User:Jess Tauber

Since 2009 I've been looking into number-theoretical underpinnings of particle counts in atomic systems, particularly shell and subshell closure (i.e. 'magic') numbers, at nuclear and electronic levels and since 2013, atomic clusters. Almost all my findings point to simple numerical combinatorics as defined by generalized Pascal Triangles. Major discoveries thus far include:

The simple harmonic oscillator (HO) models show the cleanest Pascal relationships, but even adding in other terms, as for example spin-orbit, appear merely to transform the system, as if it were some kind of matrix. This would imply that each added term dovetails to the others in a coherent fashion.

The nuclear HO, for spheres, shows pure doubled tetrahedral number magics (many have independently rediscovered this fact yet no professionals ever note it). I extended the model to biaxially deformed nuclei and showed that these just redistribute Pascal numbers according to the numerator (related to the polar semimajor axis) and denominator (related to the equatorial semimajor axis) values of the oscillator ratio. The numerator defines how many copies of a doubled triangular number go into the growing string of magic numbers, and the denominator says how many magic numbers intervene before there is a difference that is double triangular. Exceptions occur at magic sequence starts when you haven't yet cumulated the minimum number of magics to see the correct difference, and can be handled by a table of doubled natural numbers in sequences divided by the numerator.

In more realistic models of the nucleus, including spin-orbit term, organization of subshell filling is altered. The simple HO picture becomes more complex, and sequences of Pascal-related numbers are reorganized. For example in the N=1 shell of real neutrons in spherical nuclei, we find half/double square differences between the larger spin-orbit split orbital component of any L value and the smaller spin-orbit split component of L+2 value, such trend lasting up til high L values without break. In the simpler HO model for spherical nuclei such half/double square differences occur when the difference between N is 1 and all other quantum values is the same. In real nuclei, for neutrons, this latter relationship has changed so that differences now become summed pairs of every other doubled triangular number [A051890].

In the electronic system the 'physicist's' version of the periodic table is the Left-Step system of Janet, where all periods end in s2 configurations (helium and the alkaline earths). Every other alkaline earth atomic number is identical to every other tetrahedral number, with intermediates being the arithmetic means of these. This is why early chemical workers found 'triads' of like-acting elements, which led ultimately to the discovery of the periodic table in the 1860's. Counting back from the period terminals, within the periods only, using triangular numbers one always lands on positions within orbitals that have quantum number ML=0, that is the midpoints of half-orbital rows. Curiously, in another Pascal behavior, if one takes Fibonacci numbers as atomic numbers, then up to 89 (the last Fib within known elements) they ALL land on leftmost positions within orbital half-rows, where one encounters either the first singlet or first doublet electron (all orbitals fill singly before doubling up). And the singlet Fibs are all odd, while the doublet Fibs are all even. At 144 the series will be mispositioned orbitally. Related Lucas numbers map, up to 18, to rightmost positions within orbital half-rows, where we encounter the last singlet or doublet electron, with the same odd or even bias. Interestingly after 18 the periodic system has 'fixes' for the mispositionings in that for 29 (Cu) and 47 (Ag) (in the same column/group) the electronic configuration is corrected in a way that fits the Lucas trend, and 76 (Os) behaves as if a noble gas (Xe) which has the correct position.

There is much more than this, and will probably consume my free time for the remainder of my days (*sigh*). I'm utilizing and contributing to the OEIS because I want to understand better the Pascal-related sequences I keep finding and find out if the way they are distributed differentially in atomic system particle counts is itself structured and motivated mathematically. This would hopefully lead to a new theory of atomic organization that can deal with known facts from first principles (something that today is admittedly sadly lacking). Or to put it more bluntly, I HATE FUDGE FACTORS!