Talk:Magic number (physics)

Ni-28 is not the most proton rich isotope, regardless of what the cited source says.

Ni-28 has a 7/5 Proton/Neutron ratio. He-3 has a 2/1 Proton/Neutron ratio. H-1 has a 1/0 Proton/Neutron ratio.

Ni-28 is the most proton rich isotope beyond helium-3 —Preceding unsigned comment added by 70.130.130.193 (talk) 19:03, 7 April 2009 (UTC)

First Ni-28 is poorly named. Ni has 28 protons, so having Ni-28 would mean zero neutrons. Not physically possible. The article as mentioned Ni-48 as being doubly magic, with 20 neutrons and 28 protons. This is not a "real" isotope. Here, I use real to mean having a measurable half-life (or stable). Ni-49 has a half-life of ~300 ns (^[1]). Ni-48, with less neutrons stabilizing this nucleus would not be able to hold together. Ni-56 is doubly magic (28 of each, protons and neutrons). It is not stable, but its half-life (5.9 days) is orders longer than either Ni-55 (212 ms) or Ni-57 (35.60 hrs).18.189.8.62 (talk) 08:40, 3 December 2012 (UTC)

Actually Ni-48 can hold together (half-life >500 ns, theorized to be 10 ms), though I suspect this is only because it is doubly magic. Double sharp (talk) 02:14, 20 July 2014 (UTC)

Is there a generally-accepted definition of what 'magic' means, apart from the vague idea "more stable than the rest" or "with complete shells" (although the shell model is not universally accepted either)? In an article in the current issue (7) of Physik Journal (for members of the German Physics Society/Deutsche Physikalische Gesellschaft), Jan Jolie and Peter Reiter from Darmstadt state that the only criterion for a shell closure (presumably w.r.t magicity) is the energy gap between completed levels. This is backed up with data relating to 30Si and 24O (both with 16 neutrons), whereby 24O is deemed to be magic due to the larger energy gap and 30Si, with a smaller gap, is not. On this basis, 24O is doubly magic (Z=8 and N=16). It certainly seems odd that the term "magic" seems to have mean different things to different people: Everyone 'knows' what it means but there doesn't seems to be a universal definition. --TraceyR (talk) 13:07, 8 July 2009 (UTC)

Magic are such numbers of protons/neutrons which fills the corresponding shell in the corresponding energy well. Now there are corrections to the numbers in heavy nuclei, because of collective and other correlations, some energy levels from the upper shell are moved down to the previous. This is where the confusion lies. So levels from the upper shell would be filled before closing the lower shell, which means, for stable nucleus to occur the numbers should be modified to reflect that. For reference one could go to atomic physics, where some electron orbitals of higher level are filled before the last level, which is consequence of energy levels/sublevels ordering. I think it's acceptable to use the shell gap as criterion. — Preceding unsigned comment added by 78.90.43.45 (talk) 07:14, 14 July 2011 (UTC)

Several people have found connections between the nuclear (semi)magic numbers and the Pascal Triangle's values.

Because of the way spin-orbit coupling splits orbitals and the way the energy levels of these parts interact, there are two different types of (semi)magic number. They can be rationalized in pairs 2 2 : 6 8 : 14 20 : 28 40 : 50 70 : 82 112 : 126 168 : 184 240 and so on. The right member of each pair is a doubled Pascal Triangle tetrahedral number- in fact each in sequence as- 2=2x1, 8=2x4, 20=2x10, 40=2x20, 70=2x35, 112=2x56, 168=2x84, 240=2x120, etc.

The left member of each pair can be represented two ways, either as a doubled tetrahedral value with the doubled triangular value directly above it in the Triangle subtracted from it (for example 112-30=82, 168-42=126, where 30 and 42 are doubled triangular numbers), or as doubled tetrahedral numbers plus an even integer increment that comes from the doubled natural number line, where the particular number falls along the same shallow diagonal as the doubled tetrahedral numbers it adds to, such shallow diagonals also used to sum Fibonacci numbers (here doubled).

There are several more such Pascal-related mathematical behaviors in the nuclear system. Interestingly there are also Pascal mathematical behaviors in the electronic system, though these are based not on doubled, but normal single values. 69.121.117.192 (talk) 23:56, 28 September 2012 (UTC)

The double tetrahedral numbers shown above are generated by the harmonic oscillator number in spherical nuclei. It turns out that one can extend the Pascal Triangle effect over ellipsoidally deformed nuclei as well. Double tetrahedral numbers show up on the surface only for spheres, which can be thought of as default ellipsoids. In the sphere the oscillator magics have double triangular number intervals, each adding the next increment leading to the next magic double tetrahedral number.

