The Periodic Law?

[JCTheCreation]

Since The Periodic Table is fairly Standard around the world, would you consider the Periodic Law that Dmitri Mendeleev discovered in 1867 a fundamental truth?

 

Maybe a better question would be, if 2 separate groups of scientists were to begin from scratch, studying and investigating the Elements, would their Periodic Tables eventually have much the same structure, order and features do you think? 

 

Also a bonus question, Since The Periodic Table is primarily based on quantity of Protons and order of Electrons, are there a finite number of ways to list the Elements without breaking logic?

 

Thank you for your time and help! All feedback is greatly appreciated! Have a great day

 

Justin

Edited November 3, 2017 by JCTheCreation

[exchemist]

Since The Periodic Table is fairly Standard around the world, would you consider the Periodic Law that Dmitri Mendeleev discovered in 1867 a fundamental truth?

 

Maybe a better question would be, if 2 separate groups of scientists were to begin from scratch, studying and investigating the Elements, would their Periodic Tables eventually have much the same structure, order and features do you think? 

 

Also a bonus question, Since The Periodic Table is primarily based on quantity of Protons and order of Electrons, are there a finite number of ways to list the Elements without breaking logic?

 

Thank you for your time and help! All feedback is greatly appreciated! Have a great day

 

Justin

I would not consider anything in the whole of science a "fundamental truth". In science, we tend to avoid the term "truth", because in science we work with mere models of the physical world, any one of which may one day need revision in the light of new observations.

 

There have been in fact a number of variants of the layout Mendele'ev's table. Sometimes He is classed with noble gases, sometimes it sits on its own, sometimes with the alkaline earths and H is sometimes on its own and sometimes with the alkali metals.  

 

But yes I do think that any other group of scientists would probably come eventually to the same scheme. In the end, the periodicity is driven by the Aufbauprinzip applied to atoms with successively increasing nuclear charge. I can't think that, for example, the alkali metals and the alkaline earths would not be in the same groups, and similarly the halogens and the noble gases. The bonding rules in the p block would seem to sort out most of those into the familiar groups too.  

 

Where there might be more arbitrary choice would be in the transition elements, the lanthanides and actinides. But, if we allow our scientists to know the nuclear charge and the electronic configuration, I'm pretty sure they would end up classing them in a similar way.  

 

I don't think I understand your bonus question. You can list them however you like, if you make up your own rules, can't you?   

[tetrahedron]

Just found this thread. There have actually been more than a THOUSAND different layouts of the periodic table since 1869. The 'best' 2D one in my opinion (shared by many others) is the 'Left-Step' periodic table developed by the then elderly French polymath Charles Janet in the late 1920's. See https://realnumeracy.files.wordpress.com/2013/08/periodictablejanetleftstep.jpg.  Here all periods end in s-block elements. Periods add new orbitals to the left edge, so that their actual growth is s, ps, dps, fdps. In the Janet table all periods of a particular length appear TWICE, unlike the traditional table. so 1s,2s; 2p3s,3p4s; 3d4p5s, 4d5p6s; 4f5d6p7s, 5f6d7p8s.  All period lengths are half squares (or doubled squares just as much). Thus pairs of same-length periods have a square number of elements. These then sum to tetrahedral numbers with every second same-length period's s2 element's atomic number: 4,20,56,*120. Intermediate atomic numbers are the arithmetic means of the flanking tetrahedrals. (0)+4/2=2. (4+20)/2=12. (20+56)/2=38. (56+120)/2=88.  I started representing the Janet table in 3D as a tetrahedron of close-packed spheres (one per element) back in 2009- unfortunately a Russian physician beat me to the punch around 2003, and in print, no less. One can also create configurations that have absolutely no breaks in the continuity of Mendeleev's Line, the series of atomic numbers, using 'tiles', one per period, that the atomic number line segment back on itself. Everything fits perfectly. Such tiles bear a strong resemblance to Penrose tilings in 2D, though here they cover the surfaces of nested tetrahedra of close-packed spheres. Here is a pic of a period tile containing all four orbital types: https://www.facebook.com/photo.php?fbid=10155616726521346&set=gm.762705150603772&type=3&theater

 

 

Jess Tauber

Edited October 23, 2018 by pascal

[tetrahedron]

The 28 thing is interesting. One can model the charges of the fermions using vertices of a cube embedded in a reference plane through a body diagonal as a rotational axis. A body diagonal goes through the cube center and through two diametrically opposed vertices. Now we embed this diagonal in the plane- it takes two vertices with it, obviously. If you start with a third non-axis vertex in the plane and then rotate using the diagonal to an angle of arctan(sqrt27) plus or minus any multiple of 30 degrees, then normals dropped from the vertices to the plane will have different relative lengths of 0 (in the plane), 1, 2, and 3. And because the cube has 8 vertices, this will cover both matter (above the plane) and antimatter (below).  The choice of sqrt27 under the arctan is the ONLY angle that will deliver the relative lengths this way. Every other angle uses sines, the intermediates use cosines. Because of general trigonometric laws, the squares of the sines (or cosines) will all result in multiples of 1/28. The denominator is always one more than the number under the original square root sign.

