Distribution of protons in nucleus

[gildomar]

Is the most stable/likely configuration of protons in heavy nuclei that of being evenly distributed throughout the nucleus? As opposed to something like a spherical distribution?

 

[ChrisVer]

Although it is weird talking about where protons are in the nuclei, I'd guess they would prefer being somewhat near the center, where the electromagnetic repulse is less... Of course things are more complicated...

Also what is the difference to the distributions you proposed? the one means homogeneous the other means spherical :P the one doesn't cancel the other out

 

[Bill_K]

The distribution of protons in the nucleus is directly related to its charge distribution. Generally the distribution is uniform throughout most of the nucleus, while near the surface it tapers off.

 

[gildomar]

@ChrisVer - I was thinking that the protons would be scattered uniformly throughout the nucleus, as opposed to something like them being mainly near the surface, like a shell.

@Bill_K - I was thinking it was something like that. Is that mostly due to the electromagnetic interaction between the protons, given that the strong force is relatively blind to the difference between protons and neutrons?

 

[K^2]

Actually, if you are comparing protons and neutrons, you are going to find that it's neutrons that dominate the exterior. The reason is that if a proton can become a neutron and reduce overall energy of the nucleus, it's going to do so via $\beta^+$ decay. So most energetic proton will have roughly the same total energy as most energetic neutron. And because protons have repulsion energy added in, the particles with highest kinetic energy are neutrons, and so they can be found a bit further from the center of the nucleus.

You should also keep in mind that while the distributions are fairly uniform in the interior, they are also correlated. Protons and neutrons like to hang out in pairs in the interior, and there is very good evidence for larger clusters.

 

[ChrisVer]

Still this conversation scares me :p it's like we are dealing with protons as people dealed with atoms before QM... you can't say where are the protons or the neutrons in the nuclei...
what you can say is where they'd prefer to be... For example the protons even if they'd prefer to be a little bit closer to the center (to avoid EM repulsion), they are also subatomic particles and they obey Heisenberg's uncertainty principle- the more localized the more energetic...

I guess the best way to think of the nuclei is that of a nucleonic soup with a fine homogeneity... protons turn to neutrons and vice versa by a continuous interaction with puon mesons (that's one explanation of why the neutron becomes stable within the nuclei, since it always changes to proton and a proton changes to a neutron, in strong interaction characteristic times and thus faster than its weak interaction).

 

[K^2]

The pion exchange can only switch a proton and neutron places. It can't change the total number of either. And since protons and neutrons are distributed to begin with, and one proton cannot be distinguished from any other proton, you can just think of that as a contribution to propagator. It's kind of like color switching in strong interaction. It's there, but you don't have to think about it.

As for picturing particles vs picturing a homogeneous soup, former has certain advantages. Like I said, the distributions are correlated. Properly, you need to describe this with a multi-dimensional wave function. If we forget about all of the nuances of multi-particle theory, and just think about the N valence nucleons, your wave function is 3N dimensional. This is very hard to picture. Instead, you can picture various arrangement of point particles in the nucleus, and think about the state being a superposition of these. Just makes your brain hurt less. But yeah, it's all quantum.

 

[gildomar]

@K^2: Thanks for clearing that up; what I was reading didn't really explain how the two of them were distributed. As for the protons and neutrons being correlated, is that something like the weak bonding of Cooper pairs in superconductors?

@ChrisVer: I realize that the discussion sounds like we're talking about the neutrons and protons in a classical sense, but it's a little easier to talk about them in that way for the time being. But I did make sure to phrase the question at the beginning as the most likely place to find them (not where they actually are) in acknowledgment of both the uncertainty principle and the probability densities of their wavefunctions.

 

[K^2]

As for the protons and neutrons being correlated, is that something like the weak bonding of Cooper pairs in superconductors?

No, that's quite different. Cooper pairs form from identical fermions due to interaction with the lattice. Because they are fermions, they cannot be in the same exact state, and in fact, experience Pauli repulsion. As a result, a Cooper pair is a fairly "spread out" object. And not just in the sense of being delocalized, but expectation value for distance between two particles in a pair is rather large.

The pn pairs in nucleus are "tight". Again, they are still delocalized as a pair, but expectation of the distance between two particles is small. This is only possible because proton and neutron are distinguishable, and Fermi statistics does not apply. They can be in the same state, and because of isospin symmetry, they basically are.

Pretty much, the only significant similarity is that in both cases the pair has integer spin, and so behaves as a boson. Whether that last bit has any critical impact on how these pairs behave in a nucleus, I just don't know.