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
A simplified mean-field model represents a light, closed-orbital nucleus as A identical nucleons occupying a single filled orbital. Each nucleon has the same kinetic energy t, and every one of the (2A) distinct pairs of nucleons shares the same average two-body interaction energy vˉ (attractive, so vˉ<0). In the mean-field picture each nucleon moves in the field produced by the other A−1 nucleons, so its single-particle energy is ϵ=t+(A−1)vˉ.
((a)) Write the total energy E of the A-nucleon system as the sum of the kinetic energies plus the sum over distinct pairs, and show that E=2A(ϵ+t).
((b)) Show that summing the single-particle energies, Aϵ, double-counts the pairwise interaction energy, by comparing Aϵ−At with the true total interaction energy E−At.
((c)) For a nucleus with A=10 nucleons in this model, separation-energy data give a single-particle energy ϵ=−20.0MeV, and a Fermi-gas estimate of the mean kinetic energy per nucleon gives t=10.0MeV. Use the result of part (a) to estimate the binding energy B=−E, and state by what factor ∣Aϵ∣ overestimates B.