Wave Mechanics without Waves: A New Classical Model for Nuclear Reactions
| Scientific Paper | |
|---|---|
| Title | Wave Mechanics without Waves: A New Classical Model for Nuclear Reactions |
| Read in full | Link to paper |
| Author(s) | Roger A Rydin |
| Keywords | waves, nuclear reactions, compound nucleus, charged fiber model, vibration |
| Published | 2009 |
| Journal | General Science Journal |
| No. of pages | 14 |
Read the full paper here
Abstract
A comparison is made between the conventional way nuclear reactions were considered to take place circa 1960, and how they might take place using a new Classical electromagnetic model of the nucleus proposed by Charles Lucas. The old analysis used an analytic model of a compound nucleus and the linear Schrödinger wave equation to predict reaction cross sections by treating the wave solutions as quantum mechanical probabilities. Actual experimental data had to be inserted into the equation to get the response for each case. The new model considers the perturbed nucleus in terms of mechanical vibrations that lead to unstable states that foster decay to restore stability. The form of the resulting balance equation is similar to the Schrödinger wave equation, so the mechanical model has the potential to produce similar results without using waves.
Overview
Rydin presented this paper at the 16th Natural Philosophy Alliance conference (University of Connecticut, Storrs, June 2009). He writes as a nuclear engineer — Associate Professor Emeritus of Nuclear Engineering at the University of Virginia and author of a reactor theory textbook — and the paper is unusual among dissident nuclear papers in that its first half is a careful, sympathetic exposition of the orthodox compound-nucleus theory he was trained in and taught. Only after setting that out does he ask whether the same phenomenology could be reproduced by a purely mechanical model built on the charged-fiber particle structures of David L Bergman and Charles William Lucas.
The claim is deliberately modest. Rydin does not present a finished calculation or a fit to cross-section data. What he offers is a structural argument: if the nucleons of a nucleus are extended electromagnetic objects held in place by a balance of Coulomb, magnetic ("strong") and compressive ("weak") forces, then those forces can be linearised as springs, and the perturbed nucleus becomes an ordinary coupled-oscillator eigenvalue problem. The resulting equation of motion, he shows, has the same mathematical shape as the time-dependent Schrödinger equation — but its eigenvalues are mechanical resonances of actual nucleon positions rather than probability amplitudes, and nothing in it requires wave-particle duality. Hence the title: wave mechanics without waves.
The argument
What the conventional model does
Rydin summarises the circa-1960 treatment of neutron-induced reactions in two steps: formation of an excited compound nucleus, and its breakup into elastic scattering, inelastic scattering, capture with gamma emission, or fission. The incident neutron's kinetic energy in the centre-of-mass frame plus its binding energy sets the excitation energy. Because a neutron is far smaller than the target, hard collisions should be rare, so the theory invokes the de Broglie relation λ = h/p to replace the particle with "a wave train of poorly defined side-wise extent". Rydin calls this an "artifice" needed precisely because particles do interact at appreciable transverse separation, and he is similarly blunt about Coulomb barrier tunnelling, which he calls "almost magical, because the particle suddenly 'appears' on the other side of the barrier without being changed in any way."
With Schrödinger's ν = W/h the wave velocity becomes
- ν = W / √(2m(W − U)) (cm/s)
which is not the particle velocity; the group velocity recovers that. The square-well problem is then solved region by region, with the amplitude equation
- ∇2ψ + k2ψ = 0, k2 = 8π2m(W − U)/h2
whose ℓ = 0 solutions are spherical Bessel functions j0. When Ec admits an integral number of internal wavelengths across the nuclear radius R, interior and exterior solutions join with zero derivative, the amplitude is maximal, and one has a virtual level and a resonance in the cross section. Because k depends on the square root of Ec + B, levels are unevenly spaced; higher ℓ interleaves further ones. Squaring the wave function gives occupancy probability, which is "qualitatively proportional to the cross section", and the isolated resonance reduces to the Breit-Wigner single-level formula. Rydin's criticism is not that this fails but that "the whole process is a fit to data": level widths Γ are taken from measurement, not predicted.
The charged fiber alternative
The classical alternative descends from Arthur Compton's 1917–19 ring model by way of his student Winston Bostick, whose plasmoid work suggested the electron behaves as a charged toroidal fiber loop with both electric and magnetic properties. David L Bergman with Paul Wesley modelled the electron as a spinning charged ring of charge −e; Bergman modelled the proton as a ring of charge +e with different radius, and the neutron as a coplanar proton ring inside an electron ring. Charles William Lucas then packed these rings into shells, deriving the magic numbers from the idea that small shells break up and rearrange into larger, more stable ones — no attractive nuclear centre required — and fitted a semi-empirical mass formula to all ~3000 nuclides that Rydin says reproduces the binding-energy-per-nucleon curve, peaks included, within isotope mass measurement error. Boudreaux and Baxter took Bergman's nucleons into a variational force-balance code and recovered approximately correct decay energies for Be-8, Na-24 and K-40.