Thus each double triangular increment appears only once. Also, every magic number has a double triangular number interval. However, if the nucleus is deformed, the manner in which double triangular numbers contribute to each new sum changes. The wave ratio, related to the ellipsoid's axial ratio but not identical to it, is one way to define the deformation. The wave ratio for the sphere is 1:1, where normally the polar semimajor axis is defined as the numerator, and the equatorial semimajor axis as the denominator in Nilsson depictions.

It is found that the numerator determines how many copies of a double triangular number are needed in sequence. For the sphere we have one copy, and the ratio is 1:1. In the case of ratio 2:1, a prolate nucleus, we need TWO copies in sequence so 2,2,6,6,12,12,20,20,30,30... and so on, summing to 2,4,10,16,28,40,60,80,110,140... which is the correct system of magic numbers for this ratio. For 3:1 we need THREE copies. For the simple harmonic oscillator model there are no exceptions.

With oblate nuclei of wave ratio 1:2 we find instead that there is still only one copy of each double triangular number increment, but that the the double triangular number distance occurs not for EVERY magic number, but now EVERY OTHER. For 1:3 that distance is EVERY THIRD. This rule breaks down only early in the count when one has not yet cumulated enough magic numbers for the double triangular number interval to be seen. These early sequences can still be easily generated with a different rule.

When neither the numerator nor denominator of the wave ratio is 1, then we have both rules come into play. Thus for the simple harmonic oscillator the wave ratio's numerator, related to the polar semimajor axis, determines how many copies of any double triangular number interval must be utilized (multiplication), while the denominator, related to the equatorial semimajor axis, tells you how many magic numbers you need to jump to find double triangular distances (division).

When other effects such as spin-orbit coupling are added in this pattern breaks down. Even so there is still Pascal-based order remaining, but it is distributed very differently. 69.121.117.192 (talk) 13:04, 28 March 2013 (UTC)

Nuclear isomerism occurs when there is competition for relative position between low spin final orbital partials near the end of nuclear period analogues and the highest spin orbital partial belonging to the next higher period analogue. Pascal behavior is found in that the positioning itself, at least for neutrons in spheres, can be calculated simply by subtracting successive double triangular numbers from the classical spin-orbit magic numbers thusly: 28-0=28, 50-2=48, 82-6=76, 126-12=114, 184-20=164. These differences represent the count number where the intruding high spin orbital partial ENDS. One can determine where it begins by subtracting the size of the partial (so for example 1g9/2 is ten neutrons, so subtracting 10 from 48 gives 38, which is correct, while subtracting 12 from 76 gives 64, also correct, and 14 from 114 gives 100, also correct. 96.234.78.23 (talk) 18:42, 30 November 2014 (UTC)

Is 6 a nuclear magic number? Mathematically it *should* be, based on the above. But in terms of energy gap and reaction stability of carbon, it doesn't actually seem to be. Yet there is the fact that the magics are reflected in the cosmic abundances of the elements. Magic nuclei are much more commonly attested in these abundances than others. Carbon is one of the most abundant 'metals' in the universe. 69.114.46.208 (talk) 19:30, 24 January 2017 (UTC)

http://en.wikipedia.org/wiki/Electron_shell <-- Seems to be the source

"This is equivalent to saying that the outer electron shell is 'full'."

That article has tons of sources for this claim. 75.70.89.124 (talk) 07:02, 31 July 2013 (UTC)

This article does not make it obvious enough that magic numbers are computed separately for protons and neutrons. Note that the usual pictures that people draw of nuclei have a ball of separate protons and neutrons. But there is no Pauli exclusion between them, so there is no need for them to be separate. Also, ones of opposite spin don't have exclusion. Gah4 (talk) 00:55, 12 November 2016 (UTC)

