[Vo]:Introduction and shell structure patterning

[Jess Tauber]

Hi folks. I just signed up after discovering the Vortex list online. I've been studying the mathematical motivations of shell structure in atomic systems- for nuclear, electronic, and even atomic cluster shells. The first thing I realized, around 8 years ago, was that all the spherical magic numbers of nuclei in a simple harmonic oscillator model were  exactly (and only) doubled tetrahedral numbers, right out of a term-doubled Pascal Triangle, thus 2,8,20,40,70,112,168,240.. are twice 1,4,10,20,35,56,84,120... The doubling effect appears to come from pairing of spin-opposed nucleons, so ultimately if one is counting such pairs, then the system is purely tetrahedral under the harmonic oscillator. The intervals between these spherical harmonic oscillator magics are of course doubled triangular numbers, which correspond exactly to the sizes of the harmonic oscillator period analogues: 1s=2 1p=6 1d2s=12 1f2p=20 1g2d3s=30 1h2f3p=42 1i2g3d4s=56 1j2h3f4p=72 According to one mathematical physicist I contacted, the harmonic oscillator always delivers numbers of stable states which get their values from the Pascal Triangle's diagonals. WHICH diagonal depends on the dimensionality of the system you're examining. Each new dimension pushes you to the next larger diagonal. Note that the ordering of the orbitals here in the harmonic oscillator are all 'left-step', that is, starting on the left side with the highest spin orbitals and then working towards the right to the lowest. Also the orbitals are sorted for parity, either all positive (with even quantum number l values) or all negative (all odd). This is different from the electronic periodic system, where the orbitals are not sorted for parity, but alternate thereby. New magic numbers are generated when nuclei are deformed from a sphere to an ellipsoid of revolution, either prolate or oblate. Several deformation parameters have been used in plots of energy level versus deformation, including the delta-, epsilon- and beta- systems. Each has its own merits and drawbacks. However I found that just using the so-called 'oscillator ratio' delivered the best results for the harmonic oscillator. The oscillator ratio describes the polar versus equatorial extent of the matter wave, as numerator and denominator usually (though some invert these). With the default ellipsoid, the sphere, there is ONE doubled triangular number between each doubled tetrahedral magic number, and each doubled triangular number is used just ONCE to generate a magic number before moving on to the next. However, with an oscillator ratio of 2:1 (prolate), each doubled triangular number is utilized TWICE to produce a new magic number, thus the sequence 2,2,6,6,12,12,20,20,30,30....yielding harmonic oscillator magics 2,4,10,16,28,40,60,80,110,140...., which match published sources. When the oscillator ratio is 3:1, then each doubled triangular number interval is use THRICE, and so on. There are no apparent exceptions in the harmonic oscillator model. The NUMERATOR of the oscillator ratio (polar extent of the matter wave) MULTIPLIES the use of each doubled triangular number interval. When we have an oblate nucleus of oscillator ratio 1:2, then each doubled triangular number interval is used once, but there is a doubled triangular number interval between EVERY OTHER/EVERY SECOND magic number rather than between each. With ratio 1:3 there is a doubled triangular number interval between EVERY THIRD magic, and so on. In other words, we are DIVIDING the use of doubled triangular number intervals according to the DENOMINATOR (equatorial extent of the matter wave). There are exceptions to this particular rule, for mathematical reasons. When we have not yet accumulated the denominator's worth of magics the rule doesn't YET apply. For these earlier magics it turns out they are all simply double triangular numbers up to the denominator of the oscillator ratio, and then the rule 'turns on' With the above all in mind there are no exceptions to deformed magic numbers for the harmonic oscillator. The more realistic shell model of the nucleus including deformed potential wells and spin-orbit coupling adds more complication to the picture. I still haven't worked out how everything works for deformed nuclei, but for spheres the Pascal Triangle motivation remains, but is transformed. The spin-orbit magics are 2,*6,14,28,50,82,126,184.....(6 is highly controversial). They result from additions to lower harmonic oscillator-sized shells by the highest spin orbital partial from the next higher shell, of opposite parity. The spin-orbit effect splits orbital in two, a larger part and a smaller higher in energy than the large (different by 2 nucleons). I discovered that the additions (from the so-called 'intruder levels') are NOT arbitrarily sized, but coordinate size-wise so that the doubled triangular number size of period analogues/shells is preserved. Thus 1g9/2 (10 nucleons) adds to 1f2p (20) to give 30, which is the next higher doubled triangular number after 20. Similarly 1h11/2 (12) adds to 1g2d3s (30) to give 42, and so on. Earlier intruders don't actually 'intrude' according to sources I've looked at- instead they simply ride atop the previous shell in terms of energy levels. The 'home' period analogues from which the intruder levels come get decreased in size to the next lower doubled triangular number- so 1g2d3s (30) subtracts 1g9/2 (10) giving 20, etc. It also turned out that the 'depths' of intrusion/penetration into the previous shell by intruder levels was itself doubled triangular in value. Thus 1g9/2 dips 2 moves down, placing itself before the last 2 2p nucleons. 