Nuclear Models — Question 7

Question 7

An unpaired nucleon fixes the simplest shell-model spin and parity

An even-even nuclear core has $J^\pi=0^+$. One added neutron occupies an $f_{7/2}$ orbital, where f means orbital angular momentum $\ell=3$. Use a supplied magnetic factor $g_j=-.500$ and $\mu_z=g_j\mu_Nm_j$.

1. Find total spin-parity and the number of magnetic substates.

2. Find the magnetic moment projection in the stretched state $m_j=j$ and distinguish it from the angular-momentum magnitude.

Question 7 - Solution

An unpaired nucleon fixes the simplest shell-model spin and parity

(a) Method and calculation

$$J=j=7/2,\quad\pi=(-1)^\ell=(-1)^3\Rightarrow\boxed{J^\pi=7/2^-}.$$$$\boxed{2J+1=8\text{ magnetic substates}}.$$

(b) Interpretation and further calculation

$$\boxed{\mu_z(m_j=j)=(-.500)(7/2)\mu_N=-1.75\mu_N}.$$$$\boxed{|\mathbf J|=\sqrt{j(j+1)}\hbar=\frac{\sqrt{63}}2\hbar\ne j\hbar}.$$

Physical check

The paired core supplies no net angular momentum in this model. Configuration mixing and collective motion can modify real nuclear moments.