More Tetrahedral Periodic Table Findings

[tetrahedron]

Hi folks. Been a long while. But I just got a new computer and am stretching my legs.  You may remember my earlier work on nuclear shell structure from a Pascal Triangle mathematical motivational perspective.

 

All the spherical magic numbers for the simple harmonic oscillator model of the nucleus were exactly doubled tetrahedral numbers: 2,8,20,40,70,112,168...  This works because the quantum harmonic oscillator always delivers numbers of stable states which are terms from Pascal Triangle diagonals. Which diagonal is chosen depends only on the dimensionality of the system. Here the stable states are the triangular numbers (doubled because of paired opposed-spin particles). A 3D system uses the third diagonal (triangular numbers). A 2D system uses natural numbers (second diagonal), and a 1D system uses the outer diagonal of all 1's.

 

The doubled triangular number intervals tell us about the intervening shells or period analogues (which in nuclei are sorted by parity value, plus or minus). So: 1s=2, 1p=6, 1d2s=12, 1f2p=20, 1g2d3s=30, 1h2f3p=42, 1i2g3d4s=56, 1j2h3f4p=72.... and these numbers then sum to the doubled tetrahedral magics.

 

Ellipsoidally deformed nuclei have their own sets of magic numbers which also relate to the Pascal Triangle motif. We can use the oscillator ratio as a stand-in for other deformation parameters along the x-axis of the Nilsson diagram. The numerator (relating the polar extent of the matter wave) MULTIPLIES the use of doubled triangular intervals. And the denominator DIVIDES. So a prolate nucleus of oscillator (OR) 2:1 has TWO copies of the intervals used in succession to generate magics (so 2,2,6,6,12,12,20,20, 30,30) with the running sums matching published charts. And a nucleus of OR 3:1 uses THREE copies in succession, and so on.

 

An oblate nucleus  of OR 1:2 uses each interval only once, but has doubled triangular intervals not between EVERY magic, but between every SECOND magic. And one of OR 1:3 has a doubled triangular number interval between every THIRD magic, and so on. Exceptions occur before a numerator's worth of magics has been encountered, in which case we have doubled triangular magics until that point, after which the pattern shifts over to the divisional one.

 

With these three rules any sequence of deformed magics can be worked out simply on paper for any defined OR.

 

The model I used to work out these rules has all linear component 'rays' (that is energy levels change linearly over deformation). I found that each ray's slope (energy vs. deformation) were all multiples of 1/3 energy vs. deformation, in both positive and negative directions.

 

A couple of months ago I figured out that these values were all coordinated, so that when you multiplied each ray by its occupancy (how many nucleons in that level) and then summing the products, the net change was ZERO over deformation. That is, total shell energy is conserved over deformation. I'd never seen that result published in any journal or book.  And for any stated OR, there is a simple relation between the shell energies in succession vertically. So for example 1.5x2 (for 1s), 2.5x6 (for 1p), 3.5x12 (for 1d2s), 4.5x20 (for 1f2p), 5.5x30 (for 1g2d3s), 6.5x42 (for 1h2f3p), 7.5x56 (for 1i2g3d4s) and 8.5x72 (for 1j2h3f4p). I'll leave it for you to do the math here and then work out the differences between the products.

 

Looking at the more realistic spin-orbit model, I had previously determined that for the sphere, the high-spin 'intruder levels' from the next higher shell lowered in energy sufficiently to join the previous shell, with the intruder SIZES exactly those needed to expand the shell size (already a doubled triangular number in the harmonic oscillator model) to the very NEXT HIGHER doubled triangular number. So 1g9/2 (with 10 nucleons) adds to 1f2p (with 20) to give 30. 1h11/2 (12) adds to 1g2d3s (with 30) to give 42, and so on.

 

Thus at the period analogue level, the Pascal Triangle mathematical motivation is preserved, but the sequence is up-shifted one move. The resulting spherical magics all have intervals which are doubled triangular numbers PLUS 2.

