An Ordering of Neutrons

In the atomic nuclear structure hypothesized by Dr. Robert J. Moon^{1} in 1986, protons are considered to be located at the vertices of a nested structure of four of the five Platonic solids (Figure 1).
Eight protons, corresponding to the Oxygen nucleus, occupy the vertices of a cube which is the first nuclear “shell.” Six more protons, corresponding to Silicon, lie on the vertices of an octahedron which contains, and is dual to, the cube. The octahedroncube is contained within an icosahedron, whose 12 additional vertices, now totalling 26 protons, correspond to Iron. The icosahedronoctahedroncube nesting is finally contained within, and dual to, a dodecahedron. The 20 additional vertices, now totalling 46 protons, correspond to Palladium, the halfway point in the periodic table (Figure 2).
Beyond Palladium, a second dodecahedral shell begins to form as a twin to the first. After 10 of its 20 vertices are filled at Lanthanum (atomic number 56), a cube and octahedron nesting fill inside it, accounting for the 14 elements of the anomalous Lanthanide series.
Next, the icosahedron forms around the cubeoctahedron structure, completing its 12 vertices at Lead (atomic number 82), which is the stable, endpoint in the radioactive decay series. Finally the dodecahedron fills up, and the twinned structure “hinges” open, creating the instability which leads to the fissioning of uranium (Figure 3).
The completed “shells” of the Moon model, correspond to the elements whose stability is attested by their abundance in the Earth’s crust: Oxygen, Silicon, and Iron. These elements also occur at minima in the graphs of atomic volume (Figure 4), and of other physical properties (viz. compressibility, coefficient of expansion, and reciprocal melting point) as established by Lothar Meyer in the 1870s to 1880s. Palladium, which is an anomaly in the modern electronconfiguration conception of the periodic table— because it has a closed electron shell, but occurs in the middle of a period— is not anomalous in the Moon model. Further, as I discovered since my 1988 article on the subject, all four closedshell elements in the Moon model occur at maxima on the graph of paramagnetism (versus atomic number), as reported by William Draper Harkins.^{2}
The Moon model is thus consistent with much of the same experimental data which underlies the periodic table of the elements, and explains additional features not explained by the modern, electronconfiguration presentation of the periodic table. However, it seems to be inconsistent with the evidence from spectroscopy (upon which the electronconfiguration conception rests) which suggests the periods of 2, 8, 18, and 32; it is also not consistent with the older “law of octaves,” which was developed to explain the phenomena of chemical bonding, and was subsumed in Mendeleyev’s conception. 



The Moon Nuclear Model.
Christopher Sloan 
An Ordering of Neutrons
From the period of my first exposure to Moon’s nuclear model, I was of the opinion that the two apparently contradictory orderings (electron and proton) must be governed by some higher principle, which was in some way contained in the Moon conception. Moon encouraged such speculations, pointing out that the theory of electron orbits (the “extranuclear electrons,” as he insisted on calling them), had always suffered from an aseptic separation of the electron from the nucleus.
During his lifetime, I worked out an ordering principle, using the unfilled faces and edge midpoints of his nested configuration, to determine the otherwise undetermined distribution of the neutrons in the nuclei.^{3} I assumed the addition of the fifth Platonic solid (tetrahedron) as the structure of an Alpha particle at the center of the nucleus (Figure 5), and distributed the neutrons at the unoccupied edgemidpoints of the set of solids. The neutron “shells” closed at the electronshell singularities (2helium4, 10neon20, 18argon40, 36krypton84), specified in the modern periodic table (see table).
This suggested, for the first time, a relationship between the ordering of the nucleus and that of the electron shells. However, I could not see the cause for a relationship between the supposedly neutral neutrons, and the extranuclear electrons. The difficulty suggests some error of assumption, which must be contained in the oversimplification provided by the RutherfordBohr model.
Recently, I began to reexamine this problem. I was now aided by the intensive study of the AmpèreGaussWeber electrodynamics I had since carried out. The following are speculations I have examined in pursuit of this still unfinished task.
