Nuclear Models — Question 9

Question 9

Separate proton and neutron Fermi seas set a kinetic-energy scale

In a zero-temperature free Fermi-gas model of symmetric nuclear matter, total nucleon density is $.160\,\mathrm{fm^{-3}}$. Protons and neutrons each have spin degeneracy two and density half the total. Use $k_F=(3\pi^2n_s)^{1/3}$, $\hbar c=197.3\,\mathrm{MeV\,fm}$ and $mc^2=939\,\mathrm{MeV}$.

1. Find Fermi wave number and nonrelativistic Fermi kinetic energy for either species.

2. Derive mean kinetic energy from state density proportional to $k^2dk$ and evaluate it.

Question 9 - Solution

Separate proton and neutron Fermi seas set a kinetic-energy scale

(a) Method and calculation

$$\boxed{n_s=.0800\,\mathrm{fm^{-3}},\quad k_F=(3\pi^2n_s)^{1/3}=1.333\,\mathrm{fm^{-1}}}.$$$$\boxed{E_F=(\hbar c)^2k_F^2/(2mc^2)=36.83\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\langle E\rangle=\frac{\int_0^{k_F}(\hbar^2k^2/2m)k^2dk}{\int_0^{k_F}k^2dk}=\frac35E_F.$$$$\boxed{\langle E\rangle=22.10\,\mathrm{MeV}}.$$

Physical check

The species density, not total nucleon density, enters each Fermi sphere. Attractive nuclear interactions must offset this kinetic energy to produce a bound system.