Quark Shell

Quark Shell Model
A simpler approach to nuclear structure involves looking at the shells on a purely quark basis. It is theorized generally that each baryon is composed of 3 quarks, a proton has 2 up quarks and a down quark, and a neutron 2 down quarks and an up quark. Up quarks have +2/3 charge, and down quarks have -1/3. Similarly, mesons are 2 quark entities, with different rules than the baryons (and a short life to boot). The effective mass of the up quark appears to be less than the down, so its standing wave occupies more space. Additionally, quarks of both flavors come in 3 colors. A stable nucleus has to have balance in color as well as flavor. It seems likely that each color should be centered at the center of the nucleus, and 2 quarks of the same color are excluded from each others space, but quarks of differing color prefer to overlap. Baryons instantiate where quarks of all 3 colors share a volume.

It should not be assumed these quarks are billiard balls, but rather they should be viewed as wavicles moving through the net fields of their neighbors. The potential wells serve to keep the nucleus from collapsing inward, except under the extreme conditions of a black hole (where the normally trivial gravitational force finally adds up enough to unbalance the other forces). There are 3 reasonable ways a trio of quarks can be arranged: all with a common center, at the points of a triangle, and 2 at 1 center and the third separated.

Cases where a quark is in an orbit with more than the single standing wave, at least transiently, may occur. Due to the relativistic speed at which quarks are moving, the 2-wave orbit is smaller than the single wave, yielding the 2nd generation strange and charm quarks. These would typically be spherical orbits (2s) like the up and down quarks occupy (1s), but 2p orbits are perfectly legal. Stars with too much mass to remain neutron stars, that have not collapsed to black holes probably have strange/charm centers. It is possible to calculate the effective velocity of quarks in 1s orbits in free space by examining the effective mass they exhibit under low density conditions as compared to high density. That is, the index of refraction that slows quarks when they overlap can be measured by seeing how little mass they have in pions (which have 2 quarks occupying essentially the same space), compared to rho-mesons (where 2-quark overlap is much lower - near the Snell critical angle). This parallels the comparison of dark matter (where all 3 colors of quark share the same space, hence don't have any area to interact) and normal baryons (where overlap is relatively small). By examination, the independent quark would have a mass of at least 400 MeV, probably a lot more, since the rho has overlap). The index of refraction (n) also defines the maximum amount of overlap (short of complete) which 2 quarks can have - that would be the point where the angle to incident surface falls to the Snell critical theta where reflection replaces refraction. It is difficult to be sure until the exact size of the down quark is determined, since if it is a lot smaller than the up it will be bouncing around inside the up quark in the pion, rather than having close to 100% overlap. Based on the neutron/proton case (where the difference in mass is so low), the down quark has only a little bit higher effective mass there - hence only slight smaller wavelength and volume. The 3-quark index of refraction should differ from the 2-quark (in which case n² seems likely).

This model suggests baryons and mesons only exist within nuclei as virtual particles, having no stand-alone existence (except when they are apart from a nucleus). It also suggests that nuclear reactions (other than fission) primarily occur at the surface, where an up quark may combine with an electron forming a down quark and a neutrino, or a down quark and a neutrino may convert to an up quark and a lepton (probably an electron, but it depends on the flavor of the neutrino), or an up quark and an anti-neutrino can yield a positron (same note) and a down quark. Positron capture would occur if they were able to get close to the nucleus, but the electron cloud and the mutual repulsion electromagnetically between the nucleus and the positron makes this unlikely (except for isolated neutrons, or again if neutronium exists). The w boson triggers the nuclear reaction, but only when the potential exists.

Looking closer at an example nucleus (initially without the refractive meniscus), a deuteron (²H) has 6 quarks: 3 up, 3 down; 2 of each color. Assuming they are each on an axis - as in the triangle case above (for instance both reds as up on x axis: #1 & #2, both blues on y axis, an up and a down: #3 & #4, and both greens on z axis as downs: # 5 and #6) there are 12 overlaps between adjacent quarks, 2 up up (13, 23), 2 down down (45, 46), and 8 up down (14, 15, 16, 24, 25, 26, 35, 36). These in turn overlap, forming 8 triples, 4 up up down (135, 136, 235, 236), and 4 up down down (145, 146, 245, 246). Ignoring the small difference in radius, the pairs are each about 11.61% of a sphere's volume, so the triples are each about 3.04%. Each pair has 2 triples, leaving 5.53% with just 2 quarks each, and each sphere has 65.72% of its area not overlapping at all, at the free energy level. From the pions, pairs have an energy density of 135-140 MeV / up quark volume (more for the 8 mixed cases, less for the 4 pure), so that's about 147 among those, triple quark space looks to be about 45 MeV / up quark volume, so that's about 11 MeV, the rest of the energy would be from the free space of the quarks, making that 435 MeV / up quark volume. Oops, the experimental free energy is much higher. So let's look at the third solution (diquark and singleton). Now we have 4 spheres: 2 diquarks and 2 singletons. The diquarks we'll place on the y axis and the singletons (1 up, 1 down, again ignoring small differences in mass) on the x. Now there are 4 lobes of overlap, each again 11.61% of a sphere's volume and each sphere has 76.78% free. With white (3 quark) space assumed to be 45 MeV again, and 2 color space 140 MeV, that gives free space per monochrome volume 1067.4 MeV (which is in the ballpark experimentally for effective liberation energy of a quark). That would make overlap in the proton or neutron case about 23.1% (compared to the 68.3% overlap of 2 quarks in the rho case), so crossing the Snell barrier is less likely, except under extreme conditions (bye bye black holes).

