Critique of Electromagnetic Models of the Nucleus, re Older Theory
| Scientific Paper | |
|---|---|
| Title | Critique of Electromagnetic Models of the Nucleus, re Older Theory |
| Read in full | Link to paper |
| Author(s) | Roger A Rydin |
| Keywords | electromagnetic, nucleus, quantum mechanics, nuclear |
| Published | 2009 |
| Journal | General Science Journal |
| No. of pages | 11 |
Read the full paper here
Abstract
Early 1950s models of nuclei considered that they were made up solely of protons and neutrons having approximately equal sizes. A large nucleus was an approximately spherical ensemble made up of these packed entities. Nuclear reactions were treated using Quantum Mechanics, by solving the Schrödinger wave equation for wave amplitudes inside and outside a nucleus and relating these to cross sections. Beginning in the 1990s, new Classical Electromagnetic models of nuclei were developed. These are quite totally different from those in the theoretical Standard Model which uses quarks and gluons, etc. Electrons and protons were considered to be spinning charged fibers, and a neutron was considered to be a paired combination of an electron and a proton. With these new nucleon models, nuclei were crudely modeled as concentric shells of these particles placed in a 3D spatial configuration by a geometrical mapping theory called Combinatorial Geometry. This theory was able to correctly predict the "Magic Numbers" as combinations of shells filling and emptying as nucleons were added. An improved Semi-Empirical Binding Energy Formula was developed, which accurately predicts the binding energies of stable and radioactive isotopes, and also correctly predicts their spins. The Electromagnetic model was improved by making detailed spatial and directional force balances using a variational minimization technique, which predicts decay energies for various reactions.
In 2004, a new Electromagnetic model of all types of particles was developed based upon a three-level scheme of wrapping fractionally charged fibers. This new fiber model has yet to be used to redo the nucleus calculations. However, Electromagnetic models have the potential to describe nuclear reactions in terms of unstable vibrations whose equations are analogs to the Schrödinger equation. The purpose of this paper is to discuss all of these models, and to predict where the research should go next.
Overview
Roger A. Rydin, Associate Professor Emeritus of Nuclear Engineering at the University of Virginia, presented this paper to the 16th Natural Philosophy Alliance conference at the University of Connecticut in May 2009. It is a review and, as the title says, a critique — Rydin is broadly sympathetic to the classical electromagnetic programme of David L Bergman, Charles W. Lucas Jr. and Edward A Boudreaux, but he spends much of the paper listing the places where he thinks it does not yet work, and he is unusually explicit about which of its claims he cannot interpret.
The programme under review replaces the quark-and-gluon Standard Model with finite, structured, classical charges: the electron and proton are spinning charged rings or bundles of charged "fibers", and the neutron is a proton ring nested inside an electron ring. Nuclei are then assembled by balancing electric and magnetic forces between these rings, with no quantum mechanics and no separate strong or weak interaction. Rydin's structuring device is to first restate carefully what was known in 1955 from Robley Evans's The Atomic Nucleus, then to ask which of the new models survives that older body of measurement.
The argument
What the 1950s data already fixed
Rydin opens with the experimental constraints he regards as binding. Charged-particle scattering and isotope shifts give a nucleon radius parameter R0 of about 1.1 × 10−13 cm; fast-neutron scattering gives about 1.5 × 10−13 cm; the ratio is about 1.4, "so that seems to be the upper limit of size uncertainty." Hofstadter's electron scattering at 125–150 MeV showed that neutron and proton are of about the same size and that each has an internal charge distribution. The anomalous magnetic moments, +2.79 nuclear magnetons for the proton and −1.91 for the neutron, were unexplained by any 1950s theory, but Evans concluded from the neutron's moment alone that it must have internal charge structure. Rydin notes fairly that this is "not inconsistent with the Standard Model" of −1/3e and +2/3e quarks.
He also quotes Evans on nuclear shape and density: nuclei are nearly spherical and of nearly uniform charge density, the largest known quadrupole moment corresponding to a major axis only 20 per cent longer than the minor axis, with most nuclei near 1 per cent ellipticity; nuclear volume is proportional to A, so nuclear matter is essentially incompressible, with variations "only of the order of 10 per cent." Rydin draws the inference that matters for him: "the property of compression resistance implies the existence of some sort of force that resists compression." He reproduces at length Evans's section on the Nonexistence of Nuclear Electrons — the spin-and-statistics problem of N-14, positron emission, and Fermi's pair-production account — without softening it.
