pascalperiod

With the nuclear Pascal relations one can simply extend them to higher shells and build them by rule. I works pretty well. If I can get similar mileage out of the system for deformed nuclei I think most would admit this has some value. But dealing with energies rather than simple particle counts and their groupings has been difficult. While the harmonic oscillator Nilsson gives equidistant energy values at any deformation things aren't so nicely arranged in the spin-orbit and deformed potential regime. I've started playing with the Pascal Triangle's evil twin, the Leibniz Harmonic Triangle. Here instead of summing side to side and downwards, starting with ones, this Triangle utilizes fractional values as diagonal terms which sum upwards. I already have hints that something like this works for the harmonic oscillator model, now I just have to figure out if it will do for the spin orbit set. Also though I haven't dealt with it at all here, there is clear evidence that Pascal math is also behind at least some magic numbers found in atomic clusters, which are now all the rage. The ordering of rules is different than for the nucleus, but so what- as long as the system is coherent and rule-based? And by the way, physicists still have no real explanation as to why the quantum mechanical dynamics of the electron cloud differ from that of nucleons. In the nucleus, intruders are leftmost, highest-spin orbital partials which spin-orbit coupling demotes energywise enough to add to the previous period analogue. In the electronic system (at least until spin-orbit coupling really kicks in, the well-known inert pair effect) we see rightmost, lowest-spin orbital partials (the 2 s electrons) promoted up to the NEXT higher period, where they take part in bonding. This is a mirror-image effect: high-spin, left-edge, dropping vs. low-spin, right-edge, raising. I don't think these are unrelated to each other- do you? More mere coincidence? What if it turns out that the 'penetrance' of s-electrons into the next period also works via Pascal numeration? Nobody has checked yet (not even myself). And I'm pretty sure its too early yet to tell if the math has no scientific value. The reason I'm pursuing it is to see whether it might be able to help in the creation of Low Energy Nuclear Reaction (LENR) engines, assuming people researching this aren't just incompetent loons or cranks. I've already found, in mass plots of claimed production of reaction product nuclides, Lucas number and other Pascal-related patterns. It would be nice to have, beyond wishful thinking, new rich energy sources that don't produce dangerous radioactive byproducts. Jess Tauber One last thing- for all the numerical patterns I've managed to glean thus far I've used Nilsson diagrams that are termed 'representative'- they are by no means the last word, as particular details may change from nucleus to nucleus, even within core shell structure. Most nuclei are deformed, not spherical, and all the values I've taken differences from are for spherical nuclei. What this means is that in many cases the internal sequences between spherical magics is actually EXTRAPOLATED from nonspherical nuclei. So if there are any inconsistencies remember I can only use what I've got. I think I've done pretty well so far, with the wobbliness of theoretical models. Even professional nuclear physicists, when pressed, will admit that their models are approximations at best, and sometimes don't fit experimental data all that well. All that said, there is the issue of entrenched attitudes in any science- people will defend to the death the received wisdom they are comfortable with and have made a part of their own paper trail. I've gotten this from many people- even some I had previously respected. Not everyone is prepared for surprises, and we don't always have a handle on their private motivations for negativity. This goes especially for such 'icons of science' as the periodic system. Too much of a sacred cow to mess with except to add a new element here or there, of find a new magic number. I've presented much of the sequence data I have, but have left plenty more out. If I thought these would help open some eyes, along with the difference sequences, I would continue laying all of it out. But what I've seen so far isn't encouraging. So I leave you to it. I'm gone.....

