0
$\begingroup$

Consider a system with high-energy neutrinos (higher than rest mass energies of protons & neutrons) where the energy of a neutrino is specified as $E = p + sB$, $p$ being the magnitude of the momentum and $s$ its spin. $B$ is a constant external magnetic field.

I need to show using 'simple statistical mechanics' that the number density of neutrons is expected to be higher than that of protons.

I am aware that the situation is akin to a white dwarf/neutron star, and from this question I know that the electron degeneracy makes it energetically favorable for an electron capture process:

$$p+e→n+\nu_e$$

I think the 'simple statistical mechanics' that I need to use is the two-level occupation ratio:

$$ \frac{n}{p} = exp(-\frac{\Delta E}{kT}) $$

but how does one go from the information given to me to being able to apply the above?

Edit: After some more reading, I realize that I can relate $E_F$ to number densities. Assuming degenerate electron gas, which terms in the conservation laws can I relate to their Fermi energies and thus invoke the relationship to their respective number densities?

I am mildly certain electron energy/momentum terms can be replaced with their Fermi values due to the neutron star being a degenerate electron gas scenario. What else am I missing?

Improve this question
$\endgroup$
5
  • $\begingroup$ You seem to be confusing neutrinos and neutrons in places here. Neutrinos have no magnetic moment. $\endgroup$
    – Buzz ♦
    Commented Dec 22, 2020 at 15:54
  • $\begingroup$ You raise a fair point, but that is verbatim how the question was posed to me. $\endgroup$
    – AvHz
    Commented Dec 22, 2020 at 15:57
  • $\begingroup$ What can you assume as the density of the environment, since that matters? I'm also not following how you produce neutrinos with more energy than the nucleons? $\endgroup$
    – ProfRob
    Commented Dec 22, 2020 at 16:25
  • $\begingroup$ Mass density? I can take it to be of the order of neutron star density, 10^17 kg/m^3. $\endgroup$
    – AvHz
    Commented Dec 22, 2020 at 16:29
  • $\begingroup$ I reckon that the assumption being made was of ultrarelativistic neutrinos. Verbatim, the assumption I was told to make was "(higher than rest mass energies of protons & neutrons)". $\endgroup$
    – AvHz
    Commented Dec 22, 2020 at 16:53

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.