An Alternate view of Nuclear Structure

Aran David Stubbs

Summary

This is a brief description of an alternate theory of the structure of atomic nuclei.  It derives from my alternate theory of fundamental particles, but is separate from it.  In this theory, atomic nuclei are comprised of up/down diquarks plus individual up quarks and down quarks (monoquarks).  The count of monoquarks and diquarks is identical.  Each diquark binds to 1 to 6 monoquarks, and each monoquark binds to 1 to 6 diquarks, in the standard body-centered-cubic structure.  The surface is typically octahedral (which is the minimum surface for the bcc structure), except that some small nuclei have simpler structures - such as the cube of the Helium 4 nucleus. All interior monoquarks are downs, as are a few surface monoquarks. Analysis of minimum surface and statistical comparisons to actual nuclei are included, as are many illustrations.

Background

The atom was proposed by Democritus in ancient Greece for philosophic reasons, and later used by Dalton as a practical explanation of chemical behavior.   Each treated all atoms of an element as interchangeable eternal spheres.  Arrhenius discovered ions, showing atoms could be changed, and Thomson proposed isotopes which showed there were variations among the atoms of an element.  Then Rutherford showed the atom has a tiny nucleus surrounded by electrons.  The nature of the nucleus was debated, but a general consensus arose that it had protons and neutrons, bound by exchanging virtual pions.  These in turn were formed of quarks, which were bound by exchanging virtual gluons. A variety of theories have tried to explain nuclear structure, but none are fully successful.

Main Theory

An alternative view of nuclear structure is that the nucleus does not contain protons or neutrons, but that these are just small nuclei.  Instead, nuclei can be built of monoquarks and diquarks directly.  A proton is an up/down diquark plus a single up quark.   A neutron is a diquark plus a down quark.  Larger structures contain equal numbers of diquarks and monoquarks in a body-centered-cubic crystalline structure.  That structure has 2 types of spheres bound in a 3-dimensional checkerboard, with each sphere of 1 type bound to 1 to 6 spheres of the other type.  The interior monoquarks (those with 6 neighboring diquarks) are all downs, while the surface monoquarks are primarily ups (as required by Coulomb's law).  A small number of downs on the surface yields minimum dipole in the chemical sense – that is, a nearly symmetric distribution of charge.  Additionally, the surface downs allow the nucleus to approach ideal z, where changing an up to a down, or changing a down to an up both increase the energy of the resultant nucleus.

Any of the 3 color schemes is possible for a single nucleus, but a nucleus can't have monoquarks of 2 different colors. For simplicity only examples with red monoquarks are shown.

Given this general view of structure, a large number of possible nuclei can be built.  Some have large surfaces relative to their total baryon count, others less so.  Assuming that minimum surface is desirable, the simpler solutions were analyzed and patterns found.  It should be noted an absolute solution space, containing every possible arrangement is not feasible: this is what is known as a NP-complete problem, and requires essentially infinite resources to solve.  A more modest solution space, containing a practical subset was generated.

The most basic solution is a single sphere.  This has a parity violation: since a nucleus needs the same count of diquarks and monoquarks, an even number of spheres is required, so a single sphere is invalid.  Expanding this solution symmetrically produces a family, each of which has a parity violation.  Layer 2 has 6 of 1 type, and 1 of the other, etc.. It can be shown that any symmetric or eccentric solution as described below with 3 odd dimensions always has a parity violation and can be ignored.

Other simple solution do allow for valid nuclei.   Starting from a pair of spheres (1 of each type), a family of solutions can be built by expansion.  The smallest has dimensions of 1x1x2 , which can be classified as an even solution.  All symmetric and eccentric solutions can be classified as even or odd (which is useful later) by counting the number of odd dimensions.  If there are 0 or 2 odd dimensions, then the solution is even.  With 1 or 3 odd dimension, it is odd.  Again, 3 odd dimensions always have a parity violation, so are not valid.   This use of even and odd does not equate to an even or odd baryon count, but to the pattern of the family of solutions. Odd here means 1, 3, etc., spheres in a line.