Spinning charge rings
Bergman's model, adapted from A. H. Compton's 1917–1919 papers and from Winston H Bostick's plasmoid work, treats the electron as a single spinning ring of charge −e and the proton as a ring of charge +e, each sized to reproduce its measured properties, with the neutron a coplanar proton ring inside an electron ring.
Rydin names two objections and does not resolve either. First, the ring sizes are asserted to change with the external electromagnetic environment, so that "the free neutron has a considerably different size than the bound neutron"; the underlying awkwardness is that the classical electron radius exceeds the measured proton and neutron size. He offers a genuine defence here: the classical radius is defined by equating the assembly energy of charge −e to m0c2, and "if a similar definition were made for a proton, its size would not agree with the measured size" either. Second, "the spin and statistics of nuclei made up of these particles have to be rationalized away by assuming that the neutron combination acts as a single particle. While this might work for beta decay, it doesn't explain the neutrino, and the model cannot be used to explain positron decay."
The geometrical packing nucleus and the magic numbers
Lucas builds nuclei from Bergman's rings without orbital motion, arguing that "Ampere's law and Faraday's law in electrodynamics require that charged nucleons radiate energy continuously if they orbit in the nucleus", so equilibrium must come from a static balance of electric and magnetic forces at fixed radii. Protons occupy the two outermost shells to keep apart; neutrons polarise with their positive ends inward, and inner shells can be left empty.
The magic numbers are then produced as sums of shell capacities drawn from the sequence 2, 8, 18, 18, 32, 50: 2 and 8 are magic alone; 20 = 2 + 18; 28 = 2 + 8 + 18; 50 = 18 + 32 and is also a shell by itself; 82 = 32 + 50; and 126 = 50 + 32 + 18 + 18 + 8, the neutron structure Lucas assigns to Pb-208. Lucas is credited with predicting the ground-state spins of all nuclides — noting that magic and even–even nuclides have none — "while the regular shell model gets about half of the non-zero spins wrong."
Lucas also fits his own semi-empirical mass formula to all ~3000 nuclides, in the form reproduced by Rydin:
W/A = K1 − K2(nucleons in outermost shell)/A − K3Z(Z−1)A−4/3 − K4(#paired N − #paired Z)2/A − K5(#unpaired p + #unpaired n)/A
with the five terms labelled volume, surface, Coulomb, asymmetry/magic, and pairing. Rydin observes an internal tension: the leading term is proportional to A, so "in the shell model, unoccupied inner shells must not take up much space or this term would be wrong. So the shells are somewhat more mathematical than real."
Force balances, decay, and vibration
Boudreaux and Baxter let Bergman's nucleons separate and rotate to minimum energy by a variational technique, converging on Lucas's configurations and obtaining approximately correct decay energies for Be-8, Na-24 and K-40. K-40 is presented as the best case: two spin states appear as two minima in its binding-energy profile, which Rydin suggests accounts for the branching between beta-minus decay to Ca-40 and the electron-capture / positron branch to Ar-40 used in radiometric dating.
Rydin's own contribution is section 2.6. Modelling the "strong force" as electromagnetic attraction, the "weak force" as charged-fiber compression resistance, and the Coulomb force as linear springs, and writing F = ma, he reports that "the resulting balance equation is of the same form as Schrödinger's Wave Equation", so excited states become vibration eigenstates and fission becomes an instability of the old liquid-drop vibration problem. His recommendation for future work is that Lucas's newer three-level fiber particles be fed into Boudreaux's force-balance computation, with neutron polarisation and compression resistance added.
Assessment
The paper's chief virtue is its honesty. Rydin is writing inside the movement whose models he reviews, and he still writes down the objections plainly: the ring models cannot accommodate the neutrino, cannot handle positron decay, need spin-statistics "rationalized away", and depend on particle sizes that change by unstated amounts for unstated reasons. Of Lucas's claim that the neutron's fibers rearrange when bound he says simply, "I don't know how to interpret this comment, or what experimental evidence supports it." His preferred fallback — treat nucleons as black boxes with the right external electromagnetic properties and stop asking what is inside — is a defensible research posture, and his insistence that a compression-resisting force is needed for deuterium and tritium, "which have only attractive nuclear forces and no Coulomb repulsion between nucleons," is a sound piece of reasoning. His rebuttal on the classical electron radius is also correct: applying the same definition to the proton gives 2.82 fm × (me/mp) ≈ 0.0015 fm against a measured charge radius near 0.84 fm, so the definition demonstrably fails for the proton and cannot be used as a clean objection to the electron ring.
