A nucleus has central nucleon density (shared equally between protons and neutrons)…

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 nucleus has central nucleon density

\rho_0 = 0.16\,\mathrm{fm^{-3}}

(shared equally between protons and neutrons) and a measured neutron separation energy

S_n = 9.0\,\mathrm{MeV}

. ((a)) Find the Fermi wavenumber

k_F

, the Fermi energy

\epsilon_F

(take the nucleon rest energy as

mc^2 = 939\,\mathrm{MeV}

), and the depth

V_0

of the mean-field well. ((b)) Using the spin-orbit constant

C\hbar^2 = 0.55\,\mathrm{MeV}

fitted in lead, find the spin-orbit splitting

\Delta E_{so} = C\hbar^2(2\ell+1)/2

of a

1f

orbital (

\ell = 3

), and compare it with the oscillator spacing

\hbar\omega = 41A^{-1/3}\,\mathrm{MeV}

for a nucleus with

A = 56