On the construction of Kramers paired double group symmetry functions

2009 ◽  
Vol 109 (10) ◽  
pp. 2149-2167 ◽  
Author(s):  
Daoling Peng ◽  
Jianyi Ma ◽  
Wenjian Liu
Keyword(s):  
Group Symmetry ◽  
Double Group

Matrices and Wave Functions under Double-Group Symmetry

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Time Reversal ◽  
Point Group ◽  
Computational Effort ◽  
Wave Functions ◽  
Group Symmetry ◽  
Time Reversal Symmetry ◽  
Point Group Symmetry ◽  
Matrix Elements ◽  
Double Group

Symmetry is one of the most versatile theoretical tools of physics and chemistry. It provides qualitative insight into the wave functions and properties of systems, and it has also been used successfully to obtain great savings in computational efforts. In the preceding chapter we examined time-reversal symmetry, and now we turn to the more familiar point-group symmetry. We show how relativity requires special consideration and extensions of the concepts developed for the nonrelativistic case, and how time-reversal symmetry and double-group symmetry are connected. Although the techniques that incorporate double-group symmetry presented here are primarily aimed at four-component calculations, they are equally applicable to two-component calculations in which the spin-dependent operators are included at the SCF stage of a calculation. In the preceding chapter, we have shown how the use of time-reversal symmetry can lead to considerable reduction in the number of unique matrix elements that appear in the operator expressions. However, we are also interested in the overall structure of the matrices of the operators. In particular, we are interested in possible block structures, where classes of matrix elements may be set to zero a priori. If the matrices can be cast in block diagonal form, we may save on storage as well as computational effort in solving eigenvalue problems, for example. Matrix blocking will already be effected by the point-group symmetry of the molecule.

Artefact in 20Å-Resolution Electron Images of Negatively Stained Twodimensional Membrane Protein Crystals

1982 ◽  
Vol 40 ◽  
pp. 92-93
Author(s):  
Douglas L. Dorset ◽  
Barbara Moss
Keyword(s):  
Electron Scattering ◽  
Cross Correlation ◽  
Transmission Function ◽  
Group Symmetry ◽  
Asymmetric Structure ◽  
Object Model ◽  
Phase Object ◽  
Computing Systems ◽  
Resolution Electron ◽  
Plane Group

A number of computing systems devoted to the averaging of electron images of two-dimensional macromolecular crystalline arrays have facilitated the visualization of negatively-stained biological structures. Either by simulation of optical filtering techniques or, in more refined treatments, by cross-correlation averaging, an idealized representation of the repeating asymmetric structure unit is constructed, eliminating image distortions due to radiation damage, stain irregularities and, in the latter approach, imperfections and distortions in the unit cell repeat. In these analyses it is generally assumed that the electron scattering from the thin negativelystained object is well-approximated by a phase object model. Even when absorption effects are considered (i.e. “amplitude contrast“), the expansion of the transmission function, q(x,y)=exp (iσɸ (x,y)), does not exceed the first (kinematical) term. Furthermore, in reconstruction of electron images, kinematical phases are applied to diffraction amplitudes and obey the constraints of the plane group symmetry.

Structure of stacking faults in pyrite using TEM techniques

1992 ◽  
Vol 50 (1) ◽  
pp. 358-359
Author(s):  
Raja Subramanian ◽  
Kenneth S. Vecchio
Keyword(s):  
Lattice Structure ◽  
Stacking Faults ◽  
Covalent Bond ◽  
Group Symmetry ◽  
Iron Pyrite ◽  
Dislocation Glide ◽  
Partial Dislocations ◽  
Transmission Electron ◽  
Contrast Analysis ◽  
Chlorine Ions

The structure of stacking faults and partial dislocations in iron pyrite (FeS2) have been studied using transmission electron microscopy. Pyrite has the NaCl structure in which the sodium ions are replaced by iron and chlorine ions by covalently-bonded pairs of sulfur ions. These sulfur pairs are oriented along the <111> direction. This covalent bond between sulfur atoms is the strongest bond in pyrite with Pa3 space group symmetry. These sulfur pairs are believed to move as a whole during dislocation glide. The lattice structure across these stacking faults is of interest as the presence of these stacking faults has been preliminarily linked to a higher sulfur reactivity in pyrite. Conventional TEM contrast analysis and high resolution lattice imaging of the faulted area in the TEM specimen has been carried out.