 

Jess Tauber

Edited October 24, 2018 by pascal

[tetrahedron]

Oops! I misstated one of my claims- I said that matter was above the plane (so all positive positions on the number line) and antimatter was below (negative positions). Actually all the positive positions are positive charges and all the negative positions are negative charges. Sorry of the confusion. Matter and antimatter both have mixtures of positive and negative charges, in complementary distribution. Thanks for your patience.

 

Jess Tauber

[GAHD]

I dunno, there are quite a few "non standard" periodic tables.
https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=725 These guys have a bunch.
Machine learning is leading to some other "non standard" ones too. https://www.chemistryworld.com/opinion/a-3d-periodic-table/3007821.article
Some are more useful than others. https://www.meta-synthesis.com/webbook/35_pt/pt_database.php?PT_id=418

So..."no, not really"

I personally keep a fairly "standard" periodic table printout, but it's one that lists all known isotopes with percentage and their decay rates instead of an averaged mass number, shows the EV states, orbitals, etc... some of the slots are quite crammed with numbers even in 6 point font.

[tetrahedron]

According to one professional mathematical physicist I consulted, the quantum harmonic oscillator equation always delivers numbers of stable states (shells) whose sizes correspond to successive terms in Pascal Triangle diagonals. The only determining factor is the dimensionality of the system under consideration. For atomic nuclei under a simple harmonic oscillator model, and spherical shape, the shell sizes are all exactly (and only) doubled triangular numbers. Thus 1s=2, 1p=6, 1d2s=12 (10+2), 1f2p=20 (14+6), 1g2d3s=30 (18+10+2), 1h2f3p=42 (22+14+6) and so on. The magic numbers (full shells) are running sums of these doubled triangular number shell sizes, so 2,8,20,40,70,112,168,240..... The doubled triangular number interval motif is found in several other places as well, such as magic numbers for atomic clusters under a simple harmonic oscillator model, and ellipsoidally deformed nuclei under the harmonic oscillator.

 

In the periodic table, I mentioned above that in the Janet version all periods appear twice for any given length, and that the lengths were half/double squares (in terms of numbers of elements per period). The ordering of the orbitals, from highest spin on the left to lowest on the right, recapitulates in part what goes on in the nuclear system. The main difference is that in nuclei under the harmonic oscillator all orbitals in a shell have the same parity value (positive, so even quantum number ml (0,2,4...) or odd (1,3,5...), while in the electronic system odd and even alternate in the structure of a given period.

 

In a more realistic shell model for nuclei we must include, in the Hamiltonian, correction terms for spin-orbit coupling. Again for spheres this increases shell size by monotonically increasing amounts. The interesting thing, though, is that these shell size increases MAINTAIN THE DOUBLED TRIANGULAR NUMBER MOTIF. 1f2p (20 nucleons) adds 1g9/2 (10) giving 30, which is the next higher doubled triangular number after 20. 1g2d3s (30) adds 1h11/2 (12) giving 42, again the very next higher doubled triangular number. AND SO ON.

 

Anyway, these shell-size increases are due to the incorporation of the highest-spin subshell from the next higher shell being added to the one below it, because the spin-orbit effect lowers its energy.  In the traditional periodic table we like to end periods with the noble gases, because that is where the shells actually end behaviorally. But the ability to end periods with the nobles comes because said periods have added to their structures the s-orbitals from the PREVIOUS LEFT-STEP period. In other words, the LOWEST-SPIN orbital was raised in energy sufficiently to join the structure of the next HIGHER period, the direct inverse of what happens in the nucleus with spin-orbit correction.  So far as I know, I appear to be the first to note this possible connection between the two different phenomena. And it is known that spin-orbit coupling works in opposite directions between the electronic and nuclear systems.

 

So, the Janet Left-Step periodic table represents a less 'derived' format for the electronic system than the traditional table, which is more about surface chemical behaviors. And the simpler harmonic oscillator model of atomic nuclei represents a less derived version of the nuclear system, while one incorporating corrections for spin-orbit coupling is again more surface-based, on physically observable behaviors.

 

Jess Tauber

[exchemist]

According to one professional mathematical physicist I consulted, the quantum harmonic oscillator equation always delivers numbers of stable states (shells) whose sizes correspond to successive terms in Pascal Triangle diagonals. The only determining factor is the dimensionality of the system under consideration. For atomic nuclei under a simple harmonic oscillator model, and spherical shape, the shell sizes are all exactly (and only) doubled triangular numbers. Thus 1s=2, 1p=6, 1d2s=12 (10+2), 1f2p=20 (14+6), 1g2d3s=30 (18+10+2), 1h2f3p=42 (22+14+6) and so on. The magic numbers (full shells) are running sums of these doubled triangular number shell sizes, so 2,8,20,40,70,112,168,240..... The doubled triangular number interval motif is found in several other places as well, such as magic numbers for atomic clusters under a simple harmonic oscillator model, and ellipsoidally deformed nuclei under the harmonic oscillator.

