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I was wondering this: suppose you have two oxygen atoms. They will both have 8 protons and 8 neutrons in the nucleus (at least if they are the most common isotope). Now, will all those particles be arranged in the same way in both atoms? If they are, why would that be, and if not, does that affect the element's properties in any way?

But then I also thought that maybe the uncertainty principle doesn't let us even ask this question. Maybe you can't tell the particles' positions so accurately, so all you can say is that you have 8 protons and 8 neutrons all together in a small space.

So, which one is it? Can we even tell where all the particles are, and if we can, does it matter exactly how they are arranged?

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There are some forces which drive to favor certain configurations. See – Ryan Thorngren Sep 15 '12 at 2:03
@user404153 Again, the semi-empirical mass formula is not the best answer here as it works explicitly to avoid understanding the structure. Shell models address the question. – dmckee Sep 15 '12 at 2:15
The uncertainty principle doesn't ever 'not let us ask' any questions, as you can see here. Even if it did, proton/neutron arrangement isn't even nearly on that scale. A proton's RMS-charge diameter is $1.1*10^{20}$ times the plank length - 100 quintillion or 100 billion billion times the unit beyond which we can't measure. – Ehryk Sep 15 '12 at 2:17
@dmckee Thanks, I see your point. – Ryan Thorngren Sep 15 '12 at 2:24
up vote 4 down vote accepted

Javier, try looking at What is an intuitive picture of the motion of nucleons? to start.

The short answer is that it is as reasonable to say that they are identical as it is to say that the configuration of electrons in multiple atoms of a single element are identical. That is, there is a set of position (or momentum1) distributions to which they conform.

1 The position and momentum distributions turn out to be linked to each other by Fourier transformations, so information required to specify one is the same as that required to specify the other. Nuclear physicist mostly concern themselves with the momentum distributions.

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So, which one is it? Can we even tell where all the particles are, and if we can, does it matter exactly how they are arranged?

We can tell/model the distribution of the particles in the nucleus, as dmckee states, and it does matter how they are arranged as we go up the periodic table of elements. The internal arrangement/distribution give rise to the dipole and quadrupole moments of the nuclei .

Thus in aggregate the distribution of protons in the nuclei do characterize them.

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