Nuclear Models — Question 8

Question 8

Spin-orbit coupling splits a nuclear orbital into two multiplets

A nucleon with $\ell=2$ and $s=1/2$ has interaction $H_{ls}=\kappa\,\mathbf L\cdot\mathbf S/\hbar^2$, with $\kappa=-1.00\,\mathrm{MeV}$. Use $\mathbf L\cdot\mathbf S=\tfrac12[J^2-L^2-S^2]$.

1. Calculate the energy shifts for $j=5/2$ and $j=3/2$.

2. Find their separation and check that their total capacity equals the unsplit orbital capacity.

Question 8 - Solution

Spin-orbit coupling splits a nuclear orbital into two multiplets

(a) Method and calculation

$$\Delta E_j=\frac\kappa2[j(j+1)-\ell(\ell+1)-s(s+1)].$$$$\boxed{\Delta E_{5/2}=\kappa=-1.00\,\mathrm{MeV},\quad\Delta E_{3/2}=-3\kappa/2=+1.50\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{\Delta E_{3/2}-\Delta E_{5/2}=2.50\,\mathrm{MeV}}.$$$$\boxed{(2\times5/2+1)+(2\times3/2+1)=6+4=10=2(2\ell+1)}.$$

Physical check

For the negative coupling used here, the aligned j=ell+1/2 multiplet is lower. Splitting reorganizes energies without changing the total number of states.