scholarly journals Cohomology of uniserial p -adic space groups with cyclic point group

2018 ◽  
Vol 493 ◽  
pp. 79-88 ◽  
Author(s):  
Oihana Garaialde Ocaña
Keyword(s):  
Point Group ◽  
Space 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

scholarly journals Normally supportive sublattices of crystallographic space groups

2020 ◽  
Vol 76 (1) ◽  
pp. 7-23
Author(s):  
Miles A. Clemens ◽  
Branton J. Campbell ◽  
Stephen P. Humphries
Keyword(s):  
Normal Subgroup ◽  
Point Group ◽  
Group Extension ◽  
Normal Subgroups ◽  
Symbolic Form ◽  
Space Group ◽  
Space Groups ◽  
Significant Step ◽  
Crystal Class ◽  
Crystallographic Space Groups

The tabulation of normal subgroups of 3D crystallographic space groups that are themselves 3D crystallographic space groups (csg's) is an ambitious goal, but would have a variety of applications. For convenience, such subgroups are referred to as `csg-normal' while normal subgroups of the crystallographic point group (cpg) of a crystallographic space group are referred to as `cpg-normal'. The point group of a csg-normal subgroup must be a cpg-normal subgroup. The present work takes a significant step towards that goal by tabulating the translational subgroups (a.k.a. sublattices) that are capable of supporting csg-normal subgroups. Two necessary conditions are identified on the relative sublattice basis that must be met in order for the sublattice to support csg-normal subgroups: one depends on the operations of the point group of the space group, while the other depends on the operations of the cpg-normal subgroup. Sublattices that meet these conditions are referred to as `normally supportive'. For each cpg-normal subgroup (excluding the identity subgroup 1) of each of the arithmetic crystal classes of 3D space groups, all of the normally supportive sublattices have been tabulated in symbolic form, such that most of the entries in the table contain one or more integer variables of infinite range; thus it could be more accurately described as a table of the infinite families of normally supportive sublattices. For a given pair of cpg-normal subgroup and normally supportive sublattice, csg-normal subgroups of the space groups of the parent arithmetic crystal class can be constructed via group extension, though in general such a pair does not guarantee the existence of a corresponding csg-normal subgroup.

scholarly journals Relativistic spacetime crystals

2021 ◽  
Vol 77 (4) ◽  
Author(s):  
Venkatraman Gopalan
Keyword(s):  
Point Group ◽  
Minkowski Spacetime ◽  
The Other ◽  
Physical Sciences ◽  
Space Groups ◽  
Flat Spacetime ◽  
Inertial Frames ◽  
Relativistic Spacetime ◽  
Lorentz Boosts ◽  
Periodic Space

Periodic space crystals are well established and widely used in physical sciences. Time crystals have been increasingly explored more recently, where time is disconnected from space. Periodic relativistic spacetime crystals on the other hand need to account for the mixing of space and time in special relativity through Lorentz transformation, and have been listed only in 2D. This work shows that there exists a transformation between the conventional Minkowski spacetime (MS) and what is referred to here as renormalized blended spacetime (RBS); they are shown to be equivalent descriptions of relativistic physics in flat spacetime. There are two elements to this reformulation of MS, namely, blending and renormalization. When observers in two inertial frames adopt each other's clocks as their own, while retaining their original space coordinates, the observers become blended. This process reformulates the Lorentz boosts into Euclidean rotations while retaining the original spacetime hyperbola describing worldlines of constant spacetime length from the origin. By renormalizing the blended coordinates with an appropriate factor that is a function of the relative velocities between the various frames, the hyperbola is transformed into a Euclidean circle. With these two steps, one obtains the RBS coordinates complete with new light lines, but now with a Euclidean construction. One can now enumerate the RBS point and space groups in various dimensions with their mapping to the well known space crystal groups. The RBS point group for flat isotropic RBS spacetime is identified to be that of cylinders in various dimensions: mm2 which is that of a rectangle in 2D, (∞/ m ) m which is that of a cylinder in 3D, and that of a hypercylinder in 4D. An antisymmetry operation is introduced that can swap between space-like and time-like directions, leading to color spacetime groups. The formalism reveals RBS symmetries that are not readily apparent in the conventional MS formulation. Mathematica script is provided for plotting the MS and RBS geometries discussed in the work.

