Nuclear Deformation And Pascal Triangle Math

[tetrahedron]

Hi folks. Been a while. Hope all are well. In the past few weeks I've explored atomic nuclear structure and have discovered that Pascal Triangle combinatorics are behind the full-shell (i.e. 'magic') counts not only of spherical nuclei (which had been known) but also those of deformed nuclei such as oblate and prolate spheroids.

 

Surprisingly it is the axial ratio that determines the way the Pascal counts are apportioned to the growing string of magic numbers.

 

Ignoring complications due to spin-orbit splitting and coupling (which generates most of the stronger magic numbers) for spherical nuclei, the other full-shell magics are twice Pascal tetrahedral numbers 1,4,10,20,35,56,84,120..., that is 2,8,20,40,70,112,168,240. The stronger versions involving spin-orbit effects are 2,6,14,28,50,82,126,184... These can be generated (at least) three ways from double tetrahedral numbers: by subtracting the double triangular number from two rows above the double tetrahedral number (which in the apex-centered Pascal system is directly above it), by adding the double natural number in the same shallow diagonal as the double tetrahedral number to it, or by adding the double triangular number one line above to the double triangular number. All work- interestingly all three mechanisms are in a kind of triangular/hexagonal relationship geometrically within the Pascal Triangle.

 

Now Pascal combinatorics see double triangulars adding to double tetrahedrals to give the next double tetrahedral as well- thus differences between the double tetrahedral spherical nuclear magic numbers are double triangular.

 

While examining the magic numbers of biaxially deformed nuclei defined by axial ratio (so polar radius as numerator and equatorial radius as denominator) I discovered that for prolate ellipsoidal nuclei with axial ratios of N/1 the running count of full-shell/magic numbers had N numbers of double triangular numbers in sequence. That is, for 2/1 you see 2,2,6,6,12,12,20,20,30,30... For 3/1 2,2,2,6,6,6,12,12,12,20,20,20 and so on.

 

For deformed oblate ellipsoidal nuclei with axial ratios of 1/M instead we see double triangular number DIFFERENCES between every full-shell number.

 

If we think of the sphere as the default for the prolate and oblate situations, then with axial ratio of 1:1 we have 1 copy (numerator) of each double triangular number separated by one step (denominator) from the next.

 

When neither numerator nor denominator of the axial ratio are 1, then both effects complicate matters- that is one will still have N copies of each double triangular number, each copy separated by M steps in the chain of full-shell numbers. So for example if the axial ratio is 2/3, oblate, then we have two copies of each double triangular number separated from each other by 3 steps- the difference between every third magic is double triangular.

 

All this was hidden in plain sight. The reason it works has to do with the combinatorics of split orbital pieces in a map of deformation versus energy levels. And for an idealized mapping called the Nilsson diagram the relations are perfectly exact as described above, with the caveat that for M denominator differences you have to cumulate at least M magic numbers.

 

Jess Tauber

[email protected]

[Turtle]

Hi folks. Been a while. Hope all are well. In the past few weeks I've explored atomic nuclear structure and have discovered that Pascal Triangle combinatorics are behind the full-shell (i.e. 'magic') counts not only of spherical nuclei (which had been known) but also those of deformed nuclei such as oblate and prolate spheroids.

 

hi jess. nice to hear from you. alas my chem proficiency is worthy of a swift kick in the addend and my work with tetrahedral and higher tuples of triangular numbers is limited. nonetheless, here's what i can say of it for what it's worth.

 

... tetrahedral numbers 1,4,10,20,35,56,84,120...,

...

Jess Tauber

[email protected]

 

i can only find examples i worked for {1 4 10 20 35 56 84 120 165 220 286 ...} mod9 & mod11. for example 1mod9 4mod9 10mod9 20mod9 etc. and 1mod11 4mod11 10mod11 20mod11 ... my intention at the time was to find the patterns for a long sequence of moduli, i.e tetrahedral # mod{3,4,5,6,7...} but i never got around to it.

