system
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human
Are the protons and neutrons in a specific isotope always arranged in the same/similar way?

My understanding is that neutrons help glue the protons together via the residual strong force, while also helping to spread the protons out to reduce coulomb repulsion by physically separating them.  This makes a bit of sense, but from this interpretation then, it would seem to me that its not just the number of protons/neutrons in the nucleus that matter, but also (to an extent) the physical placement of the neutrons and protons with respect to one another.  It would seem that an atom with ALL of its protons clumped together in a ball that is wrapped in a shell of neutrons wouldn't be stable, despite having the correct number of protons/neutrons.  I know this is a rather extreme example, but the reason for giving it is that it makes me believe that just having the right number of protons and neutrons isn't enough.  And it also raises a bunch of other questions for me such as:  

\- How many stable configurations are there for a given atom?

\- Are protons/neutrons organized "rigidly" into a nearly spherical nucleus?  Or can the shape of the nucleus change/deform (either with time, or maybe just being statically asymmetrical)?

\- If the distribution of protons/neutrons is even slightly asymmetrical, wouldn't this give the nucleus some kind of dipole moment?

\- Is the arrangement of the protons/neutrons always the same/similar for a given isotope?

\- For extremely heavy elements that are already unstable to begin with, does variations in the distribution of protons/neutrons help to increase or decrease stability from atom to atom?

\- Do protons and neutrons move around while in the nucleus, or are they more or less fixed in place?

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Am I barking up the complete wrong tree here, and trying to push my classical views on a quantum system in a way that doesn't work?

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Any kind of info would be greatly appreciated!
gpt
Yes, the arrangement matters. Just like atoms have discrete energy levels corresponding to arrangements of electrons, nuclei do as well.

>How many stable configurations are there for a given atom?

For a given nuclide, at most one (the ground state). However many nuclei are unstable even in the ground state.

>Are protons/neutrons organized "rigidly" into a nearly spherical nucleus? Or can the shape of the nucleus change/deform (either with time, or maybe just being statically asymmetrical)?

Protons and neutrons occupy quantum-mechanical orbitals. Most nuclei are not spherical. They can have both static or dynamical deformations.

>If the distribution of protons/neutrons is even slightly asymmetrical, wouldn't this give the nucleus some kind of dipole moment?

Not a dipole moment, because that would violate parity and time reversal, but the next leading order deviation from a spherical charge distribution is an electric quadrupole moment. Non-spherical nuclei (so most nuclei) have nonzero quadrupole moments.

>Is the arrangement of the protons/neutrons always the same/similar for a given isotope?

If they’re both in the ground state, or the same excited state, yes.

>For extremely heavy elements that are already unstable to begin with, does variations in the distribution of protons/neutrons help to increase or decrease stability from atom to atom?

In any nucleus, the arrangement of protons and neutrons plays a role in their stability.

>Do protons and neutrons move around while in the nucleus, or are they more or less fixed in place?

They “move” quantum-mechanically, just like electrons in atoms.