The quoted 1950s numbers check out. The proton and neutron moments of +2.79 and −1.91 nuclear magnetons match the modern values 2.7928 and −1.9130; 1.5/1.1 is indeed about 1.4; and the magic-number arithmetic is all correct as arithmetic — 2 + 18 = 20, 2 + 8 + 18 = 28, 18 + 32 = 50, 32 + 50 = 82, and 50 + 32 + 18 + 18 + 8 = 126. One transcription slip: the largest quadrupole moment is attributed to "Lu-76", which is Evans's notation 71Lu176 for lutetium-176 (lutetium is element 71; there is no Lu-76).
But the magic-number result is weaker than it is made to sound. The capacities 2, 8, 18, 32, 50 are the 2n2 sequence of atomic electron shells, imported wholesale, and no rule is given fixing which subset of them applies to which nucleus. Fifty is reached as 18 + 32 but is also a shell in its own right; 126 needs five shells while 82 needs two, and the inner 2-shell is included for 20 and 28 but dropped for 126. With free choice of subsets from a fixed set, most integers in the range are reachable — 2 + 8 = 10, 8 + 18 = 26, 2 + 32 = 34 — so hitting 2, 8, 20, 28, 50, 82, 126 is a fit rather than a prediction until a filling rule is stated. Simple cumulative filling of 2, 8, 18, 18, 32, 50 gives 2, 10, 28, 46, 78, 128, which reproduces only two of the seven.
The mass formula deserves the same scrutiny. Its Coulomb term, K3Z(Z−1)A−4/3, is identical to the Coulomb term of the ordinary Weizsäcker formula divided by A, so nothing new is being claimed there; the surface term, being an outermost-shell count over A, scales as A−1/3 exactly as the standard surface term does. The asymmetry term genuinely differs: for even–even nuclei, (#paired N − #paired Z)2/A equals (N−Z)2/4A, which grows with A relative to the standard (N−Z)2/A2. That is testable, and it is the one place a residual table would settle the matter. The claim that the fit is accurate "to within the measurement errors of the isotope masses" is the paper's largest unsupported assertion: isotope masses are known to keV or better, while the classical semi-empirical formula has residuals of a few MeV per nucleus, and no residuals, χ2, or fitted constants are given anywhere in the paper. Likewise, "the regular shell model gets about half of the non-zero spins wrong" overstates the case; the single-particle shell model with pairing gets most odd-A ground-state spins right, and the systematic failures are concentrated in deformed regions handled by the Nilsson model. And the observation that even–even nuclei have J = 0 is not a discriminating success, since pairing gives the same result in the conventional shell model.
The deepest difficulty is one Rydin raises and then walks past. He reprints Evans's argument for the nonexistence of nuclear electrons and then endorses models whose neutron is a bound electron and proton. Beyond the N-14 statistics problem, there is a straightforward energy bound: confining an electron to a region of order 1 fm requires momentum of order ħ/Δx, and with ħc = 197 MeV·fm this is a momentum of roughly 197 MeV/c and hence an energy near 200 MeV, against an electron rest energy of 0.511 MeV and beta-decay endpoints that never exceed about 20 MeV. The same scale bites the ring geometry: a ring of charge circulating at c with angular momentum ħ/2 must have radius ħ/2mec ≈ 193 fm, some 200 times a nucleon radius. Bergman's answer, that ring sizes change elastically with the field, is exactly the step Rydin says he cannot interpret; without a stated law for how sizes change, the model has an adjustable length for every situation, and the "prediction" of nucleon properties is not constrained. See also the uncertainty principle for the general form of the confinement bound.
Two further points of contact with measurement go unmentioned. Deep inelastic electron and neutrino scattering resolves point-like, fractionally charged, spin-1/2 constituents inside the nucleon, with scaling behaviour that a rigid ring of unit charge does not reproduce; and the nucleon excitation spectrum begins at the Δ resonance about 300 MeV above the ground state, a scale that has to come from somewhere in any classical model. Finally, the closing claim that F = ma with linear springs yields "the same form as Schrödinger's Wave Equation" is asserted, not shown, and it is not quite right: a linear-spring force law gives a real, second-order-in-time classical oscillator equation, whereas the Schrödinger equation is first order in time and complex. What the two share is a discrete eigenvalue spectrum, which is a property of any bounded linear wave problem and so does not by itself establish that "Quantum Mechanics is unnecessary to explain nuclear reactions."