The vibratory model
Rydin's own contribution begins from what Lucas left out: fiber compression in bound structures (his own reading of what the weak force is) and neutron polarisation. He proposes replacing each of the three forces by springs. The magnetic "strong" attraction is effectively a contact force with a positive spring constant; the weak-force compression, which does work over a very short distance and is what stops two nucleons from simply merging, is a spring with a negative constant; Coulomb repulsion acts only between protons at multiples n·x0 of the inter-nucleon spacing x0 ≈ 3.0 × 10−13 cm.
He then walks up the light nuclei to show how spring constants might be extracted from mass differences: deuterium (collinear, strong and weak only), tritium and He-3 (coplanar triangles, magnetic attractions 50% larger by vector addition), He-4 (tetrahedral, forces doubled, very tightly bound), the absence of stable He-5, and He-6 stabilised by a paired neutron addition. Throughout, polarised neutrons contribute small net Coulomb attractions off the internucleon centrelines — his account of the spin-pairing effect. His own conclusion on this section is candid: "sorting out all of these forces to derive spring constants from this data is not easy to do."
Formally he defines a nucleon position vector P(r,t) ordering Z protons and N neutrons, a spring-constant matrix F = C + M + W (Coulomb, magnetic, weak-compression), and a diagonal intrinsic-mass matrix I. Newton's law then gives
- FP = I ∂2P/∂t2
and, with a separable damped-periodic solution P(r,t) = P(r)e−ωt, the eigenvalue problem
- [F − ω2I]P = 0.
A zero eigenvalue is the ground state; the others go as square roots of spring constants over masses and correspond to the bound and virtual levels. Complex eigenvalues give growth rates that play the role of decay constants. Light nuclei have few, widely spaced levels because they have few vibrational degrees of freedom; heavy nuclei have many closely spaced ones. Rydin further suggests that part of the measured level width may be physical rather than statistical, since a neutron can arrive in many spatial orientations, each giving a slightly different response. Beta decay becomes compression-initiated vibration of a single nucleon — a proton-electron two-body oscillation whose frequency depends only on the electron mass and the weak spring constant — which he offers as an alternative to mediation by heavy exchange bosons.
Assessment
The paper's real strength is its honesty about what it has and has not done. Rydin does not claim to have computed a cross section; he claims a structural analogy, and he states its limits himself — the linearisation "may only be valid over a small range of particle movement", the equations "do not contain enough detail to decide how the transition takes place or what new state will be chosen", and the Baxter code lacks both fiber compression and polarisation. That is a fair description of a programme sketch, and as a sketch it is coherent: coupled oscillators genuinely do produce discrete eigenfrequencies, and the observation that level density scales with the number of nucleons is a natural consequence of counting vibrational degrees of freedom rather than an assumption.
The difficulties are correspondingly large. Nothing in the paper is derived; the spring constants are conceded to be unobtainable in closed form and would need a nonlinear least-squares fit — which reinstates exactly the "fit to data" objection Rydin levels at the conventional theory, and with more free parameters, since C, M and W each carry an entry per nucleon pair. The claim that the mechanical equation is "similar to the Schrödinger wave equation" is weaker than it sounds: FP = I ∂²P/∂t² is a finite-dimensional matrix oscillator equation, second order in time, whereas the Schrödinger equation is first order in time and complex — a difference that is the source of interference and of the phase behaviour underlying the Bessel-function matching Rydin himself set out in section 1. His own equation (2), the classical wave equation with 1/ν² ∂²/∂t², is not in fact the Schrödinger equation, and the paper's identification of the two blurs the very point at issue.
There is also a question the model does not confront. Its virtual levels are resonances of nucleon positions, so a nucleus with A nucleons has of order 3A modes; observed level densities in heavy nuclei rise far faster than linearly with A at high excitation. The paper does not quote a level-density measurement against which to check this, and no cross section is computed, so the model cannot yet be confronted with the Breit-Wigner fits it aims to replace. Likewise, an electromagnetic nucleus of rings must still explain why nucleon binding energies are of order 8 MeV per nucleon while the Coulomb energy scale at 3 × 10−13 cm is of the same order — the numbers are not worked in this paper. Finally, the assertion that the mass loss on binding "implies that the gravitational effect of this mass is transmitted electromagnetically" is stated without argument and does not follow from anything preceding it.
Taken as what it is — a nuclear engineer asking whether the phenomenology he taught for decades has a classical mechanical reading — the paper is sober and useful. Taken as a replacement for wave mechanics, it is a proposal for a calculation that has not yet been done.