Families can be viewed as either progressive addition of a layer to the existing member’s surface, or subtraction of the surface to produce a smaller solution.  The later method can result in several different large solutions leading to a single smaller solution.  The large solutions are referred to hereafter as cousins to each other.   The smaller solution is described as parent to the child solutions, and the large solutions as children to the parent solution.  For symmetric and eccentric solutions, this distinction is not important, but more complex structures need the precision for clarity.

2 other simple cases are of particular interest at this point.   A solution whose core is 1x2x2 is the first useful odd solution (corresponding to the Deuteron).  A solution whose core is 2x2x2 is the second even solution (corresponding to the Alpha particle, or Helium-4 nucleus).  These 2 solutions give rise to families that are regional minimum at many points.  The 2x2x2 solution and its descendants are described hereafter as the Even Symmetric family.  The 1x2x2 solution and its descendants are the Odd near-symmetric family.  (The 1x1x2 generates an Even near-symmetric family).

Additional eccentric solutions can be generated from any small integer combinations of dimensions: 1x2x3, 2x3x3, 2x2x3, etc.. As the child solution in each case adds 2 to each dimension, cores with all non-small integers can be viewed as children nuclei instead.  So a 4x4x4 solution has a core that is 2x2x2, and is a cousin to the standard child nucleus of the 2x2x2 solution .

When the Even symmetric family is generated by minimal expansion (making every existing sphere internal, but adding no spheres not adjacent to those interior spheres), an interesting series of solutions is found.  Layer 2 has 24 surface spheres and 8 interior spheres, for a baryon count of 16.   Layer 3 has 48 surface spheres and 32 interior spheres, for a baryon count of 40.  Layer 4 has 80 surface spheres and 80 interior spheres, for a baryon count of 80.  Each of these solutions has an ideal z where the surface has 4 downs.  Layer 5, with 120 surface spheres and 160 interior spheres, needs only 2 surface downs.   Layer 6 and beyond do not correspond to any stable isotopes.   Aside from the core solution, each of these is an Octahedron, a regular solid with 8 triangular faces.  Here 4 of the faces are diquarks, and 4 monoquarks.  When 4 of the surface monoquarks are downs, they each occur in the center of a monoquark face (closest to the center of the nucleus).  With 2 or 6 surface downs, they occur on a vertex (still providing low dipole, but further from the center).  Edge length of the resulting triangles is 1 less than the layer number.  Several cases have both a 4 down surface and a 6 down surface stable isotope.

Several additional types of core structures can also be produced.  These include skew solutions, where the stretching exhibited by the eccentric solutions along a dimension instead occurs on a plane, and diagonal where the stretching occurs at an angle (through the line x=y=z).  A very large number of skew solutions have been generated, including those which are skewed in 1, 2, or all 3 planes (XY, XZ, or YZ).  Skewing can be combined with eccentricity, producing hybrid solutions (which have skewing on a plane and eccentricity on the dimension perpendicular to the plane, so XY + Z).  Each of the non-core solutions can be viewed as an octahedron as well, but no longer regular.

After a significant number of families of solutions had been generated, the results were combined through a union operation and sorted by the ratio of surface to interior for each possible baryon count.  To avoid division by zero, this was normalized by adding 8 to the sphere count of each.  This produced an interesting graph 1 2 3 4 5 6, with a zigzag lower edge (larger blue circles).  The lower edge consists of a set of line segments ending at a regional minimum. Note - green circles have 4 and 8 surface downs from lowest blue, small blue are derived solutions and those with parity violations. Pink are known isotopes, white is ideal z, and black is Dirac's limit.  Each point on the edge is a local minimum (the least surface for a complete structure with a given baryon count). It may be shown that no possible solutions are below the line connecting consecutive regional minima.

The lower edge of the solution can be viewed as ridge lines of solutions, with even ridges containing the even symmetric, and the even near-symmetric as the final 2 solutions on a ridge.  The odd symmetric and the odd near-symmetric are similarly the final 2 solutions on the odd ridges.  This immediately led to classifying the skew solutions to odd and even cases.  Skew solutions with 1 degree of skewing (2 cubes sharing an edge for 14 spheres as a core) were odd, while those with 2 degrees of skewing (either in a single plane or 2) were even.  This continued for higher degrees of skewing.  Solutions that occurred on ridge 4 were also found on ridge 6, 8, 10, etc..  Solutions on ridge 5 were also found on ridge 7, 9, 11, etc..   This greatly simplified the generation of additional solutions.

