A couple of days ago I discovered how to squeeze a Madelung-type rule out of atomic nuclear structure. In the Left-step periodic table created by the elderly French polymath Charles Janet in the late 1920's, all the periods end in s-block elements, ignoring chemical bonding behavior but reproducing (at least ideally) the sequence of introduction of new blocks in the table. Thus s, ps, dps, fdps, with each structure repeated (so 1s, 2s, 2p3s, 3p4s, 3d4p5s, 4d5p6s, 4f5d6p7s, 5f6d7p8s).
Several workers around the same time started noticing that the sums of the shell number plus the quantum number (m)l value for each of the orbitals within the Janet period structure (N+L) always had the same value:
2p= 2+1=3, 3s= 3+0=3
3p= 3+1=4, 4s= 4+0=4
3d= 3+2=5, 4p= 4+1=5, 5s= 5+0=5
4d= 4+2=6, 5p= 5+1=6, 6s= 6+0=6
4f= 4+3=7, 5d= 5+2=7, 6p= 6+1=7, 7s= 7+0=7
5f= 5+3=8, 6d= 6+2=8, 7p= 7+1=8, 8s= 7+0=8
In the atomic nucleus, on the other hand, under a simple harmonic oscillator model, we are also presented with a Left-step pattern, but with a major difference. Unlike the electronic periods, shells in the harmonic oscillator system have all their orbital components sorted for parity (either all positive (even (m)l) or negative (odd (m)l)). And unlike the electronic system, period lengths by element count are not repeated. Rather the total number of orbitals within the shell are repeated.
So 1s, 1p (both with only one orbital); 1d2s, 1f2p (with two orbitals); 1g2d3s, 1h2f3p (with three orbitals); 1i2g3d4s, 1j2h3f4p (with four orbitals).
I found that the Madelung-type rule for the harmonic oscillator nucleus with 2N+L works just fine, thus:
Edited by pascal, 22 February 2019 - 09:58 AM.