About neutron decay in neutron stars

[tonyxon22]

Hi,
I was reading about free neutron beta decay the other day and it came to me the idea of neutron stars. As I understand, neutron stars are held together by gravity instead of strong force interaction (which I think is the mechanism that gives stability to neutron in common nuclei). So one would expect that neutrons forming a neutron star would decay into protons.
I read a similar thread about this. In this thread someone mentions that the electron chemical potential is what makes neutrons stable.
https://www.physicsforums.com/threads/neutron-decay-at-neutron-star.665571/
Unfortunately that thread it is already closed, so I kind of reopen it here by asking:
a) Why are there electrons in a neutron star? I saw a diagram of the structure of a neutron star on Wikipedia and it shows the outer crust to be composed of ions and electrons. However, the article does not mention anything about that. How do the electrons stay in the outer crust without repulsing each other into the universe?
b) What would be, with a little more detail, this electron chemical potential mechanisms that is referred? How does it act on the neutrons?
Thanks and best regards,

 

[ChrisVer]

a) where would the electrons be after the neutron-degeneracy prevents the gravitational collapse? just think that the neutron star is formed right when the electron-degeneracy of the white dwarf is replaced by the neutron degeneracy due to the core collapse. Obviously what could hold the electrons together are the strong magnetic fields and gravity. Also these electrons are highly degenerate.

b) In general the chemical potential for those electrons will have to be very small, almost zero, because of the highly-degenerate state they are into - they won't leave the Fermi Sea.

 

[tonyxon22]

Well thanks! I understand that gravity along a strong magnetic field that leaves the electrons in degenerate state is enough for keeping them together...
However, you mentioned that due to exactly that fact, their chemical potential has to be very small... And it is still enough for providing stability to the neutrons?
Regards,

 

[ChrisVer]

Well I may be wrong on my guess for the chemical potential.
But what lead me in the idea to say that it is very small, is because I want the neutrons to be "shielded" somehow. If the chemical potential is small, then the objects are in equilibrium. So if neutrons are to decay into protons and electrons at a rate X, then the electron will be captured to form a new neutron at the same rate X. So the stability is achieved. The same somehow works for the neutrons in the Big Bang Nucleosynthesis theory. When the equilibrium is broken, the neutrons will decay into protons (reaching the densities ratio $\frac{n_n}{n_p} \approx 0.3$ at around the BBN era) or later on will be captured and shielded into "deuterium", "tritium" and so on...
If you allow for a non-vanishing chemical potential, then you change those rates, and you make one-way interactions more favorable.

 

[ChrisVer]

In particular the decay:
$n + \nu_e \leftrightarrow p + e^-$
will be happening on both ways (left/right arrow)

 

[ChrisVer]

Also I may be wrong. When I think about it, I don't see where the produced electron be... all the electron states are occupied.

 

[tonyxon22]

I don’t think that the neutron star is in equilibrium but the neutrons are constantly decaying into protons and electrons and then reappearing in the inverse process with the same rate. I prefer to think that the decay does not happen at all, even doe I don’t understand how the stability of the neutron is possible

 

[ChrisVer]

Then maybe you'll like my last message. By time I also tend to think this is better.
$n \rightarrow pe \bar{v}_e$ cannot happen because electron states are all occupied.

 

[tonyxon22]

So that is the reason? The neutrons don't decay because of no electron states available?

 

[ChrisVer]

So it seems. What would happen to the creared electron with no free state to occupy? Even if the decay happened in a suppressed way (because of the fermi-dirac distribution function in $\Gamma$ ) the electron would immediately be captured by the proton as the rest electrons did before it at the neut.star formation

 

[mfb]

So that is the reason? The neutrons don't decay because of no electron states available?

Yes. And if there are empty accessible states, they get filled quickly.
Both directions together give cooling as some neutrinos can escape.

 

[snorkack]

Look at, say, the contrast between rhenium 187 atom and rhenium 187 nucleus. Or the atoms and nuclei of dysprosium and holmium 163.

Why are there electrons in a neutron star? Why don´t neutrons decay?

If you had a neutron star consisting of only neutrons, then yes - neutrons would decay. They would decay because there are no electrons in the neutron star, and all electron states are free.
However, as the low-lying electron states fill (and some neutrons turn into protons), the electrons would be emitted into high-lying states. These electrons would escape the neutron star... and leave protons behind.
So the neutron star acquires a net positive charge and a positive electric potential. This hampers the further escape of electrons alone into infinity.
Alternatively, the electron might escape together with a proton. This is not significantly hindered by electric field, but the escape of proton is hampered by gravity field.
So, the low-lying states of electrons inside and around neutron star get filled. Eventually, all electron states are full to the level where the neutron decay produces no energy. At which point neutron beta decay would stop.
(Actually, it would go on because neutrons and protons also interact with each other. But the net result is still that there would be mostly neutrons, and a small amount of protons and electrons).

 

[nikkkom]

a) Why are there electrons in a neutron star?

If you would magically remove all electrons (and protons) from neutron star, its neutrons now can decay.

As they decay and produce protons and electrons, electrons fill available energy levels, starting from the lowest level, and eventually they fill all levels up to the one where electrons are so energetic that they move at relativistic speeds and their energy is exactly the difference between neutron and proton mass.

After this moment, remaining neutrons can't decay: the resulting electron would have insufficient energy to reach the lowest free energy level.

In the real NS, electrons are present from the formation time (they are survivors from white dwarf stage - the "neutronization" doesn't consume all of them) and neutrons were never able to decay for the reason explained above.

 

[tonyxon22]

Many thanks to all of you. I’m glad that we analyzed deeper the beta decay in a neutron star situation. Actually many of your comments took me to read and investigate about other related subjects and I will soon come back with more questions.
Thanks again and best regards =)