Unfortunately, the refractive meniscus (the amount of extra volume shared by 2 refracting spherical waves caused by the bending of the beam by the refraction) can't be ignored overall. In the deuteron case, the overlapping areas are 90° apart, so their meniscuses don't overlap. In the alpha particle case (and all larger nuclei), there are 3 or more overlaps per quark, with a planar separation of 60°. Ignoring the meniscus, the quarks would only be separated by .71 diameters, but that would again give overlap of 11.61%, so each would be 65.17% free, but the actual amount is much more. The meniscus in the deuteron is then similarly non-zero, so the actual volume of overlap is more, and the free space is less (so the free mass is higher). From looking at the alpha particle (He-4 nucleus), mean separation looks to be about .76 quark diameters, so assuming spherical regions, we are about 12.5% overlap in the deuteron, and the free energy is around 1078 MeV. (Note- the meniscuses are not identical in size, the diquark has a meniscus intruding into its space roughly twice the volume of the meniscus intruding into the single quark's space). This suggests a meta-stable triangular case of 1981 MeV for 3 quarks (ie 1043 MeV extra energy added to a neutron, which is the ceiling before a meson shower is expected). See illustration 5 for meniscus at the sweet point (where the radius is minimal).

Larger nuclei would tend to form with planar layers of diquarks alternating with layers of singletons. As down quarks are more massive (and have lower charge intensity), they would tend to accumulate internally, with the up quarks on the surface (so the charge would be on the surface in the quark shell model). Typical arrangements of the spheres would be hexagonal, since this is the tightest packing, with layers offset so a trio on 1 layer has a single quark above its center. Small nuclei could form eccentric layers, with down quarks and up quarks splitting the sheet evenly (and slightly offset - see diagram 4) - but solutions with the down quarks internal look better analytically.

Points at which the inner block is a simple set of integers (4,4,4 for Calcium 40, or 6,7,8 for Uranium 238) and the outer layers are octahedral are more stable than their peers. The next such is 6,8,8 for Hassium 280 - although the other even isotopes through 288 of Hassium also look good. Oddly spherical forms are LESS stable than the octahedron. Cropping an extra pair of spheres (1 di + 1 up) off the edge of the original box changes the atomic number without changing the neutral core, so some "Magic Numbers" of neutrons with many stable isotopes can occur. Similarly, when the surface has a 2x3 or 2x4 block, a pair can be trimmed off leading to an isotope with the same atomic number, but 1 (or n) fewer neutrons.

Fission is the shearing off at a plane layer, so 1 of the 2 child nuclei gets the pole through central (biggest) layer (and possibly beyond), the other gets the opposite pole through the next-to-center layer, and both chunks adjust into a more-spherical shape, often discarding small chunks (neutrons, mesons, and alpha particles) during stabilization. Alpha emission is a special form of fission in this model, where a 12-quark chunks splits off, and the balance of the nucleus adjusts towards a more spherical form. (It is these adjustments that result in gamma radiation).

The octahedrsl quark shell model could be disproved by sufficient analysis of mean field theory, once the actual count of charges on the surface for various nuclei is determined. If computations assuming all the charge is at the surface are significantly worse than those assuming charge is distributed uniformly, the model would be dead. If such computations are significantly better, it is strongly supported. Disproving this model would leave the eccentric forms, but since charge is more dispersed they are harder to disprove.

Other quark shell models have also been proposed. In some of these the charge is distributed throughout the nucleus. Some models have baryons instantiated in clumps, often pairs. Others assume the baryons are physical, but arrange themselves so that the quarks line up as close to the spherical case as possible (with holes here and there, and a rough surface where a baryon sticks up). Chiral (handed) fields can be analyzed as soliton and hadron models. Other models imply that additional flavors of quarks may be introduced (either at the surface or deeper within) at least transiently. Reactions leading to strange, charmed, top, or bottom quarks from either up or down quarks would be required to support these theories. Behavior under extreme conditions would also need to be examined (especially the big bang). If the strange is 2nd level to the downs 1st, and beauty is 3rd, additional level would exist. Since the energy rises so rapidly among these three (as velocity approaches C asymptotically) 4th and higher level would have huge energy per quark.

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