New Period Relation Linking Electronic And Nuclear Levels

[tetrahedron]

Hope everyone is well.

 

I found another new type of relation in the periodic table, this one relating the nuclear and electronic levels.

 

During the first half of the 19th century, chemists realized that elements with similar properties often (though not always) fell into patterns where their measured atomic weights were in rough 'triads'. That is, if A were the lightest, B intermediate in atomic weight, and C the heaviest, then in terms of their atomic weights, (A+C)/2=B.

 

Such relationships formed much of the basis for the eventual discovery of the periodic relation itself starting around 1860, by several workers, once discrepancies in the determination of atomic were finally resolved and agreed upon by chemists.

 

From a quantum standpoint, the periodic table should end periods with the s block elements, with new orbital types added successively leftwards in every other new period: s s ps ps dps dps fdps fdps for elements up to atomic number 120.

 

Yet behaviorally the system acts as if it were instead s sp sp sdp sdp sfdp sfdp which is the standard depiction. Endless arguments have been fought about which depiction is primary, and better for educational purposes.

 

Let's suppose we keep to the quantum depiction which keeps orbital in their introduced order so they end in elements with electronic configuration s2, that is, helium and the alkaline earths.

 

These have atomic numbers (0),2,4,12,20,38,56,88,120,170,220... We can see that 2 is the arithmetic mean of 4 and 0, 12 of 20 and 4, 38 of 56 and 20, 88 of 120 and 56, and 170 of 220 and 120, and so forth.

 

The numbers (0),4,20,56,120,220... are every other Pascal Triangle tetrahedral number. The intermediate s2 elements, however, differ from the intermediate tetrahedral numbers 1,10,35,84,165 by monotonically increasing amounts 1,2,3,4,5...

 

In the spherical nucleus on the other hand we have two different sets of 'magic' numbers which define analogues of the noble gases found in the electronic system.

 

The first set is the DOUBLED tetrahedral numbers 2,8,20,40,70,112,168,240. The doubling appears to be due to the fact that nucleons pair spinwise immediately, and don't wait til orbital lobes are filled singly before doubling up, as see in in the electronic system. The second set contains most of the more familiar magic numbers, 2,6,14,28,50,82,126,184... which, above 20, are more stable than the double tetrahedral magics.

 

The first set involves the harmonic oscillator only, while the second includes a spin-orbit term.

 

Now, if you remember above I wrote that every other s2 atomic number is identical to every other tetrahedral number, which relates to the harmonic oscillator. The arithmetic means of successive pairs give the intermediate s2 atomic numbers.

 

But nobody apparently has previously taken the arithmetic means of the intermediate s2 atomic numbers themselves. Because if they had, they would have noticed, as I did only today, that they are half-values of every other spin-orbit magic number in the nucleus!!! So 7 (vs. 14), 25 (vs. 50), 63 (vs. 126), 129 (vs. 258) etc. Remember that the nucleons fill shells using DOUBLED Pascal values, while the electronic system uses SINGLE.

 

What does it all mean? First off, it means that the PERIODS in the electronic system ALTERNATE their associations regarding harmonic oscillator and spin-orbit magic numbers. This is fascinating because the same exact thing happens in the proton filling scheme, and of course the protons link directly to the electrons.

 

If we ignore the effects of intruder levels from higher period analogues (such analogues organized by splitting parity groups here while the electronic system intersperses them) into lower ones, then we have s/p HO, ds/fp SO, gds/hfp HO, igds/jhfp SO so far as can be explored by following rays in Nilsson diagrams.

 

Why would there be such an alternation? Neutrons don't use it in THEIR shell filling, since again abstracting out the intruders, they use all spin-orbit orders. But there has to be some sort of correlation between the charged atomic particles, and this is one of them (albeit by a doubling relation for the nucleons where each period analogue of a given size comes once, yet single relation for the electrons where each period of a given size comes twice).

 

Some things, however, are not the same. The sign of the spin-orbit effect is opposite between nucleons versus electrons. There is no trace of downward-shifted high quantum number l energy levels as intruders in the electronic system. Instead apparently we have UPWARD shifted low quantum number l energy levels, leading to, at least in the earlier parts of the electronic period table, the s block elements appending themselves to the next period, giving us classical behavior that led to the currently popular depictions. S electrons are higher energy than f or d electrons and so are outside of them, and they thus oxidize first before them.

 

Eventually however other effects start to muck up this nice pattern- differential shielding leading to atomic contractions which start to shift atomic behaviors allied with relativistic velocity shifts for innermost electrons. But I think they may be in lock step with the Pascal math! Later elements exhibit, at least in the p block, the 'inert-pair' effect (from spin-orbit coupling) which basically takes the two s electrons out of the picture to greater and greater extents from the reactive behavior of the elements in these periods.

 

If I'm right then at extreme atomic numbers even more orbitals (or their spin-orbit parts) would be shifted upwards in energy- extrapolations of the electronic system show that by the 9th period the gfdps order is largely caput. Do calculated models fit my Pascal-based mathematical predictions?

 

Stay tuned!

 

Jess Tauber

[Rade]

Hello, I greatly enjoyed your post. I have discovered another relationship of nuclear shells and electronic orbits that may relate to your discussion.

 

Consider the first electronic orbital, 1s. It contains two electrons (e-) when filled, thus:

 

1s = (e-) + (e-)

 

There are two ways this orbital can be filled if the (e-) are associated with protons (p+) and neutrons (n)as isotopes:

 

1s filled = [(np+)(e-)] + [(np+)(e-)]

 

or

 

1s filled = [(e-) + (+pnp+) + (e-)]

 

In the first filled case, the two electrons are associated with two deuterium isotopes or a single helium-4 isotope

In the second filled case, the two electrons are associated with a single helium-3 isotope

 

==

 

The 1s orbital is unfilled also in two ways associated with protons and neutrons as isotopes:

 

1s unfilled = [(np+)(e-)]

or

 

1s unfilled = [(nnp+)] + (e-)]

 

In the first unfilled case, the single electron is associated with a single deuterium isotope

In the second unfilled case, the single electron is associated with a single tritium isotope

 

==

 

All known isotopes of nature can be formed using the above selection rules that relate: (1) nuclear shells that contain a few fundamental nucleon isotopes, with (2) electronic orbitals. Single valence electrons in outer energy orbits are represented by the 'unfilled' quantum possibilities.

 

note: edits made

Edited October 8, 2013 by Rade

[Aethelwulf]

More chemistry, rather than quantum, yet to be honest, both have strong equivalence.

[Rade]

More chemistry, rather than quantum, yet to be honest, both have strong equivalence.

Do you have an interest in working on possible mathematical representation of how nuclear shells with positive charge from protons (p+), within isotopes, associate with electronic orbitals (e-) at quantum level ? Edited October 8, 2013 by Rade