On the optimum set of polarization measurements for the scattering of two indentical fermions to determine the helicity amplitudes

The polarization asymmetries related to usual polarizations combine at least 4 different helicity amplitudes. The introduction of ±45° inclined transverse polarizations allows us to obtain the complete set of new polarization asymmetries combining only up to 3 helicity amplitudes. There are no further polarization asymmetries depending on a smaller number of helicity amplitudes. These expressions are most economical to determine the helicity amplitudes from observed data. Some of them are suitable to study especially the spin-flip effects. We give a complete Table of all such polarization asymmetries.     

Quantization of collective regular structures of particles

The theory of many-body systems constitutes one of the most complex part of quantum mechanics. Closed results are not readily available and approximations become extremely involved. It is, therefore, of considerable interest that the quantization of certain simple collective motions of particles can describe the energies of atoms, nuclei and particle models rather well. We describe here the most elementary examples of such structures.Dedicated to H. Walther, a great scientist and who demonstrated among many other things the regular shapes of atomic particles, the subject of this essay     

Center-of-mass motion of a system of relativistic Dirac particles

The relativistic center-of-mass motion for a system ofN fermions can be exactly separated because of the linearity of the Dirac operators in momenta which is not possible for quadratic Klein-Gordon particles. The covariant equations derived from Maxwell-Dirac field theory are considered. The center-of-mass equation is still a 4 ^N -component spinor equation. We solve these equations for two- and three-body systems, as well as the relative motion for the non-interacting case, and discuss the quantum numbers and identification of eigenstates and eigenvalues. The results apply for both bound and scattering states. Dedicated to the Third Centenary of the Publication of Principia: Corollary IV.... and therefore the common center of gravity of all bodies acting upon each other (excluding external actions and impediments) is either at rest, or moves uniformly in a right line. Is. Newton, Philosophiae Naturalis Principia Mathematica (S. Pepys, Julii 5, 1686, Londini)     

Synthesis and amino acid extraction abilities of chiral calix[4]arene triamides containing amino alcohol units

Herein the synthesis and extraction abilities of new d-/l-phenylalaninol substituted p-tert-butylcalix[4]arene triamide derivatives (3 and 4) towards amino acids are reported. These compounds (3 and 4) have been easily synthesized via aminolysis of p-tert-butylcalix[4]arene trimethylester (2) with d-/l-phenylalaninol in methanol-toluen solvent system at one step. The extraction properties of the prepared chiral calix[4]arene triamide derivatives (3 and 4) towards some selected amino acid methylesters are studied by liquid–liquid extraction. Results show that these chiral calix[4]arene triamide derivatives (3 and 4) exhibited a good affinity towards all amino acid species without any remarkably discrimination.     

Nonperturbative quantum electrodynamics: The Lamb shift

The nonlinear integro-differential equation, obtained from the coupled Maxwell-Dirac equations by eliminating the potential A, is solved by iteration rather than perturbation. The energy shift is complex, the imaginary part giving the spontaneous emission. Both self-energy and vacuum polarization terms are obtained. All results, including renormalization terms, are finite.     

The einstein a-coefficient of spontaneous emission: a relativistic calculation in the Heisenberg representation

We present a simple approach to the relativistic calculation of the rates of spontaneous emission starting from the Heisenberg picture formula for the power radiated by a charged particle undergoing acceleration, and evaluate atomic decay rates using relativistic Dirac-Coulomb wavefunctions. The spin of the electron, embedded in its relativistic wavefunction, is shown to correctly provide the two polarization states of the emitted radiation. We discuss selection rules and calculate the Hydrogen 2P → 1S transition rate, among others, to be Γ=(6.2650±0.0007)×10⁸ s^?1 in good agreement with the full field theory calculation as well as with experiment.     

Bis(2‐pyridyl­methanol)­bis­(saccharinato)­zinc(II) and ‐cadmium(II) at 120 K: three‐dimensional structures containing both N‐and O‐coordinated ambi­dentate saccharinate ligands

The title complexes [M(sac)₂(mpy)₂] [sac is saccharinate (C₇H₄NO₃S) and mpy is 2‐pyridyl­methanol (C₆H₇NO)], with M = Zn^II and Cd^II, are isostructural and consist of neutral mol­ecules. The Zn^II or Cd^II cations are octahedrally coordinated by the two neutral mpy and two anionic sac ligands. The mpy ligand acts as a bidentate donor through the amine N and hydroxyl O atoms. The sac ligands exhibit an ambidentate coordination behaviour; one is N‐coordinated and the other is O‐coordinated within the same coordination octahedron. The crystal packing is determined by C—H?O‐type hydrogen bonding, as well as by weak py–py and sac–sac aromatic π–π‐stacking interactions.     