 

In the periodic table, I mentioned above that in the Janet version all periods appear twice for any given length, and that the lengths were half/double squares (in terms of numbers of elements per period). The ordering of the orbitals, from highest spin on the left to lowest on the right, recapitulates in part what goes on in the nuclear system. The main difference is that in nuclei under the harmonic oscillator all orbitals in a shell have the same parity value (positive, so even quantum number ml (0,2,4...) or odd (1,3,5...), while in the electronic system odd and even alternate in the structure of a given period.

 

In a more realistic shell model for nuclei we must include, in the Hamiltonian, correction terms for spin-orbit coupling. Again for spheres this increases shell size by monotonically increasing amounts. The interesting thing, though, is that these shell size increases MAINTAIN THE DOUBLED TRIANGULAR NUMBER MOTIF. 1f2p (20 nucleons) adds 1g9/2 (10) giving 30, which is the next higher doubled triangular number after 20. 1g2d3s (30) adds 1h11/2 (12) giving 42, again the very next higher doubled triangular number. AND SO ON.

 

Anyway, these shell-size increases are due to the incorporation of the highest-spin subshell from the next higher shell being added to the one below it, because the spin-orbit effect lowers its energy.  In the traditional periodic table we like to end periods with the noble gases, because that is where the shells actually end behaviorally. But the ability to end periods with the nobles comes because said periods have added to their structures the s-orbitals from the PREVIOUS LEFT-STEP period. In other words, the LOWEST-SPIN orbital was raised in energy sufficiently to join the structure of the next HIGHER period, the direct inverse of what happens in the nucleus with spin-orbit correction.  So far as I know, I appear to be the first to note this possible connection between the two different phenomena. And it is known that spin-orbit coupling works in opposite directions between the electronic and nuclear systems.

 

So, the Janet Left-Step periodic table represents a less 'derived' format for the electronic system than the traditional table, which is more about surface chemical behaviors. And the simpler harmonic oscillator model of atomic nuclei represents a less derived version of the nuclear system, while one incorporating corrections for spin-orbit coupling is again more surface-based, on physically observable behaviors.

 

Jess Tauber

Hmm, but your last points are the most important. The Periodic Table is devised for use by chemists, who are interested in comparing and contrasting the physical and chemical behaviour of elements, rather than just their electronic structures. Hence the standard format, which exists in a handful of minor variants only. 

 

By the way, why do you talk of a "harmonic oscillator" model for the atom? Or do you mean spherical harmonics?

[tetrahedron]

The harmonic oscillator model for atomic nuclei is indeed based on spherical harmonics, but has also been extended to include deformations to ellipsoids of revolution, both prolate and oblate. See https://application.wiley-vch.de/books/info/0-471-35633-6/toi99/www/struct/struct.pdf. Fig. 16 shows the Nilsson-style plot of energy levels over deformation, which I used in my analyses. Not only did I extend the chart into greater deformations on both sides, I also did so into higher energies than shown in this particular chart.  Interestingly at infinite deformations (-3.0 oblate and +1.5 prolate on the delta deformation parameter scale) each magic number has an infinite number of orbital components (the straight rays you see in the plot).  This same plot was also the basis of my discovery that individual shell energies were conserved over deformation. Each shell adds new components as deformation grows, and this makes up for the fact that the single particle energy differences between shells get smaller and smaller. All coordinated and compensatory, which is really surprising. Nobody seems to have noticed this before I did. And this conservation principle works for the more realistic and complex plots under the spin-orbit model as well. 

 

Jess Tauber

[exchemist]

The harmonic oscillator model for atomic nuclei is indeed based on spherical harmonics, but has also been extended to include deformations to ellipsoids of revolution, both prolate and oblate. See https://application.wiley-vch.de/books/info/0-471-35633-6/toi99/www/struct/struct.pdf. Fig. 16 shows the Nilsson-style plot of energy levels over deformation, which I used in my analyses. Not only did I extend the chart into greater deformations on both sides, I also did so into higher energies than shown in this particular chart.  Interestingly at infinite deformations (-3.0 oblate and +1.5 prolate on the delta deformation parameter scale) each magic number has an infinite number of orbital components (the straight rays you see in the plot).  This same plot was also the basis of my discovery that individual shell energies were conserved over deformation. Each shell adds new components as deformation grows, and this makes up for the fact that the single particle energy differences between shells get smaller and smaller. All coordinated and compensatory, which is really surprising. Nobody seems to have noticed this before I did. And this conservation principle works for the more realistic and complex plots under the spin-orbit model as well. 

 

Jess Tauber

Hang on, you are talking about nuclear structure, not atomic structure at all.