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>

scholarly journals Multiplicity, Parity and Angular Momentum of a Cooper Pair in Unconventional Superconductors of D4h Symmetry: Sr2RuO4 and Fe-Pnictide Materials

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1435
Author(s):  
Victor G. Yarzhemsky
Keyword(s):  
Angular Momentum ◽  
Cooper Pair ◽  
Point Group ◽  
Group Symmetry ◽  
Cooper Pairs ◽  
Group Approach ◽  
Angular Momentum Projection ◽  
Space Group ◽  
Unconventional Superconductors ◽  
Even Pairs

Sr2RuO4 and Fe-pnictide superconductors belong to the same point group symmetry D4h. Many experimental data confirm odd pairs in Sr2RuO4 and even pairs in Fe-pnictides, but opposite conclusions also exist. Recent NMR results of Pustogow et al., which revealed even Cooper pairs in Sr2RuO4, require reconsideration of symmetry treatment of its SOP (superconducting order parameter). In the present work making use of the Mackey–Bradley theorem on symmetrized squares, a group theoretical investigation of possible pairing states in D4h symmetry is performed. It is obtained for I4/mmm , i.e., space group of Sr2RuO4, that triplet pairs with even spatial parts are possible in kz direction and in points M and Y. For the two latter cases pairing of equivalent electrons with nonzero total momentum is proposed. In P4/nmm space group of Fe- pnictides in point M, even and odd pairs are possible for singlet and triplet cases. It it shown that even and odd chiral states with angular momentum projection m=±1 have nodes in vertical planes, but Eg is nodal , whereas Eu is nodeless in the basal plane. It is also shown that the widely accepted assertion that the parity of angular momentum value is directly connected with the spatial parity of a pair is not valid in a space-group approach to the wavefunction of a Cooper pair.

scholarly journals Symmetries and anomalies of (1+1)d theories: 2-groups and symmetry fractionalization

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Matthew Yu
Keyword(s):  
Spectral Sequence ◽  
Global Symmetries ◽  
Group Symmetry ◽  
Low Dimension ◽  
The One ◽  
Group Symmetries

Abstract We investigate the interactions of discrete zero-form and one-form global symmetries in (1+1)d theories. Focus is put on the interactions that the symmetries can have on each other, which in this low dimension result in 2-group symmetries or symmetry fractionalization. A large part of the discussion will be to understand a major feature in (1+1)d: the multiple sectors into which a theory decomposes. We perform gauging of the one-form symmetry, and remark on the effects this has on our theories, especially in the case when there is a global 2-group symmetry. We also implement the spectral sequence to calculate anomalies for the 2-group theories and symmetry fractionalized theory in the bosonic and fermionic cases. Lastly, we discuss topological manipulations on the operators which implement the symmetries, and draw insights on the (1+1)d effects of such manipulations by coupling to a bulk (2+1)d theory.

Raman-Spektren von SnJ4, GeJ4, TiBr4 und TiJ4 in flüssiger und kristalliner Phase / Raman Spectra of SnJ4, GeJ4, TiBr4, and TiJ4 in the Liquid and Crystalline Phase

1970 ◽  
Vol 25 (12) ◽  
pp. 1374-1381 ◽  
Author(s):  
W. Kiefer ◽  
H. W. Schrötter
Keyword(s):  
Raman Spectra ◽  
Ruby Laser ◽  
Detection System ◽  
Point Group ◽  
Visible Region ◽  
Group Symmetry ◽  
Pure Liquid ◽  
Electronic Detection ◽  
Polarization Studies ◽  
First Time

The Raman spectra of four molecules absorbing in the visible region (SnJ4, GeJ4, TiBr4, and TiJ4) are presented. They were excited with a quasi-continuous ruby laser and recorded with a special electronic detection system. Except for TiJ4, complete Raman spectra of crystal powder pellets could be obtained for the first time. The assignment reported by previous authors was confirmed by accurate polarization studies of solutions or pure liquid. The assignment is also in the solid state possible on the basis of Td point group symmetry. The fundamental vibrations of TiJ4 in solutions are: ν1 (A1) =162, ν2 (E) =51, ν3 (F2) =319 and ν4 (F2) Y = 67 cm-1

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