The Mirror: Mother of All Symmetries in Crystals

2020 ◽  
Vol 12 (3) ◽  
pp. 289-313
Author(s):  
Mohammad Abdul Wahab
Keyword(s):  
Mirror Symmetry ◽  
Point Group ◽  
Prima Facie ◽  
Fundamental Symmetry ◽  
Combination Scheme ◽  
Unit Cells ◽  
Diffraction Patterns ◽  
Group Symmetries ◽  
The Given ◽  
Conventional Unit

Mirror symmetry is found to be the only fundamental symmetry in crystalline solids because all other symmetries, such as rotation, inversion, rotoreflection, rotoinversion and translational periodicity can be easily derived from suitable combinations of mirrors. Similarly, the point group symmetries can also be derived from the same. The mirror combination scheme is found to work in accordance with the principle of Wigner-Seitz cells and Brillouin Zones (and not with the conventional unit cells as proposed by Bravais), where the zone boundaries of a Brillouin zone represent different sets of Bragg planes obtained from diffraction pattern of the given crystal, while the diffraction of given crystal takes place in terms of decreasing interplanar spacing in reciprocal space. Because the Wigner Seitz cells, the Brillouin zones and the diffraction patterns possess defined origin and exhibit spherical symmetry, they cannot have translational symmetry of any kind (microscopic or macroscopic). Results obtained on the basis of this concept help us to remove the existing ambiguities in crystallography and make the crystal structure determination simple. Further, prima facie the diffraction patterns are found to take care of the proposed 'systematic absences' arising due to the so called lattice centering, glide planes and screw axes without actually taking them into consideration. This newly and first discovered concept is expected to explain all other complicated or less understood issues related to crystallography.

scholarly journals Spherical polar Fourier assembly of protein complexes with arbitrary point group symmetry

2016 ◽  
Vol 49 (1) ◽  
pp. 158-167 ◽  
Author(s):  
David W. Ritchie ◽  
Sergei Grudinin
Keyword(s):  
Protein Complexes ◽  
Complex Structure ◽  
Point Group ◽  
Arbitrary Point ◽  
Group Symmetry ◽  
Test Cases ◽  
Point Group Symmetry ◽  
Group Symmetries ◽  
The Given ◽  
Ab Inito

A novel fast Fourier transform-basedab initodocking algorithm calledSAMis presented, for building perfectly symmetrical models of protein complexes with arbitrary point group symmetry. The basic approach uses a novel and very fast one-dimensional symmetry-constrained spherical polar Fourier search to assemble cyclicCnsystems from a given protein monomer. Structures with higher-order (Dn,T,OandI) point group symmetries may be built using a subsequent symmetry-constrained Fourier domain search to assemble trimeric sub-units. The results reported here show that theSAMalgorithm can correctly assemble monomers of up to around 500 residues to produce a near-native complex structure with the given point group symmetry in 17 out of 18 test cases. TheSAMprogram may be downloaded for academic use at http://sam.loria.fr/.

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.

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.

Dyadic Obligations over Complex Actions as Deontic Constraints in the Situation Calculus

2020 ◽  
Author(s):  
Jens Claßen ◽  
James Delgrande
Keyword(s):  
Artificial Intelligence ◽  
Deontic Logic ◽  
Action Theory ◽  
Prima Facie ◽  
Situation Calculus ◽  
Artificial Agents ◽  
Planning Techniques ◽  
Course Of Action ◽  
Reasoning About Action ◽  
The Given

With the advent of artificial agents in everyday life, it is important that these agents are guided by social norms and moral guidelines. Notions of obligation, permission, and the like have traditionally been studied in the field of Deontic Logic, where deontic assertions generally refer to what an agent should or should not do; that is they refer to actions. In Artificial Intelligence, the Situation Calculus is (arguably) the best known and most studied formalism for reasoning about action and change. In this paper, we integrate these two areas by incorporating deontic notions into Situation Calculus theories. We do this by considering deontic assertions as constraints, expressed as a set of conditionals, which apply to complex actions expressed as GOLOG programs. These constraints induce a ranking of "ideality" over possible future situations. This ranking in turn is used to guide an agent in its planning deliberation, towards a course of action that adheres best to the deontic constraints. We present a formalization that includes a wide class of (dyadic) deontic assertions, lets us distinguish prima facie from all-things-considered obligations, and particularly addresses contrary-to-duty scenarios. We furthermore present results on compiling the deontic constraints directly into the Situation Calculus action theory, so as to obtain an agent that respects the given norms, but works solely based on the standard reasoning and planning techniques.

scholarly journals Random k-Body Ensembles for Chaos and Thermalization in Isolated Systems

Entropy ◽  
2018 ◽  
Vol 20 (7) ◽  
pp. 541 ◽  
Author(s):  
Venkata Kota ◽  
Narendra Chavda
Keyword(s):  
Hermite Polynomials ◽  
Point Group ◽  
Majorana Fermions ◽  
Many Body ◽  
Boson Systems ◽  
Random Matrix Ensembles ◽  
Isolated Systems ◽  
Group Symmetries ◽  
Statistical Relaxation

Embedded ensembles or random matrix ensembles generated by k-body interactions acting in many-particle spaces are now well established to be paradigmatic models for many-body chaos and thermalization in isolated finite quantum (fermion or boson) systems. In this article, briefly discussed are (i) various embedded ensembles with Lie algebraic symmetries for fermion and boson systems and their extensions (for Majorana fermions, with point group symmetries etc.); (ii) results generated by these ensembles for various aspects of chaos, thermalization and statistical relaxation, including the role of q-hermite polynomials in k-body ensembles; and (iii) analyses of numerical and experimental data for level fluctuations for trapped boson systems and results for statistical relaxation and decoherence in these systems with close relations to results from embedded ensembles.

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