 

in my found examples, for mod9 the residues have a repeating pattern after 27 values and for mod11 a repeating pattern after 11 values. i'll have to do a little work to give you the mod9 pattern values as i have it as a color code and not numeric, but the repeating pattern of residues of tetrahedral numbers mod11 is {1 4 10 9 2 1 7 10 11 11 11}

 

if that doesn't throw any light on anything for you i hope at least it throws no shadows. fascinating work you're doing fo shizzle and thanx 4 sharing.

 

ps i think the correct mod9 pattern is {1 4 1 2 8 2 3 3 3 4 7 4 5 2 5 6 6 6 7 1 7 7 5 2 9 9 9 }. you will want to check me on that.

Edited October 28, 2012 by Turtle

[tetrahedron]

Got some more progress to report- hope you all had a good Turkey-Day.

 

I've described above how Pascal numbers work with regard to the simple and idealized theoretical Nilsson model of nucleon summing in atomic nuclei. This model is unrealistic in that it lacks a spin-orbit term as well as modifications of the nuclear potential well having to do with the idea that internal nucleons are completely surrounded by others (so feel a relatively uniform field), those on the surface are overly exposed, as it were, to the elements (so not so well covered field-wise), while those in between are around half-covered.

 

When these two terms are added to the equation describing the nucleus at different axial ratios all the straight rays from the simpler Nilsson diagram start to curve- more than this some curve up, some down depending on the parity value (s,d,g,i... one way, p,f,h,j... the other. And at some points certain now curved rays take crazy courses bucking the overall trends from their spherical sources. Weird stuff.

 

I started playing with orbital intervals for neutrons for the spherical nuclei from this more realistic model (their words) and began finding unexpected patterning.

 

Three different such Pascal-related mathematical patterns emerged which help to pin down the relative locations of all the orbitals in the cumulation of neutrons in such spherical nuclei.

 

First was that in many cases differences between like orbitals (same quantum number l (L) value, different n (energy level) utilized the same set of numbers: 2,6,14,26,42,62,86,114,146,182... (note the palindromic effect on final numerals within each term, which I just noticed myself as I write this!). This sequence is defined by 4tri+2 (or 2(2tri+1)), where tri is a triangular number. Not all the differences worked this way, but most did. I'm now beginning to understand that the ones that don't work have had their orbital parts rearranged (and in a coherent fashion at that!).

 

The second pattern that showed itself was that the absolute number value of the last nucleon in each orbital sequence for the smaller spin-split component (splits being 2/0 for s, 4/2 for p, 6/4 for d, 8/6 for f and so forth, with label XN-1 for each (X=s,p,d,f... and N being the split numbers at left) for the vast majority of forms (the ones not rearranged) were double tetrahedral numbers minus double triangular numbers.

 

And the third mathematical pattern I discovered had to do with distances between lower and higher spin-split components from different orbitals which for most ended up being half/double square numbers.

 

For spherical nuclei and only neutron counts considered here (proton counts having (as yet to me) no obvious such patterning), these three relations are sufficient to place the vast majority of the orbital pieces into their proper order.

 

However, it also became clear that there are clashes between these relations as mathematical motivations. The rearrangements that ruined some of the 4tri+2 differences also created the proper half/double square differences. Some sort of hierarchical priority? I'm starting to think that what is going on here is an at least two-step derivation, which I'll have to reconstruct going backwards to see whether it works.

 

And it may be that the proton count system is another variation on this theme, either having gone much further (or not far enough) with the derivations, or perhaps by re-ordering the rules themselves. Such things go on in human languages all the time (and a very active professional subfield was created to study it around 20 years ago, called Optimality Theory, where balance is achieved between forces that can either compete or cooperate).

