Symmetry

2004 ◽  
Author(s):  
Robert E. Newnham
Keyword(s):  
Single Crystals ◽  
Point Group ◽  
Three Dimensional ◽  
Unit Cell ◽  
Translational Symmetry ◽  
Identity Operator ◽  
Space Groups ◽  
Rotation Axes ◽  
Point Groups ◽  
And Inversion

All single crystals possess translational symmetry, and most possess other symmetry elements as well. In this chapter we describe the 32 crystallographic point groups used for single crystals. The seven Curie groups used for textured polycrystalline materials are enumerated in the next chapter. We live in a three-dimensional world which means that there are basically four kinds of geometric symmetry operations relating one part of this world to another. The four primary types of symmetry are translation, rotation, reflection, and inversion. As pictured in Fig. 3.1, these symmetry operators operate on a point with coordinates Z1, Z2, Z3 and carry it to a new position. By definition, all crystals have a unit cell that is repeated many times in space, a point Z1, Z2, Z3 is repeated over and over again as one unit cell is translated to the next. A mirror plane perpendicular to one of the principal axes is a two-dimensional symmetry element that reverses the sign of one coordinate. Rotation axes are one-dimensional symmetry elements that change two coordinates, while an inversion center is a zero-dimensional point that changes all three coordinates. In developing an understanding of the macroscopic properties of crystals, we recognize that the scale of physical property measurements is much larger than the unit cell dimensions. It is for this reason that we are not concerned about translational symmetry and work with the 32 point group symmetries rather than the 230 space groups. This greatly simplifies the structure–property relationships in crystal physics. Aside from the identity operator 1, there are only four types of rotational symmetry consistent with the translation symmetry common to all crystals. Fig. 3.2 shows why. Parallelograms, equilateral triangles, squares, and hexagons will pack together to fill space but, pentagons symmetry axes are found in crystals. This is the starting point for generating the 32 crystal classes. When taken in combination with mirror planes and inversion centers, these four types of rotation axes are capable of forming 32 self-consistent three-dimensional symmetry patterns around a point. These are the so-called 32 crystal classes or crystallographic point groups.

Kristallographische Daten von Quecksüber(I)-acetat

1965 ◽  
Vol 122 (1-6) ◽  
pp. 156-158
Author(s):  
H. Puff ◽  
G. Lorbacher ◽  
B. Skrabs
Keyword(s):  
Single Crystals ◽  
Unit Cell ◽  
Space Groups ◽  
Monoclinic Unit Cell ◽  
Probable Space

Abstract From single crystals of Hg2(CH2COO)2 a monoclinic unit-cell with a = 5.184, 6 = 5.962, c = 12.16, Å, β = 100.05° was established which contains two formula units. The probable space groups are C3 2h-C2/m, C3 3-Cm or C3 2-C2.

scholarly journals Point-group symmetry detection in three-dimensional charge density of biomolecules

2019 ◽  
Vol 36 (7) ◽  
pp. 2237-2243
Author(s):  
Cyril F Reboul ◽  
Simon Kiesewetter ◽  
Dominika Elmlund ◽  
Hans Elmlund
Keyword(s):  
Charge Density ◽  
Single Particle ◽  
Statistical Tests ◽  
Point Group ◽  
Three Dimensional ◽  
Group Symmetry ◽  
Point Group Symmetry ◽  
Density Maps ◽  
Point Groups ◽  
Density Map

Abstract Motivation No rigorous statistical tests for detecting point-group symmetry in three-dimensional (3D) charge density maps obtained by electron microscopy (EM) and related techniques have been developed. Results We propose a method for determining the point-group symmetry of 3D charge density maps obtained by EM and related techniques. Our ab initio algorithm does not depend on atomic coordinates but utilizes the density map directly. We validate the approach for a range of publicly available single-particle cryo-EM datasets. In straightforward cases, our method enables fully automated single-particle 3D reconstruction without having to input an arbitrarily selected point-group symmetry. When pseudo-symmetry is present, our method provides statistics quantifying the degree to which the 3D density agrees with the different point-groups tested. Availability and implementation The software is freely available at https://github.com/hael/SIMPLE3.0.

