Magic Numbers Derivation from Variable Phase Nuclear Model
| Scientific Paper | |
|---|---|
| Title | Magic Numbers Derivation from Variable Phase Nuclear Model |
| Read in full | Link to paper |
| Author(s) | Xavier Borg |
| Keywords | Magic numbers, nuclear model, electromagnetic mass, tetrahedral nucleus, dimensions |
| Published | 2006 |
| Journal | General Science Journal |
| No. of pages | 16 |
Read the full paper here
Abstract
A physical model based on a simplex structure is used to describe the nuclear structure. The paper details how one may easily obtain mathematical sequences, based on hyper dimensional geometry, for all known nuclear magic numbers. It also shows that mass has both real and imaginary components, as described in the variable phase nuclear model, a concept that would eliminate various anomalies present in the standard model. This model is supported by experimental results by the same author.
Overview
Xavier Borg of Blaze Labs Research published this paper in electronic form on 1 February 2006. Its target is one of the oldest unexplained regularities in nuclear physics: the "magic numbers" 2, 8, 20, 28, 50, 82, 126, at which nuclei become anomalously stable. Borg's complaint is that the shell model names these numbers but does not generate them — they are recovered only by inserting a spin-orbit coupling term chosen to make the counting come out right. He proposes instead that they follow from geometry: nucleons occupy the nodes of stacked tetrahedra, and the magic numbers are read off from the triangular and tetrahedral number sequences.
The second half of the paper pushes further, and this is where it departs most sharply from the standard account. Borg argues that the tetrahedral picture works only up to Z = 20, and that beyond that the structure can be understood only as a three-dimensional projection of a higher-dimensional object — a "hypertetrahedron" or simplex. On this reading the nuclear Mass defect is not binding energy converted away but the apparent loss of volume when two four-dimensional objects overlap in a three-dimensional projection, so that mass acquires a real and an imaginary component and the "phase" of nuclear matter varies with the projection angle. The strong force and the mesons invented to carry it are, in his view, artefacts of a picture that lacks structure.
The argument
What the standard model does not explain
Borg opens with a survey: the shell, liquid-drop, cluster, Moon's and double-tetrahedron models each assume a different phase of matter for the nucleus — gas, liquid, semi-solid, platonic solid and tetrahedral solid respectively — and each successfully describes some selected properties while none gives a comprehensive description. "Most of the characteristics of the different phases are mutually exclusive."
He then lists the evidence for geometrical structure, drawn from the standard literature: enhanced abundance of elements for which Z or N is magic; the stable end-members of the natural radioactive series all having a magic number of protons or neutrons; sharply lower neutron absorption cross-sections at N = magic; the binding energy of the last neutron peaking at a magic number and dropping sharply for the next one added; electric quadrupole moments near zero for magic nuclei; and higher first-excitation energies for closed shells. He adds that peaks and dips in binding energy repeat every fourth nucleon, and that nuclei containing an exact number of alpha particles (2P+2N) are more tightly bound than their neighbours — pronounced for the lightest nuclei but still perceptible up to A = 28.
Two further observations carry the geometrical argument. Nuclei with even numbers of protons and neutrons are more stable than odd ones, which Borg reads as requiring "an even number of vertices" — and no regular polyhedron has an odd number of vertices. And the magic numbers appear in the three most abundant elements: hydrogen (1+1 = 2), helium (2, 2) and oxygen (8, 8).
He quotes Maria Goeppert-Mayer's own account of the discovery at length, then makes his objection: "Visualizing the densely packed nucleus in terms of orbits and shells seems much less plausible than the corresponding shell model for atomic electrons." Nucleons in orbit ought to be colliding continuously. His answer is that they are not orbiting at all but seated in the fixed nodes of a standing-wave frame — in which case the Pauli exclusion principle becomes unnecessary, having been "devised in the first place due to the lack of information about the geometrical structure of the electron shells."
Tetrahedral stacking and spin
The simplest stable close-packed three-dimensional structure has four nodes: a tetrahedron. Four nucleons fill it, and two protons plus two neutrons is exactly helium-4, the alpha particle. Borg notes that this is not new — Linus Pauling's 1964 notebooks used tetrahedral stacking, and relativistic quark confinement and coloured-quark-exchange calculations have been found consistent with a tetrahedral nucleus. It also explains why radionuclides shed alpha particles rather than single nucleons.