 

I found that the shell energy in the spin-orbit model seems to be also conserved (harder to tell because Nilsson mappings are only partially theory based and partially experimentally determined, with no first-principle support).

 

Anyway, I'm back and feeling good. Hope you've all been busy. :)

 

Jess Tauber

[email protected]

[exchemist]

Hi folks. Been a long while. But I just got a new computer and am stretching my legs.  You may remember my earlier work on nuclear shell structure from a Pascal Triangle mathematical motivational perspective.

 

All the spherical magic numbers for the simple harmonic oscillator model of the nucleus were exactly doubled tetrahedral numbers: 2,8,20,40,70,112,168...  This works because the quantum harmonic oscillator always delivers numbers of stable states which are terms from Pascal Triangle diagonals. Which diagonal is chosen depends only on the dimensionality of the system. Here the stable states are the triangular numbers (doubled because of paired opposed-spin particles). A 3D system uses the third diagonal (triangular numbers). A 2D system uses natural numbers (second diagonal), and a 1D system uses the outer diagonal of all 1's.

 

The doubled triangular number intervals tell us about the intervening shells or period analogues (which in nuclei are sorted by parity value, plus or minus). So: 1s=2, 1p=6, 1d2s=12, 1f2p=20, 1g2d3s=30, 1h2f3p=42, 1i2g3d4s=56, 1j2h3f4p=72.... and these numbers then sum to the doubled tetrahedral magics.

 

Ellipsoidally deformed nuclei have their own sets of magic numbers which also relate to the Pascal Triangle motif. We can use the oscillator ratio as a stand-in for other deformation parameters along the x-axis of the Nilsson diagram. The numerator (relating the polar extent of the matter wave) MULTIPLIES the use of doubled triangular intervals. And the denominator DIVIDES. So a prolate nucleus of oscillator (OR) 2:1 has TWO copies of the intervals used in succession to generate magics (so 2,2,6,6,12,12,20,20, 30,30) with the running sums matching published charts. And a nucleus of OR 3:1 uses THREE copies in succession, and so on.

 

An oblate nucleus  of OR 1:2 uses each interval only once, but has doubled triangular intervals not between EVERY magic, but between every SECOND magic. And one of OR 1:3 has a doubled triangular number interval between every THIRD magic, and so on. Exceptions occur before a numerator's worth of magics has been encountered, in which case we have doubled triangular magics until that point, after which the pattern shifts over to the divisional one.

 

With these three rules any sequence of deformed magics can be worked out simply on paper for any defined OR.

 

The model I used to work out these rules has all linear component 'rays' (that is energy levels change linearly over deformation). I found that each ray's slope (energy vs. deformation) were all multiples of 1/3 energy vs. deformation, in both positive and negative directions.

 

A couple of months ago I figured out that these values were all coordinated, so that when you multiplied each ray by its occupancy (how many nucleons in that level) and then summing the products, the net change was ZERO over deformation. That is, total shell energy is conserved over deformation. I'd never seen that result published in any journal or book.  And for any stated OR, there is a simple relation between the shell energies in succession vertically. So for example 1.5x2 (for 1s), 2.5x6 (for 1p), 3.5x12 (for 1d2s), 4.5x20 (for 1f2p), 5.5x30 (for 1g2d3s), 6.5x42 (for 1h2f3p), 7.5x56 (for 1i2g3d4s) and 8.5x72 (for 1j2h3f4p). I'll leave it for you to do the math here and then work out the differences between the products.

 

Looking at the more realistic spin-orbit model, I had previously determined that for the sphere, the high-spin 'intruder levels' from the next higher shell lowered in energy sufficiently to join the previous shell, with the intruder SIZES exactly those needed to expand the shell size (already a doubled triangular number in the harmonic oscillator model) to the very NEXT HIGHER doubled triangular number. So 1g9/2 (with 10 nucleons) adds to 1f2p (with 20) to give 30. 1h11/2 (12) adds to 1g2d3s (with 30) to give 42, and so on.