1h11/2 dips down 6 move. 1i13/2 dips down 12 moves, and so on. This works without apparent exception for neutrons, but for protons an expected placement of 12 moves (so 114 from 126) instead gives 20 (106 from 126).  I'm not sure of this reference. In any case such jumps seem to be allowed by the math, from the shell structure itself. Whether there are other alternative intrusion positions is unclear. The general rule of intruder placement is striking, and interesting, but unexplained. Except for the 106 placement for protons, EVERY INTRUDER positions itself AFTER THE THIRD ORBITAL PARTIAL OF THE PREVIOUS SHELL. Thus until we actually have three such orbital partials intruders may not be possible. As mentioned in passing above, I haven't really been able to derive a simple mathematically-based rule for deformed nuclei under the spin-orbit model, but this may be due to poor graphical resources on the one hand (which don't seem to extend far enough in deformation in either the prolate or oblate directions) and the correction terms of the nuclear Hamiltonian itself (which are ad hoc to fit data rather than from first principles, something well known to professionals in the field). I have found that some of the same things that are found in the harmonic oscillator model get preserved in 'mutated' form in the spin-orbit model. For example in the latter each energy level is represented by a straight line, or 'component ray' as it is termed in some sources. All these rays converge at regular fixed points at particular oscillator ratios. At +1.5 and -3.0 deformation in the delta deformation system we have infinited deformation (prolate and oblate respectively), and every convergence point/node has an infinite number of component ray contributions.  Also the slopes of these component rays work in a highly regular way, with bundles of these converging rays all having slopes that are 1/3 delta/h-bar, omega-bar differences from their nearest neighbors. It looks like the spin-orbit plots (Nilsson diagrams) also show component ray convergences but they are sporadic and the infinite deformations are 'off-stage' in published plots. The highest-spin orbital partials of the larger part of split orbitals are nearly linear (most others are not and some exhibit rather complex curves) and seem to converge at around -2.7 epsilon and 1.5 h-bar, omega-bar. The latter energy value is the same as what we see with the harmonic oscillator plot, though the deformation values are in different scales (epsilon for the spin-orbit, delta for the harmonic oscillator). It would be very useful to have all these plots in the same system, ideally the oscillator ratio. But you take what you can get. The electronic system is also based on Pascal Triangle combinatorics. If one organizes the electronic periods with the s-block elements on the left we get the standard periodic table, but if we put them on the RIGHT, we reproduce the 'left-step' periodic table first devised by an elderly French polymath named Charles Janet from the late 1920's. It organizes periods such that the higher quantum number l is leftwards, and the lower rightwards. With this table periods all end in s2 electronic configurational elements (helium and the alkaline earths). This ignores the chemical behavior but does respect the actual PHYSICAL organization of periods. In this system each such 'Janet' period pairs for size: 1s/2s:  2p3s/3p4s:  3d4p5s/4d5p6s:   4f5d6p7s/5f6d7p8s. Interestingly every other period here ends in atomic numbers which are every other tetrahedral number: 4, 20, 56, *120, and the intermediate period ending numbers are the arithmetic means of these (period duals). Remember that each period is in length both a half and a doubled square number 2,8,18,32,50... (half 4,16,36,64,100, doubled 1,4,9,16,25...). Intervals between tetrahedral numbers are triangular numbers, and sums of pairs of nearest neighbor triangulars are all squares: 1+3=4, 3+6=9, 6+10=16, 10+15=25..... The facts above allow one to create three-dimensional periodic tables (solids) which are tetrahedral in shape, using close packed spheres each representing one element. There are several good arrangements each of which has good symmetry on the one hand or preserves the continuity of Mendeleev's Line (the linear ordering of elements). Such mathematically motivated shell structure based on Pascal Triangle combinatorics may also be seen with atomic clusters which are hemispherical in shape and bound to a substrate on their 'flat' side, but I won't bore you here with the details. Anyway, the above is the introduction to my recent work. I'm very interested in table-top nuclear reactor potential, and particularly in the spin-orbit magic number 6, which would indicate carbon. Carbon is heavily utilized in stellar nucleosynthesis in stars more massive than our sun, in the CNO cycle (absorbing hydrogens along the way, C transforms to N which transforms to O and then finally regenerating C catalytically, spewing out He and lots of energy). Several organizations appear to be very interested in exploiting this cycle, which unfortunately on the face of it requires much more robust temperature and pressure conditions to achieve fusion. With the special position of carbon in the shell system it MAY be possible somehow to work around this caveat, and get systems to work at near-terrestrial conditions. I'm hoping some of you might have some thoughts on this matter. Thanks for your attention. Jess Tauber goldenra...@earthlink.net