Pascal Triangle diagonal numbers

[yahganlang]

Hi folks. Since people have already discussed tetrahedral and triangular numbers from the Pascal Triangle in nuclear physics, in peer-reviewed journals, this isn't a new idea.

In the electronic system one can also find these numbers instantiated. In the left-step periodic table by Charles Janet (1928-9), shifted periods end with alkaline Earth's, on the right, and all orbital blocks defined by quantum number l decrease widthwise and increase heightwise as one moves rightwards. This table is based purely on quantum number combinations and ignores other factors such as relativistic effects, and related phenomena such as spin-orbit coupling, Aufbau shell-filling anomalies, etc. All periods are dual (i.e. same length occurring twice, while the traditional kluge table fails here). Because electronic magic numbers are half/double squares, dual counts are squares (of even integers).

It turns out that every other alkaline Earth atomic number (4,20,56,120...) is identical to every other tetrahedral number from the Pascal Triangle. Intermediate alkaline Earth atomic numbers are means of tetrahedral number pairs, and the basis of the 'triads' that led to the discovery of the periodic relation in the first place. Tetrahedral numbers are (0), 1, 4, 10, 20, 35, 56, 84, 120... Intermediate tetrahedral numbers (1,10,35,84...) differ from the intermediate alkaline Earth atomic numbers (2,12,38,88...) by intervals that increase as integers/natural numbers (1,2,3,4...).

It also is found that, counting leftwards from the alkaline Earth's using intervals equal to the triangular numbers from the Pascal Triangle one lands, without exception, on elements whose quantum number ml=0. Triangular numbers are (0), 1, 3, 6, 10, 15, 21, 28, 35..., and intervals between them increase as the integers/natural numbers. Tetrahedral numbers are the running sums of triangular numbers.

Perhaps relatable is the fact that pairs of contiguous triangular numbers are squares. The relevant tetrahedral numbers are the running sums of squares of even integers. For ex. 56 = 4+16+36.

In the nucleus it seems further that doubled triangular number intervals can be found in several different reading frames between particular positions, shifted up or down or changed in size, or both, not just to those involving the classical magic and semimagic numbers.

This is very interesting, and possibly relates to the splitting of the nucleus into core, outer core, mantle, and free spherons as in Pauling's work.

Thoughts?

Yahganlang

 

[alxm]

This is very interesting

I disagree. I think it's trivial and uninteresting. You basically found a pattern, and one which was already known, namely the periodic table.
There's no new information or any kind of physical insight. As it were, the physical causes of that pattern have already been given a rigorous physical http://pubs.acs.org/doi/abs/10.1021/ed056p714" anyway.

and possibly relates to the splitting of the nucleus into core, outer core, mantle, and free spherons as in Pauling's work.

Why? Because both involve magic number patterns? Pauling didn't event nuclear 'magic numbers', and his ideas on nuclear physics were never taken seriously.
He also had wrong ideas about vitamin megadoses (which he continued to pursue even after his own results showed no effect), about anesthetics, and all sorts of stuff.
Basically all his science work outside physical/quantum/structural chemistry was a load of bunk, so I don't know why anyone would pay attention to it. (Nobody really is, just look at the citations for his nuclear physics work)

As for the idea of a relationship between nuclear and electronic shell filling, that's a pretty straightforward idea that has already been thought of many times before.
And it just doesn't work because the physical system of nucleons is so different from the electrons. To quote Cook's Models of the Atomic Nucleus:

It bears repeating that, in contrast to the closure of electron shells, no single criterion of magic stability produces the textbook magic numbers,
and indeed no unique set of criteria produces only the magic numbers and not other numbers

[Emphasis in the original]

 

[yahganlang]

How about these patterns, based on idealized block patterns (all even quantum number l + following all odd quantum number l) and their leftmost, rightmost margins:

Intervals between nucleon numbers (again, in the idealized system outlined above):

Rightmost-Rightmost

330-240= 90
249-168= 72
168-112= 56
112- 70= 42
70- 40= 30
40- 20= 20
20- 8= 12
8- 2= 6
2- 0= 2
0- 0= 0

These differences are twice the triangular numbers from Pascal's Triangle: 45,36,28,21,15,10,6,3,1,(0)

Leftmost-Rightmost

258-168= 90
184-112= 72
126- 70= 56
82- 40= 42
50- 20= 30
28- 8= 20
14- 2= 12
6- 0= 6
2- 0= 2
0- 0= 0

Rightmost-Leftmost

330-258= 72
240-184= 56
168-126= 42
112- 82= 30
70- 50= 20
40- 28= 12
20- 14= 6
8- 6= 2
2- 2= 0

The anomaly here is the Leftmost-Leftmost. In this case we have instead double triangular numbers + 2. Three of four identical intervals, one differing by 2.

This seems to go way beyond standard depictions of the nuclear shell relationships. Or you've seen this already, hmm?

Yahganlang

 

[cgk]

Numerology is not a generally accepted way of getting ahead in science.

Note that for any pattern of numbers you can always construct some algorithm for coming up with these numbers. And even if you found a simple algorithm of this sort, you'd still have learned nothing at all about the physics behind them.

In short: There is no point in engaging in this kind of activity.

 

[yahganlang]

You're absolutely right- numerology has no place in science- in fact, numbers are useless, as are lawful patterns of them. They tell you nothing. Opaque equations are better. The periodic relation emerged full blown from the head of Zeus, and the triad relations used for organizing elements in the first half of the 19th century were meaningless diversions, as is any search for regularities.

Indeed the now traditional depiction of the periodic table looks pleasingly like a city-scape, with nice gaps and misplacements, much better than the regular alteration in block dimensions as in the left-step table.

The Pascal Triangle? Pshaw! A child's toy, beneath consideration by professionals. What WAS I thinking!?

Yahganlang

 

[Vanadium 50]

Which sounds like the end of this thread.