 

In any case this is where my work stands at the moment. I'll probably post actual numbers to show how this all works; perhaps some of you might have further insights into what is going on. Later I've got to try to attack nonspherical configurations as well. The crazy rays mentioned above with regard to the curved-ray Nilsson model may mean more reordering goes on. One way of thinking about this is as a multidimensional sliding tile puzzle, where the linear reordering makes more sense (orbital pieces able to go around each other rather than butting heads in an abstract space).

 

Jess Tauber

goldenratio @ earthlink . net

[Turtle]

Got some more progress to report- hope you all had a good Turkey-Day.

...

First was that in many cases differences between like orbitals (same quantum number l (L) value, different n (energy level) utilized the same set of numbers: 2,6,14,26,42,62,86,114,146,182... (note the palindromic effect on final numerals within each term, which I just noticed myself as I write this!). This sequence is defined by 4tri+2 (or 2(2tri+1)), where tri is a triangular number. ...

...

In any case this is where my work stands at the moment. I'll probably post actual numbers to show how this all works; perhaps some of you might have further insights into what is going on.

...

Jess Tauber

goldenratio @ earthlink . net

 

substituting for tri in your expression, the expression for triangular numbers in the variable n, where n is the ordinal of triangular numbers.

 

triangular numbers: [n(n+1)]/2 or (n^2+n)/2

your expression: 4tri+2 (or 2(2tri+1))

substituting: 4[(n^2+n)/2]+2

simplifying: 2n^2+2n+2 (or 2(n^2+n+1)

 

checking...

n=orbital

0 2

1 6

2 14

3 26

...

:thumbs_up

 

technically there is no 0^th triangular number and the 1^st & 2^nd triangular numbers are trivially polygonal. nevertheless, the expression 2(n^2+n+1) generates your list of orbital differences and works the squares into the mix.

 

that's all i see. love the palindrome. turkey was good. :thumbs_up

[tetrahedron]

After dozens of unanswered emails to professional nuclear physicists I decided to just phone one and see what he had to say about issues pertaining to the Nilsson diagrams I use to determine the mathematical patterns I keep finding.

 

It turns out that though I need diagrams extended for deformation on both sides of the x axis such diagrams do not exist. In fact the diagrams that do exist currently utilize values that may have several different causes distorting the spin-orbit paired nucleon component rays near to the center with 0 deformation (1:1 axial ratio).

 

In the simple Nilsson diagram without spin-orbit component in the equation (straight rays) everything comes together under the Pascal math umbrella- there are NO exceptions. What I really need is an equivalent for the model that includes the spin-orbit component, but without the further distortions of ray paths brought in by other factors, in order to see just how well the Pascal math survives the transformation.

 

Already I've found that patterns in the former are shifted relative to those in the latter. For the simple straight-ray model full-shell values are all related to single additions or subtractions of double triangular intervals. When the orbitals end up split due to spin-orbit effects in the more realistic model with curved rays, then intervals now pertain not to full shells anymore but to subshells. And the intervals are composed not of single additions or subtractions of double triangular numbers, but binary combinations of them (for example one set of differences are half/double square numbers, and as we know sums of nearest-neighbor triangulars are squares, so doubled they become half/double squares).

 

It is all very weird, yet for the most part mathematically consistent. In some places there are exceptions, but these appear to be rule-based as well, as if there were two competing patterns, one being the major theme, the other the minor.

 

Another issue I found for online versions of the Nilsson diagrams is that they each have different scaling. What the heck is THAT for??? Same x, different y scaling, or the opposite, with different sheets from the same system. As if someone didn't want to make things easy for laypeople. So, since I don't have a scanner or access to programs that can both rescale from graphic data and then extrapolate off the edges I have to do this all by hand. Have to get myself a French curve too. Grrrr Arrrgh!

 

Jess Tauber

[Turtle]

...

