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Grzesik J.A.,Allwave Corporation
Annals of Nuclear Energy

A simple bilinear functional F is introduced on behalf of the Milne subcritical problem with replication parameter 0≤c≤1. This functional depends upon two arguments, respectively intended to be the neutron flux and its adjoint, and is stationary about the true solution pair where, in addition, it vanishes. The stationarity and null value can then be united as a basis for the demand that F continue to vanish even when flux and adjoint are both approximated by just the two modes from the discrete eigenvalue spectrum, a representation akin to what is known as the asymptotic portion of the neutron flux, and one which is clearly incapable of matching interface boundary conditions. The stationarity of F, however, renders it tolerant of such boundary defect, as a result of which one can expect the persisting null demand, F=0, to yield the best possible value for the ratio of the two discrete mode amplitudes. We go on to implement this program, and find as its outcome that the optimum amplitude ratio is determined as one preferred solution of a simple quadratic equation. With that solution in hand, it is an easy step then to a computation of the linear extrapolation length λ. We follow through with a numerical embodiment of these ideas, obtaining the discrete, real and positive eigenvalue ν0 on the run via a Newton-Raphson tangent encroachment root hunt. With sufficient start-up care the Newton-Raphson root hunt proves here to be exceedingly rapid, and it, together with the quadratic underpinning, provides for λ a string of values that differ by less than 0.5% from those found in the classic compendium on neutron transport from the pens of Case, de Hoffmann, and Placzek. In particular, we are able to bypass in this way, and with quite elementary tools indeed, a known canonical machinery of far greater weight and sophistication, be it based upon the Wiener-Hopf method, or else upon flux decomposition along both discrete and singular eigenfunction modes. To our way of thinking, such a simple alternative is aesthetically pleasing in its own right, and both provides a measure of confirmation to, and is itself checked by, the more formidable apparatus. © 2014 The Authors. Published by Elsevier Ltd. Source

Grzesik J.A.,Allwave Corporation
PIERS 2010 Cambridge - Progress in Electromagnetics Research Symposium, Proceedings

One embodiment of a plasma fusion reactor, in vogue a number of years ago at the Lawrence Livermore National Laboratory, employed a cylindrical chamber having its ends capped by massive, Yin-Yang (Y-Y) magnetic coils serving as barriers against charged particle escape.Such Y-Y coils, by their very geometry, require opposed current flow in close proximity, circumstance which summons forth a dilatational magnetic pressure raising device disintegration to the level of a calamitous possibility. And, while such Y-Y fragmentation is surely not a welcome design outcome, nevertheless it does invite a preliminary analysis as to its potential violence, analysis which enjoys besides a modicum of theoretical interest by virtue of making relevant a scenario of electrodynamics in an expanding cavity. With these dual aims in mind, we had many years ago undertaken the study of the very simplest of such expanding cavity situations, namely, the growing interstitial (vacuum) wafer separating two massive metallic plates undergoing a symmetric flight from one another. Quick penetration into the heart of this problem was provided by the observation that, on the one hand, a quasistatic (QS) field computation would surely suffice, while, on the other, that a moving boundary condition (MBC) could be fashioned in lowest relativistic order by combining laboratory-frame electric E and magnetic B fields, and the boundary velocity v; and thence requiring that the effective tangential electric field ∼ v̂×{E + v×B} vanish upon both plate boundaries. In this process, a secondary computation of E was bootstrapped upon a primary, QS one for B via Faraday's law, whereby the obligatory time derivative of the latter was implicitly tethered to the dynamic evolution of its underlying separation parameter η(t): Under this viewpoint there easily emerged the invariance against time of the product of B by η (or else of current I by η) leading to a simple differential equation for the dynamical evolution of the net separation η (t); and, in particular, to the identification of a characteristic time scale τ suggesting a most vigorous magnet disintegration. This aspect of the work has been previously reported in summary form,and is set out anew here for the purpose of building an intuitive, heuristic base concerning field evolution within the primitive, expanding wafer cavity now at hand. A heuristic base of this sort is far too coarse to account for field retardation effects due to signal transit at finite light speed c: We remove this defect by returning to the Faraday/Ampμere equations in their primitive form and subjecting them first to Fourier transformation in coördinate z along the direction of cavity expansion perpendicular to magnet walls. Such transformation embraces the entire interval -∞ < z < ∞ and, as such, submits to a null-field attitude which regards the field, in both its electric E and magnetic B manifestations, as being zero exterior to the expanding wafer, i.e., |z| > η (t)/2. Due deference must of course be paid, in the form of Dirac delta function sources placed at z = ±η(t)/2; to the radiation emanating from surface current density ±I(t) flowing on cavity walls. Elimination of either field transform leads then to a simple harmonic differential equation in time t having a source gauged by I(t). Its solution is readily gotten in a form that allows inverse Fourier transformation to proceed smoothly and, in particular, to identify a retarded signal emission time t* < t as gauged from either plate which obeys the intuitively pleasing condition c(t - t*) = {η(t) + η(t*)} /2: All in all one confronts at this point a relatively simple pattern of connections between the field and its source I as reckoned at retarded times t* suitably structured so as to track upper/lower plate emissions, connections which succumb at length to an a posteriori enforcement of the non-relativistic limit η(t) < c so as to recover anew the key I× invariance previously inferred during the prelude of approximate, QS/MBC analysis. Source

