As far as I understand , a particular sub-shell is filled with either protons or neutrons, $2*(2l+1)$ number of them, and never both together since protons and neutrons fill up levels separately in the shell model.So, are the magic numbers (2,8,20...) achieved by filling either of the energy levels corresponding to protons or neutrons? What if proton energy levels are filled( eg. $(1s)^2$ - 2 protons) and neutron sub-shells are not ( eg. $(1s)^2 + (1p)^4$ - 6 neutrons)? Will we still get the stability corresponding to magic number 2?

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    $\begingroup$ Related: physics.stackexchange.com/q/323350 $\endgroup$ – PM 2Ring Jan 23 '19 at 15:21
  • $\begingroup$ That does not really clear things up for me. $\endgroup$ – D.Mason Jan 23 '19 at 15:43
  • $\begingroup$ I wasn't implying that it answered your question, just that it has relevant info. FWIW, your example of 2 protons & 6 neutrons, $\mathrm{^8He}$ is much more stable than its neighbours, $\mathrm{^7He}$ & $\mathrm{^9He}$. It has a half-life of 119 milliseconds, they have half-lives around $10^{-21}$ seconds. See en.m.wikipedia.org/wiki/Isotopes_of_helium#List_of_isotopes $\endgroup$ – PM 2Ring Jan 23 '19 at 18:25
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    $\begingroup$ A glance at the usual charts of nuclide stability is enough to show that there is a band of stability around $A/Z \approx 2$ at low mass and if you follow it out to high mass you see the ratio dropping a bit. Are you asking what sets the stability criteria? Something else? $\endgroup$ – dmckee --- ex-moderator kitten Jan 26 '19 at 7:15
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    $\begingroup$ @D.Mason The answers I linked previously explain that the proton & neutron levels get filled approximately equally, and if they are too out of balance then nuclear reactions occur to achieve that balance. But for heavier elements stability requires a higher neutron : proton ratio. And for the heaviest elements complete stability is impossible. $\endgroup$ – PM 2Ring Jan 26 '19 at 10:05

Yes, a magic number is achieved when there is a magic number of either protons or neutrons. In your example of p=2 protons and n=6 protons ($^8$He), we might then expect expect it to have "magic" properties such as a preference for a spherical shape and large gap in energy between the ground state and the first excited state.

For nuclei with a magic number of protons and a magic number of neutrons (for instance $^{16}$O with p=8 and n=8) these "magic" properties are further enhanced and we call them "doubly magic nuclei".

Magic numbers are useful references when thinking about the properties of nuclei but there are many subtleties to consider. For instance, in the $^8$He case you mention, there are n=6 neutrons. This number is not "magic" but it is enough to close the sub-shell level (p$_{3/2}$) which can give it more stability compared to $^7$He and $^9$He (also because of the pairing effect which provides more binding when an even number of protons or neutrons is present). Additional complications are that some magic numbers are stronger than others, and that as we study nuclei further from stability the magic numbers appear to change.

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