In examining solutions with 3 dimensions of skewing, these were found to be cousins to solutions with 1 less degree of skewing in each plane.  So solution Skew 1, 1, 1 layer 2 has a core that is 2x2x2, just as solution Skew 0, 0, 0 layer 2 does. (Skew 0, 0, 0 is the even symmetric solution viewed as the origin of the skew group of families).

The even near symmetric solution at first appears to be the best fit on the even ridges.  It has a lower ratio of surface to interior than the even symmetric does.  However looking at an actual model, it is apparent the solutions have high dipole.  For instance layer 4 has a surface with 25 monoquarks and 25 diquarks,and an interior with 19 of each.  But, the monoquarks are all in a single hemisphere, and the diquarks are in the other hemisphere.  Since the up has a charge of 2/3 each, and the diquark has a charge of 1/3, the monoquark hemisphere has twice the charge of the diquark hemisphere.   This could be addressed by having a third of the surface monoquarks as downs (which does happen for the first 3 layers), but this leads to Chlorine 44 at layer 4, which is a z of 17, much less than the actual ideal z of about 20. The shortfall increases rapidly at higher layers (for instance Lanthanum 231 at layer 7 with z of 57, where 90 is ideal.)

Complete structures are a small fraction of the possible solutions that can be generated.  Solutions derived from complete structures (either by adding a few sphere, or trimming a few spheres) produce a much larger solution space.  Returning to the original graph, these can be shown as producing V’s of solution points adjacent to each local minimum. Trimmed solutions have the same surface sphere count as the complete structure, but fewer interior spheres.  Enlarged solutions have the same interior (at least near the complete structure), but more surface.  Trimmed and enlarged solutions from an Even near-symmetric solution may have enough less dipole to be best fit for a point, even if the near-symmetric solution proper is not.

A significant number of models of the solutions corresponding to best fit for a baryon count, and some nearly best have been generated (in Sketchups by Google, or physically with molecule models).  These may be helpful in visualizing what is meant by eccentric or skew.  A large pair of tables of solutions has also been generated: the first grouped by family and the second sorted by baryon count.   Each is linked to individual solution pages for all the reasonable structures, and many not so reasonable.  Additionally, 2 tables of isotopes with links to the individual solution pages were generated.  The standard form (n vs z), and a compact form (n-z vs n-2z) each gives some added insight to relationships among solutions. 3 smaller tables give only good complete structures, an overall, an even ridge, and an odd ridge table. These also have links to structure detail information. Detail pages often have images generated from the sketchups, or photographs of the molecule models. In those cases where 1 or more sketchups have been rendered either from the complete structure or some derivative thereof, a link to the sketchup is also included, as are links to parents, children, cousins, and in a few cases twins (S244 is a twin to S006, with identical counts for each complete structure pair).

Graphs showing the solution spaces in relation to the known isotopes 1 2, and possible future discoveries are also available.  The graph of the difference between minimum surface for a baryon count and actual known isotopes is particularly revealing, showing no known stable isotopes requiring more than 8 surface downs, and no stable isotope with a baryon count under 176 needing a greater than minimum surface.  All summary information includes a rough estimate of ideal z for baryon count.  It is most visible against the compact isotope table, showing a band of viable isotopes both known and undiscovered.

References

M. Gell-Mann. Phys. Lett. 8, 214 (1964).

P.W. Higgs. Phys. Lett. 12, 132 (1964).

D.B. Lichtenburg, L.J. Tassie, P.J. Keleman. Quark-Diquark Model of Baryons and SU(6). Phys. Rev. 167, 1535-1542 (1968).

J.E.Horvath. Stable Diquark Matter. Phys. Lett. B 3-4, 412-416 (1992).

R. Rapp, E. Shuryak, I Zahed. Chiral crystal in cold QCD matter at intermediate densities? Phys. Rev. D 63, 034008 (2001).

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