 

The Periodic Table is irrelevant to nuclear structure. It arises from atomic structure, i.e. the Aufbau Principle applied to electrons in the atom.

 

You seem to be talking about something totally different. 

[tetrahedron]

Actually the periodic table is NOT irrelevant to nuclear structure. If you start from the Janet Left-Step periodic table, then all periods end with elements with s2 electronic configurations- that is helium and the alkaline earths. Every other period (all second same length periods of 'duals') ends with an atomic number that is a tetrahedral number: 4, 20, 56, *120. And all the intermediate atomic numbers, following the old 'triad' rule from the first half of the 19th century that led to the discovery of the periodic law in the first place, are the arithmetic means of the flanking tetrahedral numbers, so starting from a default 0, (0+4)/2=2; (4+20)/2=12; (20+56)/2=38; (56+120)/2=88.  Now, in the simpler harmonic-oscillator only model of the nucleus, there is no parallel to the dual occurrence of same-length periods in the Left-Step periodic table. All period analogues in nuclear shells for spheres here contain all same-parity orbital components, so all odd or all even quantum number ml components. Each shell here contains a doubled triangular number of nucleons: 1s=2; 1p=6; 1d2s=12, 1f2p=20; 1g2d3s=30; 1h2f3p=42; 1i2g3d4s=56; 1j2h3f4p=72. Running sums of these period analogues (shells) give rise to the doubled tetrahedral magic numbers 2,8,20,40,70,112,168,240....  Well it turns out then that we can align these against the period terminals of the Janet Left-Step periodic table. Every other magic number is twice every other Janet period terminal atomic number. So 2x Janet 4=8 spherical magic. 2xJanet 20=40 spherical magic. 2xJanet 56= 112 spherical magic. And 2xJanet 120=240 spherical magic. The two systems, one based on single tetrahedral numbers and the other based on doubled tetrahedral numbers, are in register.

 

In both these systems the largest-spin orbital components are on the left, and successive orbital components have smaller and smaller spin.  One big difference is that when we add spin-orbit coupling to the nuclear Hamiltonian, the highest-spin orbital partial from the next higher shell joins the structure of the previous shell. This doesn't happen in the electronic system. In fact, spin-orbit effect works in reverse between the two systems. In the electronic system, the SMALLEST-SPIN orbital component from the previous shell joins the structure of the NEXT shell, which is why we traditionally end periods with the noble gases rather than continue on to the next s2 configuration elements, as would be the case with the Janet table. Until my analysis I don't think anyone had considered that we would have mirror-image effects between these two systems.

 

Another difference is that under the spin-orbit model, all orbitals are split, with the larger part being lowered in energy, and the smaller part raised. This causes the components to change relative positions in the structure of a period analogue (shell) relative to that of the simple harmonic oscillator model. We don't see this in the electronic system usually. However I've started wondering whether something along these lines can show up in the chemical behavior.

 

Jess Tauber

[exchemist]

Actually the periodic table is NOT irrelevant to nuclear structure. If you start from the Janet Left-Step periodic table, then all periods end with elements with s2 electronic configurations- that is helium and the alkaline earths. Every other period (all second same length periods of 'duals') ends with an atomic number that is a tetrahedral number: 4, 20, 56, *120. And all the intermediate atomic numbers, following the old 'triad' rule from the first half of the 19th century that led to the discovery of the periodic law in the first place, are the arithmetic means of the flanking tetrahedral numbers, so starting from a default 0, (0+4)/2=2; (4+20)/2=12; (20+56)/2=38; (56+120)/2=88.  Now, in the simpler harmonic-oscillator only model of the nucleus, there is no parallel to the dual occurrence of same-length periods in the Left-Step periodic table. All period analogues in nuclear shells for spheres here contain all same-parity orbital components, so all odd or all even quantum number ml components. Each shell here contains a doubled triangular number of nucleons: 1s=2; 1p=6; 1d2s=12, 1f2p=20; 1g2d3s=30; 1h2f3p=42; 1i2g3d4s=56; 1j2h3f4p=72. Running sums of these period analogues (shells) give rise to the doubled tetrahedral magic numbers 2,8,20,40,70,112,168,240....  Well it turns out then that we can align these against the period terminals of the Janet Left-Step periodic table. Every other magic number is twice every other Janet period terminal atomic number. So 2x Janet 4=8 spherical magic. 2xJanet 20=40 spherical magic. 2xJanet 56= 112 spherical magic. And 2xJanet 120=240 spherical magic. The two systems, one based on single tetrahedral numbers and the other based on doubled tetrahedral numbers, are in register.

