[seqfan] Madelung-like rules for atomic nuclei

[Jess Tauber]

Hello again, folks. In recent weeks I've been working on Madelung-like
rules that concern shell structures in atomic nuclei. One of the most
important relations in the electronic periodic table (celebrating its 150th
anniversary this year) is the so-called Madelung rule. In the Left-Step
Periodic Table variant, developed by the then elderly French polymath
Charles Janet in the late 1920's, all the periods end with s-block elements
rather than with noble gases (though He belongs to both groups). This
ignores chemical reactivity but faithfully reproduces the order in which
new orbital types are introduced into the system, thus 1s, 2s, 2p3s, 3p4s,
3d4p5s, 4d5p6s, 4f5d6p7s, 5f6d7p8s.

Only a few years later several workers noticed that summing the shell
quantum number N with the azimuthal angular momentum quantum number (M)L
resulted in the same sums for all orbitals within the same Janet period.

Thus, for example in 4d5p6s we have (4+2)=6, (5+1)=6, and (6+0)=6. The sums
increase monotonically through the table as we go from period to period, so
from 1 through 8.

Last week I discovered that for simpler spherical atomic nuclei under a
harmonic-oscillator-only model there was a very similar relation, but there
the sum was 2N+L rather than N+L as in the electronic shells.

The spherical shells sequence is: 1s,1p, 1d2s, 1f2p, 1g2d3s, 1h2f3p,
1i2g3d4s, 1j2h3f4p....
With 2N+L we then have:
1s= 2(1)+0=2
1p=2(1)+1=3
1d2s= 2(1)+2=4; 2(2)+0=4
1f2p= 2(1)+3=5; 2(2)+1=5
1g2d3s= 2(1)+4=6; 2(2)+2=6; 2(3)+0=6
1h2f3p= 2(1)+5=7; 2(2)+3=7; 2(3)+1=7
1i2g3d4s= 2(1)+6=8; 2(2)+4=8; 2(3)+2=8; 2(4)+0=8
1j2h3f4p= 2(1)+7=9; 2(2)+5=9; 2(3)+3=9; 2(4)+1=9

As such nuclei in this simpler model are deformed into ellipsoidal shapes
the orbital constitution of the shells is altered in regular ways. One way
of representing the deformation is by the so-called oscillator ratio (the
ratio of the matter wave in the polar direction (numerator) versus that in
the equatorial direction (denominator).

It turns out that for oscillator 2:1 (prolate ellipsoid) the Madelung-like
relation is N+L, and for 4:1 (hyperdeformed prolate ellipsoid) the relation
is N+2L.

Nuclei can also be deformed to oblate ellipsoids of revolution with
oscillator ratios where the denominator is greater than the numerator in
numerical value. I haven't been able to work out the Madelung-like
relations here. I'm hoping that some of you might have the mathematical
skills to figure this out. Thoughts? Thanks.