11 is a Lucas number- and the s block is ambivalent because all first members of HALF blocks are identical to last members. And I never stated that ALL first and last members of orbital half blocks belonged to the Fib or Luc sets. What I did write was that the Fib and Luc numbers, with the exceptions already pointed out in the Luc series, mapped to the block positions, not vice versa. Since 123 (Luc) isn't anywhere near a half block terminal (either half) that trend should end. And 144 similarly (for Fib). But then again it may be impossible to create elements with atomic numbers this big. Even if coincidental its all a damned big coincidence. I've been able to find nothing like it in the nuclear system. And 34, Se, is p4, the first member of the second half of its orbital block, which means it obeys the mapping perfectly. Right? Jess Tauber

Iron is so common because radioactive nickel, 28 (magic) protons (and neutrons?) decays into it. Nickel is actually the most tightly bound nucleus, not iron (specifically the rarer nickel-62). Carbon nucleosynthesis, by the way, is an odd duck, requiring the coming together of THREE alphas under specific conditions- and has been the subject of endless anthropic-principle discussions. But even if carbon seems to be an exception (remember carbon-14, despite its 8 neutrons, decays pretty well back into the nitrogen from which it springs), one exception doesn't destroy the entire numerical pattern. Look at the chemical periodic table. You can easily predict electron configurations based on where in the chart an element appears, except for the about 20 anomalous electronic configurations that have been discovered. By the way, if one really wants to get all New Agey, consider the following discovery. If taken as atomic numbers, Fibonacci numbers (up to 89, that is, within the known atomic system) ALL pattern as FIRST members of their orbital blocks, without exception, even (as in the case for Ac) they have anomalous configurations for their positions. Interestingly, ALL the ODD Fibs are in the FIRST HALF of their orbital block, where only one electron occupies a lobe (or not a lobe, for H), while ALL the EVEN Fibs are in the SECOND HALF of their orbital block, where two electrons begin to occupy each lobe. Furthermore, similar (though less perfect) patterning occurs when related Lucas numbers are taken as atomic numbers, but now instead of FIRST positions in orbital half-rows, we have a mapping to FINAL positions in same, and with the same odd/even bias to first or second half rows. Unlike FIb numbers, Lucas numbers get out of step after 18. Yet even here, within known elements, the system has behavioral 'fixes'. So, we have 29 and 47, copper and silver respectively, and in the same column, no less, both having anomalous electronic configurations which steal one electron from an earlier filled s orbital, donating it to the nearly filled d-orbital, completing it (so s1,d10 even though the position demands s2,d9). Then osmium, atomic number 76, often ACTS as if it were xenon, which does have a filled orbital. It also turns out, that if one builds up a table of generalized Fibonacci sequences (where the second number is 1, but we vary the initial), then the first handful of numbers in each sequence relate either directly or indirectly to quantum number specifications in the electronic (but also nuclear) orbital system. I don't write the rules, but clearly they have either been declared by fiat, or evolved on their own. Jess Tauber

Carbon, with 6 protons and 6 neutrons, is the fourth most abundant element in the universe, according to online sources. Given that magics were derived in part from such abundance data to say that 6 has nothing to do with this trend seems to me a bit off track. Some of the magic numbers are deemed semi-magic in certain contexts (certainly the later double tetrahedral harmonic oscillator magics fit the bill here). And we know that some doubly magic nuclei are pretty unstable. So perhaps we need a better DEFINITION of what is magic rather than simply large energy gaps, since they do form a mathematically analyzable SYSTEM. Jess Tauber

The number 34 may well be magic, but it is NOT spherical (the neutrons and protons are quite imbalanced and the nucleus has 'dripline' issues), and my scheme applies ONLY to spheres (real and hypothetical). In fact I haven't yet had much luck in determining whether the Pascal patterning exists in deformed nuclei, since the Nilsson charts I utiliize (from Lawrence Berkeley Labs) don't extend very far on either side of the sphere deformationally- no straight lines easy to extrapolate, and curves representing component rays sometimes to littlle twisty dances that make me wonder if even a French curve will do at the margins. Add to this the fact that they all use epsilon deformation which isn't the same as delta (which is used for the harmonic oscillator chart). For all I know realistic nonspherical (deformed) magic numbers may not follow precise deformation values as one