Instantaneous Configuration Research Articles

Geomagnetic field distortions and their effects on radiation belt particles

The study of trapped‐particle motion in the earth's magnetic field is usually geared toward building the foundation of radiation‐belt dynamics (particle injection, acceleration, diffusion, and loss). In this article we turn the situation around and use trapped‐particle motion to explore the properties of the geomagnetic field itself. One of the quadrupole terms in the main geomagnetic field is found to contribute to a north‐south ‘shear distortion’ of the particle‐drift shells, whereas one of the octupole components causes a longitude‐dependent radial deformation and associated ‘drift‐shell splitting.’ The collective action of all higher multipoles on trapped‐particle motion is then used to analyze the ‘true’ anomalies or distortions of the internal geomagnetic field that are independent of the quadrupole‐related eccentricity of the main dipole. These ‘true’ anomalies must originate in upper‐mantle or crustal perturbations that lie relatively near the earth's surface on both sides of the mid‐Atlantic ridge; they influence trapped‐particle drift shells only where the latter have their closest approach to the earth (South Atlantic and South African areas). The quadrupole and octupole perturbations, on the other hand, obviously originate deep in the earth's core. The most adequate space and particle coordinate systems are introduced, and their validity is briefly analyzed. In the final part of this review, we discuss the effects of external magnetospheric currents. A time‐dependent symmetric ring current causes drift shells to be displaced radially, with associated particle acceleration; magnetopause currents introduce a day‐night asymmetry, causing shell splitting. Time dependence of these effects introduces further complications; an invariant particle coordinate system is discussed that can be used for particle‐flux representations that are independent of the particular instantaneous field configuration (provided that the latter is known and that all changes occur slowly, i.e., adiabatically).

Calculation of the Eulerian angles and large amplitude vibrational coordinate corresponding to an arbitrary instantaneous molecular configuration

Calculation of the Eulerian angles and large amplitude vibrational coordinate corresponding to an arbitrary instantaneous molecular configuration

Extension of the methods of limit analysis to the investigation of the plastic flow of bodies with variable geometry

Making use of the fact that the limit state does not depend on the loading history, the author proposes that limit analysis theory be used to investigate plastic flow processes associated with a change in initial geometry by representing the deformation process as a continuous sequence of limit states defined by a given instantaneous configuration.

Motion analysis of rigid-link mechanism by gradient optimisation on an analogue computer

A general method is presented for simulating the motion of planar mechanisms on an analogue computer. It is based on a continuous solution of the loop equations describing the instantaneous mechanism configuration using a gradient optimisation technique. Examples of the application of the method to a four-bar eccentric turning block and to a two-pivot Stephenson linkage are given.

Molecular Structure of XeF6. II. Internal Motion and Mean Geometry Deduced by Electron Diffraction

The distribution of internuclear distances in gaseous XeF6 exhibits unusually diffuse XeF6 bonded and F–F geminal nonbonded peaks, the latter of which is severely skewed. The distribution proves the molecule cannot be a regular octahedron vibrating in independent normal modes. The instantaneous molecular configurations encountered by the incident electrons are predominantly in the broad vicinity of C3υ structures conveniently described as distorted octahedra in which the xenon lone pair avoids the bonding pairs. In these distorted structures the XeF bond lengths are distributed over a range of approximately 0.08 Å with the longer bonds tending to be those adjacent to the avoided region of the coordination sphere. Fluorines suffer angular displacements from octahedral sites which range up to 5° or 10° in the vicinity of the avoided region. Alternative interpretations of the diffraction data are developed in detail, ranging from models of statically deformed molecules to those of dynamically inverting molecules. In all cases it is necessary to assume that t1u bending amplitudes are enormous and correlated in a certain way with substantial t2g deformations. Notwith-standing the small fraction of time that XeF. spends near Oh symmetry, it is possible to construct a molecular potential-energy function more or less compatiable with the diffraction data in which the minimum energy occurs at Oh symmerty. The most notable feature of this model is the almost vanishing restoring force for small t1u bending distortions. Indeed, the mean curvature of the potential surface for this model corresponds to a υ4 force constant F44 of 10−2 mdyn/Å or less. Various rapidly inverting non-Oh structures embodying particular combinations of t2g and t1u deformations from Oh symmetry give slightly better radial distribution functions, however. In the region of molecular configuration where the gas molecules spend most of their time, the form of the potential-energy function required to represent the data does not distinguish between a Jahn–Teller first-order term or a cubic V445 term as the agent responsible for introducing the t2g deformation. The Jahn–Teller term is consistent with Goodman's interpretation of the molecule. On the other hand, the cubic term is found to be exactly analogous to that for other molecules with stereochemically active lone pairs (e.g., SF4, ClF3). Therefore, the question as to why the XeF6 molecule is distorted remains open. The reported absence of any observable gas-phase paramagnetism weighs against the Jahn–Teller interpretation. The qualitative success but quantitative failure of the valence-shell–electron-pair-repulsion theory is discussed and the relevance of the “pseudo-Jahn–Teller” formalism of Longuet-Higgins et al. is pointed out. Brief comparisons are made with isoelectronic ions.