 

Thus at the period analogue level, the Pascal Triangle mathematical motivation is preserved, but the sequence is up-shifted one move. The resulting spherical magics all have intervals which are doubled triangular numbers PLUS 2.

 

I found that the shell energy in the spin-orbit model seems to be also conserved (harder to tell because Nilsson mappings are only partially theory based and partially experimentally determined, with no first-principle support).

 

Anyway, I'm back and feeling good. Hope you've all been busy. :)

 

Jess Tauber

[email protected]

I don't think you can mean the Periodic Table, since that has nothing to do with nuclear structure, being based on the electronic structure of atoms: the Aufbauprinzip, according to which the electrons in successive elements fill the electron orbital states in order of their energy, staring with the lowest.  Each state can accept two electrons of opposite spin, which leads to the following numbers of electrons in successive shells: 2; 2+6=8; 2+6+10=18;2+6+10+14=32; and so on. These numbers do not fit your series of course. 

 

If what you mean to refer to is not the Periodic Table but the series of nuclear states in succeeding elements, i.e. according to the nuclear shell model, here: https://en.wikipedia.org/wiki/Nuclear_shell_model  then I cannot comment, not having studied nuclear stability. 

Edited September 8, 2017 by exchemist

[exchemist]

Actually maybe you can help me with a question about nuclear physics: is there in fact anything like a periodic table of nuclei? Or is it this that you are trying to develop, for the first time? 

[arkain101]

I did a little research to become familiar with some of the terms you used here in your opening post. However, this did not help me to see or understand what your main point is here. Basically your talking about a numbers game that applies to models of atoms, more specifically, atomic nuclei. However, I can see that it does not have to apply only to atomic nuclei and could apply to other models as well.

I think you need to highlight your main points here in order to get more useful feedback from others, because it reads like a grocery list and not something of coherent structure.

Also, you could supply some links that explain some of the terms you used so that people can be sure they are understanding what you are referring to.

I think you may find this post useful in your efforts to formulate mathematical theories:

Post #17

http://www.scienceforums.com/topic/30400-an-attempt-of-quantum-gravity/?do=findComment&comment=350944

[tetrahedron]

One could make something like a periodic table for nuclei, at least for the individual proton or neutron shells.

 

The first two period analogues consist of 1s and 1p levels, with 2 and 6 members respectively. Unlike the electronic shells in the Left-Step PT arrangement by Janet (from circa 1929, built with the new quantum theory in mind), nuclear period analogues do not contain the same number of particles, as witnessed by the 1s and 1p levels, which appear once each. Instead, paired nuclear period analogues contain the same number of ORBITALS. 

 

So 1s and 1p contain 1 orbital each.

1d2s and 1f2p contain 2 orbitals each.

1g2d3s and 1h2f3p contain 3 orbitals each

1i2g3d4s an 1j2h3f4p contain 4 orbitals each.

 

Remember we're still considering the harmonic oscillator-only model here, with doubled tetrahedral magic numbers for the sphere.

 

When intruders from the next higher shells are introduced, they come from orbitals with opposite parity to the harmonic oscillator period analogue they're introduced into.

 

Anyway, I've drawn such 'nuclear' periodic tables many times. The protonic and neutronic orders of orbital partials within the period analogues are different from each other. The neutron system has a completely consistent structural motif, while the protonic system alternates between that of the neutronic system on the one hand, and the harmonic oscillator system on the other.