climbs the energy scale on the y-axis. Time will tell, IF can get my hands on adequate charts. In any case there are likely to be a very large number of deformed magic numbers in sequences we haven't seen yet, as more and more nuclei are synthesized. There are a couple of other Pascal related patterns within the sphere I haven't mentioned. One has to do with half/double square differences between levels of same parity but from opposite cuts of the spin orbit scheme. Examples: 1s1/2 ending at 2 vs. 1d3/2 ending at 20, difference 18 1p3/2 ending at 6 vs. 1f5/2 ending at 38, difference 32 1d5/2 ending at 14 vs. 1g7/2 ending at 64, difference 50 1f7/2 ending at 28 vs. 1h9/2 ending at 100, difference 72 At higher quantum number l this seems to break down but I don't know whether this is real or a failure of the Nilsson model at high spin, as shown in the charts. It is also interesting that the sequence BEGINS at 18. From a purely mathematical standpoint it makes one wonder whether virtual negative l values might have physical meaning here, which would generate in their ordering the half/double squares 0,2, and 8. A second phenomenon has to do with numerical differences between final level counts such as 1s1/2 versus 2s1/2, 3s1/2, 4s1/2 and similarly for other orbital types. Though there are a couple of anomalous differences (several having to do with the ambivalence of intruder levels with nuclear isomers), the vast majority of such differences belong to the following sequence: 2,6,14,26,42,62,86,114,146,182,222... This is no random sequence, either. Each term is the sum of pairs of EVERY OTHER DOUBLED TRIANGULAR NUMBER. So 0+2=2, 0+6=6, 2+12=14, 6+20=26, 12+30=42, 20+42=62, 30+56=86, 42+72=114, 56+90=146, 72+110=182, 90+132=222... I mentioned early in my posts that the differences between spin-orbit magic numbers is double triangular plus 2, so 2,4,8,14,22,32,44,58,74,92... I discovered quite by accident while doing research on the math behind these two sequences that they actually occupy the triangular number diagonal positions, parallel to opposite sides, of the SAME modified Pascal Triangle. Quite a coincidence for those who believe that everything above is just that. In fact the first sequence relates to the numbers of intersectional zones created by ellipses, while the second deals with the numbers of intersectional zones created by circles. I have no idea what this implies physically, if anything at all. But it is still a striking result. Jess Tauber

There are TWO distinct sets of nuclear magic numbers covering most of the magic ground. The first consists of the double tetrahedral numbers: 2,8,20,40,70,112,168. These are the ones that fit perfectly the Pascal trends. Note that 2,8,20 are the first three of the 'seven most widely recognized magic numbers'. They also occur BEFORE the intruder phenomenon comes into play, intruders being those extra counts that bulk up the double tet magics to the spin-orbit magics. The second set can be gleaned from the first quite easily, in more than one way. The numbers pair thusly, with the spin-orbit on the left and the double tet on the right: 2,2 (identical) 6,8 (difference 2) 14,20 (difference 6) 28,40 (difference 12) 50,70 (difference 20) 82,112 (difference 30) 126,168 (difference 42) 184,240 (difference 56) Notice that the differences here are all DOUBLE TRIANGULAR NUMBERS, without exception. OTOH, lets organize in differently, with double tet first and spin-orbit second, but again in sequence. 0,2 (difference 2) 2,6 (difference 4) 8,14 (difference 6) 20,28 (difference 8) 40,50 (difference 10) 70,82 (difference 12) 112,126 (difference 14) 168,184 (difference 16). Note that now the differences are all double natural numbers, which in the (term-doubled) Pascal Triangle occupy the diagonal immediately prior to that of the double triangular numbers. The differences are also the sizes of the intruder levels that would come down to the prior period analogue. For the 'intruders' before 1g9/2 (10 nucleons), the spin-orbit effect appears to be incapable of lowering the constituent nucleons enough to actually 'intrude' (that is successfully compete with the existing energy levels in the lower period analogue- they end up merely appended to the end of it, which gives you the missing magic numbers. For protons there appear to be other magic numbers not appearing in these two sets, such as 114. You can get this as well from the intruder phenomenon, since one of the intruders ends at this number. But this shell then leaves out the final 12 nucleons that would bring it up to 126. I'm not yet aware as to whether other shells have similar behavior which would end shells at the terminations of other intruder levels. Jess Tauber