scholarly journals Crystal structure of the coordination polymer [FeIII2{PtII(CN)4}3]

2015 ◽  
Vol 71 (1) ◽  
pp. i1-i2
Author(s):  
Maksym Seredyuk ◽  
M. Carmen Muñoz ◽  
José A. Real ◽  
Turganbay S. Iskenderov
Keyword(s):  
Crystal Structure ◽  
Coordination Polymer ◽  
Point Group ◽  
Three Dimensional ◽  
Title Complex ◽  
Group Symmetry ◽  
Point Group Symmetry ◽  
Rotation Axes ◽  
Square Planar ◽  
Cyanide Ligands

The title complex, poly[dodeca-μ-cyanido-diiron(III)triplatinum(II)], [FeIII2{PtII(CN)4}3], has a three-dimensional polymeric structure. It is built-up from square-planar [PtII(CN)4]2−anions (point group symmetry 2/m) bridging cationic [FeIIIPtII(CN)4]+∞layers extending in thebcplane. The FeIIatoms of the layers are located on inversion centres and exhibit an octahedral coordination sphere defined by six N atoms of cyanide ligands, while the PtIIatoms are located on twofold rotation axes and are surrounded by four C atoms of the cyanide ligands in a square-planar coordination. The geometrical preferences of the two cations for octahedral and square-planar coordination, respectively, lead to a corrugated organisation of the layers. The distance between neighbouring [FeIIIPtII(CN)4]+∞layers corresponds to the lengtha/2 = 8.0070 (3) Å, and the separation between two neighbouring PtIIatoms of the bridging [PtII(CN)4]2−groups corresponds to the length of thecaxis [7.5720 (2) Å]. The structure is porous with accessible voids of 390 Å3per unit cell.

scholarly journals Crystal structures of Sr(ClO4)2·3H2O, Sr(ClO4)2·4H2O and Sr(ClO4)2·9H2O

2014 ◽  
Vol 70 (12) ◽  
pp. 510-514 ◽  
Author(s):  
Erik Hennings ◽  
Horst Schmidt ◽  
Wolfgang Voigt
Keyword(s):  
Point Group ◽  
Three Dimensional ◽  
Group Symmetry ◽  
Water Molecules ◽  
Point Group Symmetry ◽  
Rotation Axes ◽  
Dimensional Network ◽  
Tricapped Trigonal Prism ◽  
Trigonal Prismatic Coordination ◽  
Solid Liquid

The title compounds, strontium perchlorate trihydrate {di-μ-aqua-aquadi-μ-perchlorato-strontium, [Sr(ClO4)2(H2O)3]n}, strontium perchlorate tetrahydrate {di-μ-aqua-bis(triaquadiperchloratostrontium), [Sr2(ClO4)4(H2O)8]} and strontium perchlorate nonahydrate {heptaaquadiperchloratostrontium dihydrate, [Sr(ClO4)2(H2O)7]·2H2O}, were crystallized at low temperatures according to the solid–liquid phase diagram. The structures of the tri- and tetrahydrate consist of Sr2+cations coordinated by five water molecules and four O atoms of four perchlorate tetrahedra in a distorted tricapped trigonal–prismatic coordination mode. The asymmetric unit of the trihydrate contains two formula units. Two [SrO9] polyhedra in the trihydrate are connected by sharing water molecules and thus forming chains parallel to [100]. In the tetrahydrate, dimers of two [SrO9] polyhedra connected by two sharing water molecules are formed. The structure of the nonahydrate contains one Sr2+cation coordinated by seven water molecules and by two O atoms of two perchlorate tetrahedra (point group symmetry ..m), forming a tricapped trigonal prism (point group symmetrym2m). The structure contains additional non-coordinating water molecules, which are located on twofold rotation axes. O—H...O hydrogen bonds between the water molecules as donor and ClO4tetrahedra and water molecules as acceptor groups lead to the formation of a three-dimensional network in each of the three structures.

