scholarly journals Consistency Relations of an Extension Polycyclic Free Abelian Lattice Group by Quaternion Point Group

2021 ◽  
Vol 1988 (1) ◽  
pp. 012071
Author(s):  
Siti Afiqah Mohammad ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Point Group ◽  
Lattice Group ◽  
Abelian Lattice

On the Abelianization of a Torsion Free Crystallographic Group

2014 ◽  
Vol 70 (1) ◽  
Author(s):  
Nor'ashiqin Mohd Idrus ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim ◽  
Rohaidah Masri
Keyword(s):  
Fundamental Group ◽  
Riemannian Manifolds ◽  
Point Group ◽  
Crystallographic Group ◽  
Finite Point ◽  
Lattice Group ◽  
Bieberbach Group ◽  
Trivial Center ◽  
Free Abelian Groups ◽  
Torsion Free

A torsion free crystallographic group, which is also known as a Bieberbach group is a generalization of free abelian groups. It is an extension of a lattice group by a finite point group. The study of n-dimensional crystallographic group had been done by many researchers over a hundred years ago. A Bieberbach group has been characterized as a fundamental group of compact, connected, flat Riemannian manifolds. In this paper, we characterize Bieberbach groups with trivial center as exactly those with finite abelianizations.  The abelianization of a Bieberbach group is shown to be finite if the center of the group is trivial.

scholarly journals The Exterior Square of a Bieberbach Group with Quaternion Point Group of Order Eight

2020 ◽  
pp. 1-7
Author(s):  
S.A. Mohammad ◽  
N.H. Sarmin ◽  
H.I. Mat Hassim
Keyword(s):  
Mathematical Method ◽  
Point Group ◽  
Mathematical Structure ◽  
Mathematical Representation ◽  
Crystallographic Group ◽  
Finite Point ◽  
Bieberbach Group ◽  
Abelian Lattice ◽  
Torsion Free ◽  
Exterior Square

A Bieberbach group is defined to be a torsion free crystallographic group which is an extension of a free abelian lattice group by a finite point group. This paper aims to determine a mathematical representation of a Bieberbach group with quaternion point group of order eight. Such mathematical representation is the exterior square. Mathematical method from representation theory is used to find the exterior square of this group. The exterior square of this group is found to be nonabelian. Keywords: mathematical structure; exterior square; Bieberbach group; quaternion point group

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.

Friedel'S law breakdown and interpretation of stacking fault images in tetrahedral semiconductors

1987 ◽  
Vol 45 ◽  
pp. 318-319
Author(s):  
Rob. W. Glaisher ◽  
A.E.C. Spargo
Keyword(s):  
Relative Phase ◽  
Point Group ◽  
Binary Compound ◽  
Stacking Fault ◽  
Fold Axis ◽  
Atom Pair ◽  
Tetrahedral Semiconductors ◽  
Group Symmetries ◽  
Thin Crystal ◽  
Phase Differences

Images of <11> oriented crystals with diamond structure (i.e. C,Si,Ge) are dominated by white spot contrast which, depending on thickness and defocus, can correspond to either atom-pair columns or tunnel sites. Olsen and Spence have demonstrated a method for identifying the correspondence which involves the assumed structure of a stacking fault and the preservation of point-group symmetries by correctly aligned and stigmated images. For an intrinsic stacking fault, a two-fold axis lies on a row of atoms (not tunnels) and the contrast (black/white) of the atoms is that of the {111} fringe containing the two-fold axis. The breakdown of Friedel's law renders this technique unsuitable for the related, but non-centrosymmetric binary compound sphalerite materials (e.g. GaAs, InP, CdTe). Under dynamical scattering conditions, Bijvoet related reflections (e.g. (111)/(111)) rapidly acquire relative phase differences deviating markedly from thin-crystal (kinematic) values, which alter the apparent location of the symmetry elements needed to identify the defect.

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

TEM study of lanthanum aluminate

1990 ◽  
Vol 48 (4) ◽  
pp. 1060-1061
Author(s):  
C.Y. Yang ◽  
Z.R. Huang ◽  
Y.Q. Zhou ◽  
C.Z. Li ◽  
W.H. Yang ◽  
...  
Keyword(s):  
Domain Boundary ◽  
Point Group ◽  
Optical Microscope ◽  
Dark Field ◽  
Zone Axis ◽  
Superconducting Film ◽  
Ion Milling ◽  
Lanthanum Aluminate ◽  
Ring Diameter ◽  
Diffraction Patterns

Lanthanum aluminate(LaAlO3) single crystal as a substrate for high Tc superconducting film has attracted attention recently. We report here a transmission electron microscopy(TEM) study of the crystal structure and phase transformation of LaAlO3 by using Philips EM420 and EM430 microscopes. Single crystals of LaAlO3 were investigated first by optical microscope. Stripe-shaped domains of mm size are clearly seen(Fig.1a), and 90° domain boundary is also obvious. TEM specimens were prepared by mechanical grinding and polishing followed by ion-milling.Fig.lb shows μm size stripe domains of LaAlO3. Convergent beam electron diffraction patterns (CBED) from single domain were taken.Fig. 2a and Fig. 2c are [001] zone axis patterns which show a 4mm symmetry, and the (200) dark field of this zone axis gives 2mm symmetry(fig.2b). Therefore the point group of this crystal is either 4/mmm or m3m. The projection of the first order Laue zone(FOLZ) reflections on zero layer (fig. 2c) shows that the unit cell is face centered. A tetragonal unit ceil is chosen, with a=0.532nm and c=0.753nm, c being determined from the FOLZ ring diameter.

