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.

Cruel Consciousness

2016 ◽  
Vol 71 (1) ◽  
pp. 89-114
Author(s):  
Jeanette Samyn
Keyword(s):  
Nineteenth Century ◽  
Natural World ◽  
John Ruskin ◽  
Men And Women ◽  
Moral Standards ◽  
Human Ethics ◽  
Ecological Thought ◽  
The Difference ◽  
The Way

Jeanette Samyn, “Cruel Consciousness: Louis Figuier, John Ruskin, and the Value of Insects” (pp. 89–114) This essay examines two opposing theories of consciousness and value in relation to nineteenth century entomology. In The Insect World (1868), the French popularizer of science Louis Figuier extends consciousness to aesthetically unappealing and seemingly cruel insects such as parasites by attributing to them sociality and industry. With little recourse to theological or conventional moral standards, Figuier ascribes value to parasites—on account of their consciousness, which aligns their experience with human sentience, and also because of their role as environmental mediators. In this view, he subtly paves the way for a biocentric approach to the natural world that remains controversial today. John Ruskin, meanwhile, brings up popular entomology (epitomized, he says, by Figuier’s text) as a complicated counter to his own views on labor and aesthetics in his letters to the working men and women of England, Fors Clavigera (1871–84). Questioning the contemporary “instinct” for the study of parasites—and despite recent associations of Ruskin with ecological thought—Ruskin takes pains in these letters to uphold the difference between human and nonhuman life. In his efforts to limit consciousness to the most valuable and difficult of human labors, however, he engages seriously with the implications of proto-parasitological thought for human ethics.

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.

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 Methods of space-group determination – a supplement dealing with twinned crystals and metric specialization

2015 ◽  
Vol 71 (10) ◽  
pp. 916-920 ◽  
Author(s):  
Howard D. Flack
Keyword(s):  
Single Crystals ◽  
Bravais Lattice ◽  
Space Group ◽  
Space Groups ◽  
Lattice Type ◽  
Unknown Structure ◽  
Direct Use

Tables for the determination of space group for single crystals, twinned crystals and crystals with a specialized metric are presented in the form of a spreadsheet for use on a computer. There are 14 tables, one for each of the Bravais-lattice types. The content of the tables is arranged so that at the intersection of rows, displaying the conditions for reflection, and of columns, displaying the Laue and crystal classes, one finds those space groups compatible with the observed Bravais-lattice type, the conditions for reflection and the Laue and crystal classes. The tables are intended to be of direct use to an experimentalist working with an unknown structure.

scholarly journals Different teaching way to space group and symmetry operation: polyhedral vs ball

2014 ◽  
Vol 70 (a1) ◽  
pp. C1389-C1389
Author(s):  
Jian Zhou ◽  
Hejing Wang
Keyword(s):  
Teaching Practice ◽  
Symmetry Plane ◽  
Symmetry Operation ◽  
The Other ◽  
Space Group ◽  
Fundamental Knowledge ◽  
Crystallographic Symmetry ◽  
Position Shift ◽  
Symmetry Operations

To introduce the regulations of space group combining with a symmetry operation, put an orientation ball at a position shift away from the lattice tops is a good way [1]. However, based on the fundamental knowledge of "lattice", it often occurs that the tops of a lattice "should be" the positions of "atom balls" thought by most beginnings in teaching practice. This "thought" leads them never deduce out those regulations in symmetry operations and often misleads a wrong conclusion. As a beginning one wishes watching movies and pictures instead of mathematical deduction or vector calculation. It easily arises that a lattice has eight tops with atom balls. This "idea" lets the orientation balls shifting away from the lattice tops become difficult to understand. Nevertheless, the balls with a sign of "comma" in the middle are also difficult to understand that they can stand for a certain orientation because ball is circle. "Tops" and "directions" are two troubles in learning crystallographic symmetry and symmetry operations for those beginnings. How to guide them to overcome the two fences is an important step that will lead those beginnings to a never understanding status, on one hand, or let them understand throughout all regulations of space group(s) combining with a symmetry operation on the other. From teaching practice, a polyhedral at lattice tops could overcome both difficulties at position and in orientation. First, a polyhedral is always in orientation, even it is a cubic. This is easily understood. Secondly the centre of a polyhedral could easily meet with the tops of a lattice; it lets students easily understand "a lattice has eight tops occupied – a natural thought by beginnings". This way let them easily understand and deduce all regulations in crystallographic symmetry operations, such as a body-centred lattice combining with a symmetry plane (m) produces n symmetry operation at 1/4t, etc. see figures below.

scholarly journals Symmetry determination following structure solution inP1

2008 ◽  
Vol 41 (6) ◽  
pp. 975-984 ◽  
Author(s):  
L. Palatinus ◽  
A. van der Lee
Keyword(s):  
Structure Factor ◽  
Symmetry Operation ◽  
Symmetry Analysis ◽  
New Method ◽  
Complete Space ◽  
Space Group ◽  
Structure Solution ◽  
Symmetry Operations

A new method for space-group determination is described. It is based on a symmetry analysis of the structure-factor phases resulting from a structure solution in space groupP1. The output of the symmetry analysis is a list of all symmetry operations compatible with the lattice. Each symmetry operation is assigned a symmetry agreement factor that is used to select the symmetry operations that are the elements of the space group of the structure. On the basis of the list of the selected operations the complete space group of the structure is constructed. The method is independent of the number of dimensions, and can also be used in solution of aperiodic structures. A number of cases are described where this method is particularly advantageous compared with the traditional symmetry analysis.