It would be helpful to simply LIST the stable or nearly-stable isotopes with just magic numbers of neutrons. 2, 8, 20, 28, 50, 82, and 126. Easy to do, too, but it's not so easy to make it complete.
These start the list: He-3, N-15, S-38, Cl-37, Ar-38, K-39, all stable.
Ti-50, V-51 (its only stable isotope), Cr-52, Fe-54.
Rb-87 with a very long half-life, Sr-88, Y-89 (its only stable isotope), Zr-90, and Mo-92.
These are the elements helium (two stable isotopes), nitrogen (two stable isotopes), sulfur (four stable isotopes), chlorine (just two), argon (just three), potassium (just two), titanium (five stable isotopes), vanadium (just one), chromium (four stable isotopes), iron (four stable isotopes), rubidium (one stable isotope Rb-85), strontium (four stable isotopes), yttrium (just one), zirconium (four stable isotopes), and molybdenum (six stable isotopes, but Mo-92 being the lightest one).
I will let you other weenies figure out the ones with 82 and 126 neutrons. I would have thought that there was a magic number of neutrons associated with thorium-232, uranium-238, and plutonium-242, but this is not so.
The number 20 + 20 = 40 is an interesting one, but it is not a magic one.
Ca-40, with 20p and 20n, is very, very stable for its atomic mass number.
K-40, with 19p and 21n, decays three ways, into Ar-40 (18p + 22n) or Ca-40 (20p + 20n), by electron capture or positron emission.
Also, the two nearby isotopes, K-39 and K-41, are both stable, but Ca-39 and Ca-41 is radioactive, despite having 20 protons. Ar-39, Ar-41, and Sc-41 are all radioactive. The most common isotope of argon on the Earth is Ar-40, but in the Universe it is Ar-36 (18p + 18n).

47.215.183.159 (talk) 02:02, 4 September 2017 (UTC)

I believe that the large amount of Ar-40 in the earth's atmosphere is due to decay of K-40. Also, elements with nice oxides tend to be more abundant in the earth's crust than ones that don't. (Oxides float.) I am not, then, suprised that it is different from the rest of the universe. Gah4 (talk) 05

51, 4 September 2017 (UTC)

I listed the elements to cover the proton magic numbers. Feel free to add a list of isotopes with magic neutron numbers. --mfb (talk) 14:52, 4 September 2017 (UTC)

Th and U have semi-closed shells, at least theoretically, though since actinide nuclei are not spherical the applicability of the unaltered shell model is a little suspect. Double sharp (talk) 15:06, 3 December 2017 (UTC)

Recently more nuclides were added to a paragraph on double magic nuclides. I believe that is fine, but also that we should separate stable (or almost stable) from unstable ones. Ca-40 and Ca-48 have a 1e28s or so half life, close enough to stable for most of us. The extreme nickel and tin nuclides, while more stable than many, are not very stable. Gah4 (talk) 23:16, 8 April 2019 (UTC)

Looking over this article, a number of rather minor mistaken terminology uses, omissions, and inconsistencies are noticeable throughout. I just added 56Ni to the list of doubly magic nuclei (see discussions above as to whether this list may really be useful), but it was strange to see the exotic species 48Ni and 78Ni without the long-known 56Ni in such a list. "Double magic" seems less preferred than "doubly magic" in the literature (a quick internet search for nuclear physics shows more than five times the results for the latter than the former). 4He is called "most abundant (and stable)" lending a superlative to a binary fact of beta-stability ("tightly-bound" would be correct, but "most stable" is not sensible). Another misleading term is "proton-rich isotope" which is a misnomer; an isotope is defined (in part) by its proton number; therefore, while "neutron-rich isotope" and "neutron-deficient isotope" are sensible terms, "proton-rich isotope" is not meaningful (e.g., all isotopes of nickel have 28 protons by definition). 48Ca is (as noted above) not in fact stable, but very long lived (in contrast to what is here). Contrast this with 208Pb, which is stated as "the heaviest stable nuclide"; while in fact true, 209Bi is very long lived which leads to an inconsistency for the treatment of long-lived isotopes. The lone (and incomplete paragraph) on hassium feels rather biased (considering other omissions) and hence out of place over a decade after its discovery; note that the article does not delve into the concept of shifting magic numbers far from stability, and as it stands the article does not support Z=108 nor N=162 as magic numbers. I would like to clean up these points in a few weeks' time, if there is no extended discussion to oppose these wide-ranging but relatively simple updates. DAID (talk) 20:04, 14 May 2019 (UTC)

I like to use nuclide over isotope, but the latter is commonly used, where you note that it should not be. Isotope might be the WP:COMMONNAME, though I am not sure about that. Gah4 (talk) 21:57, 25 September 2024 (UTC)

This is a Compound (linguistics) and has an etymology. Catchpoke (talk) 15:33, 12 July 2021 (UTC)

In normal cases alpha decay energy decreases as neutron number increases, but we know that there are huge increases when passing N = 82 and N = 126. Actually, the energy increase is present around every magic number. Data copied from NNDC.