Already I've found that patterns in the former are shifted relative to those in the latter. For the simple straight-ray model full-shell values are all related to single additions or subtractions of double triangular intervals. When the orbitals end up split due to spin-orbit effects in the more realistic model with curved rays, then intervals now pertain not to full shells anymore but to subshells. And the intervals are composed not of single additions or subtractions of double triangular numbers, but binary combinations of them (for example one set of differences are half/double square numbers, and as we know sums of nearest-neighbor triangulars are squares, so doubled they become half/double squares).

 

looking at "sums of nearest-neighbor triangulars are squares" now, i'm not sure i did know that. mmmm...well...i guess i know it now. :thumbs_up thanks in any regard! :smart:

 

so if by "double triangular" you mean simply 2 x triangular (two times a triangular), i can give some perspective. (if not, disregard this bit.)

the set of numbers 2 x triangular are called oblong numbers, and they are the sums of the even natural numbers. (aka pronic numbers, rectangular numbers, and heteromecic numbers)

Pronic Number @ Wikipedia

 

i have attached my own graphic representation using gnomons below. parallelagramic?

 

It is all very weird, yet for the most part mathematically consistent. In some places there are exceptions, but these appear to be rule-based as well, as if there were two competing patterns, one being the major theme, the other the minor.

 

Another issue I found for online versions of the Nilsson diagrams is that they each have different scaling. What the heck is THAT for??? Same x, different y scaling, or the opposite, with different sheets from the same system. As if someone didn't want to make things easy for laypeople. So, since I don't have a scanner or access to programs that can both rescale from graphic data and then extrapolate off the edges I have to do this all by hand. Have to get myself a French curve too. Grrrr Arrrgh!

 

Jess Tauber

 

there is no royal road to geometry. ~ euclid

Edited December 16, 2012 by Turtle

[tetrahedron]

Thanks much for this- it may turn out to be quite useful!

 

Jess Tauber

[Turtle]

Thanks much for this- it may turn out to be quite useful!

 

Jess Tauber

 

you're welcome. :)

 

armed with your consecutive-triangulars-summed-***-square factoid, i constructed a gnomonic demonstration -if not proof- of it. can you say rhombigramic?

Edited December 16, 2012 by Turtle

[tetrahedron]

One of my tetrahedral periodic table (electronic) mappings involves skewed rhombi of this type, that is, bent up along the minor axis to a tetrahedral dihedral angle. I had not thought of these as containing summed triangular numbers per se (though I did recognize that they contained triangular faces of next nearest size). Bonk on the head- Bonk! Bonk! (with apologies to Miri). And of course rhombi of close packed spheres contain square numbers of spheres. It is as if one had a mapping (square) from the diagonals of the (2,1) sided Pascal Triangle forced to live under the rules of the classical (1,1) sided version, where triangular numbers dominate the 2-dimensioned diagonal. Wonder what would happen if you took pentagons from the (3,1) sided Pascal Triangle 2D diagonal and forced to use square motifs...hmmm.

 

Anyway in the nucleus I found double/half square number differences between subshell counts that had the same differences of quantum numbers n and l (that is the absolute number line positions of the last neutrons in blocks of neutrons separated by substantial gaps in the number line real-estate). When you just visually examine these block positions on the number line you don't see any regularity- too confusing. Only when you actually take differences between positions do the regularities start to become clearer, and even then most of the 2Tri+/-2Tri differences weren't so obvious. Needed a table of them to be sure.

 

Jess Tauber

[Turtle]

One of my tetrahedral periodic table (electronic) mappings ... snip

 

When you just visually examine these block positions on the number line you don't see any regularity- too confusing. Only when you actually take differences between positions do the regularities start to become clearer, and even then most of the 2Tri+/-2Tri differences weren't so obvious. Needed a table of them to be sure.

 

Jess Tauber

 

hi pascal! well nows, as we were talking mappings elsewhere, your number-line-too-confusing can be brought to a focus with a spiral array. it is just this goal that led me to [re]invent the square *** Ulam spiral. trying to map unruly non-polygonal numbers on the line don't you know. so then it seemed to me to take the next logical steps and make [equilateral]triangle arrays and hexagon arrays. [a plethora of other spiral arrays exist as well.]