Objectcy and Agency

2018 ◽  
Author(s):  
Denis M. Walsh
Keyword(s):  
Adaptive Evolution ◽  
Natural World ◽  
Modern Synthesis ◽  
Scientific Theories ◽  
Object Theory ◽  
Theory Of Evolution ◽  
Agent Theory ◽  
Comprehensive Account ◽  
The Difference ◽  
The Way

Organisms are like nothing else in the natural world. They are agents. Methodological vitalism is a view according to which the difference that organisms make to the natural world cannot be captured wholly if we treat them as mere objects. Understanding agency calls for a different kind of theory, an agent theory. Most of our scientific theories are object theories. The modern synthesis theory of evolution is a prominent example of object theory. Being the way it is, it cannot countenance the contribution to evolution that organisms make as agents. A comprehensive account of adaptive evolution requires an agent theory.

Group actions as applied to symmetry breaking in hexagonal crystals with space group P63/m

1984 ◽  
Vol 62 (11) ◽  
pp. 1152-1173
Author(s):  
Rose M. Morra ◽  
Robin L. Armstrong
Keyword(s):  
Phase Transition ◽  
Structural Phase ◽  
Group Actions ◽  
Order Parameters ◽  
High Symmetry ◽  
Space Group ◽  
Space Groups ◽  
Symmetry Space ◽  
Low Symmetry ◽  
Symmetry Space Group

A procedure is outlined that allows one to predict the possible low-symmetry space groups for a commensurate phase transition associated with a broken symmetry. By considering each irreducible representation of the high-symmetry space group and using a subduction procedure involving the theory of group actions, one may determine a complete set of representative order parameters and their little groups. Each little group can be identified as a possible space group for the low-symmetry phase. The method is used to deduce all possible broken symmetries of space group P63/m that can result from a phase transition associated with one of the symmetry points in the Brillouin zone. As a specific application of the resulting tables of possible order parameters and associated low-symmetry space groups, changes in the X nuclear quadrupole resonance spectrum due to structural phase transitions in hexagonal AX3 crystals with the high-symmetry space group P63/m are considered.

scholarly journals Determination of Chirality of the Chiral Space Groups with Two-fold Screw Axis

2014 ◽  
Vol 70 (a1) ◽  
pp. C1700-C1700
Author(s):  
Kazuhiko Ishikawa ◽  
Masahito Tanaka ◽  
Motoo Shiro ◽  
Toru Asahi
Keyword(s):  
Optical Activity ◽  
Solid Material ◽  
Symmetry Operation ◽  
Linear Birefringence ◽  
Screw Axis ◽  
Space Group ◽  
Space Groups ◽  
Molecular Arrangement ◽  
The Absolute ◽  
Absolute Structure

Crystals are classified according to their symmetry operations into 230 space groups. A two-fold screw axis, 21, a symmetry operation, is frequently found in the crystal, especially non-centrosymmetric organic crystals. The two-fold screw axis shows a chiral character when molecules in the crystal tilt against the screw axis as shown in Figure. The crystals belonging to a chiral space group with two-fold screw axis such as a P21or P212121exhibit two different types with opposite chirality. However, although the two crystals are not identical in a molecular arrangement, we cannot distinguish them with the present notation of space group. Recently, the idea of determining the handedness of two-fold screw axis have been successfully proposed using hierarchical interpretation of the crystals.(I. Hisaki, T. Sasaki et al., 2012) Nevertheless, the issue on a notation to distinguish chiral crystals with two-fold screw axis still remains unsettled as far as we know. Therefore, we attempt to propose a novel notation for the crystals belonging to chiral space groups with two-fold screw axis. We focus on the relationship between the absolute structure and optical activity of the crystals. We have selected alanine crystals, which belong to P212121, as model crystals to discuss the notation. We have determined the absolute structure of the alanine crystals by X-ray diffraction and have measured their optical activity with Generalized High Accuracy Universal Polarimeter (G-HAUP).(M. Tanaka, N. Nakamura et al., 2012) G-HAUP is an apparatus that can measure simultaneously the linear birefringence, linear dichroism, circular birefringence, i.e., optical activity, and circular dichroism in any solid material. These experimental results have successfully correlated the absolute structure to the optical activity of the alanine crystals.

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