 

In both these systems the largest-spin orbital components are on the left, and successive orbital components have smaller and smaller spin.  One big difference is that when we add spin-orbit coupling to the nuclear Hamiltonian, the highest-spin orbital partial from the next higher shell joins the structure of the previous shell. This doesn't happen in the electronic system. In fact, spin-orbit effect works in reverse between the two systems. In the electronic system, the SMALLEST-SPIN orbital component from the previous shell joins the structure of the NEXT shell, which is why we traditionally end periods with the noble gases rather than continue on to the next s2 configuration elements, as would be the case with the Janet table. Until my analysis I don't think anyone had considered that we would have mirror-image effects between these two systems.

 

Another difference is that under the spin-orbit model, all orbitals are split, with the larger part being lowered in energy, and the smaller part raised. This causes the components to change relative positions in the structure of a period analogue (shell) relative to that of the simple harmonic oscillator model. We don't see this in the electronic system usually. However I've started wondering whether something along these lines can show up in the chemical behavior.

 

Jess Tauber

This seems to be just numerology. And I can't make any sense out of your comments on spin-orbit coupling. 

 

There does not seem to be any science here. I think I'll leave it. 

[tetrahedron]

Number THEORY is not numerology. And numerology usually has a metaphysical or supernatural component. You see anything remotely resembling such a thing in any of my claims. I have noticed such name-calling to be simply a knee-jerk reaction to any results not in standard textbooks or other received wisdom. So let me get this straight. Even with firmly established theoretical or pretheoretical blinders professionals could not POSSIBLY have missed any phenomenon as simple as this. That's blatant hubris. So let me take you through it the way I would a child.

 

Spherical magic numbers for nuclei in a pure harmonic oscillator model are 2,8,20,40,70,112,168,240.....  This sequence is published in many books and papers, despite its never having actually been observed. That's because its based on the theory of the harmonic oscillator.  Yet you'll never find any reference to this same sequence being identical to 2x tetrahedral numbers 1,4,10,20,35,56,84,120..... The professionals didn't make the connection. I seem to have been the first to do so, many decades after the first models appeared.

 

Now please examine Fig. 17 at https://application.wiley-vch.de/books/info/0-471-35633-6/toi99/www/struct/struct.pdf. This figure reproduces the single-particle energy levels for harmonic oscillator nuclei, over a range of deformations in either direction. At deformation 1:1 we have the sphere, which is the default ellipsoid. The numbers printed at this place on the x-axis are the spherical magic numbers, as listed above.

 

Take a look now at the magic numbers at 2:1 on the x-axis. 2:1 means we have a prolate super-deformed ellipsoid whose long axis (polar) is twice the size of its short axis (equatorial). This sequence is 2,4,10,16,28,40,60,80,110,140,182...

 

No one to date has looked at such deformed magic sequences from any perspective other than energy levels. I looked at them numerically. Lets take a look at the numerical differences between these magic sequences. For the sphere, differences are 2-0=2, 8-2=6, 20-8=12, 40-20=20, 70-40=30, 112-70=42, 168-112=56, 240-168=72.  The differences,  2, 6, 12, 20, 30, 42, 56, 72.... are twice 1, 3, 6, 10, 15, 21, 28, 36...., which are the triangular numbers. Believe it or not, I have had on multiple occasions had to point this out to naysayers, who even deny that these numbers are any recognizable sequence. 

 

The reason for these magic numbers is because these are also the sizes of the individual harmonic oscillator shells. 1s=2, 1p=6, 1d2s=10+2=12, 1f2p=14+6=20. 1g2d3s=18+10+2=30, 1h2f3p=22+14+6=42, 1i2g3d4s= 26+18+10+2=56, 1j2h3f4p=30+22+14+6=72.....

 

See anything metaphysical or supernatural here?

 

Now lets look at the magics for oscillator ratio (deformation) 2:1 (prolate):  2,4,10,16,28,40,60,80,110,140,182..   Differences again are 2-0=2, 4-2=2, 10-4=6, 16-10=6, 28-16=12, 40-28=12, 60-40=20, 80-60=20, 110-80=30, 140-110=30, 182-140=42.

 

So the resulting set of differences are 2,2,6,6,12,12,20,20,30,30,42...... In other words, the SAME set of doubled triangular numbers, but each difference appearing TWICE in sequence, each time generating the next magic number at this deformation. I am the first to discover this simple relationship, or at least to mention it online.

 

One can do the same type of analyses at any deformation expressed as a ratio of natural numbers (integers). For 3:1, each doubled triangular number interval appears THREE TIMES in sequence to generate a new magic number. And for 4:1 FOUR TIMES. And so on. That sure looks like a RULE, doesn't it? The numerator (polar extent of the matter wave) of the oscillator ratio is IDENTICAL to the number of times we use any particular doubled triangular number interval between magic numbers at that oscillator ratio (deformation). PERIOD. At least for prolate nuclei.

