The email I sent to the circulation list concerning the shell energy conservation:
Hi folks. On page 20 of https :// application .wiley -vch.de/
books/info/0-471-35633-6/toi99/www/struct/struct.pdf 'Appendix Nuclear Structure', which I generally use to analyze shell model data graphically, is the harmonic oscillator plot of energy levels versus deformation.
As some of you may remember, all the spherical magic numbers in this simple model are exactly (and only) doubled tetrahedral numbers from a term-doubled Pascal Triangle. The doubling comes apparently because in fact we are counting spin-opposed PAIRS of nucleons.
The intervals between these counts are from period analogues, with orbitals of all the same parity (alternating positive and negative). The sizes of such period analogues are always here doubled triangular numbers in size, thus:
1s=2
1p=6
1d2s=12
1f2p=20
1g2d3s=30
1h2f3p=42
1i2g3d4s=56
1j2h3f4p=72
The energy levels in the harmonic oscillator plot at the site above are all straight line 'component rays' (I'm not sure how generally that term is used, but I've seen it in several papers and books). As such they have fixed slopes.
The orbitals are still split here, but in the sphere NOT differentiated by energy value (unlike in the spin-orbit model).
Slopes are (in terms of h-bar, omega-bar (energy) versus delta (deformation parameter, as defined in the paper):
1s1/2 0/3; *1s-1/2 -3/3 (hypothesized empty level to keep math consistent)
1p3/2 1/3; 1p1/2 -2/3
1d5,2 2/3; 1d3/2 -1/3; 2s1/2 -4/3; 2s-1/2 -7/3
1f,7/2 3/3; 1f5/2 0/0; 2p3/2 -3/3; 2p1/2 -6/3
1g9/2 4/3; 1g7/2 1/3; 2d5/2 -2/3; 2d3/2 -5/3; 3s1/2 -8/3; 3s-1/2 -11/3
1h11/2 5/3; 1h9/2 2/3; 2f7/2 -1/3; 2f5/2 -4/3; 3p3/2 -7/3; 3p1/2 -10/3
1i13/2 6/3; 1i11/2 3/3; 2g9/2 0/3; 2g7/2 -3/3; 3d5/2 -6/3; 3d3/2 -9/3; 4s1/2 -12/3; 4s-1/2 -15/3
1j15/2 7/3; 1j13/2 4/3; 2h11/2 1/3; 2h9/2 -2/3; 3f7/2 -5/3; 3f5/2 -8/3, 4p3/2 -11/3; 4p1/2 -14/3
You can see from the above that each move to the right, within any period analogue and between contiguous suborbitals within the all-harmonic-oscillator model, is a change of -3/3 slope. And all moves downward between the highest slopes are all +1/3.
On a hunch, yesterday I worked out that when level occupancy is factored in (that is, how many particles are actually in each level), then something very interesting happens.
Multiplying the slope by its occupancy to give a product, and then adding all the slopes within any shell to give a sum, yields result of ZERO. Within the harmonic oscillator model there appear to be no exceptions to this new rule.
Thus the TOTAL ENERGY of any level is conserved despite the deformation, even though INDIVIDUALLY each component ray varies linearly, but they all do so in a coordinated manner. This reminds me strongly of J, the total spin, in spin-orbit systems. And I wonder whether it relates at all to conjugate variables.