[seqfan] The family atomics...
Jess Tauber
goldenratio at earthlink.net
Sun May 12 07:05:50 CEST 2013
Hello folks. I just joined and wanted to give you a short(ish) intro to my work on sequences in atomic systems.
I took my first brief steps into this area back in the Disco era, when I discovered that the periodic table could be recast as a rough tetrahedron (completing and centering each orbital block over the next by dimensions). Other things intervened (to say the least) and in 2009 I found that someone had independently rediscovered my tetrahedral relationship (and even improved on my version: see www.perfectperiodictable.com).
Over the next years, with help from others, I began to see that there were very interesting numerical relationships hidden within the structure of the electronic system of atoms, and also began to extend my analysis (essentially method of common differences, would you believe!) to nuclear systems. Just recently I also found that the method works for atomic clusters.
What pops out are generalized Pascal Triangle sequences, every which way. It had already been known that Pascal rows/horizontals governed NMR peak numbers and their intensities. Many had also independently discovered that the magic numbers of nucleons for the simple harmonic oscillator were doubled tetrahedral numbers, for spherical nuclei (the most stable kind). I extended this last year to cover all biaxial ellipsoidal nuclei, which just redistribute doubled Pascal numbers by reference to the oscillator ratio. The polar dimension (generally the numerator) tells you how many copies of a doubled triangular number go into the running sum that generates magics, and the equatorial dimension (generally the denominator) tells you how many magics intervene between differences which are double triangular. Exceptions at the beginnings of magic sequences, when the minimum number of magics has yet to cumulate, can be handled by a table of doubled natural numbers cut into chunks according to the oscillator ratio equatorial number.
More realistic models of nuclei include other factors which cut the Pascal numbers even further and into amazingly complex patterns over different ellipsoidal deformations. I've only been able so far to succeed partially in characterizing these systems for the sphere- and here we find many other sequences, some of which are also in the OEIS (either as such or halved). Differences between magics or submagics are based on particular mappings between quantum numbers, and only now am I able to label them for what they are mathematically.
There are many such patterns at all three levels of atomic relations, some quite unexpected. For example I appear to be the first to notice, in the electronic system, that Fibonacci and Lucas numbers pattern nonrandomly and nonarbitrarily as atomic numbers. Fib numbers up to 89 ALL map to leftmost/first positions in orbital half-rows, as first singlet or doublet electron occurrences, with odd Fibs singlets and even Fibs doublets. A mirror pattern occurs for Lucas numbers, which as atomic numbers map, up to 18, to rightmost/last positions in orbital half-rows, with the same odd/even bias. 29 and 47 act 'as if' they were correctly positioned though they are one move left of where they *should* be, via anomalous electronic ground state configurations, and 76 acts like a noble gas, though it is not, nobles also having filled half-orbitals. So there are behavioral 'fixes'. Very strange stuff, but it establishes that we also need to consider Pascal shallow diagonal values.
And generalized Fib numbers first few digits actually contain, directly or indirectly, information on all four quantum numbers. Three start-offset variants of Fib itself code s,p,d, and Luc codes f blocks, with each further generalized Fib coding the next orbital type.
How far all this goes will take a while to analyze. People are also working on Golden Mean-based structures in DNA, in protein- some speculate that brainwaves work this way, and there is also evidence that languages also structure along these lines.
I say lines because we appear not be limited to just Phi- the other Metallic Means seem to be implicated in neutron/proton numbers in whole nuclei (if one includes 1.000... as the 'first', which is controversial).
Anyway, these are the things that make me very interested in sequences in general, and I've used the OEIS before, and am feeling it will become a much bigger part of my research life in the years to follow. Thanks for your patience.
Jess Tauber
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