 

Now lets take a look at the oblate deformed nuclei. For an oscillator ratio of 1:2 (twice as wide as long), the magic numbers in Fig.17 are: 2,6,14,26,44,68,100,140,190....Differences between these are 2-0=2, 6-2=4, 14-6=8, 26-14=12, 44-26=18, 68-42=24, 100-68=32, 140-100=40, 190-140=50....  Note that the resulting set of differences, 2,4,8,12,18,24,32,40,50... themselves have differences 2-0=2, 4-2=2, 8-4=4, 12-8=4, 18-12=6, 24-18=6, 32-24=8, 40-32=8, 50-40=10. Each second difference is a doubled natural (even) number.  And each such secondary difference appears TWICE in sequence.  Now, on a whim a decade ago I tried looking at differences between every OTHER magic numbers here:

14-2=12, 26-6=20, 44-14=30, 68-26=42, 100-44=56, 140-68=72, 190-100=90. ALL THESE DIFFERENCES are doubled triangular numbers. AGAIN.

 

As I explored my extrapolated Fig.17 chart (since nobody else seems to have done it) I started running into an issue on the oblate side. The earliest magic numbers in sequences weren't fitting into this pattern of differences. Up to now, the numerator of the oscillator ratio defined how we took the numerical differences. For the sphere, as the default ellipsoid of oscillator ratio 1:1, there was one single use of each doubled triangular number interval between each magic number, and such doubled triangular number intervals appeared between every single magic in sequences. But with an oscillator ratio of 1:2, these intervals appeared between every SECOND MAGIC. It then turned out that for an oscillator ratio of 1:3 they appeared between every THIRD MAGIC, and so on. But not at the beginnings of each magic sequence. In fact, until there were as many magics as the denominator of the oscillator ratio, the magics themselves were doubled triangular number in value, without exception. After there are as many magics as the oscillator ratio denominator, the rule higher up began, where that denominator defines the size of the jump between magics needed to generate a doubled triangular number difference.  IN OTHER WORDS, THE OSCILLATOR RATIO'S DENOMINATOR DIVIDES THE SYSTEM, JUST AS THE NUMERATOR MULTIPLIES USE OF THE INTERVALS. Its just that these systems don't like to deal with remainders.  With oscillator ratio numerator multiplication of doubled triangular number usage generating magics, denominator division of same, and the switch to doubled triangular number magics at early points in the oblate deformations as above, there are no exceptions to these rules, and anyone can, with pencil and paper, lay out strings of magic numbers for any defined oscillator ratio, without even the need of a calculator. I don't think that anyone else worked all this out before. At least not in print.

 

Is this numerology? And as to the accusation that there is no science here, consider the famous Rydberg formula describing atomic spectral lines: https://en.wikipedia.org/wiki/Rydberg_formula

 

I didn't run any experiments myself, made no physical observations. I used PUBLISHED professional data and existing, accepted models of nuclear systems, and just analyzed the numbers further. How is this not SCIENCE?

 

Jess Tauber

Edited October 30, 2018 by pascal

[exchemist]

Number THEORY is not numerology. And numerology usually has a metaphysical or supernatural component. You see anything remotely resembling such a thing in any of my claims. I have noticed such name-calling to be simply a knee-jerk reaction to any results not in standard textbooks or other received wisdom. So let me get this straight. Even with firmly established theoretical or pretheoretical blinders professionals could not POSSIBLY have missed any phenomenon as simple as this. That's blatant hubris. So let me take you through it the way I would a child.

 

Spherical magic numbers for nuclei in a pure harmonic oscillator model are 2,8,20,40,70,112,168,240.....  This sequence is published in many books and papers, despite its never having actually been observed. That's because its based on the theory of the harmonic oscillator.  Yet you'll never find any reference to this same sequence being identical to 2x tetrahedral numbers 1,4,10,20,35,56,84,120..... The professionals didn't make the connection. I seem to have been the first to do so, many decades after the first models appeared.

 

Now please examine Fig. 17 at https://application.wiley-vch.de/books/info/0-471-35633-6/toi99/www/struct/struct.pdf. This figure reproduces the single-particle energy levels for harmonic oscillator nuclei, over a range of deformations in either direction. At deformation 1:1 we have the sphere, which is the default ellipsoid. The numbers printed at this place on the x-axis are the spherical magic numbers, as listed above.

 

Take a look now at the magic numbers at 2:1 on the x-axis. 2:1 means we have a prolate super-deformed ellipsoid whose long axis (polar) is twice the size of its short axis (equatorial). This sequence is 2,4,10,16,28,40,60,80,110,140,182...

 

No one to date has looked at such deformed magic sequences from any perspective other than energy levels. I looked at them numerically. Lets take a look at the numerical differences between these magic sequences. For the sphere, differences are 2-0=2, 8-2=6, 20-8=12, 40-20=20, 70-40=30, 112-70=42, 168-112=56, 240-168=72.  The differences,  2, 6, 12, 20, 30, 42, 56, 72.... are twice 1, 3, 6, 10, 15, 21, 28, 36...., which are the triangular numbers. Believe it or not, I have had on multiple occasions had to point this out to naysayers, who even deny that these numbers are any recognizable sequence. 

