Geometrical Angles Research Articles

Invariant-Angle Coherent States for the Singular Oscillator and Geometrical Phases and Angles

Exact coherent states describing the invariant-angle variables for the time-dependent singular oscillator are constructed. Through the use of these coherent states we show how to derive the nonadiabatic Hannay's angle from nonadiabatic Berry's phase and also we get the exact classical evolution from the quantum evolution for the time dependent singular oscillator.

Three-Center, Two-Electron Systems. Origin of the Tilting of Their Substituents

The origin of the tilting of the H-substituents of 14 parent three-center, two-electron (3c−2e) systems, C2H3+, C2H42+, C2H5+, B2H3-, B2H4, B2H5-, Be2H42-, Si2H3+, Si2H42+, Si2H5+, Al2H3-, Al2H4, Al2H5-, and Mg2H42-, is analyzed. The mixing of the σ and π orbitals underlies the upward tilt of these hydrogen substituents. Except for the ethyl cation and the ethylene dication, all 3c−2e sytems have all their hydrogens on one side of a plane that contains both heavy elements. These elements thus have inverted geometries. It is shown that the geometrical tilt angles between the X−Ht and the X−X bonds of all the 3c−2e electron systems (anions, cations, dianions, dications, and neutrals containing eight different elements of the first and second rows) correlate linearly with the electronegativities of the heavy elements. The electronic structures of these systems are also reported. They highlight the convex curvature of the X−X bonds. All the investigated 3c−2e systems are either minima or transition structures...

Ermakov systems, exact solution, and geometrical angles and phases.

Ermakov systems are pairs of coupled, time-dependent nonlinear dynamical equations possessing a joint constant of motion. We show how to derive the Ermakov system from nonharmonic oscillators. We present a detailed study of Ermakov systems from a classical and quantum point of view. Finally the nonadiabatic Hannay's angle and Berry's phase for the system are calculated along with its adiabatic limit.

Spectral width of type 2 coherent echoes at large magnetic aspect angles

The spectral characteristics of smoothly varying type 2 signals were studied using two 50‐MHz CW radio links. The echoes observed came from two closely located scattering volumes where the geometrical aspect angles of observations were about 10° for both links but the azimuthal angles differed by 88°. The events reported here were selected when the spectra on both links were wide, with a comparable spectral width, but the mean Doppler shift was almost zero on one link and very high, sometimes in excess of the ion‐acoustic velocity, on the other link. Highly shifted Doppler spectra were observed to be more skewed than the spectra with low shifts. The skewness was negative for negative Doppler shifts and positive for positive Doppler shifts in accordance with previous radar observations at large magnetic aspect angles. The variation of spectral width with the radial velocity or with the magnetic disturbance level exhibited two regions of linear dependence, with a larger slope at small values and a much smaller slope at higher ones. These different slopes likely reflect the change in the rate of the irregularity amplitude increase with the electric field.

Observations of 50-MHz type-II coherent echoes from within the polar cap

Abstract. 50-MHz type-II coherent echoes at geometrical aspect angles of 11.5° have been observed in the northern polar cap during pre-noon hours. The echoes had an unusually large Doppler width of 1200-1400 m s-1 and were well correlated with strong magnetic disturbances in the range 500-1000 nT. The dependence of intensity, spectral width and skewness versus radial velocity were similar to those known from previous experiments at lower latitudes and at small aspect angles. It is concluded that echo onset was due to the combination of several factors, including a highly conducting ionosphere, the presence of a very intense electric field, and strong radar wave refraction.

Aspect angle dependence of the radar aurora Doppler velocity

The aspect angle dependence of the Doppler velocity of radar auroral echoes was studied by means of a comparison between the signals received simultaneously on four 50‐MHz CW links. In most of the events, the observed Doppler velocity decreased with increasing aspect angle but the rate of decrease was lower than that predicted by linear theory. The velocity decrease was about a factor of 0.4 between geometrical aspect angles of 1.5° and 4.0° and 0.6 to 0.7 between 3.2° and 5°. A model for the Doppler velocity behavior of the radar auroral signals is proposed, combining both propagation and scattering effects. Allowance is made for “ray spreading” arising from processes like refraction and diffraction during propagation. In addition, height integration over the scattering volume is taken into account.

