In the Fermi-gas model of nuclear matter

PHYSICS · Multi-part · Fermi University Physics · Nuclear Physics · The nuclear force and nuclear models · The nuclear force and nuclear models

The Nucleus Resolved, Ch The shell model: a mean field, a spin-orbit term and the magic numbers

Problem

In the Fermi-gas model of nuclear matter, nucleons of one kind (protons or neutrons, treated separately) fill momentum states up to a Fermi wavenumber kFk_F. The number of such nucleons with wavenumber magnitude between kk and k+dkk+\mathrm{d}k, in a volume VV, is dN(k)=Vπ2k2 dk,\mathrm{d}N(k) = \frac{V}{\pi^2}k^2\,\mathrm{d}k, which already includes the two spin states per momentum value. ((a)) Using this density of states, derive the mean kinetic energy per nucleon in the filled Fermi sea, ⟨ϵkin⟩=35ϵF,\langle\epsilon_{\rm kin}\rangle = \frac{3}{5}\epsilon_F, where ϵF=ℏ2kF2/2m\epsilon_F=\hbar^2k_F^2/2m is the non-relativistic Fermi energy. ((b)) Taking ϵF=37 MeV\epsilon_F = 37\,\mathrm{MeV}, appropriate to nuclear matter at the central density ρ0=0.16 fm−3\rho_0 = 0.16\,\mathrm{fm^{-3}}, calculate the total kinetic energy of the nucleons in 40Ca^{40}\mathrm{Ca} (Z=N=20Z=N=20), treating protons and neutrons as two independent, symmetric Fermi gases each filled to the same ϵF\epsilon_F.

Related theory

Browse the problem library