Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics

In quantum theory, symmetry has to be defined necessarily in terms of the family of unit rays, the state space. The theorem of Wigner asserts that a symmetry so defined at the level of rays can always be lifted into a linear unitary or an antilinear antiunitary operator acting on the underlying Hilbert space. We present two proofs of this theorem which are both elementary and economical. Central to our proofs is the recognition that a given Wigner symmetry can, by post-multiplication by a unitary symmetry, be taken into either the identity or complex conjugation. Our analysis often focuses on the behaviour of certain two-dimensional subspaces of the Hilbert space under the action of a given Wigner symmetry, but the relevance of this behaviour to the larger picture of the whole Hilbert space is made transparent at every stage.

Cubic and ultimate groups of complete symmetry

It is shown that the systems of definite actions described by polar and axial tensors of the second rank and their combinations during the superposition of their elements of complete symmetry with the elements of complete symmetry of the "grey" cube, result in 11 cubic crystallographical groups of complete symmetry. There are 35 ultimate groups (i.e., the groups having the axes of symmetry of infinite order) in complete symmetry of finite figures. 14 out of these groups are ultimate groups of symmetry of polar and axial tensors of the second rank and 24 are new groups. All these 24 ultimate groups are conventional groups since they cannot be presented by certain finite figures possessing the axes of symmetry {Mathematical expression}. Geometrical interpretation for some of the groups of complete symmetry is given. The connection between complete symmetry and physical properties of the crystals (electrical, magnetic and optical) is shown.

The change of complete symmetry of the crystals during the phase transitions in ferroelectrics and ferromagnetics

The extension of the superposition principle of the symmetries (P. Curie principle of symmetry) for the case of complete symmetry is given. The enumeration of all crystallographical groups of complete symmetry is presented, the number of elements having complete symmetry for each class of the crystals being indicated. The change of complete symmetry of the crystals under the phase transitions is obtained by superimposing the elements of complete symmetry of polar or axial vectors on the one hand, and the elements of complete symmetry of the crystals on the other. The tables of complete symmetry changes for the cubic, rhombic, monoclinic and triclinic crystals during the ferroelectric and ferromagnetic phase transitions are given.

Symmetry and shape issues in nanostructure growth

Symmetry plays a key role in dictating the equilibrium morphology of crystals. However, several growth morphologies that deviate from the point group symmetry are routinely observed under several different growth conditions. In this article, we present a summary of symmetry-breaking mechanisms that are operative for crystals grown from the vapour phase as well as those formed as a result of wet chemical synthesis. This understanding is crucial for rationalizing the variety of morphologies observed during nanocrystal synthesis and also providesa rational framework for the synthesis of anisotropic nanostructures.

A Simple and General Strategy for the Design of Fluorescent Cation Sensor Beads

Chenodeoxycholic acid based PET sensors for alkali metal ions have been immobilized on Merrifield resin and on Tentagel. The fluorescence of the sensor beads is enhanced upon binding the cations. The modular nature of the sensor allows designing different sensors based on this concept.

Reversible cation/anion extraction from K2La2Ti3O10: Formation of new layered titanates, KLa2Ti3O9.5 and La2Ti3O9

A new soft-chemical transformation of layered perovskite oxides is described wherein K2O is sequentially extracted from the Ruddlesden-Popper (R-P) phase, K2La2Ti3O10 (I), yielding novel anion-deficient KLa2Ti3O9.5 (II) and La2Ti3O9 (III). The transformation occurs in topochemical reactions of the R-P phase I with PPh4Br and PBu4Br (Ph = phenyl; Bu = n-butyl). The mechanism involves the elimination of KBr accompanied by decomposition of PR4+ (R = phenyl or n-butyl) that extracts oxygen from the titanate. Analysis of the organic products of decomposition reveals formation of Ph3PO, Ph3P, and Ph-Ph for R = phenyl, and Bu3PO, Bu3P along with butane, butene, and octane for R = butyl. The inorganic oxides II and III crystallize in tetragonal structures (II: P4/mmm, a = 3.8335(1) angstrom, c = 14.334(1) angstrom; III: /4/ mmm, a = 3.8565(2) angstrom, c = 24.645(2) angstrom) that are related to the parent R-P phase. II is isotypic with the Dion-Jacobson phase, RbSr2Nb3O10, while III is a unique layered oxide consisting of charge-neutral La2Ti3O9 anion-deficient perovskite sheets stacked one over the other without interlayer cations. Interestingly, both II and III convert back to the parent R-P phase in a reaction with KNO3. While transformations of the R-P phases to other related layered/three-dimensional perovskite oxides in ion-exchange/metathesis/dehydration/reduction reactions are known, the simultaneous and reversible extraction of both cations and anions in the conversions K2La2Ti3O10 reversible arrow KLa2Ti3O9.5 reversible arrow La2Ti3O9 is reported here for the first time.