Observations of 50-MHz type-II coherent echoes from within the polar cap

50-MHz type-II coherent echoes at geometrical aspect angles of 11.5° have been observed in the northern polar cap during pre-noon hours. The echoes had an unusually large Doppler width of 1200-1400 m s-1 and were well correlated with strong magnetic disturbances in the range 500-1000 nT. The dependence of intensity, spectral width and skewness versus radial velocity were similar to those known from previous experiments at lower latitudes and at small aspect angles. It is concluded that echo onset was due to the combination of several factors, including a highly conducting ionosphere, the presence of a very intense electric field, and strong radar wave refraction.

Measurements and computations of electromagnetic fields in electric power substations

The magnetic fields generated by a typical distribution substation were measured and calculated based on a computer model which takes into account currents in the grounding systems, distribution feeder neutrals, overhead ground wires and induced currents in equipment structures and ground grid loops. Both measured and computed results indicate that magnetic fields are significantly influenced by ground currents, as well as induced currents in structures and ground system loops. All currents in the network modelled were computed, based on the measured currents impressed at the boundary points (ends of the conductor network). The agreement between the measured and computed values is good. Small differences were observed and are attributed mainly to uncertainties in the geometry of the network model and phase angles of some of the currents in the neutral conductors which were not measured in the field. Further measurements, including more accurate geometrical information and phase angles, are planned. >

Enhancement of the reflectron TOF focusing energy range

The requirements for the geometrical dimensions and ion packet deflection angles to achieve a maximum resolution of the mass-reflection with a field-free gap introduced between its decelerating and reflecting gaps are considered. The results obtained demonstrate the capability of the device proposed to make TOF analysis of the ion packets with large ion energy spread. It is also shown that, if by some reasons, the energy range of the ions entering the routine reflection can be predetermined, the regime with slightly detuned deceleration voltage is preferable, as it provides better mass resolution.

A rephasing-invariant analysis of the KM unitarity triangle in neutral-B decays

Three typical neutral-B decays Bd0 to psi KS, pi + pi -, and Bs0 to rho 0KS, which are usually suggested to probe the three angles of the KM unitarity triangle via their CP-violating asymmetries are analysed rephasing-invariantly. Considering the penguin contribution to Bd0 to pi + pi -, the author calculates the CP-violating angle at the O( lambda 2) level and remark its deflection from the previous O( lambda ) result. A comparison is also given between the geometrical unitarity angles and the corresponding process-dependent ones.

RAMM — A new procedure for theoretical conformational analysis of carbohydrates

The development is described of a new molecular-mechanics procedure which is designed for investigation of the molecular structure and for the prediction of minimum-energy conformations of flexible saccharide systems. In place of the optimization of energy with respect to the Cartesian coordinates of the atoms, used in conventional molecular mechanics, the minimization is carried out on internal geometrical parameters, namely, bond lengths, bond angles, and torsional angles. For this purpose, the non-derivative method of conjugate direction based on the Powell-Zangwill algorithm is used, resulting in a considerable computational advantage for large systems. Another principal feature of the procedure is the algorithm for a determination of the energetically favored orientations of pendant groups, based on a random-walk technique in a given conformation, independent of the initial starting geometry. The performance of the program RAMM using this procedure has been illustrated by comparing (ϕ, ψ) maps obtained with different approaches utilizing the MM2 set of potential functions for a model disaccharide. It is shown that the shape of the (ϕ, ψ) maps and the relative energies of conformers are significantly affected by orientation of pendant groups.

The crevice model of bubble nucleation

The crevice model for heterogeneous nucleation of bubbles in water in response to a decreasing liquid pressure is studied. The model neglects gas-diffusion effects and is therefore more suited for acoustic than for flow cavitation. It is argued that previous work has overlooked the essential requirement of unstable growth of the interface in the crevice. As a consequence, the available results are incorrect in some cases. Another feature of the model which is considered is the process by which the interface moves out of the crevice. It is concluded that, depending on circumstances, the conditions for this step may be more stringent than those for the initial expansion of the nucleus inside the crevice. Some numerical examples are given to illustrate the complex behavior of nuclei, depending of geometrical parameters, gas saturation, contact angles, and other quantities.

Geometric angles in cyclic evolutions of a classical system.

A perturbative method, using Lie transforms, is given for calculating the Hannay angle for slow, cyclic evolutions of a classical system, taking into account the finite rate of change of the Hamiltonian. The method is applied to the generalized harmonic oscillator. The classical Aharonov-Anandan angle is also calculated. The interpretational ambiguity in the definitions of geometrical angles is discussed.