On the description of anisotropic workhardening

A model of workhardening is proposed which generalizes the known rules of isotropic and kinematic workhardening by introducing the concept of a ‘field of workhardening moduli.’ This field is defined by a configuration of surfaces of constant workhardening moduli in the stress space. For any loading history the instantaneous configuration can be determined by calculating the translation and expansion or contraction of all surfaces; the material behaviour can thus be determined for complex loading paths, in particular for cyclic loadings. Several examples for a plane stress state are presented.

Cluster Development Method in the Quantum Mechanics of Many Particle System, I

In order to investigate the many particle system by the variational principle, the cluster develop­ ment method propcsed by Jastrow is sys~cmatically formulated. It is a dircct generalization of the Hartree method, in which two particle functions are included in the trial function to represent the correlations between particlcs. The cnergyxpectation value is expar..ded in a series, each term of which is expressed in cluster integra's. It is found that the series convergcs rapidly when the cor· relations are marked only fot· small particle separations, so that this method is expected to be useful especially for the system of particles interacting through short range potentials with repulsive cores. § I. Introduction The Hartree method ·for a many particle system, in which each particle is supposed to move independently 2.nd to be influenced by other p2.rticles only through their average field of force, has been an importa.nt approximation a.nd successful in various problems. However, in some cases where the correlations between particles play an important role, this simple approximation is no longer valid. In one case, when the range of the interaction between particles is long compared with the mean distance between them, many particles may be coupled togeth:::r to move in some organized mode of motion as a whole, as is seen in the typical example of the plasma oscillation in- a dense electron gas. The recent study by Tomonaga 1l made the mechanism of this so-called collective motion quite clear_ In :>.nothe: case, when the interaction is strong but of short range, the actual potential determi11cd by the instantaneous configuration of the other . particles fluctuates from the average one considerably, and the image of the independent particle motion becomes far from the truth. Such a situation occurs especially when the interaction potential has a repulsive core at short distance as in liquid helium .or in the atomic nucleus. (In the latter case the existence of the repulsive core seems very likely from the analysis of the nucleon-nucleon scattering/) the binding energy differ~nce between H'l and He3, 3l and so forth.) In fact, if .we apply the Hartree method to these systems, the rc3ulting average potential diverges to infinity. Actually, the wave function of the system must vanish in the region of the configuration space where at least two particles arc fotmd. within the hard core range. In such cases we have to take into· account this strong short range correlation in some way.

Transverse oscillations in travelling strings

The behaviour of transverse standing waves is examined for a uniform string which is pulled over two smooth supports under constant speed and tension. A harmonic spectrum of resonance frequencies is found which, for small ratios of the speed of travel to the propagation velocity of transverse waves, are only slightly lower than those of a string with clamped ends. The frequency response to a simple harmonic disturbance and the envelope of the oscillations are similar to those of a clamped string, but the phase of oscillation varies from point to point so that the instantaneous configuration of the string is not a sine curve. The conclusions remain valid in the presence of damping.

On the Transitional Phenomena of Glass

A new interpretation on the glassy state, which includes the explanation of the transitional phenomena, is proposed on the basis of the superposition of the two modes of motion, one of which is the network deformation and another is the vibration around the instantaneous configuration of network. To formulate this idea, a new parameter, that is the “network temperature”, is introduced and the thermodynamical treatment is performed, with the independent derivation of the equation of motion. Finally, some experimental data are checked with this theory and its validity is discussed.

Some Studies Concerning Rotating Axes and Polyatomic Molecules

The theory of small vibrations when the potential energy is invariant under the rotation-displacement group is developed. The results are compared with the Brester-Wigner theory of the normal coordinates, and it is shown that the use of these coordinates implies the use of a particular (normal) system of rotating axes whose construction is given. It is shown that when the motion of a normal molecule is referred to these axes, those terms of the Hamiltonian which are linear in the angular momenta will be especially small and of the same order of magnitude as the quadratic terms (Casimir's condition). When the amplitude of one or more of the normal vibrations becomes large, this is no longer true of the normal axes; this will always be the case when one of the normal frequencies is small compared to the others, as has been noted by other writers. The normal axes are not the principal axes of inertia of the instantaneous configuration of the system, and certain conclusions recently published by the author are wrong for that reason.