Why are protons allowed in the nucleus?

[Andrew Wright]

TL;DR

I'm told that electrons can't collapse into the nucleus because they would be at rest, letting them have no uncertainty in position or momentum. But what about protons in the nucleus?

Feynman said in one of his lectures that electrons can't just fall down into the nucleus since the uncertainty principal wouldn't let them have a known position and momentum at the same time. The problem I have with this is that protons seem quite happy in the nucleus. Is my objection fair? Why/Why not? Thanks.

 

[Dale]

Summary:: I'm told that electrons can't collapse into the nucleus because they would be at rest, letting them have no uncertainty in position or momentum. But what about protons in the nucleus?

Feynman said in one of his lectures that electrons can't just fall down into the nucleus since the uncertainty principal wouldn't let them have a known position and momentum at the same time. The problem I have with this is that protons seem quite happy in the nucleus. Is my objection fair? Why/Why not? Thanks.

It is a valid concern. The big difference between the protons and the electrons (in this context) is their mass and the force.

The uncertainty relationship is between position and momentum. For the same momentum, a proton has much lower speed and therefore much much lower KE. So the minimum energy configuration for a proton is a confinement to a much smaller region than for an electron.

 

[Andrew Wright]

Does this mean that protons also cannot fall down into the centre of the nucleus?

 

[Andrew Wright]

I'm guessing that protons might not be attracted to the centre of the nucleus in the same way an electron might be.

 

[DaveC426913]

Protons aren't bound to the nucleus by gravity, so they have nowhere to fall to.
They are bound by the much stronger Strong Nuclear Force.

 

[PeterDonis]

I'm guessing that protons might not be attracted to the centre of the nucleus in the same way an electron might be.

Of course not. The nucleus has positive charge, so any individual proton is repelled by that. But there is also the strong interaction confining the protons. The size of the nucleus is a balance between those two forces; but that balance also has to take into account the uncertainty principle as it applies to the position and momentum of nucleons.

 

[PeterDonis]

Protons aren't bound to the nucleus by gravity, so they have nowhere to fall to.

I don't think the term "fall" was meant to imply gravity. Gravity on electrons in atoms is also negligible compared to the electrostatic interaction with the nucleus. By "fall" for electrons Feynman was referring to their electrostatic attraction to the nucleus.

 

[Andrew Wright]

Doesn't the strong force have a kind of "centre-of-gravity" inside the nucleus?

 

[Dale]

Does this mean that protons also cannot fall down into the centre of the nucleus?

More like there is a limit to how small a nucleus can be based on the forces involved and the uncertainty principle. This limit is much smaller than the corresponding limit for the electrons based on the relative strength of the forces and the different masses.

 

[PeroK]

Does this mean that protons also cannot fall down into the centre of the nucleus?

Since gravity plays no part, I'm not sure the concept of "falling" into the centre of the nucleus is relevant. The electrons are bound to the nucleus by the attraction of positive and negative charges.

The nucleus has a shell structure with the protons and neutrons occupying distinct energy levels. See here, for example:

http://hyperphysics.phy-astr.gsu.edu/hbase/Nuclear/shell.html

 

[Andrew Wright]

Doesn't the strong force have a kind of "centre-of-gravity" inside the nucleus?

I guess the strong force doesn't appear to always act from the same place like gravity and perhaps electric forces. So, I'm thinking there is no "centre" that attracts stuff.

 

[bob012345]

I guess the strong force doesn't appear to always act from the same place like gravity and perhaps electric forces. So, I'm thinking there is no "centre" that attracts stuff.

It is misleading to think of forces at the sub-atomic level like we think of forces in the ordinary world. They are very different. As I understand it, they aren't actually forces in the usual sense at all but rather interactions.

 

[etotheipi]

I guess the strong force doesn't appear to always act from the same place like gravity and perhaps electric forces. So, I'm thinking there is no "centre" that attracts stuff.

Gravitational and electric forces are central forces. Nuclear forces have non-central components.

 

[Andrew Wright]

There's this weird thing where the strong force suddenly stops acting over larger distances. I often wondered if it was a perfect cut-off or whether it just diminished very quickly.