Anomalous NMR behavior of meso compounds with remote stereogenic centers on addition of chiral shift reagent or chiral solvating agent

A problem has arisen in using chiral shift reagents (CSR) and chiral solvating agents (CSA) to determine meso and racemic forms of diastereoisomers in which the stereogenic centers of the molecules are separated by achiral spacers. It is found that NMR signals of both meso and racemic forms of diastereoisomers may exhibit doubling on addition of CSR/CSA, which means that unequivocal assignments cannot be made without characterizing the effects for separate meso and racemic forms; this is particularly important for additions of CSR/CSA at relatively low concentrations, which always result in the splitting of some NMR signals of diastereoisomers. The phenomenon is demonstrated in the (31)P NMR spectra of meso and racemic forms of three spermine-bridged gem-disubstituted cyclotriphosphazatrienes, 1a-c, and compared with analogous achiral molecules, the per-substituted spermine-bridged cyclotriphosphazatrienes 2a-d. As expected, only one set of (31)P NMR signals was observed for the achiral compounds 2a-d, even on addition of CSA. Two sets of (31)P NMR ABX multiplets corresponding to meso and racemic diastereoisomers were observed for compounds 1a-c; on addition of CSA, the signals of at least one of the multiplets for each compound separated into more than the expected groups of three lines with an intensity distribution of 2:1:1. To understand this phenomenon, the meso and racemic forms of 1a and 1b and the meso form of 1c have been separated and characterized by X-ray crystallography. On addition of CSA to the racemic forms of 1a and 1b, the (31)P NMR spectrum shows the expected doubling of signals, but, unexpectedly, the same is observed for each of the meso forms of 1a-c. Analogous results using both CSA and CSR have been obtained for the meso and racemic forms of the diastereoisomeric piperazine-bridged macrocyclic-phosphazene compound, 3, whereas no effect was observed for the two meso forms of the doubly bridged macrocyclic-phosphazene compound 4. The phenomenon of doubling of the (31)P NMR signals of the meso form of singly bridged cyclotriphosphazatrienes, 1a-c and 3, is explained by consideration of the equilibrium in solution of independent complexation of a chiral ligand with molecules that have two chiral cyclophosphazene moieties separated by an achiral spacer group. The results show that the stereogenicity of such diastereoisomeric molecules in solution cannot be characterized unequivocally by NMR measurements on addition of either CSR or CSA.     

Sensitive Flow-Injection Electrochemical Determination of Hydrogen Peroxide at a Palladium Nanoparticle-Modified Pencil Graphite Electrode

A novel flow-injection amperometric method was proposed for the sensitive and enzymeless determination of hydrogen peroxide based on its electrocatalytic reduction at a palladium nanoparticle-modified pretreated pencil graphite electrode in a laboratory-constructed electrochemical flow cell. Cyclic voltammograms of the unmodified and modified electrodes were recorded in pH 7.0 phosphate buffer containing 0.10 M KCl at a scan rate of 50?mV s^?1 for the investigation of electrocatalytic reduction of hydrogen peroxide at the palladium nanoparticle-modified pretreated pencil graphite electrode. Cyclic voltammograms of the pretreated pencil graphite electrode revealed an irreversible oxidation peak and a weak reduction peak of hydrogen peroxide at +1100?mV and –450?mV vs. an Ag/AgCl/KCl saturated reference electrode. However, the reduction of hydrogen peroxide was observed at –100?mV with an increase in current in the cyclic voltammograms of the palladium nanoparticle-modified pretreated pencil graphite electrode compared to the unmodified electrode. These results indicate that the palladium nanoparticle-modified pretreated pencil graphite electrode exhibits efficient electrocatalytic activity for the reduction of hydrogen peroxide. A linear concentration range was obtained between .01 and 10.0?mM hydrogen peroxide with a detection limit of 3.0 µM from flow injection amperometric current–time curves recorded in pH 7.0 phosphate buffer at –100?mV and a 2.0?mL min^?1 flow rate. The novelty of this work relies on its use of a laboratory-constructed flow cell constructed for the pencil graphite electrode using these inexpensive, disposable, and electrochemically reactive modified electrodes for the amperometric determination of hydrogen peroxide in a flow injection analysis system.     

Students’ mathematical work on absolute value: focusing on conceptions,errors and obstacles

ZDM – Mathematics Education - This study investigates students’ conceptions of absolute value (AV), their performance in various items on AV, their errors in these items and the...