Grzesik J.A.,Allwave Corporation
IEEE Antennas and Propagation Society, AP-S International Symposium (Digest)

Far-field reconstruction from near-field data acquired over apertures of finite, practically realizable size is examined with point electric dipoles serving as primitive sources that retain some vestige of physical credibility. A fairly rigorous analytic apparatus and its numerical implementation reveal that the far-field patterns so attained are inevitably marred by gain ripples, and thus compromised as to their utility. These ripples, however, are an artifact of the fiction that wave propagation is entirely unburdened by dissipation. Indeed, in the special case of a horizontally aligned dipole, we are able to show quite unequivocally that gain ripples along its broadside and, by extension, along all other pattern directions, duly subside when wave dissipation is admitted into play. © 2015 IEEE. Source

Chang F.C.,Allwave Corporation
IEEE Antennas and Propagation Magazine

An efficient technique is developed to simplify computations in the field of vector analysis. The evaluation of vector algebraic and differential operations becomes more simple and straightforward by simply transforming the vector operations into matrix operations. The matrix operations are especially useful when there are mixed coordinate basis involved in the vector operations. © 2012 IEEE. Source

Grzesik J.A.,Allwave Corporation
AIP Advances

We revisit the problem of de Haas-van Alphen (dHvA) diamagnetic susceptibility oscillations in a thin, free-electron film trapped in a synthetic harmonic potential well. A treatment of this phenomenon at zero temperature was announced many years ago by Childers and Pincus (designated hereafter as CP), and we traverse initially much the same ground, but from a slightly different analytic perspective. That difference hinges around our use, in calculating the Helmholtz free energyF, of an inverse Laplace transform, Bromwich-type contour integral representation for the sharp distribution cutoff at Fermi level μ. The contour integral permits closed-form summation all at once over the discrete orbital Landau energy levels transverse to themagnetic field, and the energy associated with the in-plane canonical momenta k x and k z. Following such summation/integration, pole/residue pairs appear in the plane of complex transform variable s, a fourth-order pole at origin s=0, and an infinite ladder, both up and down, of simple poles along the imaginary axis. The residue sum from the infinite pole ladder automatically engenders a Fourier series with period one in dimensionless variable μ/ω (with effective angular frequency ω suitably defined), series which admits closedform summation as a cubic polynomialwithin any given periodicity slot. Such periodicity corresponds to Landau levels slipping sequentially beneath Fermi level μ as the ambient magnetic field H declines in strength, and is manifested by the dHvA pulsations in diamagnetic susceptibility. The cöexisting steady contribution from the pole at origin has a similar cubic structure but is opposite in sign, inducing a competition whose outcome is a net magnetization that is merely quadratic in any given periodicity slot, modulated by a slow amplitude growth. Apart from some minor notes of passing discord, these simple algebraic structures confirm most of the CP formulae, and their graphic display reveals a numerically faithful portrait of the oscillatory dHvA diamagnetic susceptibility phenomenon. The calculations on view have a merely proofof-principle aim, with no pretense at all of being exhaustive. The zero-temperature results hold moreover the key to the entire panorama of finite-temperature thermodynamics with T>0. Indeed, thanks to the elegant work of Sondheimer andWilson, one can promote the classical,Maxwell-Boltzmann partition function Z (s), via an inverse Laplace transform of its ratio to s2, directly into the required, Fermi-Dirac Helmholtz free energy F at finite temperature T > 0. While the underlying cubic polynomial commonality continues to bestow decisive algebraic advantages, the evolving formulae are naturally more turgid than their zero-temperature counterparts. Nevertheless we do retain control over them by exhibiting their retrenchment into precisely these antecedents. So fortified, we undertake what is at once both a drastic and yet a simpleminded step of successive approximation, a step which clears the path toward numerical evaluation of the finite-temperature diamagnetic susceptibility. We are rewarded finally with a persistent dHvA periodicity imprint, but with its peaks increasingly flattened and its valleys filled in response to temperature rise, all as one would expect on physical grounds. Copyright © 2012 Author(s). Source

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