 

[Nugatory]

Gravitational and electric forces are central forces.

That's true in the classical limit. It is not at all clear that that description applies at the scale of an atomic nucleus.

 

[PeterDonis]

Doesn't the strong force have a kind of "centre-of-gravity" inside the nucleus?

No. At most, there might be an approximation where each nucleon could be considered a "center" of interaction. But it's probably better, as others have said, to not think of the strong interaction as a typical "central force" at all.

 

[DrChinese]

There's this weird thing where the strong force suddenly stops acting over larger distances. I often wondered if it was a perfect cut-off or whether it just diminished very quickly.

The below description is a gross simplification, as the issues involved are quite complex. The strong force does not have an exact analogy to the electromagnetic force or the gravitational force, which in comparison are somewhat easier to understand (although a fuller understanding gets very complex too).

-----------------

The strong force is mediated by massive virtual gluons (primarily inside a neutron or a proton, a/k/a nucleons). Between nucleons in the nucleus, the strong force is mediated by virtual mesons (quark-antiquark pairs). Their mass effectively limits their "reach", as their lifetimes are quite short.

As a result, the strong force acts within particle structures that are on the order of 10^-15 meters or less ( a few femtometers). Anything further away is essentially immune to the strong force. So a helium nucleus is about this size, and in that area the strong force attracts the positively charged protons together much more strongly than the electromagnetic charge can push them apart. If you try to push 2 protons together, they will repel until you can confine them within the femtometer size range that the strong force manifests itself.

Technically it is not a perfect cut-off, but it may as well be once you get a few femtometers out. The probability of an exchange becomes so near zero that it can be ignored.

Again, this is a simplification. Read up on this yourself to get a better idea of how this works. Keywords: Strong force, quark, gluon, meson, color confinement.

 

[PeterDonis]

massive virtual gluons

Do you just mean that the gluons are (generally considered to be) off shell? Gluons (at least on-shell ones) are massless.

 

[DrChinese]

Do you just mean that the gluons are (generally considered to be) off shell? Gluons (at least on-shell ones) are massless.

I always understood that virtual gluons accounted for most of the effective mass/energy of the nucleus, with the quarks themselves only accounting for a small percentage. (Of course, I am sure that anything to do with the color force can get complicated pretty quick.) Perhaps "energetic" gluons would be a more accurate description.

 

[PeterDonis]

I always understood that virtual gluons accounted for most of the effective mass/energy of the nucleus

A better term might be "energy stored by the strong interaction".

For one thing, I don't think the perturbation theory approach works very well for this case, and perturbation theory is the only basis we have for "virtual particles" as a concept.

For another, I think I've seen descriptions in which this effective mass/energy is described as kinetic energy of the quarks, which are confined (by the strong interaction) in a small volume and therefore, by the uncertainty principle, have a large kinetic energy. Which then brings in valence quarks vs. "sea" quarks, and other further complications.

 

[snorkack]

van der Waals forces also drop off much faster than monopole-monopole forces, but again with no sharp cutoff.
How about an atom consisting of a proton, a neutron and a tauon? The tauon unlike an electron or even muon is more massive than either proton or neutron, but unlike them does not undergo strong interaction. What is on average nearer to proton - the more massive tauon or the strongly interacting neutron?

 

[PeterDonis]

What is on average nearer to proton - the more massive tauon or the strongly interacting neutron?

The average distance from the center of an atomic orbital varies inversely with the mass of the particle in the orbital, so a tauon, with a mass about 3.5 million [correction: 3.5 thousand] times that of an electron, would have an average distance from the center about 3.5 million [correction: 3.5 thousand] times smaller than the Bohr radius of $5.3 \times 10^{-11}$ meters. That comes to about $1.5 \times 10^{-14}$ meters, or about 100 times smaller [correction: 10 times larger] than the size of the nucleus.

 

[snorkack]

The average distance from the center of an atomic orbital varies inversely with the mass of the particle in the orbital, so a tauon, with a mass about 3.5 million times that of an electron,

IIRC, it is more like 3.5 thousand times.