1.1 PROTONS CONSIDERED AS AMPERE MAGNETIC MOLECULES
In 1821, AndréMarie Ampère adopted the hypothesis, suggested by his friend and colleague Augustin Fresnel, that there existed within matter, tiny circular loops of electrical current, which, like the macroscopic ones he had studied, acted like magnets. If the current loop were thought of as the “equator” of an imaginary sphere, the north and south poles of the sphere would be the north and south poles of the magnet which the current loop produced. Ampère supposed that magnetization of a piece of iron consisted of alignment of these “magnetic molecules” in parallel columns, so as to magnify their effect.
I wondered what would result if we considered the Moon nucleus as constructed of magnetic molecules of positive charge, each one representing a proton. I supposed that each proton consisted of a tiny ring of electrical current. If one arranges four hoops into the configuration which produces the 12 vertices of a cuboctahedron at the intersection points of the hoops (Figure 6), it is seen that the poles of the four hoops (taken as “equators”) point to the 8 spherical triangles produced by the configuration. The centers of these 8 triangles correspond to the 8 vertices of a cube. The “magnets” thus lie along the four axes which connect the 4 pairs of diagonally opposite vertices of a cube.
The cube is thus magnetically “polarized.” Can the magnetic poles find a stable position in this configuration? By proper choice of polarities, it will be seen that they can. We will first examine this stability, and then turn to the question of what we mean by the magnetic poles. In the original Moon model, the protons are represented as singularities at the vertices of the various solids. In the variation under investigation here, the protons are rings of current, while the vertices of the solids are now magnetic poles produced by them. Let us turn, first, to the possibility of stable arrangements.
Consider a perspective cube drawn on a sheet of paper. On the top face, assign north poles to two diagonally opposite vertices (that is, along the diagonals of a face), and south poles to the other pair (Figure 7). The vertices which lie diagonally opposite these (that is, along the long diagonals which pass through the cube) must have the opposite magnetic polarity. Consider any vertex bearing a south polarity. It will be seen that its three nearest neighbors (distant by the length of an edge) all carry a north polarity, while its three more distant neighbors (distant by the length of a face diagonal) carry the south polarity.
Now, consider the diagonally opposite vertex (that is, the one along the long diagonal that passes through the cube). It is a north pole, while its nearest neighbors are south, and its more distant neighbors are north. The attractions and repulsions in opposing directions being equal and mutually opposite, it is seen that the magnetic dipole we have examined will have no moment of rotation. The same holds for the other three magnetic dipoles which connect the remaining vertices of the cube.
Thus, a set of four insulated, circular loops of copper wire, constrained so as to remain around a common center, and electrified, would find a steady state in the mooted configuration, or even be induced to take up such a configuration. I have thought of building such a model using slotted hoops as forms for the circular wire loops.
Stability of Octahedron
Now turn to the octahedron. The configuration is simpler. It can be constructed from three rings, placed in three mutually orthogonal directions (Figure 8a). In this case, the six vertices produced by the intersections, correspond to the six poles produced by the three magnetic molecules. The “magnets” may thus be represented by a drawing of three mutually orthogonal axes (Figure 8b). It is readily seen that no arrangement, in which north and south poles are opposite each other, will produce a stable configuration; rather a rotational moment will always be imparted to the magnets.
Such would be the end of any attempt at a linear modelling, one derived merely from properties of the solids taken one at a time. However, the nesting of the solids in Moon’s nucleus, produces a multiple connectivity that now makes the octahedron possible. To see it, we must now superimpose upon the drawing of the perspective cube, the six poles of the octahedron, one standing above the center of each face of the cube (Figure 9). It is now seen that, in relationship to the nearest underlying face of the cube, the pole of the octahedron, whether north or south, is balanced, being mutually attracted in two opposite directions, and mutually repelled in another pair of opposite directions perpendicular to the first.
Thus, when we consider the poles of the octahedron, by themselves, they are unstable. However, when the octahedron is circumscribing the cube, we see that any given octahedral pole is considerably closer to the corners of the underlying cube than to the nearest adjacent pole of the octahedron. Since the force of a single magnetic pole upon another, falls off as the square of the distance, it will be seen that the effect of the nearer cubic poles is three times as great as that of the adjacent octahedral ones.