 

Jess Tauber

[tetrahedron]

Oh and there IS a relation between the nuclear and electronic systems, mathematically, when both systems are rationalized as left-step tables (for the harmonic oscillator). The electronic period terminals in the left-step periodic table of Charles Janet (circa 1929, which was inspired by the then new quantum mechanics rather than surface chemical behavior), are all from s2 elements (helium plus the alkaline earths). In this depiction (which can be found on many pages online), each orbital length occurs TWICE in sequence:

 

1s

2s

2p3s

3p4s

3d4p5s

4d5p6s

4f5d6p7s 

5f6d7p8s

 

Because each period's length is a half square, the total of equal length 'duals' (which is what professional chemists call them) is a square: 4,16,36,64.  These then sum to give tetrahedral numbers 4,20,56,120. The intermediate s2 element atomic numbers are 2,12,38,88, which are the arithmetic means of the flanking tetrahedral numbers (starting from 0).  Anyway this allows one to recast the left-step system as a tetrahedron of close-packed sphere, each sphere representing one element. There are several good ways to patten the periodic system over the volume of the tetrahedron.

 

Back in 2009 I worked out the simplest, most symmetrical model. Each dual period is laid out as a rhombus of close packed spheres, with 4 s spheres in the center surrounded by 12 p spheres, surrounded again by 20 d spheres, and finally by 28 f spheres. Then the entire rhombus is bent up along its shorter axis to a tetrahedral dihedral angle. resulting in so-called 'skew' rhombi,and the sequence of resulting tetrahedra nest, like a Russian doll. In fact, it turned out that I was beaten to the punch at least by 6 years, by a Russian: see http://weise.symmetry-us.com/files/2016/07/Fig-6.jpg.

 

There are other arrangements that are my own alone, which allow for an unbroken chain of elements within any dual, and with only slight gaps between them.

 

Anyway, remember that the tetrahedral numbers which terminate duals in the Janet table are 4,20,56,120. Well it turns out that these are also HALF the sizes of every other spherical nuclear magic number in the harmonic oscillator number, whose magic sequence is 2,(  8  ) ,20,(40),70,(112), where the numbers in question have the parentheses about them. Even the as yet unsynthesized element with atomic number 120, which ends the next dual and is also a tetrahedral number, is HALF the size of the next harmonic oscillator spherical magic number, 240.

 

It turned out (something I was told by a famous university mathematical physicist who wants to remain anonymous because of prevailing prejudices against supposed 'numerological' work) that THE QUANTUM HARMONIC OSCILLATOR SYSTEM ALWAYS DELIVERS NUMBERS OF STABLE STATES WHICH ARE TERMS OF PASCAL TRIANGLE DIAGONALS. Here the individual shells/period analogues have the stable state numbers, not the magics. And the particular diagonal of the Pascal Triangle chosen depends solely on the dimensionality of the system being defined.

 

Jess Tauber

Edited September 9, 2017 by pascaltriangle

[tetrahedron]

A couple of years back I got permanently banned from another online physics discussion list, not for bad behavior of any kind, but because I had the temerity to continue this topic when I found new results from my graphical analyses of published Nilsson diagrams (which map energy levels of nucleons versus deformation). Posters touting 10000 posts and more claimed that my numbers were a) nonsense numerology, and b  )

 that science already knew everything I was claiming. I don't know how both these complaints can be true at the same time. All I've done is look at lists of known data and extracted patterns from them. That's all. Its similar in flavor to what Rydberg did in the 1880's before the rise of quantum mechanics. Aside from a couple of online postings from the past 10 years I can find NO text or article references to the Pascal Triangle connection. If all physicists know this stuff they picked it up by osmosis. One can see the math without recognizing its identity.

 

Jess Tauber

Edited September 9, 2017 by pascaltriangle

[tetrahedron]

Watch what happens when you multiply the spherical harmonic oscillator energies times their occupancies (how many nucleons in that level) to give total shell energies:

 

0.5 x 0    = 0

1.5 x 2    = 3

2.5 x 6    = 15

3.5 x 12  = 42

4.5 x 20  = 90

5.5 x 30  = 165

6.5 x 42  = 273

7.5 x 56  = 420

8,5 x 72  = 612

9.5 x 90  = 855

 

Then the differences between these total shell energies are, respectively, 3, 12, 27, 48, 75, 108, 147, 192, 243, which when divided by 3, become 1, 4, 9, 16, 25, 36, 49, 64, 81, or, the squares of the natural numbers. You won't find this result in any published journal article or book, because nobody ever explored the internal workings of the Nilsson charts before, except on the most shallow level. Similar trends exist for deformed harmonic oscillator energy levels, but they are skewed a bit. Still highly regular, however.  And interestingly, these trends appear to be preserved as well for spin-orbit nuclei, despite what looks like a crazy-quilt of energy levels on the surface.