Systematic prediction of new inorganic ferroelectrics in point group 4

1999 ◽  
Vol 55 (4) ◽  
pp. 494-506 ◽  
Author(s):  
S. C. Abrahams
Keyword(s):  
Crystal Structure ◽  
Point Group ◽  
Inorganic Crystal Structure Database ◽  
Space Group ◽  
Space Groups ◽  
Group 4 ◽  
Structure Database ◽  
Inorganic Crystal ◽  
Crystal Structure Database

The latest release of the Inorganic Crystal Structure Database contains a total of 87 entries corresponding to 70 different materials in point group 4. The structures reported for 11 materials in space group P4 satisfy the criteria for ferroelectricity, as do four in P41, one each in P42 and P43, 12 in I4, including seven that form three families, and another three in I41. Three previously known ferroelectrics were also listed in I4 and one in I41. In addition, the listing for point group 4 contains 22 entries for nonferroelectric materials and three with misassigned space groups. Among the newly predicted ferroelectrics in point group 4, assuming the validity of the underlying structural reports, are Ce5B2C6, modulated NbTe4, Na3Nb12O31F, Ca2FeO3Cl, K4CuV5O15Cl, TlBO2, CrOF3, PbTeO3, VO(HPO3)(H2O).3H2O, MgB2O(OH)6, β-tetragonal boron, CuBi2O4, WOBr4, Na8PtO6, SbF2Cl3, Ba1.2Ti8O16, Ni[SC(NH2)2]4Cl2, Ca2SiO3Cl2, the mineral caratiite, NbAs, β-NbO2 and Ag3BiO3.

Procedures for Point and Space Group Analysis

1989 ◽  
Vol 47 ◽  
pp. 488-489
Author(s):  
M. Tanaka
Keyword(s):  
Point Group ◽  
Group Analysis ◽  
Convergent Beam Electron Diffraction ◽  
Special Issue ◽  
Zeroth Order ◽  
Space Group ◽  
Space Groups ◽  
Flow Charts ◽  
Crystal Space ◽  
Convergent Beam

Description of general procedures for point- and space-group determination by convergent-beam electron diffraction(CBED) soon appears in a special issue of J. Electron Microsc. Tech. The essence of them was already given in the flow charts in ref.2. We here demonstrate the procedures using an example of La2 CuO4 - 6, which is the end member of a 40K class superconductor La2-x Mx CuO4 - 6 (M = Ba, Sr and Ca). The CBED method has utilized to date only dynamical extinction(GM lines) appearing in zeroth-order Laue-zone(ZOLZ) reflections for the space-group determination. We emphasize that the use of GM lines appearing in higher-order Laue-zone(HOLZ) reflections makes it easy to determine crystal space-groups.The substance was already known to belong to the orthorhombic system with the lattice parameters of a=5.3548Å, b=5.4006Å and c=13.1592Å. Fig. 1(a) shows a CBED pattern taken at the [001] incidence, Fig.l(b) being the central part of Fig.l(a). The whole pattern has a symmetry 2mm. This indicates that the crystal has two mirror symmetries perpendicular to each other and that the point group is mmm or mm2.

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.

scholarly journals The space groups of point group C 3: some corrections, some comments

2002 ◽  
Vol 58 (5) ◽  
pp. 893-899 ◽  
Author(s):  
Richard E. Marsh
Keyword(s):  
Point Group ◽  
Crystallographic Data ◽  
Cambridge Structural Database ◽  
Cambridge Crystallographic Data Centre ◽  
Space Group ◽  
Space Groups ◽  
Data Centre ◽  
Group Assignment ◽  
Structural Database

A survey of the October 2001 release of the Cambridge Structural Database [Cambridge Structural Database (1992). Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge, England] has uncovered approximately 675 separate apparently reliable entries under space groups P3, P3_1, P3_2 and R3; in approximately 100 of these entries, the space-group assignment appears to be incorrect. Other features of these space groups are also discussed.

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.

Crystal Structure Analysis of Cu-Zr Alloys by Convergent Beam Diffraction

1981 ◽  
Vol 39 ◽  
pp. 354-355
Author(s):  
A. F. Marshall ◽  
J. W. Steeds ◽  
D. Bouchet ◽  
S. L. Shinde ◽  
R. G. Walmsley
Keyword(s):  
Crystal Structure ◽  
Point Group ◽  
Intermetallic Phase ◽  
Three Dimensional ◽  
Crystal Structure Analysis ◽  
Ion Milling ◽  
Composition And Structure ◽  
Unit Cells ◽  
Sputter Deposited ◽  
Convergent Beam

Convergent beam electron diffraction is a powerful technique for determining the crystal structure of a material in TEM. In this paper we have applied it to the study of the intermetallic phases in the Cu-rich end of the Cu-Zr system. These phases are highly ordered. Their composition and structure has been previously studied by microprobe and x-ray diffraction with sometimes conflicting results.The crystalline phases were obtained by annealing amorphous sputter-deposited Cu-Zr. Specimens were thinned for TEM by ion milling and observed in a Philips EM 400. Due to the large unit cells involved, a small convergence angle of diffraction was used; however, the three-dimensional lattice and symmetry information of convergent beam microdiffraction patterns is still present. The results are as follows:1) 21 at% Zr in Cu: annealed at 500°C for 5 hours. An intermetallic phase, Cu3.6Zr (21.7% Zr), space group P6/m has been proposed near this composition (2). The major phase of our annealed material was hexagonal with a point group determined as 6/m.

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