 

well, long story short, i mapped the diagonal {1 4 10 20 35 56 84 120 165 220 286 364 455 560 680 816 969 ...} from Pascal's triangle onto my heaxagon spiral array for you. see attached. wolfram alpha computational engine gives the equation a_n =1/6(n³+6n²+11n+6)

 

while this pattern looks visually confusing to me, the hexagon array has some very specific mathematical qualities that put the confusion in a manageable context that the number line cannot. plots of triangular numbers are over in the non-fig thread. let me know if you want other Pascalonal trianglenal diagonal plots.

 

that's all. :coffee_n_pc:

 

PS added the mapping on my triangle spiral array. (notable on the triangle array in this orientation; all left-pointing triangles are odd, all right-pointing are even. ) will make a plot on the square array when i get remastered.

 

PPS :lol: found a short sloppy bass ackwards square master of days gone mold. only goes to a little ove 600, but it's better than a poke in the eye with a sharp stick.

Edited March 6, 2013 by Turtle

[tetrahedron]

Thanks! I've made some headway since posting here last, simplifying the numerical dependencies in the models of 'realistic' nuclear counts.

 

My primary focus has been for neutrons only in spherical nuclei. The simple harmonic oscillator plus ellipsoidal deformation model yields straight component rays in a regular angular array where it is easy to analyze the mappings of magic numbers and the way they pattern based on the Pascal Triangle trends.

 

However real nuclei also suffer from additional factors (such as spin-orbit coupling and deformed potential) that distort the pattern to produce curved rays, some going up, some going down, and depressed energies differentially expressed depending on shell and subshell (in other words quantum number effects).

 

Two major patterns relating to the Pascal math emerged for the realistic neutron in sphere mapping (Nilsson diagram). Both depend on the final nucleon in that part of the subshell and its numerical value in the count.

 

Because of spin-orbit coupling subshells are no longer unitary, and they get divided in twain plus a single nucleon exchange. So s orbital's 2 nucleons get split into 1:1 and then one goes to the other side, yielding 2:0. p orbital's 6 nucleons becomes 4:2, d's 10 becomes 6:4 and so on. And while in the simpler straight ray model without spin-orbit coupling these cut subsections remain side by side, in the realistic model space develops between many of them filled by other suborbital parts. A crazy quilt.

 

It was within this seemingly opaque-to-numerical-analysis sequence that I started to find the patterning I've posted about earlier.

 

One pattern involves the subshell subpart terminal counts between like-labeled components differing only by quantum number n. So for example 1s,2s,3s, 1p,2p,3p. This is the one that has intervals that relate to summed double triangular number pairs where the chosen numbers are every other double triangular number (including 26,42,62,86,114, etc.). The sequence of differences that apply is shifted upstream as one climbs total count values, because it has to, as new orbital types are introduced with each shell and distances grow. That it is as regular as it is is a total surprise, despite many exceptions.

 

The other pattern involves intervals between terminals of the same quantum number n, but different l, but of the same parity. Parity here means positive for even l, negative for odd l, and relates to the upward vs. downward curviness of the mappings of rays in the Nilsson diagram.

Previously I'd found that for n=1 only that differences between the larger spin-split part of the lower l value and the smaller part of the next higher l value had double/half square intervals between terminal counts (so for example between 1p3/2 and 1f5/2 (where the fraction's numerator is one less than the total nucleons in that group, so 3/2>4 count and 5/2>6 count). Each shift upward in the sequence yielded the next double/half square, until distortions very high up in the system rendered the trend ineffective. Again, though, that it works as long as it did is as surprising as my much earlier discovery that Fibonacci and Lucas numbers had electronic system mappings much further than one would expect.