Microdiffraction analysis of precipitate crystallography and morphology in Al alloys

1990 ◽  
Vol 48 (4) ◽  
pp. 954-955
Author(s):  
B.C. Muddle ◽  
G.R. Hugo
Keyword(s):  
Point Group ◽  
Hexagonal Lattice ◽  
Intersection Point ◽  
Saed Pattern ◽  
Point Symmetry ◽  
Hexagonal Symmetry ◽  
Precipitate Crystal ◽  
The Matrix ◽  
The Subject ◽  
Electron Microdiffraction

Electron microdiffraction has been used to determine the crystallography of precipitation in Al-Cu-Mg-Ag and Al-Ge alloys for individual precipitates with dimensions down to 10 nm. The crystallography has been related to the morphology of the precipitates using an analysis based on the intersection point symmetry. This analysis requires that the precipitate form be consistent with the intersection point group, defined as those point symmetry elements common to precipitate and matrix crystals when the precipitate crystal is in its observed orientation relationship with the matrix.In Al-Cu-Mg-Ag alloys with high Cu:Mg ratios and containing trace amounts of silver, a phase designated Ω readily precipitates as thin, hexagonal-shaped plates on matrix {111}α planes. Examples of these precipitates are shown in Fig. 1. The structure of this phase has been the subject of some controversy. An SAED pattern, Fig. 2, recorded from matrix and precipitates parallel to a <11l>α axis is suggestive of hexagonal symmetry and a hexagonal lattice has been proposed on the basis of such patterns.

The crystallography characteristic of (Mn,Fe)S single crystal

1990 ◽  
Vol 48 (4) ◽  
pp. 898-899
Author(s):  
Jiang Xishan
Keyword(s):  
Single Crystal ◽  
Cast Steel ◽  
Point Group ◽  
Central Area ◽  
Hexagonal Crystal ◽  
Growth Step ◽  
Left And Right ◽  
Relationship Of ◽  
Fcc Structure ◽  
New Discovery

This paper reports the growth step pattern and morphology at equilibrium and growth states of (Mn,Fe)S single crystal on the wall of micro-voids in ZG25 cast steel by using scanning electron microscope. Seldom report was presented on the growth morphology and steppattern of (Mn,Fe)S single crystal.Fig.1 shows the front half of the polyhedron of(Mn,Fe)S single crystal,its central area being the square crystal plane,the two pairs of hexagons symmetrically located in the high and low, the left and right with a certain, angle to the square crystal plane.According to the symmetrical relationship of crystal, it was defined that the (Mn,Fe)S single crystal at equilibrium state is tetrakaidecahedron consisted of eight hexagonal crystal planes and six square crystal planes. The macroscopic symmetry elements of the tetrakaidecahedron correpond to Oh—n3m symmetry class of fcc structure,in which the hexagonal crystal planes are the { 111 } crystal planes group,square crystal plaits are the { 100 } crystal planes group. This new discovery of the (Mn,Fe)S single crystal provides a typical example of the point group of Oh—n3m.

Hydrothermal Synthesis and Solid-State Laser Refrigeration of Ytterbium-Doped Potassium Lutetium Fluoride (KLF) Microcrystals

2020 ◽  
Author(s):  
Xiaojing Xia ◽  
Anupum Pant ◽  
Xuezhe Zhou ◽  
Elena Dobretsova ◽  
Alex Bard ◽  
...  
Keyword(s):  
Hydrothermal Synthesis ◽  
Solid State ◽  
Information Science ◽  
Point Group ◽  
Solid State Laser ◽  
Lanthanide Ions ◽  
Group Symmetry ◽  
Luminescence Spectra ◽  
Crystalline Phases ◽  
State Laser

Fluoride crystals, due to their low phonon energies, are attractive hosts of trivalent lanthanide ions for applications in upconverting phosphors, quantum information science, and solid-state laser refrigeration. In this article, we report the rapid, low-cost hydrothermal synthesis of potassium lutetium fluoride (KLF) microcrystals for applications in solid-state laser refrigeration. Four crystalline phases were synthesized, namely orthorhombic K<sub>2</sub>LuF<sub>5</sub> (Pnma), trigonal KLuF<sub>4</sub> (P3<sub>1</sub>21), orthorhombic KLu<sub>2</sub>F<sub>7</sub> (Pna2<sub>1</sub>), and cubic KLu<sub>3</sub>F<sub>10</sub> (Fm3m), with each phase exhibiting unique microcrystalline morphologies. Luminescence spectra and emission lifetimes of the four crystalline phases were characterized based on the point-group symmetry of trivalent cations. Laser refrigeration was measured by observing both the optomechanical eigenfrequencies of microcrystals on cantilevers in vacuum, and also the Brownian dynamics of optically trapped microcrystals in water. Among all four crystalline phases, the most significant cooling was observed for 10%Yb:KLuF<sub>4</sub> with cooling of 8.6 $\pm$ 2.1 K below room temperature. Reduced heating was observed with 10%Yb:K<sub>2</sub>LuF<sub>5</sub>

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