 

[PeterDonis]

IIRC, it is more like 3.5 thousand times.

Oops, yes, I put in an extra factor of a thousand. I'll edit my post to correct.

 

[snorkack]

But now going on to more substantial things.
The average distance to centre of orbital goes down with inverse of the mass. But there is nothing important at the centre of the orbital, just an empty point of space.
When it is the tauon that is massive and the nucleus that is light, the distance between tauon and nucleus is also constrained by the mass of nucleus.
If neutron stays closer to proton than the tauon does, this cannot be because the tauon is lighter than neutron (it is the neutron that is lighter). It could only be because neutron is subject to strong force which is stronger than the electrostatic force that attracts the tauon.

 

[PeterDonis]

there is nothing important at the centre of the orbital, just an empty point of space.

I did not use the average distance to the center of the orbital because anything "important" was at that particular point. I used it because it's the best we can do at estimating the effective "size" of the orbital, in order to compare it with the effective size of the nucleus.

When it is the tauon that is massive and the nucleus that is light, the distance between tauon and nucleus is also constrained by the mass of nucleus.

I don't understand what you mean here.

If neutron stays closer to proton than the tauon does, this cannot be because the tauon is lighter than neutron (it is the neutron that is lighter).

Nobody ever said it was so I don't understand what you're arguing against here.

It could only be because neutron is subject to strong force which is stronger than the electrostatic force that attracts the tauon.

The strong force being stronger than the electrostatic force is indeed why the nucleus is smaller than the effective size of the tauon's orbital in this case, yes.

 

[snorkack]

I did not use the average distance to the center of the orbital because anything "important" was at that particular point. I used it because it's the best we can do at estimating the effective "size" of the orbital, in order to compare it with the effective size of the nucleus.
I don't understand what you mean here.

Distance from charge to centre of orbital is not the best estimate for the effective distance between charge and nucleus to compare with the effective size of the nucleus, because what is relevant is the full distance between charge and nucleus. Which consists of two arms - from charge to fulcrum, and from fulcrum to nucleus. When the charge is massive compared to nucleus, as is the case with tauon, both arms are important, and the sum of both arms is the better comparison to the size of the nucleus than just the arm fulcrum to tauon.

 

[DrChinese]

A better term might be "energy stored by the strong interaction".

For one thing, I don't think the perturbation theory approach works very well for this case, and perturbation theory is the only basis we have for "virtual particles" as a concept.

For another, I think I've seen descriptions in which this effective mass/energy is described as kinetic energy of the quarks, which are confined (by the strong interaction) in a small volume and therefore, by the uncertainty principle, have a large kinetic energy. Which then brings in valence quarks vs. "sea" quarks, and other further complications.

Mass of 2 ups quarks and 1 down quark (the confined quarks of a proton) = 9 MeV
Mass of fully-dressed proton = 938 MeV

Difference due to "what you said" = 929 MeV. Yeah, pretty complicated, pretty quick.

 

[PeterDonis]

Difference due to "what you said" = 929 MeV. Yeah, pretty complicated, pretty quick.

Yes, and it gets even more complicated when you realize that even the 9 MeV attributed to the rest masses of the valence quarks is really due to things like electroweak symmetry breaking and the Higgs interaction, assuming our current best model of that is correct, which it might not be.

 

[PeterDonis]

Distance from charge to centre of orbital is not the best estimate for the effective distance between charge and nucleus to compare with the effective size of the nucleus

The "nucleus" is not one thing; it's multiple particles. Unless you are considering a hydrogen-1 nucleus, but in that case the center of the orbital is where the proton is, so your objections don't even apply to that case in the first place.

For nuclei consisting of multiple nucleons, the average effective distance from the particle in an orbital (electron or tauon or whatever) to any nucleon is indeed the distance to the center of the orbital. If you want to deal with the tauon's attraction to individual nucleons instead of just a "nucleus", then you are talking about a different model altogether which is beyond what is being discussed in this thread.

what is relevant is the full distance between charge and nucleus. Which consists of two arms - from charge to fulcrum, and from fulcrum to nucleus.

I have no idea what you are talking about here. Do you have a reference?