groups ◽  
Symmetry Operations

Algorithms are presented for three-dimensional crystallographic space groups, handling tasks such as the generation of symmetry operations, the characterization of symmetry operations (determination of rotation-part type, axis direction, sense of rotation, screw or glide part and location part), the determination of space-group type [identified by the space-group number of the International Tables for Crystallography (Dordrecht: Kluwer Academic Publishers)] and the generation of structure-seminvariant vectors and moduli. The latter are an algebraic description of allowed origin shifts, which are important in crystal structure determination methods or for comparing crystal structures. The space-group type determination produces a change-of-basis matrix which transforms a given space-group representation to the standard one according to the International Tables for Crystallography. The algorithms were implemented and tested using the SgInfo library. The source code is free for non-commercial applications.

Non-centrosymmetric racemates: space-group frequencies and conformational similarities between crystallographically independent molecules

2000 ◽  
Vol 56 (4) ◽  
pp. 715-719 ◽  
Author(s):  
Bjørn Dalhus ◽  
Carl Henrik Görbitz
Keyword(s):  
Cambridge Structural Database ◽  
Asymmetric Unit ◽  
Space Group ◽  
Space Groups ◽  
Common Space ◽  
Molecular Superposition ◽  
Structural Database ◽  
The Common ◽  
The Mean ◽  
Independent Molecules

DL-Allylglycine (DL-2-amino-4-pentenoic acid, C5H9NO2) yields crystals with Pca21 symmetry and two crystallographically independent yet pseudo-inversion-related enantiomers. The distribution among the common space groups of other crystalline racemates with more than one molecule in the asymmetric unit has been established. The conformational similarities between crystallographically independent enantiomers in 114 non-centrosymmetric racemates were quantified using the r.m.s. deviation for a molecular superposition. The analysis shows that in the majority of crystals the conformations of the crystallographically independent molecules are very similar with mean r.m.s. deviation = 0.190 Å. In almost 80% of the structures the mean r.m.s. deviations is in the interval 0–0.2 Å. It is estimated that racemates constitute 23% of the centrosymmetric organic structures in the Cambridge Structural Database.

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.

SPGGEN: a computer program for retrieval of space-group information in several settings and generator-containing space-group symbols

2016 ◽  
Vol 49 (4) ◽  
pp. 1370-1376 ◽  
Author(s):  
Uri Shmueli
Keyword(s):  
Computer Program ◽  
Dimensional Space ◽  
Three Dimensional ◽  
General Setting ◽  
Detailed Comparison ◽  
Space Group ◽  
Space Groups ◽  
Group Number ◽  
Group Information ◽  
Three Dimensional Space

A brief outline of the algorithm for the derivation of a space group is followed by a detailed description of the explicit space-group symbols here employed. These space-group symbols are unique insofar as they contain explicitly the generators of the space group dealt with. Next, the implementation of the above in a computer program,SPGGEN, is briefly discussed and the options presented by the program are outlined. Briefly, these options are (i) conventional derivation of the space group from an explicit symbol, including a user-defined one; (ii) such derivation from the conventional space-group number only; (iii) introduction of a general setting into the derivation; (iv) introduction of a Cartesian setting into the derivation; and (v) treatment of some non-conventional settings of orthorhombic space groups. This is followed by a detailed comparison withInternational Tables for Crystallography, Vol. A, and by examples of the output ofSPGGEN. A complete tabulation of the explicit three-dimensional space-group symbols is readily accessed.

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.

