Hi folks. I just joined. I study mathematical patterning in electronic and nuclear shell systems, and have for nearly a decade. There seems to be a pervasive Pascal Triangle motivation for much of it, due to the fact that the quantum harmonic oscillator always delivers numbers of stable states whose values come from Pascal Triangle diagonals- which diagonal depends only on the dimensionality of the system.
So consider the harmonic oscillator shell model for spherical nuclei. The magic numbers are 2,8,20,40,70,112,168..., which turn out to be exactly, and only, doubled tetrahedral numbers from a term-doubled Pascal Triangle. Doubling comes from spin-opposition, so we're counting pairs of nucleons. The shell sizes are all doubled triangular numbers:
1s=2
1p=6
1d2s=12
1f2p=20
1g2d3s=30
1h2f3p=42
1i2g3d4s=56
1j2h3f4p=72
It also turns out that for ellipsoidally deformed harmonic oscillator nuclei, the numarator and denominator of the oscillator ratio, which detail the polar and equatorial extents of the matter wave, determine how these doubled triangular number intervals work between magics. The oscillator ratio (OR) can stand in for the usually chosen deformation parameters (beta, delta, epsilon).
The sphere, as default ellipsoid, has OR 1:1. It has ONE copy of each doubled triangular number interval between each magic, and each doubled triangular number interval is used only once to generate magics. But for a prolate nucleus of OR 2:1, each interval is used TWICE, generating a magic each time, thus 2,2,6,6,12,12,20,20.... resulting in new magics 2,4,10,16,28,40,60,80... which match published lists. For OR 3:1, each interval is used THRICE, and so on. The numerator MULTIPLIES the use of each doubled triangular number interval.
For oblate nuclei, the denominator of the OR DIVIDES the system. So for an OR of 1:2, there is one doubled triangular number interval between EVERY SECOND magic, and for 1:3 between every THIRD, and so forth. The only exceptions occur at the beginnings of magic sequences, when the series haven't yet accumulated the denominator's worth of magics. In such cases the magics are themselves doubled triangular numbers.
It also turns out that total shell energy, under the simple harmonic oscillator model, is conserved over deformation. As each component of the shells varies its energy over deformation, this means that they are all coordinating their relative energies in some manner.
The more realistic model of shell structure incorporating spin-orbit couplings shifts the spherical magic numbers, by monotonically increasing amounts, due to so-called 'intruder' levels adding their nucleons to the harmonic oscillator shells. The new magics become 2,*6,14,28,50,82,126,184... Interestingly the sizes of the intruders are coordinated such that they increase the size of the harmonic oscillator shells (already doubled triangular numbers) to the very next higher doubled triangular numbers. Thus 1g9/2(10 nucleons) adds to 1f2p(20) to give total 30. And 1h11/2 (12) adds to 1g2d3s (30) to give 42. And so on.
Moreover, the depths of insertion of intruders are also doubled triangular numbers- 1g9/2 is 2 nucleons deep into the previous shell, 1h11/2 is 6, 1i13/2 is 12, 1j15/2 is 20, at least for neutrons. There is some ambiguity for protons, with 20 found when 12 is expected, as if the system were anticipating the trend somehow (and because the shell structure allows it).
It also looks like conservation of total shell energies also work under the spin-orbit model.
You won't find any of this described in professional papers and books. I guess everyone already just understands it all tacitly.
The electronic system (in terms of the periodic system) also shows Pascal Triangle motivation. Organized so that orbital introduction is the primary structural motif, rather than chemical behavior, has the s-block on the RIGHT after the p-block. It places He with the other s2 elements. This is the Left-Step Periodic Table described by the elderly French polymath Charles Janet in 1928, following upon then new-fangled quantum mechanics. The organization ends periods not with noble gases, but with all the s2 elements. It seems to have escaped most workers that every other atomic number of the s2 column are tetahedral numbers (4,20,56,*120), in fact, every other tetrahedral numbers. And all the intermediate atomic numbers are the arithmetic means of the flanking tetrahedrals (so 'triads' as discovered by Dobereiner in the early 19th century). This allows the periodic system to be rationalized as a tetahedron of close-packed spheres, each representing one element. Rhombi (which contain square numbers of spheres, just like 'duals' of same-length periods), may be 'skewed' to tetrahedral dihedral angle, and then stacked like Russian dolls, leading to ever larger tetrahedra. And if the skew rhombi are bisected, one may tesselate the tetrahedral surface allowing complete continuity of Mendeleev's line (the numerical sequence of atomic numbers). There are no other simple geometrical models which can maintain such continuity- yet for the tetrahedral model there are *8* variants that work. Researchers have found similar Pascal Triangle motivation for harmonic oscillator models of hemipsherical atomic clusters, so the applicability could be substantially larger than even outlined above.