scholarly journals Crystal structure of chain silicate Cs3LuSi3O9

2021 ◽  
Vol 77 (12) ◽  
Author(s):  
Hiromitsu Kimura ◽  
Hisanori Yamane
Keyword(s):  
Crystal Structure ◽  
Single Crystals ◽  
Title Compound ◽  
Three Dimensional ◽  
Unit Cell ◽  
Single Chain ◽  
Orthorhombic Space Group ◽  
Space Group ◽  
Chain Silicate ◽  
A Chain

A caesium lutetium(III) silicate, Cs3LuSi3O9, was synthesized by heating a pelletized mixture of Cs2CO3, Lu2O3 and SiO2 at 1273 K. Single crystals of the title compound were grown in a melted area of the pellet. Cs3LuSi3O9 is a single-chain silicate (orthorhombic space group Pna21) with a chain periodicity of six and is isostructural with Cs3 RE IIIGe3O9 (RE = Pr, Nd and Sm–Yb). The two symmetry-dependent [Si6O18]12− chains in the unit cell lie parallel to the [011] direction. The Lu3+ ions are octahedrally coordinated by O atoms of the silicate chains, generating a three-dimensional framework. Cs+ ions are located in the voids in the framework.

scholarly journals Color-Coding Point Group & Space Group Diagrams and other Mathematical Journeys

2014 ◽  
Vol 70 (a1) ◽  
pp. C1275-C1275
Author(s):  
Maureen Julian
Keyword(s):  
General Position ◽  
Point Group ◽  
Three Dimensional ◽  
Irreducible Representations ◽  
Color Coding ◽  
Space Group ◽  
Point Groups ◽  
Symmetry Operations

Color clarifies diagrams point group and space group diagrams. For example, consider the general position diagrams and the symbol diagrams. Symmetry operations can be represented by matrices whose determinant is either plus one or minus one. In the former case there is no change of handedness and in the latter case there is a change of handedness. The general position diagrams emphasis this information by color-coding. The symbol diagrams are a little more complicated and will be demonstrated. The second topic is a comparison of the thirty-two three-dimensional point groups with their corresponding 18 abstract mathematical groups. The corresponding trees will be explored. This discussion leads into the topic of irreducible representations.

Cryo-Electron Microscopy and Image Processing Methods for Studying the Ribonucleoprotein Vault

1998 ◽  
Vol 4 (S2) ◽  
pp. 976-977
Author(s):  
L. B. Kong ◽  
L. H. Rome ◽  
P. L. Stewart
Keyword(s):  
Software Package ◽  
Point Group ◽  
Three Dimensional ◽  
Particle Orientation ◽  
Major Vault Protein ◽  
Ribonucleoprotein Complexes ◽  
Cryo Electron Microscopy ◽  
Point Groups ◽  
Ribonucleoprotein Vault ◽  
Particle Images

Vaults are highly conserved ribonucleoprotein complexes found in thousands of copies per cell in various tissues. The major vault protein, pi 04, accounts for over 70% of the particle mass and has been recently found to be involved in cancer cell multidrug resistance. We have utilized cryoelectron microscopy (cryo-EM), and three-dimensional image reconstruction to generate a 3-D structure of the vault particle. Individual particle images were isolated using the QVIEW software package, which simultaneously performs density exclusion, planar background subtraction, and application of an elliptical mask. The IMAGIC-5 software package was used for the subsequent particle orientation and reconstruction steps (Fig. I).A symmetry self-search test was used to measure the signal strength for a variety of point groups and the highest correlation was found for the cyclic 8-fold point group. We have utilized the sinogram correlation algorithms in IMAGIC-5 to find the Euler angles of both 1 μm and 2 μm defocus particle images.