129.104.241.231 (talk) 18:10, 25 September 2024 (UTC)

^135,137Ba, ¹³⁹La, ¹⁴⁹Sm, and ^151,153Eu are beta-stable. One may expect that the intermediate beta-stable nuclides with A = 141, 143, 145, and 147 are ¹⁴¹Ce, ¹⁴³Pr, ¹⁴⁵Nd, and ¹⁴⁷Pm, and the correct result would give a surprise (only ¹⁴⁵Nd is correct and the others would undergo β^-). This must have some connection to the closed shell N = 82. 169.155.234.214 (talk) 18:56, 23 December 2024 (UTC)

I call it "141/2/3/6/7 abnormality", as the nuclides with the lowest energies among isobars A = 141,142,143,146,147 are not ¹⁴¹Ce, ¹⁴²Ce, ¹⁴³Pr, ¹⁴⁶Nd, and ¹⁴⁷Pm, but are ¹⁴¹Pr, ¹⁴²Nd, ¹⁴³Nd, ¹⁴⁶Sm, and ¹⁴⁷Sm.

Major abnormality occurs for A = 142 and 143: ${\displaystyle Q_{2\beta ^{-}}({}^{142}\mathrm {Ce} )=1416.5\,\mathrm {keV} }$, ${\displaystyle Q_{\beta ^{-}}({}^{143}\mathrm {Pr} )=933.9\,\mathrm {keV} }$;

Minor abnormality occurs for A = 146 and 147: ${\displaystyle Q_{2\beta ^{-}}({}^{146}\mathrm {Nd} )=70.7\,\mathrm {keV} }$, ${\displaystyle Q_{\beta ^{-}}({}^{143}\mathrm {Pr} )=224.1\,\mathrm {keV} }$.

So we see that beta-stability for A = 146 and 147 is biased towards nuclides with much lower alpha stability:

${\displaystyle Q_{\alpha }({}^{146}\mathrm {Sm} )-Q_{\alpha }({}^{146}\mathrm {Nd} )=Q_{2\beta ^{+}}({}^{146}\mathrm {Sm} )-Q_{2\beta ^{+}}({}^{142}\mathrm {Nd} )=}$ (a negative number with small absolute value) - (a negative number with large absolute value) >> 0 (in fact ${\displaystyle ((-70.7)-(-1416.5))\,\mathrm {keV} =1345.8\,\mathrm {keV} }$);

${\displaystyle Q_{\alpha }({}^{147}\mathrm {Sm} )-Q_{\alpha }({}^{147}\mathrm {Pm} )=Q_{\beta ^{+}}({}^{147}\mathrm {Sm} )-Q_{\beta ^{+}}({}^{143}\mathrm {Nd} )=}$ (a negative number with small absolute value) - (a negative number with large absolute value) >> 0 (in fact ${\displaystyle ((-224.1)-(-933.9))\,\mathrm {keV} =709.8\,\mathrm {keV} }$).

But why beta-stability for A = 142 and 143 is biased towards nuclides with much greater alpha stability? Because there is no abnormality for A = 138 and 139!

${\displaystyle Q_{\alpha }({}^{142}\mathrm {Nd} )-Q_{\alpha }({}^{142}\mathrm {Ce} )=Q_{2\beta ^{+}}({}^{142}\mathrm {Nd} )-Q_{2\beta ^{+}}({}^{138}\mathrm {Ce} )=}$ (a negative number) - (a positive number) << 0 (in fact ${\displaystyle ((-1416.5)-692.7)\,\mathrm {keV} =-2109.2\,\mathrm {keV} }$);

${\displaystyle Q_{\alpha }({}^{143}\mathrm {Nd} )-Q_{\alpha }({}^{143}\mathrm {Pr} )=Q_{\beta ^{+}}({}^{143}\mathrm {Nd} )-Q_{\beta ^{+}}({}^{139}\mathrm {Ce} )=}$ (a negative number) - (a positive number) << 0 (in fact ${\displaystyle ((-933.9)-278.9)\,\mathrm {keV} =-1212.8\,\mathrm {keV} }$). 129.104.241.88 (talk) 10:07, 6 January 2025 (UTC)

N = 82 closed shell is the main culprit of the alpha-instability of some lanthanide nuclides with N = 84～88. Z = 82 and N = 126 closed shells, together, kills every nuclide with Z ≥ 84 and N = 127～136.

Strangly enough, those magic numbers have also impacts on the beta-stability of neutron numbers above them. There are no beta-stable odd-even nuclides with N = 84 or 128, for example. ~2025-36573-71 (talk) 13:39, 26 November 2025 (UTC)