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.

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.

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.

Symmetry breaking in convergent-beam diffraction

1989 ◽  
Vol 47 ◽  
pp. 480-481
Author(s):  
D.J. Eaglesham
Keyword(s):  
Symmetry Breaking ◽  
Point Group ◽  
X Ray Diffraction ◽  
Multiple Diffraction ◽  
Space Groups ◽  
Atomic Displacements ◽  
Small Probe ◽  
Phase Sensitive ◽  
Convergent Beam

Convergent Beam Electron Diffraction is now almost routinely used in the determination of the point- and space-groups of crystalline samples. In addition to its small-probe capability, CBED is also postulated to be more sensitive than X-ray diffraction in determining crystal symmetries. Multiple diffraction is phase-sensitive, so that the distinction between centro- and non-centro-symmetric space groups should be trivial in CBED: in addition, the stronger scattering of electrons may give a general increase in sensitivity to small atomic displacements. However, the sensitivity of CBED symmetry to the crystal point group has rarely been quantified, and CBED is also subject to symmetry-breaking due to local strains and inhomogeneities. The purpose of this paper is to classify the various types of symmetry-breaking, present calculations of the sensitivity, and illustrate symmetry-breaking by surface strains.CBED symmetry determinations usually proceed by determining the diffraction group along various zone axes, and hence finding the point group. The diffraction group can be found using either the intensity distribution in the discs

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.

scholarly journals The determination of point groups from imprecise molecular geometries

2021 ◽  
Author(s):  
Peter J. Knowles
Keyword(s):  
Symmetry Breaking ◽  
Point Group ◽  
Simple Function ◽  
Coordinate Frame ◽  
New Approach ◽  
Measure Of Symmetry ◽  
Point Groups

AbstractWe present a new approach for the assignment of a point group to a molecule when the structure conforms only approximately to the symmetry. It proceeds by choosing a coordinate frame that minimises a measure of symmetry breaking that is computed efficiently as a simple function of the molecular coordinates and point group specification.

Crystallography and Diffraction

2021 ◽  
pp. 220-276
Author(s):  
Kannan M. Krishnan
Keyword(s):  
Rotational Symmetry ◽  
Reciprocal Lattice ◽  
Real Space ◽  
Crystalline Materials ◽  
Space Groups ◽  
Ewald Sphere ◽  
Point Groups ◽  
Diffraction Patterns ◽  
Point Line ◽  
Periodic Arrangement

Crystalline materials have a periodic arrangement of atoms, exhibit long range order, and are described in terms of 14 Bravais lattices, 7 crystal systems, 32 point groups, and 230 space groups, as tabulated in the International Tables for Crystallography. We introduce the nomenclature to describe various features of crystalline materials, and the practically useful concepts of interplanar spacing and zonal equations for interpreting electron diffraction patterns. A crystal is also described as the sum of a lattice and a basis. Practical materials harbor point, line, and planar defects, and their identification and enumeration are important in characterization, for defects significantly affect materials properties. The reciprocal lattice, with a fixed and well-defined relationship to the real lattice from which it is derived, is the key to understanding diffraction. Diffraction is described by Bragg law in real space, and the equivalent Ewald sphere construction and the Laue condition in reciprocal space. Crystallography and diffraction are closely related, as diffraction provides the best methodology to reveal the structure of crystals. The observations of quasi-crystalline materials with five-fold rotational symmetry, inconsistent with lattice translations, has resulted in redefining a crystalline material as “any solid having an essentially discrete diffraction pattern”

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