Symmetry and space Groups

2010 ◽  
Author(s):  
Jenny Pickworth Glusker ◽  
Kenneth N. Trueblood
Keyword(s):  
Fundamental Property ◽  
Natural World ◽  
Symmetry Operation ◽  
Bravais Lattice ◽  
High Symmetry ◽  
Space Group ◽  
Space Groups ◽  
The Difference ◽  
Symmetry Operations ◽  
The Way

A certain degree of symmetry is apparent in much of the natural world, as well as in many of our creations in art, architecture, and technology. Objects with high symmetry are generally regarded with pleasure. Symmetry is perhaps the most fundamental property of the crystalline state and is a reason that gemstones have been so appreciated throughout the ages. This chapter introduces some of the fundamental concepts of symmetry—symmetry operations, symmetry elements, and the combinations of these characteristics of finite objects (point symmetry) and infinite objects (space symmetry)—as well as the way these concepts are applied in the study of crystals. An object is said to be symmetrical if after some movement, real or imagined, it is or would be indistinguishable (in appearance and other discernible properties) from the way it was initially. The movement, which might be, for example, a rotation about some fixed axis or a mirror-like reflection through some plane or a translation of the entire object in a given direction, is called a symmetry operation. The geometrical entity with respect to which the symmetry operation is performed, an axis or a plane in the examples cited, is called a symmetry element. Symmetry operations are actions that can be carried out, while symmetry elements are descriptions of possible symmetry operations. The difference between these two symmetry terms is important. It is possible not only to determine the crystal system of a given crystalline specimen by analysis of the intensities of the Bragg reflections in the diffraction pattern of the crystal, but also to learn much more about its symmetry, including its Bravais lattice and the probable space group. As indicated in Chapter 2, the 230 space groups represent the distinct ways of arranging identical objects on one of the 14 Bravais lattices by the use of certain symmetry operations to be described below. The determination of the space group of a crystal is important because it may reveal some symmetry within the contents of the unit cell.

Alkoxistannate, II [1] Tri(tert-butoxi)alkalistannate(II): Darstellung und Strukturen / Alkoxistannate, II [1] Tri(tert-butoxi)alkalistannates(II): Synthesis and Structures

1986 ◽  
Vol 41 (9) ◽  
pp. 1071-1080 ◽  
Author(s):  
M. Veith ◽  
R. Rosier
Keyword(s):  
Orthorhombic Crystal ◽  
Triclinic Space Group ◽  
Space Group ◽  
Space Groups ◽  
Average Value ◽  
Trigonal Bipyramidal ◽  
Oxygen Coordination ◽  
The Common ◽  
The One ◽  
Oxygen Bond

Abstract Tri(tert-butoxi)alkalistannates (M(O'Bu)3Sn, M = Li, Na, K, Rb, Cs) are obtained by reaction of alkali-fert-butanolates with tindi-tert-butoxide. If M equals Li or Na (1, 2) molecular com pounds are formed, which consist of two formula units. 1 crystallizes in a m onoclinic cell (space group P21/c; a = 966.5(3), b = 1819(1), c = 1014(1) pm. β = 107.1(1)°, Z = 4); 2 is triclinic (space group P1̄; a = 1041(1), b = 2046(1), c = 1033(1) pm. a = 92.3(2), β = 118.6(1), y = 108.3(3)° and Z = 4). The molecules 1 and 2 are closely related structurally despite their different space groups. The common structural feature is a Sn2O6M2 cage, which is built of two seco-norcubane Sn2M2O , units, sharing a M2O2 four-membered ring. Characteristic distances are: 1: Sn-O = 209.3, L i-O = 193.0 and 211.6 pm, 2: Sn-O = 210.5, Na-O = 227.4 and 240.7 pm. The tert/-butoxistannates of K, Rb and Cs (3, 4, 5) all crystallize in the orthorhombic crystal system, space group P212121 (cell constants of 3: a = 1907(1), b - 1060(1), c - 896(1) pm, Z = 4). Contrary to the lithium- and sodium derivates 3, 4 and 5 have a polymeric structure. The one dimensional polymer consists of distorted trigonal bipyramidal SnO3M “cages” (substituted at the oxygen atoms by tertbutyl groups), which align in a way to allow the metal atom to have a five-fold oxygen coordination. The tin atoms have trigonal pyramidal coordination. While the tin-oxygen bond lengths are essentially invariant (average value 206.7 pm ), the potassium-oxygen distances range from 256.4(5) pm to 318.8(6) pm

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.

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

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 .

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