He then gives a physical account of Spin. A shell is complete when the number of nodes equals the number of vertices of a spherical platonic solid; complete shells have zero spin. Incomplete shells cannot form a stationary standing wave, and the resulting motion of the pattern is the spin. His analogy is an oscilloscope: at input frequency equal to the timebase frequency f one wave stands still on the screen; at 2f, two stand still; in between, the pattern drifts across the screen. A two-spoke wheel under a stroboscope behaves the same way. Hydrogen has non-zero spin because it has half the four nodes needed for the smallest spherical tetrahedron, while the alpha particle, which completes it, has zero spin.
Deriving the sequence
Borg cites two number series. The nth triangular number is TRI(n) = (n/2)(n + 1), giving 1, 3, 6, 10, 15, 21, 28, 36, 45, 55…; the nth tetrahedral number is TETRA(n) = (n/6)(n + 1)(n + 2), giving 1, 4, 10, 20, 35, 56, 84, 120, 165, 220…; and SQR(n) = TRI(n−1) + TRI(n) = n2.
Two empirical facts are then imposed. First, nuclei up to Z = 20 have equal numbers of protons and neutrons, and beyond that the balance is lost. Second, the binding-energy curve shows the nucleus building up as a double structure, so each tetrahedral level is completed in pairs before the next begins. Up to Z = 20 the structure is a perfectly symmetrical pair — "in electromagnetic terms, a perfect dipole", and a perfect quadrupole once the neutron structure is added.
For n ≤ 3 the double tetrahedron closes at twice the tetrahedral number:
- Magic(n) = 2·TETRA(n) = (n/3)(n + 1)(n + 2), giving 2, 8, 20
Beyond Z = 20 the two tetrahedra "hinge together or share the same space, much like covalent bonds are known to share the same orbits in chemistry" — the second tetrahedron is inverted beneath the first, its base-adjacent triangular layer occupying the space of the first one's. The shared layer is TRI(n−1), counted once instead of twice:
- Magic(n) = 2·[TETRA(n) − TRI(n−1)] = (n/3)(n + 1)(n + 2) − n(n−1) = (n/3)(n2 + 5)
which gives 28, 50, 82, 126, 184. Borg's table sets this out level by level:
| Level n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| TETRA(n) | 1 | 4 | 10 | 20 | 35 | 56 | 84 | 120 |
| TRI(n−1) | — | — | — | 6 | 10 | 15 | 21 | 28 |
| Magic(n) | 2 | 8 | 20 | 28 | 50 | 82 | 126 | 184 |
The whole known sequence is reproduced, and 184 — widely expected as the next magic number — falls out without further adjustment. The bonding layer, "as chemists would call it, is always one layer above the base of the tetrahedron." Borg identifies the doubly-counted layers with the binding energy or mass deficiency, and interprets it electrically: for Z ≤ 20 the dipole is purely resistive, and beyond that the overlap makes it reactive, "a situation analogous to the real and apparent electric power in reactive loads we learn in electrical theory."
The quantum shell capacity and the inert gases
Slicing a stacked tetrahedron into two-level layers, the nucleons per slice are TRI(2n) + TRI(2n−1), which by the square-number identity equals (2n)2 = 4n2. Taking protons as half of that gives
- Zmax = ½·4n2 = 2n2
— the familiar 2, 8, 18, 32 for shells K, L, M, N. Borg's reading is that each principal quantum shell simply is a two-level slice of the tetrahedral stack, so "the electron structure is closely related or simply a direct effect of the nuclear structure", by the same logic that lets an antenna's far-field pattern be deduced from its dipole structure.
Extending this, a pair of tetrahedral stacks whose topmost tetrahedra overlap in the same space gives the sequence 2, 8, 8, 18, 18, 32, 32 — matching the conventional s, sp, sp, spd, spd, spdf, spdf subshell order. Cumulating gives 2, 10, 18, 36, 54, 86, 118: the inert gases, including element 118. Again the overlap of the top tetrahedron is taken as evidence of a higher-dimensional entity.
Variable phase and imaginary mass
The closing argument is by analogy. Two three-dimensional spheres projected onto a plane appear as two circles; as the projection angle changes they overlap, and the total shadow area falls, until an observer confined to the plane cannot tell whether there are one or two. If density is defined as mass per unit area, that observer records a "missing mass".