 

The reason for these magic numbers is because these are also the sizes of the individual harmonic oscillator shells. 1s=2, 1p=6, 1d2s=10+2=12, 1f2p=14+6=20. 1g2d3s=18+10+2=30, 1h2f3p=22+14+6=42, 1i2g3d4s= 26+18+10+2=56, 1j2h3f4p=30+22+14+6=72.....

 

See anything metaphysical or supernatural here?

 

Now lets look at the magics for oscillator ratio (deformation) 2:1 (prolate):  2,4,10,16,28,40,60,80,110,140,182..   Differences again are 2-0=2, 4-2=2, 10-4=6, 16-10=6, 28-16=12, 40-28=12, 60-40=20, 80-60=20, 110-80=30, 140-110=30, 182-140=42.

 

So the resulting set of differences are 2,2,6,6,12,12,20,20,30,30,42...... In other words, the SAME set of doubled triangular numbers, but each difference appearing TWICE in sequence, each time generating the next magic number at this deformation. I am the first to discover this simple relationship, or at least to mention it online.

 

One can do the same type of analyses at any deformation expressed as a ratio of natural numbers (integers). For 3:1, each doubled triangular number interval appears THREE TIMES in sequence to generate a new magic number. And for 4:1 FOUR TIMES. And so on. That sure looks like a RULE, doesn't it? The numerator (polar extent of the matter wave) of the oscillator ratio is IDENTICAL to the number of times we use any particular doubled triangular number interval between magic numbers at that oscillator ratio (deformation). PERIOD. At least for prolate nuclei.

 

Now lets take a look at the oblate deformed nuclei. For an oscillator ratio of 1:2 (twice as wide as long), the magic numbers in Fig.17 are: 2,6,14,26,44,68,100,140,190....Differences between these are 2-0=2, 6-2=4, 14-6=8, 26-14=12, 44-26=18, 68-42=24, 100-68=32, 140-100=40, 190-140=50....  Note that the resulting set of differences, 2,4,8,12,18,24,32,40,50... themselves have differences 2-0=2, 4-2=2, 8-4=4, 12-8=4, 18-12=6, 24-18=6, 32-24=8, 40-32=8, 50-40=10. Each second difference is a doubled natural (even) number.  And each such secondary difference appears TWICE in sequence.  Now, on a whim a decade ago I tried looking at differences between every OTHER magic numbers here:

14-2=12, 26-6=20, 44-14=30, 68-26=42, 100-44=56, 140-68=72, 190-100=90. ALL THESE DIFFERENCES are doubled triangular numbers. AGAIN.

 

As I explored my extrapolated Fig.17 chart (since nobody else seems to have done it) I started running into an issue on the oblate side. The earliest magic numbers in sequences weren't fitting into this pattern of differences. Up to now, the numerator of the oscillator ratio defined how we took the numerical differences. For the sphere, as the default ellipsoid of oscillator ratio 1:1, there was one single use of each doubled triangular number interval between each magic number, and such doubled triangular number intervals appeared between every single magic in sequences. But with an oscillator ratio of 1:2, these intervals appeared between every SECOND MAGIC. It then turned out that for an oscillator ratio of 1:3 they appeared between every THIRD MAGIC, and so on. But not at the beginnings of each magic sequence. In fact, until there were as many magics as the denominator of the oscillator ratio, the magics themselves were doubled triangular number in value, without exception. After there are as many magics as the oscillator ratio denominator, the rule higher up began, where that denominator defines the size of the jump between magics needed to generate a doubled triangular number difference.  IN OTHER WORDS, THE OSCILLATOR RATIO'S DENOMINATOR DIVIDES THE SYSTEM, JUST AS THE NUMERATOR MULTIPLIES USE OF THE INTERVALS. Its just that these systems don't like to deal with remainders.  With oscillator ratio numerator multiplication of doubled triangular number usage generating magics, denominator division of same, and the switch to doubled triangular number magics at early points in the oblate deformations as above, there are no exceptions to these rules, and anyone can, with pencil and paper, lay out strings of magic numbers for any defined oscillator ratio, without even the need of a calculator. I don't think that anyone else worked all this out before. At least not in print.

 

Is this numerology? And as to the accusation that there is no science here, consider the famous Rydberg formula describing atomic spectral lines: https://en.wikipedia.org/wiki/Rydberg_formula

 

I didn't run any experiments myself, made no physical observations. I used PUBLISHED professional data and existing, accepted models of nuclear systems, and just analyzed the numbers further. How is this not SCIENCE?

 

Jess Tauber

If it does not help to account for or predict observed natural phenomena, then it is not science. 

 

What observable effects does this playing with numbers predict or explain, please? 