Quasicrystals Perspectives and Potential Applications

MRS Bulletin ◽  
1997 ◽  
Vol 22 (11) ◽  
pp. 34-39 ◽  
Author(s):  
Daniel J. Sordelet ◽  
Jean Marie Dubois
Keyword(s):  
Window Glass ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Unit Cell ◽  
Translational Symmetry ◽  
Integer Number ◽  
Unit Cells ◽  
Potential Applications ◽  
Incommensurate Structures ◽  
Three Dimensional Space

For decades scientists have accepted the premise that solid matter can only order in two ways: amorphous (or glassy) like window glass or crystalline with atoms arranged according to translational symmetry. The science of crystallography, now two centuries old, was able to relate in a simple and efficient way all atomic positions within a crystal to a frame of reference in which a single unit cell was defined. Positions within the crystal could all be deduced from the restricted number of positions in the unit cell by translations along vectors formed by a combination of integer numbers of unit vectors of the reference frame. Of course disorder, which is always present in solids, could be understood as some form of disturbance with respect to this rule of construction. Also amorphous solids were naturally referred to as a full breakdown of translational symmetry yet preserving most of the short-range order around atoms. Incommensurate structures, or more simply modulated crystals, could be understood as the overlap of various ordering potentials not necessarily with commensurate periodicities.For so many years, no exception to the canonical rule of crystallography was discovered. Any crystal could be completely described using one unit cell and its set of three basis vectors. In 1848 the French crystallographer Bravais demonstrated that only 14 different ways of arranging atoms exist in three-dimensional space according to translational symmetry. This led to the well-known cubic, hexagonal, tetragonal, and associated structures. Furthermore the dihedral angle between pairs of faces of the unit cell cannot assume just any number since an integer number of unit cells must completely fill space around an edge.

scholarly journals Chaotic mixing in crystalline granular media

2019 ◽  
Vol 871 ◽  
pp. 562-594 ◽  
Author(s):  
Régis Turuban ◽  
Daniel R. Lester ◽  
Joris Heyman ◽  
Tanguy Le Borgne ◽  
Yves Méheust
Keyword(s):  
Granular Media ◽  
Point Group ◽  
Three Dimensional ◽  
Chaotic Mixing ◽  
Mean Flow ◽  
Symmetry Point Group ◽  
Crystalline Structures ◽  
Space Groups ◽  
Pore Networks ◽  
Bcc Lattice

We study the Lagrangian kinematics of steady three-dimensional Stokes flow over simple cubic (SC) and body-centred cubic (BCC) lattices of close-packed spheres, and uncover the mechanisms governing chaotic mixing in these crystalline structures. Due to the cusp-shaped sphere contacts, the topology of the skin friction field is fundamentally different to that of continuous (non-granular) media, such as open pore networks, with significant implications for fluid mixing. Weak symmetry breaking of the flow orientation with respect to the lattice symmetries imparts a transition from regular to strong chaotic mixing in the BCC lattice, whereas the SC lattice only exhibits weak mixing. Whilst the SC and BCC lattices posses the same symmetry point group, these differences are explained in terms of their space groups. This insight is used to develop accurate predictions of the Lyapunov exponent distribution over the parameter space of mean flow orientation. These results point to a general theory of mixing and dispersion based upon the inherent symmetries of arbitrary crystalline structures.

scholarly journals The chromatic symmetry of twins and allotwins

2019 ◽  
Vol 75 (3) ◽  
pp. 551-573 ◽  
Author(s):  
Massimo Nespolo
Keyword(s):  
Point Group ◽  
Symmetry Operation ◽  
Space Groups ◽  
Different Types ◽  
Point Groups ◽  
Full Symmetry

The symmetry of twins is described by chromatic point groups obtained from the intersection group {\cal H}^* of the oriented point groups of the individuals {\cal H}_i extended by the operations mapping different individuals. This article presents a revised list of twin point groups through the analysis of their groupoid structure, followed by the generalization to the case of allotwins. Allotwins of polytypes with the same type of point group can be described by a chromatic point group like twins. If the individuals are all differently oriented, the chromatic point group is obtained in the same way as in the case of twins; if they are mapped by symmetry operation of the individuals, the chromatic point group is neutral. If the same holds true for some but not all individuals, then the allotwin can be seen as composed of twinned regions described by a twin point group, that are then allotwinned and described by a colour identification group; the allotwin is then described by a chromatic group obtained as an extension of the former by the latter, and requires the use of extended symbols reminiscent of the extended Hermann–Mauguin symbols of space groups. In the case of allotwins of polytypes with different types of point groups, as well as incomplete (allo)twins, a chromatic point group does not reveal the full symmetry: the groupoid has to be specified instead.

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