(If the length of the edge of the cube is taken as 1, the edge of the circumscribing octahedron is 3/2. The distance of the octahedral pole to the nearest cubic one is 2/3 the altitude of the face triangle of the octahedron, or 6/2. Hence, the ratio of the squares of the distance from the nearest cubic vertex to the nearest octahedral vertex stands as 1:3).
Thus, three Ampère magnetic molecules in octahedral formation would possess a considerable stability, when circumscribing the arrangement of four magnetic molecules in the configuration that produces poles at the eight vertices of a cube. They would not possess such stability when not in the nested ordering. This also would suggest why the cube must precede the octahedron in the Moon configuration.
Progression to 5Fold Symmetries
Let us next consider the circumscribing icosahedron and dodecahedron as a group. To move from the octahedron to icosahedron, the nested model must pass through a singularity, the same one examined in Johannes Kepler’s famous “SixCornered Snowflake,” that is the transition from the sixfold to fivefold symmetry. In Moon’s configuration, the six vertices of the octahedron are placed along the altitude lines of 6 of the 20 triangular faces of the icosahedron, at a distance defined by the golden section.
In my recent consideration of protons as Ampère current loops, I chose to make a variation on this. My arrangement is built around the longrange relationship between the inner cube and the outer dodecahedron. The cube in my configuration, is oriented so as to align itself with one of the five possible cubes whose 12 edges can be inscribed within the 12 faces of the dodecahedron. In that case, the cube’s vertices align with 8 of the 20 dodecahedral vertices, and the latter are designated a polarity such that the relationship to the nearest underlying cubic vertex is attractive.
This arrangement of the cube determines that the octahedron, unlike in Moon’s arrangement, is fitted such that its 6 vertices sit at the midpoints of 6 of the 30 edges of the icosahedron. It will then be seen that the polarities can be assigned, such that each of the 6 octahedral vertices are held in place by 4 nearby vertices of opposing polarity, two of these are the nearest adjacent vertices of the icosahedron, and two are the nearest adjacent vertices of the dodecahedron, which lie on an axis perpendicular to the justmentioned icosahedral pair. (A threedimensional model makes this very clear).
In such a fashion, the completed structure of four nested solids achieves a stability, based on the multiple connectivity of magnetic pole relationships ranging from inner cube to outer dodecahedron, which each of the figures by themselves (the cube excepted), would not possess. The criterion of stability rests entirely upon two features: the hypothesis of the magnetic molecule first asserted by Ampère, and the geometric properties inherent in the structure. It will be seen however, that despite this pretty stability, more problems arise in considering such a structure as a model for the nucleus.
1.1a The Problem of Doubling
On the assumption that each of the above described rings corresponds to a single charged particle, the entire construction so far results in only half the necessary charges. As the relative abundancies and many other properties indicate, nature favors the evennumbered elements. The fact that the nuclear dipole and quadrupole moments measure zero for all evennumbered elements, has suggested that there is a pairing of charges in the nucleus.
It is easy to imagine each of the current rings we have constructed as pairing, with one rotating in the opposite sense to the other, thus eliminating the magnetic polarity of every other element. However, such a solution would eliminate the original basis for the construction, which was to provide a reason, in the laws of electrodynamics, for the nested arrangement of the Moon model.
One might overcome this drawback simply by supposing that each ring is capable of carrying two charges rotating in the same direction, thus doubling its magnetic strength. However, in such a case the evennumbered elements would no longer be magnetically neutral. I have not given up this possibility, because the cause of the measured nuclear dipole magnetic moment may not correspond with the magnetism that results from this putative variation on the Moon model. However, in the course of contemplating the problem, another variation occurred to me, which I will now describe.
1.2 PROTONS CONSIDERED AS OSCILLATING WEBER PAIRS
Wilhelm Weber’s “Sixth Memoir of 1871”^{4} is, in part, a response to an ideologically based criticism by Hermann Helmholtz, charging that Weber’s Fundamental Law of Electrical Force violates the principle of conservation of energy. Weber’s reply was so thorough and devastating that James Clerk Maxwell was forced to withdraw his earlier support for Helmholtz on this matter. But the piece is far more than a refutation of Helmholtz’s insistent carping. It is one of the few truly revolutionary documents of theoretical physics. In the course of development of the “Memoir,” Weber derives from his Fundamental Law of Electrical Force a critical length, below which the repulsion of like particles is reversed. The socalled “Coulomb barrier” is overcome, and the particles, rather than repelling each other, experience an attraction.