Solution cation-binding properties of segmented polyethylene oxide-A C-13 NMR spectroscopic study

Two segmented polyethylene oxides, SPEO-3 and SPEO-4, were prepared using a novel transetherification methodology. Their structures were confirmed by H-1 and C-13 NMR spectroscopy. The complexation of these SPEO's with alkali-metal ions in solution was investigated by C-13 NMR spectroscopy. The mole-fraction method was used to determine the complexation ratio of SPEO with LIClO4 at 25 degrees C, which showed that these formed 1:1 (polymer repeat unit/salt) complexes. The association constant, K, for the complex formation was calculated from the variation of the chemical shift values with salt concentration, using a standard nonlinear least-square fitting procedure. The maximum change in chemical shift (Delta delta) and the K values suggest that both SPEO-3 and SPEO-4 formed stronger complexes with lithium salts than with sodium salts. Unexpectedly, the K values were found to be different, when the variation of delta of different carbons was used in the fitting procedure. This suggests that several possible complexed species may be in equilibrium with the uncomplexed one. Structurally similar model compounds were also prepared and their complexation studies indicated that all of them also formed 1:1 complexes with Li salts. Interestingly, it was observed that the polymers gave higher K values suggesting the formation of more stable complexes in polymers when compared to the model analogues. (C) 2000 John Wiley & Sons, Inc.

A.C. electrical conductivity of pure and doped potassium perchlorate

A.C. electrical conductivity of potassium perchlorate (KP) has been measured in the temperature range 25�325°C at frequencies ranging from 50�500 Hz using an automated technique. The results are interpreted in terms of a novel mechanism involving Schottky defects in the anion sublattice and Frenkel defects in the cation sublattice. Theconductivity behavior of KP is compared with literature data on similar low-symmetry systems containing polyatomic ions.

X-ray diffraction from single Al6CuLi3 grains showing five-fold symmetry

We present here the detailed results of X-ray diffraction from single quasicrystals of Al6CuLi3. X-ray precession photographs taken down the two-, three- and five-fold axes along with rotation and zero-level Weissenberg photographs are shown. Preliminary analysis of the diffraction data rules out the twin hypothesis.

An X-ray and neutron diffraction study of cation substituted TlSr2CuO5

The structures of TlSr(2−x)LaxCuO(5+δ), with x=0.5, 0.75 and 1, and Tl.5Pb0.5Sr2CuO(5+δ) have been examined with X-ray and neutron powder Rietveld refinement. They are isostructural (P4/mmm) with the corresponding thallium-barium cuprate having one Cu-O layer with Cu3+ ions in octahedral coordination with oxygen (structure type 1201). The influence of cation substitution and disorder on the structure and superconducting properties of these phases have been investigated. La3+ substitution for Sr2+ stabilises the structure and reduces Cu3+, permitting superconductivity, while Pb2+ substitution for Tl3+ only stabilises the structure, without reducing Cu3+.

Symmetry preservation during radiation damage

An examination of radiation-damage processes consequent to high-energy irradiation in certain ammonium salts studied using ESR of free radical together with the structural information available from neutron diffraction studies shows that, other factors being equal/nearly equal, symmetry-related bonds are preserved in preference to those unrelated to one another by any symmetry.

Structure, dielectric, and magnetic properties of Sr2TiMnO6 ceramics

Sr2TiMnO6, a double perovskite associated with high degree of B-site cation disorder was investigated in detail for its structural, magnetic, and dielectric properties. Though x-ray powder diffraction analysis confirms its cubic structure, first order Raman scattering and infrared reflectivity spectra indicate a breaking of the local cubic symmetry. The magnetization study reveals an anomaly at 14 K owing to a ferrimagnetic/canted antiferromagneticlike ordering arising from local Mn-O-Mn clusters. Saturated M-H hysteresis loops obtained at 5 K also reflect the weak ferromagnetic exchange interactions present in the system and an approximate estimation of Mn3+/Mn4+ was done using the magnetization data for the samples sintered at different temperatures. The conductivity and dielectric behavior of this system has been investigated in a broad temperature range of 10 to 300 K. Intrinsic permittivity was obtained only below 100 K whereas giant permittivity due to conductivity and Maxwell-Wagner polarization was observed at higher temperatures. X-ray photoemission studies further confirmed the presence of mixed oxidation states of Mn and the valence band spectra analysis was carried out in detail. (C) 2010 American Institute of Physics. doi: 10.1063/1.3500369]

Growth Kinetics of ZnO Nanocrystals in the Presence of a Base: Effect of the Size of the Alkali Cation

Following an earlier study (J. Am. Chem Soc. 2007, 129, 4470) describing a very unusual growth kinetics of ZnO nanoparticles, we critically evaluate here the proposed mechanism involving a crucial role of the alkali base ion in controlling the growth of ZnO nanoparticles using other alkali bases, namely, LiOH and KOH. While confirming the earlier conclusion of the growth of ZnO nanoparticles being hindered by an effective passivating layer of cations present in the reaction mixture and thereby generalizing this phenomenon, present experimental data reveal an intriguing nonmonotonic dependence of the passivation efficacy on the ionic size of the alkali base ion. This unexpected behavior is rationalized on the basis of two opposing factors: (a) solvated cationic radii and (b) dissociation constant of the base.

K1-xLaxCa2-xNb3O10, a Layered Perovskite Series with Variable Interlayer Cation Density, and LaCaNb3O10, a Novel Layered Perovskite Oxide with No Interlayer Cations

A series of layered perovskite oxides of the formula K1-xLaxCa2-xNb3O10 for 0 < x ≤ 1.0 have been prepared. All the members are isostructural, possessing the structure of KCa2Nb3O10. The interlayer potassium ions in the new series can be ion-exchanged with protons to give H1-xLaxCa2-xNb3O10. The latter readily forms intercalation compounds of the formula (CnH2n+1NH3)1-x LaxCa2-xNb3O10, just as the parent solid acid HCa2Nb3O10. The end member LaCaNb3O10 containing no interlayer cations is a novel layered perovskite oxide, being a n = 3 member of the series An-1BnX3n+1.