 

In the past week I've also found that there is a pattern for n=2, but shifted from double/half square differences by 6. I don't yet have enough information about higher n values as a series because they simply don't exist in Nature, but could be modeled mathematically (something that would require more computation than I'm used to). It may be that there is an idealized trend between successive n's here.

 

For an idealized system just the two patterns may be sufficient to predict the positions in the string of all neutrons in spherical nuclei. I say idealized because in reality nuclei don't actually grow this way, along the set of numbers in an undeformed sphere. In fact, as they grow nucleon by nucleon they do become ellipsoidally deformed, careening from oblate to prolate to sphere and so forth in a regular fashion, because each addition changes the system of magic numbers that are most stable, and the internal shell constituency changes as well. Nucleons have no inherent loyalty to one shell or subshell- they can change these. And its a dynamic thing as well- even the shape of the nucleus wobbles, there are Coriolis forces, and such.

 

So what I'm planning on doing next is tracking the deformation/magic number/terminal count trajectories of growing nuclei (which is complexified by the fact you have both protons and neutrons together and their (sub)shell configurations can affect each other!) to see if it gives a more perfect Pascal system than within any one fixed deformation. This will only take, say, 50 years or so....

 

Jess Tauber

[email protected]

[tetrahedron]

Does anyone know of natural (or theoretical) instances of dimensional projection/reduction mathematics? I'm working on a hunch that this might be what is going on in atomic nuclei.

 

Here's why- The harmonic oscillator magic numbers are all double tetrahedral numbers, for spherical nuclei. Then for biaxially deformed ellipsoidal nuclei double Pascal Triangle numbers more generally (naturals, triangulars) come together to create magics in such a way that the oscillator frequency ratio (which maps in a regular way to the deformation) determines the magic sequence.

 

It turns out that more realistic magic numbers on the sphere, where spin orbit effects come in, are already present on the oblate deformed side of the Nilsson mapping (deformation on x-axis, energy on y-axis), distributed in a regular fashion along the oscillator frequencies (that is, -1/0, 0/1, 1/2, 2/3, 3/4, 4/5 and so forth, getting ever closer to the zero deformation of the sphere while also getting larger and larger).

 

But nuclei can also have triaxial (and higher) level symmetrical deformations. If all or some of these can also project their particular magic sequence preferences simultaneously onto the sphere, then perhaps the problem of the weird sequences of actual nuclei might be solved without all the fudge factors, from first principles.

 

Jess Tauber

[email protected]

[tetrahedron]

And yet more Pascal madness ensues. I've been looking for a way to extend my triangular numbering system to higher N-axial ellipsoids, since real nuclei can assume more states than just rotationally symmetrical biaxial ones.

 

Anyway, what I found (which is probably common knowledge to the likes of yourselves) is that for binary combinations of axes producing oscillatory ratios each successive N produces a DOUBLE TRIANGULAR NUMBER of such binaries.

 

This is fairly interesting. At least for the biaxials, in the harmonic oscillator model, these oscillator ratios completely determine the way Pascal numbers associate to produce stable 'magic' full shell numbers. Now it seems even how many different 2-way combinations we have is determined, again through Pascal numbers.

 

So a HIERARCHY of Pascal math. Add this to the fact that the doubled Pascal numbering system appears to also work at three atomic levels: the nucleus, the electronic cloud, and atomic clusters. Another hierarchy. And in each case the particular featural primitives are differently ranked.

 

Thus a question arises as to any possible systematicity to all these levels in different fictive or real dimensions.

 

And the crazy quilt of 'realistic' numerical values (as opposed to the idealized harmonic oscillator, etc.) may be due to looking at the whole system from a particular vantage point (or points), so we see contributions from other levels off kilter. It would all make sense from a multidimensional God's Eye vantage, beyond the capacity of us mere lowly mortals. Yet math can tell us what we need to know, if we have access to enough pieces of the puzzle, despite our disadvantage.

 

Jess Tauber