[seqfan] Re: Madelung-like rules for atomic nuclei

[jean-paul allouche]

Hi, coincidentally I have seen today a paper on the ArXiv
namely https://arxiv.org/abs/1902.10752, that seems
to share [vaguely] the same flavor as your question. Honestly
I am not sure it can help, but who knows.

best wishes
jean-paul

Le 28/02/2019 à 22:24, Jess Tauber a écrit :
> 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.
>
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