Past exam of the particle and nuclear physics course of the University of Cambridge / 2017 / 1 / b

The separate proton and neutron shell capacities are $2j+1=2,4,2,6,\dots$. Filled shells and paired particles have zero net spin and magnetic dipole moment. The single-particle nuclear magnetic moment is $\mu =g_{j}j{\mu }_N$, with orbital factors $g_{\ell }^p=1$, $g_{\ell }^n=0$ and $s=1/2$.

For nitrogen, the eight neutrons form a closed shell. The seventh proton occupies $1p_{1/2}$, so $\ell =1$, $j=1/2$ and the Landé g-factor givesHence the nuclear shell model predictsIts single-particle spin and parity is $1/2^{-}$. The negative moment results from the proton spin being predominantly opposed to the total angular momentum in this $j=\ell -1/2$ level.

For oxygen, eight protons and the first eight neutrons are paired. The ninth neutron occupies $1d_{5/2}$, with $\ell =2$, $j=5/2=\ell +1/2$. Since its orbital factor is zero, $g_j=g_n/(2\ell +1)=-3.826/5$. ThereforeThese are ideal single-particle predictions in units of the nuclear magneton, not fits incorporating nuclear configuration mixing.