Deconvolution of neutron depth profiling spectra

Iterative methods for determining deconvolued depth profiles from measured neutron depth profiling (NDP) energy spectra are presented. The methods account for energy broadening caused by system noise, energy straggling, multiple small angle scattering, and geometrical acceptance angles. Examples of the methods are given for known and unknown profiles of helium-3 and nitrogen-14.

The twin formation of tugtupite, a contribution

SummaryTugtupite, space group 1, belongs morphologically to the crystal class 2m. The paper describes pseudotrigonal contact triplets recently found as freely developed individuals on two specimens donated to the Geological Museum, Copenhagen. The contact triplets, described as a symmetric twin, have the composition planes, which are also the twin planes, (10) and (01); the pseudo-threefold axis, along which the crystals are pronouncedly elongate, is [111]. The calculated reentrant angles on the two sides of the pseudotrigonal prism, the protruding angle on the third side, and the ‘central misfit’, expressed as the geometrical angles, are 178° 41′, 181° 20′, and 2° 39′ respectively. Tetragonal crystals forming pseudotrigonal twins are rare and it is emphasized that the pseudotrigonal habit has become apparent only because: the faces developed on individual I and on individuals II and III are not the same; and the proportion between the lengths of the traces of the faces (01) individual I and (01) individuals II and III in a plane perpendicular to the pseudo-threefold axis and the traces in the same plane of all other faces constituting the pseudo-threefold prism are 2:1.

The electronic structure of CIS and TRANS nitrous acids

The variations of the geometrical (bond lengths and angles), electronic (dipole moment magnitude and orientation) and chemical (acidity) data which are experimentally observed when going from the cis to the trans isomer of nitrous acid are fully reproduced within the framework of semi-empirical CNDO/2, INDO and MINDO/2 approaches.

A new set of puckering coordinates for four-membered ring molecules

A set of two independent internal coordinates “polar angles” are proposed which allow description of arbitrary finite bond-bending puckering deformations of four-ring molecules with constant bond length. In particular for four-ring systems of the type xyyx and xxxx, whose conformations have symmetry C 2 v - C s , D 2 h - C 2 v and D 4 h - D 2 d , respectively, a rather complete set of relations expressing geometrical parameters leke bond angles, puckering angles, torsional angles, height and diagonals by the polar angles and their implications are discussed. Using Born-Oppenheimer approximation, a qualitative illustration of the ring-bending puckering potential function as expressed by polar angles is given.

Ligand field splitting in seven coordinated NbF 7 2− -type complexes

The splittings ofd-orbitals for seven coordinated complexes of the type NbF7 2− is calculated using a point charge model. The perturbed energies are expressed as a function of λ =G 2/G 4. The results are very sensitive on assumed geometrical angles of the molecule.

APPROXIMATE DETERMINATION OF THE INCOMPRESSIBLE FLOW REGION IN FRONT OF A BLUNT BODY IN HYPERSONIC FLOW

It would have been surprising indeed if the shock for a general body of revolution at an angle of attack ag would be of the same shape as the axisymmetric shock of this body turned by an angle vj = ag — a (that is, at an angle a), as is the case for the cone. By coordinating the shock characterized by <r and K at angle of attack a to a body at angle ag, a more complex relationship would be expected. The relationship indicated by Eq. (11) says that the curvature on the wind side is smaller by 1 — corr (corr proportional to a) than the curvature K, and the curvature on the shadowside is correspondingly larger by 1+ corr, but otherwise the equation for the pressure is the same as that for a shock a ± a in axisymmetric flow. If the shock of a sphere cone with cone half angle 6 + ag is compared to that of a sphere cone with cone half angle 6, both in axisymmetric flow, it is seen that the curvature of the larger sphere cone is indeed smaller. The result of the present investigation is not a solution of the problem but a suggestion to replace the problem by an axisymmetric one with a body 6 ± ae. The whole investigation is restricted to small a, and, if the afterbody is conical, it is suggested that the conical relation between the effective and the geometrical angles of attack be used [see Eq. (8) inRef.3]. Comparison of measurements for sphere cones at angle of attack with such measurements for equivalent sphere cones at zero angle of attack shows excellent agreement with each other and with numerical calculations according to Eqs. (1-11), if based on the axisymmetric shock of the equivalent body.