I'm sure you must be aware that the Nilsson Hamiltonian contains corrections for spin-orbit coupling as well as deformed potential well. These two terms have 'constants' (which are not exactly constant since they keep shifting as one ascends shell structure) that have never been derived from first principles. Rather they are reverse engineered from experimental data to force the system to spit out known numbers. Is this all we're capable of doing? My Pascal patterning has the potential to inspire a first-principles alternative to the current perturbational Nilsson Hamiltonian. Certainly THAT's something worth having? Jess Tauber

Hi folks- I'm new here. For the past couple of years now I've been teasing out new mathematical patterning within the known sequences of orbital and orbital partials in harmonic oscillator and spin-orbit models of nuclear shell structure. My project actually started back in the late 1970's when I first noticed that one could cut out and stack the 4 orbital based blocks from the periodic table in 3D, leading to a distorted tetrahedron if one extended the atomic numbers to 120. Many years later I found online work by an independent rediscoverer who had found the same relationship and had improved upon what I'd seen. www.perfectperiodictable.com One of the improvements was the use of close-packed spheres, though in his model these represented 2 elements each. I eventually found ways to get the same packing of the periodic table into a regular tetrahedron with one element per sphere, which makes things more coherent. The 2D Left-Step table arrangement of Charles Janet (discovered in the late 1920's) follows the known order of introduction of new quantum l numbers from left to right (so finally f,d,p,s). This goes against chemist's precepts about where periods should terminate (in the LSPT all s2), but generally matches what physicists expect. In any case in the LSPT all periods pair for length: 2,2,8,8,18,18,32,32... Note that each period length is a half/double square number, and taken together as pairs (or 'duals') they sum to full squares. Such stacking of squares justifies the tetrahedral motif, especially in light of Pascal Triangle diagonals and how their terms behave. I eventually started exploring the shell structure in atomic nuclei. For the simple harmonic oscillator model spherical magic numbers are identical, without exception, to the sequence of (doubled) tetrahredral numbers from the Pascal Triangle: 2,8,20,40,70,112,168 (being twice tetrahedral 1,4,10,20,35,56,84...). Amazingly I've had quite a few online geniuses from various discussion lists deny this! But then it takes all kinds, doesn't it? I took an online chart for the harmonic oscillator (which has energy versus deformation on y and x axes respectively, and straight rays for the energies) and extended it upwards and sideways by extrapolating the rays. From this I was able to show that the changing sequences of magic numbers at different deformations from the sphere are always based on Pascal numeration. In fact the oscillator ratio itself determines how they shift. For any prolate nucleus with oscillator ratio N/1 the numerator N tells you how many copies of doubled triangular numbers sum to the next magic number at that deformation. The sequence of doubled triangular numbers is 2,6,12,20,30,42,56,72,90,110,132... For the default ellipsoid/spheroid, the sphere, the oscillator ratio is 1:1. Here exactly ONE copy of each doubled triangular number sums to the next magic number, which is a doubled tetrahedral number, in keeping with the way such numbers operate in the Pascal Triangle (though of course with values doubled- such doubling apparently due to the fact that nucleons pair immediately, unlike the situation in the electron cloud, where, as the system continues to build, electrons first singly occupy orbital 'lobes' til all are filled in any one orbital, and only then start to double up in them). So 0+2=2; 2+6=8; 8+12=20; 20+20=40; 40+30=70; 70+42=112; 112+56=168.... and so on. For oscillator ratio 2:1 (prolate), the harmonic oscillator generates TWO copies of each double triangular number, each copy addition summing to a magic at this deformation: 0+2=2; 2+2=4; 4+6=10; 10+6=16; 16+12+28; 28+12=40: 40+20=60: 60+20=80; 80+30=110; 110+30=140; etc. This matches exactly the sequence of magic numbers for harmonic oscillator model nuclei of oscillator ratio 2:1 as published in many sources. For oscillator ratio 3:1 (prolate) THREE copies of each doubled triangular number sum, successively, to the next magic number, for 4:1 FOUR copies. There don't seem to be any exceptions based on published sources, or on my extrapolated Nilsson-style chart of energy versus deformation for the harmonic oscillator. This multiplication of summing numbers had already been discovered by earlier professional workers, but the identity of the numbers so summing as doubled triangular numbers seems to be mine alone. On the oblate side things get even more interesting. Here when the oscillator ratio has shape 1/M, instead of multiplying the number of appearances of each doubled triangular number going into magics we have instead to DIVIDE! For example, if we have an oscillator ratio of 1:2, then the double triangular difference between magics will appear between EVERY OTHER magic number rather