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My understanding is that the strong interaction between nucleons, while very strong, is much weaker than the strong interaction between quarks within a nucleon. So even though an atomic nucleus in an incredibly complicated strongly coupled QCD mess, there is still a sense in which the nucleons retain their individual identities, and we can talk about their individual locations within the atomic nucleus.

I believe that the strong interaction is equally strong between protons and neutrons, and the main difference between them is the existence of an electrostatic repulsion between protons (which is much weaker than the attractive strong interaction that holds them together). It seems reasonable to me that in the ground state of an atomic nucleus, the protons would spread out symmetrically as far apart as possible in order to reduce their electrostatic repulsion energy.

A standard soccer ball has 32 symmetrically arranged faces - 12 pentagons and 20 hexagons. None of the pentagons touch each other.

Magnesium has a slightly radioactive isotope magnesium-32 with 12 protons and 20 neutrons, and a relatively long (by nuclear standards) half-life of 86 milliseconds.

Putting all of the observations together:

Do magnesium-32 nuclei look like soccer balls?

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there is still a sense in which the nucleons retain their individual identities

Sort of. Don’t forget that nucleons are indistinguishable particles. Your nuclear state must be anti symmetric against the exchange of the proton in “this” orbital with the proton in “that” orbital.

we can talk about their individual locations within the atomic nucleus.

Nope, absolutely not. Like electrons in an atom, nucleons in a nucleus are described by wavefunctions which fill the nuclear volume. A soccer-ball or buckyball model of a nucleus is at least as wrong as the planetary electron orbits of the Bohr model.

A useful exercise for a quantum system is to find the kinetic energy associated with its particles, and from there compute their de Broglie wavelength. Nucleons emphatically fill the nucleus, and electrons emphatically fill the atom. However, at the soot-friendly temperatures where buckyballs and nanotubes form, it is reasonable to talk about carbon atoms being localized within their lattice.

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