In exploring the possibilities of motion of electrical particles as a consequence of this law, Weber recognizes two cases in which molecular states of aggregation of particle pairs may lawfully occur. In one such state, two particles of unlike charge may rotate around one another. Supposing, as he does, that one of the charged particles is considerably heavier than the other, the representation corresponds to the view of the hydrogen atom, not generally accepted until half a century later (and based on far less firm foundation). Weber describes this state of aggregation of two dissimilarly charged particles as representing an Ampèrian molecular current.
This conception forms the basis for his attempt to construct the periodic table on the basis of the laws of electrodynamics as hinted at in the “Sixth Memoir,” (Section 16. Applicability to Chemical Atomic Groups), and later developed in his last memoir Determinations of Electrodynamic Measure: particularly in respect to the Connection of the Fundamental Laws of Electricity with the Law of Gravitation.^{5}
There also exists a second molecular state of aggregation, according to Weber. In this case, two like particles will form an oscillating pair, containing themselves within the bounds of the critical minimal length. According to Weber’s derivation, the two particles, starting from a distance of separation just below the critical length, accelerate towards each other on a straight line, pass one another at a maximal velocity, then decelerate to zero velocity as they reach the boundary point of the critical minimal length, at which point the process repeats itself. (It seems necessary that the two material particles literally pass through one another, a difficulty which is overcome by use of Louis deBroglie’s conception of particlewave.)
1.3 APPLICATION TO THE MOON MODEL
I have imagined this latter derivation of Weber’s to be the state of aggregation of proton pairs, whose straightline trajectories are aligned along the diagonal axes of the nested Platonic solids of the Moon nuclear model. Thus, the oscillating pairs orient in the same direction as the magnetic dipoles of the previously examined hypothesis. In this case, the orientation of the pairs would not appear to be guided by the leastaction configurations of magnetic poles which we constructed in the first case. One would rather have to assume that the arrangement expresses an underlying spatial geometry, or results from some other cause not yet adduced.
We may also briefly consider the trajectory of an extranuclear electron guided by a Weber pair of oscillating protons. A moving charge produces a magnetic effect in the shape of a circle in a plane perpendicular to the direction of its motion. The strength of the magnetic effect is proportional to the velocity of the charge. Let us picture the magnetic effect by allowing the radius of a surrounding circle to designate its strength at that point in the charge’s trajectory.
In the Weber molecular aggregate, the two charges are accelerating towards the centerpoint where they meet. Thus, an axial crosssection of the surrounding circles would show an exponential curve increasing from both ends towards the center (Figure 10). It would look much like an axial slice of a pseudosphere. An extranuclear electron moving in this field would tend to circle the axis of the proton pair, and at the same time be carried along the axis by the motion of the proton, the resultant being a spiral path. One would expect that an electron pair would follow opposing spiral trajectories around this pseudospherelike object, guided by the frequency of oscillation of the paired protons.
The motion of the electron, itself a charged particle, along this trajectory would now produce a magnetic dipole aligned with the axis of the proton pair. That weaker dipole, created as an effect of the motion of the extranuclear electron, would have precisely the same alignment as that produced under the first conception (of the Ampère magnetic molecule) and might thus be thought of as providing the structure to the nucleus.
Yet, the continued stability of the nucleus under condition of being stripped of the extranuclear charge (ionization) casts doubt on such a construction. It were possible, however, to imagine the justdescribed trajectories as being those of the nuclear electrons. While these are now usually conceived as bound with a positive charge into a small sphere (the neutron), it were possible to explore other, more probable configurations. The strict, but as yet unexplained, determination of the isotope variations by addition of neutrons, suggests that the neutron must be a necessary part of the nuclear structure, not an arbitrary addition. Such explorations remain to be carried out. 