than between EVERY magic as in the sphere, the default oblate ellipsoid. For oscillator ratio 1:3 the double triangular difference will appear between EVERY THIRD magic, and so on. A complication arises at sequence beginnings because we haven't yet accumulated enough magics to perform this little trick- it turns out that the earliest magic numbers here are all double triangular in and of themselves. In fact, the further towards extreme oblate shape we go more and more magics in turn are double triangular, until at infinite oblate deformation ALL magics are double triangular. When neither the oscillator ratio's numerator or denominator are 1 then BOTH effects- numerator multiplication of double triangular components and division by denominator come into play. This system is entirely self-consistent and there don't appear to be exceptions other than those detailed above. There are two very remarkable boundaries at the edges of deformation for the harmonic oscillator-only nucleus. In the delta deformation parameter these occur at -3.0 delta (oblate) and +1.5 delta (prolate). At these positions rays representing energies converge to such an extent that each magic number sums an infinite number of them. Published charts don't show these, as they are 'offstage'- you'll only see them by extrapolating such charts, though one might predict their appearance from the general formula for deformation in the delta system. Anyway, the above pretty much sums up the Pascal relations for the simple harmonic oscillator model. In the more 'realistic' model including spin-orbit and deformed potential well corrections to the Hamiltonian the pattern of Pascal numbers is quite different, but still present. The predicted spin-orbit magic numbers, for neutrons in spheres, are: 2,6,14,28,50,82,126,184... Here the inter-magic differences are all double triangular+2. The addition of 2 nucleons to the expected double triangular numbers is caused, where it actually exists, by the 'intruder' levels from any one period analogue- the highest spin orbital partial native to a period analogue drops out to the previous period analogue, while the one from the next higher period analogue comes in from above. Each highest spin partial is 2 particles larger than the one below, and this is the source of the difference between the spin-orbit magics and the harmonic oscillator magics, for spheres. I've also determined that the DEPTHS of intrusion (caused ultimately by energy drops coming from the alignment of particle spin with orbit) are themselves double triangular. 50-2=48 for 1g9/2, where 48 is the total particle number where the orbital partial terminates, so starting after 38. 82-6=76 for 1h11/2 similarly, so starting after 64. 126-12=114 for 1i13/2, starting after 100. 184-20=164 for 1j15/2, starting after 148. These ranges match those stated by Goeppert-Mayer and Jensen. There are NO actual intrusions before 39 which mirrors the fact that the double triangular number difference would be 0 before this point. One can also get the intruder initiation boundaries by subtracting double triangular numbers from the harmonic oscillator magics: 40-2=38; 70-6=64; 112-12=100; 168-20=148 etc. There are further Pascal effects. The basic period analogue size from the harmonic oscillator is always in and of itself a doubled triangular number, just because of parity sorting: s=2 p=6 ds=10+2=12 fp=14+6=20 gds=18+10+2=30 hfp=22+14+6=42 igds=26+18+10+2=56 jhfp=30+22+14+6=72 It turns out, in a weird kind of mathematical coordination, that adding the next highest orbital partial from the next period analogue yields a new period analogue size that is the NEXT HIGHER doubled triangular number: 2+4 (p3/2)=6 6+6 (d5/2)=12 12+8 (f7/2)=20 20+10 (g9/2)=30 30+12 (h11/2)=42 42+14 (i13/2)=56 56+16 (j15/2)=72 Similarly, SUBTRACTION of the highest spin orbital partial from a double triangular length period analogues gives the NEXT LOWER double triangular number. When both these effects occur together in the same shell, then all that remains is the double triangular+2 difference between shells, giving the known spin-orbit magic numbers for spheres. In fact, if one actually draws a value-doubled Pascal Triangle, then the doubled triangular number values in the triangular number diagonal are directly adjacent to, in the same row, the length values of the highest spin orbital partials that intrude into them. All the findings above are observations of mathematical regularities (gleaned by taking differences in one of the simplest mathematical operations available) within known or theoretical sequences discovered or devised by professionals. I have no real explanation as yet for why things pack the way they do here- thus no crazy crank theory to be easily debunked. One cannot simply dismiss this as 'numerology' (as many have attempted to do since it rankles in their guts). Clearly there is something worth further exploration going on. Though I suspect many of you will be reaching for the Pepto.... Jess Tauber goldenratio@earthlink.net