Borg asks the reader to raise this by one dimension. Two four-dimensional hyperspheres projected onto a three-dimensional "screen" appear as two separate spheres when far apart, but as the phase angle increases they merge into one, and the observer sees "a 'mass defect', as Einstein called it, accompanied with a respective increase in internal or binding energy." In the macroscopic world, he says, the same effect shows as a decrease in density and a change of phase from solid to liquid to gas to plasma and finally to vacuum. He concludes that tetrahedral stacking is "just a limited projection in 3D, of a higher dimensional entity — a hypertetrahedral stack", that "if we define mass as a 3D entity, it will always be a shadow effect", and that spin and angular momentum are "direct effects of the imaginary components" of mass. Nuclear matter therefore has no fixed phase: like an electrical component that is neither purely resistive nor purely reactive, the nucleus is neither purely solid nor purely Vacuum energy.
He also states his objections to the "bunch of grapes" picture directly: it treats elementary particles as points of zero dimension, "which results in infinite energy when electric field energy is taken into account"; it needs a hypothesised strong force and an invented class of exchange particles to hold the protons against mutual repulsion, a hypothesis that "has in fact never been proved"; and it gives "no hint to the existence of the nuclear magic numbers and no way to predict, or even explain the existence of its obvious shell structure." Against the probabilistic reading of the electron cloud he sets the fact that crystal lattices "do not build up in random shapes, but in very specific shapes."
Assessment
The arithmetic core of the paper is its genuine achievement and can be checked in a minute. Both formulae are correct as stated: (n/3)(n+1)(n+2) gives 2, 8, 20 for n = 1, 2, 3, and (n/3)(n2+5) gives 28, 50, 82, 126, 184 for n = 4 through 8. The whole magic sequence is reproduced by two short expressions with no free parameters, and the predicted continuation, 184, is the value most independent shell-model calculations also give for the next neutron closure. That is a real and pleasing regularity, and the derivation of Zmax = 2n2 from a two-level triangular slice is an elegant piece of combinatorics. The paper is also well aimed: the shell model really does obtain 28, 50, 82 and 126 only after a spin-orbit term is added by hand, and Borg is right that a scheme generating them from a single structural rule would be worth having.
The central difficulty is that the derivation is a fit, not a physical argument. The break at n = 3 is not derived from anything in the model; it is imposed from outside as "fact no.1", the observed departure from N = Z. Given a break at a chosen point and a free choice of which layer to subtract, two low-order polynomial forms can be tuned to seven integers without much difficulty — and Borg does not test the fit against anything the sequence does not already contain. Nothing in the paper predicts a nuclear property that was not used to construct it: no binding energies, no separation energies, no level orderings, no quadrupole moments, no decay systematics. By contrast the shell model, for all the inelegance of its spin-orbit insertion, delivers spins, parities and magnetic moments across the chart. Borg's own claim that the model explains spin is qualitative only: the oscilloscope analogy tells us that incomplete shells rotate, but yields no value for any nucleus's spin, and does not explain why nucleons carry spin ½ individually.
Several steps are asserted rather than demonstrated. That "the physical structure must have an even number of vertices" because even-even nuclei are more stable does not follow — even-even stability is a pairing phenomenon that survives in nuclei far from any closed shell. The claim that the exclusion principle becomes unnecessary once nucleons sit at fixed nodes conflicts with the measurement it was introduced to explain, atomic spectra: the alkali doublets and the Zeeman structure require two states per orbital, which a purely geometrical node count does not supply. And the identification of the mass defect with a projection artefact runs against the fact that the missing mass is not merely apparent but measured as released energy — the 28.3 MeV binding energy of helium-4 is recovered as kinetic energy and radiation in fusion, not as a change in an observer's viewing angle. The paper never confronts this: a shadow that shrinks does not heat a reactor.
The dimensional argument is the weakest part. The projection analogy is vivid but does no work: it establishes that a lower-dimensional observer would be confused, not that we are one. No test is offered that would distinguish a hypertetrahedral nucleus from a three-dimensional one, and the phrase "mass has both real and imaginary components" is never given an operational meaning — no experiment is proposed in which the imaginary part could be measured, and no equation in the paper contains it. The claim that changes of phase from solid to liquid to gas to plasma reflect increasing hyperdimensional overlap is offered without any quantitative link to latent heats or to anything else measurable. Finally, the appeal to "the sharing of the top tetrahedron" as evidence for higher dimensions is circular, since the sharing was introduced in the first place to make the counting work.
What remains, and it is not nothing, is the numerical observation. If the magic numbers really are 2·TETRA(n) below the N = Z break and 2·[TETRA(n) − TRI(n−1)] above it, that coincidence deserves an explanation whether or not Borg's is the right one — much as the Titius–Bode relation deserved examination before it was understood as a resonance artefact. The paper is best read as posing that question sharply rather than as answering it.