[tetrahedron]

In the spin-orbit model of atomic nuclei mapping of single particle energy levels against ellipsoidal deformation results in a crazy-quilt of shifting slopes, quantum number string label exchanges, and other effects. In the simpler harmonic oscillator-only model all these energy changes are linear, with 'component rays' (each representing orbital partials) having fixed slopes that change monotonically for components of each shell.  The spin-orbit model appears to be a variant on this theme. For example, the highest-spin component in the harmonic oscillator model for each shell has slopes (in terms of h-bar omega-bar over delta deformation units) of: 1s1/2=0/3, 1p3/2=1/3, 1d5/2=2/3, 1f7/2=3/3, 1g9/2=4/3,. 1h11/2=5/3, 1i13/2=6/3, 1j15/2=7/3,. etc. It turns out that the numerators of all these slopes are equal to the quantum number ml value for the orbitals so labeled. My discovery.  In the spin-orbit model, the highest-spin pair in the larger spin-split orbital has an identical slope to the harmonic oscillator versions. More than this, if one extrapolates these nearly straight rays to lower energy values and greater oblate deformation, they all converge at 1.5 h-bar omega-bar at -2.7 epsilon deformation (see Figs. 4 through 14 at the same URL noted in my earlier post). Nobody seems to have noticed this either. Well one of the rays didn't fit the trend- 1g9/2 9/2[404]. It was off slightly. From this I was able to detect an calculation error in the math used to create the charts at this point. When this was corrected then total shell energy, as already calculated by me on the basis of the component rays, ended up conserved across deformation, just like it was for the other shells.

 

I already mentioned how doubled triangular number intervals are multipled in prolate deformed nuclei in the harmonic oscillator model. By tracking the various rays to their convergence points in the chart, and identifying them in terms of their quantum number values, I was able to make up rules that predicted similar identities for higher shells. I've also done this in the spin-orbit model tracking individual single-particle rays. You can use this type of relation to show how many different types of interactions show up between single particle rays: They can end up parallel but not crossing. They can cross and end up parallel. They can cross without ending up parallel, and they can repel and not end up parallel. Not only can you show how many ray pairs of each type there are in each shell (increases monotonically), but you can also show how their quantum number strings change within and between shells. All from playing with numbers.

 

Jess Tauber

Edited October 30, 2018 by pascal

[exchemist]

In the spin-orbit model of atomic nuclei mapping of single particle energy levels against ellipsoidal deformation results in a crazy-quilt of shifting slopes, quantum number string label exchanges, and other effects. In the simpler harmonic oscillator-only model all these energy changes are linear, with 'component rays' (each representing orbital partials) having fixed slopes that change monotonically for components of each shell.  The spin-orbit model appears to be a variant on this theme. For example, the highest-spin component in the harmonic oscillator model for each shell has slopes (in terms of h-bar omega-bar over delta deformation units) of: 1s1/2=0/3, 1p3/2=1/3, 1d5/2=2/3, 1f7/2=3/3, 1g9/2=4/3,. 1h11/2=5/3, 1i13/2=6/3, 1j15/2=7/3,. etc. It turns out that the numerators of all these slopes are equal to the quantum number ml value for the orbitals so labeled. My discovery.  In the spin-orbit model, the highest-spin pair in the larger spin-split orbital has an identical slope to the harmonic oscillator versions. More than this, if one extrapolates these nearly straight rays to lower energy values and greater oblate deformation, they all converge at 1.5 h-bar omega-bar at -2.7 epsilon deformation (see Figs. 4 through 14 at the same URL noted in my earlier post). Nobody seems to have noticed this either. Well one of the rays didn't fit the trend- 1g9/2 9/2[404]. It was off slightly. From this I was able to detect an calculation error in the math used to create the charts at this point. When this was corrected then total shell energy, as already calculated by me on the basis of the component rays, ended up conserved across deformation, just like it was for the other shells.

 

I already mentioned how doubled triangular number intervals are multipled in prolate deformed nuclei in the harmonic oscillator model. By tracking the various rays to their convergence points in the chart, and identifying them in terms of their quantum number values, I was able to make up rules that predicted similar identities for higher shells. I've also done this in the spin-orbit model tracking individual single-particle rays. You can use this type of relation to show how many different types of interactions show up between single particle rays: They can end up parallel but not crossing. They can cross and end up parallel. They can cross without ending up parallel, and they can repel and not end up parallel. Not only can you show how many ray pairs of each type there are in each shell (increases monotonically), but you can also show how their quantum number strings change within and between shells. All from playing with numbers.

 

Jess Tauber

So the physically observable implications are what, exactly? 

[tetrahedron]

As one develops more adequate Nilsson mappings of energies versus deformations, the finding of conserved individual total shell energies can guide creation of new models. We'll know where to expect to see the appearances of particular single particle levels. These models depend on experimental measurements for confirmation. So far we don't have one based on first principles. Rather the correction terms have numerical coefficient 'constants' which aren't constant at all, but change from shell to shell, apparently without any overarching mathematical motif that can be recognized. The constant goes up, then it comes down, as we move from shell to shell progressively. I'm aiming at a first-principles model.

 

Jess Tauber