 

Jess Tauber

Edited September 10, 2017 by yahganlang1958

[exchemist]

A couple of years back I got permanently banned from another online physics discussion list, not for bad behavior of any kind, but because I had the temerity to continue this topic when I found new results from my graphical analyses of published Nilsson diagrams (which map energy levels of nucleons versus deformation). Posters touting 10000 posts and more claimed that my numbers were a) nonsense numerology, and b  )

 that science already knew everything I was claiming. I don't know how both these complaints can be true at the same time. All I've done is look at lists of known data and extracted patterns from them. That's all. Its similar in flavor to what Rydberg did in the 1880's before the rise of quantum mechanics. Aside from a couple of online postings from the past 10 years I can find NO text or article references to the Pascal Triangle connection. If all physicists know this stuff they picked it up by osmosis. One can see the math without recognizing its identity.

 

Jess Tauber

Reported for absurdly blatant sock puppetry.

[tetrahedron]

In fact, if your divide the total shell energies themselves by 3, the sequence becomes 0,1,5,14,30,55,91,140,204... You can be forgiven for not recognizing this sequence. These numbers are found on the 'sister' generalized Pascal Triangle with sides (2,1). This same Triangle's shallow diagonals have terms which in one direction sum to give up-shifted Fibonacci numbers on one side, and Lucas numbers on the other. The sequence in question, however, is the diagonal which stands in place of the tetrahedral number diagonal in the classical Pascal Triangle. They are also sums of successive pairs of tetrahedral numbers.

 

Jess Tauber

[Turtle]

...

Back in 2009 I worked out the simplest, most symmetrical model. Each dual period is laid out as a rhombus of close packed spheres, with 4 s spheres in the center surrounded by 12 p spheres, surrounded again by 20 d spheres, and finally by 28 f spheres. Then the entire rhombus is bent up along its shorter axis to a tetrahedral dihedral angle. resulting in so-called 'skew' rhombi,and the sequence of resulting tetrahedra nest, like a Russian doll. In fact, it turned out that I was beaten to the punch at least by 6 years, by a Russian: see http://weise.symmetry-us.com/files/2016/07/Fig-6.jpg ...

 

Jess Tauber

Hi Jess.   Can you elaborate on the close-packing lattice, inasmuch as a rhombus is a 2-d figure and spheres are 3-d. I have been messing about with building kites based on different solids including Fuller's favorite the 'vector equilibrium, aka cuboctahedron. Fuller fully embraced the cuboctahedron as the 'shape of everything' more or less.

Anyway, in poking into polytopes I do recall a solid with rhombic faces, the rhombic dodecahedron, which is the dual of the cuboctahedron.

 

As to tetrahedrons nesting as in space-filling by more than one solid, tetrahedrons do indeed fill space when nested among octahedrons in 2 different ways.

Tetrahedral-octahedral honeycomb

 

Gyrated tetrahedral-octahedral honeycomb

 

Edited July 14, 2018 by Turtle

[exchemist]

Reported for absurdly blatant sock puppetry.

I owe the poster an apology. Due to not having read all these long posts, I had missed that the sock puppet was only created to overcome a technical glitch. 

[tetrahedron]

In my post from a couple of days ago describing the 3x square number difference for shell energies I accidentally wrote:  <<Similar trends exist for deformed harmonic oscillator energy levels>>. I MEANT to write 'for SPIN-ORBIT energy levels.

Sorry for any confusion.

 

Jess Tauber