Nuclear magnetic moment
Last update: April 22, 2024
A nucleus consists of one or more nucleon. Each nucleon carry two types of magnetic moment: orbital magnetic moment $\mu_l$ and spin magnetic moment $\mu_s$
\[\begin{align} \mu_l &= -g_l {e \over 2m_e} l \\ \mu_s &= -g_s {e \over 2m_e} s \end{align}\]Nuclear magnetic moment vector $\vec{\mu} = g_l \vec{l} + g_s \vec{s}$
Nuclear magnetic moment $\mu$ \(\begin{align} \mu &= \langle (l,s),j,m_j=j | \mu_z | (l,s),j,m_j=j \rangle \\ &= \langle (l,s),j,m_j=j | \vec{\mu} \cdot \hat{j} | (l,s),j,m_j=j \rangle \end{align}\)
We have
\({\mu \over \mu_N} = {g_s\left[j(j+1)+s(s+1)-l(l+1)\right] + g_l \left[ j(j+1)+l(l+1)-s(s+1) \right] \over 2(j+1)}\) where $\mu_N={e \hbar \over 2m_p}$ is nuclear magneton (do not confuse with Bohr magneton $\mu_B={e \hbar \over 2m_e}$). Since the paired nucleons will cancel out each other, we only need to calculate the unpair ones
\[{\mu \over \mu_N} = \begin{cases} g_l \,l + {1\over2} g_s &\quad j=l+{1\over2} \\ {j\over j+1} \left[ g_l\, (l+1) - {1\over2}g_s \right] &\quad j=l-{1\over2} \end{cases}\]${\mu \over \mu_N}$ value table
| orbit | proton | neutron |
|---|---|---|
| $0g_{7/2}$ | 3.007 | 0.5417 |
| $1d_{5/2}$ | 3.851 | -0.9765 |
| $0h_{11/2}$ | 7.2415 | -1.155 |
| $1d_{3/2}$ | 0.8304 | 0.5829 |
| $2s_{1/2}$ | 1.782 | -1.005 |
More accurate calculation
PhysRevC.71.044317 calculates effective $g$ factor using nuclear shell model. Take $0g_{7/2}$ for an example. The table IV shows
| proton | neutron | |
|---|---|---|
| $g_s$ | 5.587 | -3.826 |
| $g_l$ | 1 | 0 |
| $\delta g_s$ | -2.19 | 1.933 |
| $\delta g_l$ | 0.113 | -0.05 |
The new $g_l \rightarrow g_{l_\text{eff}}=g_l + \delta g_l$ and $g_s \rightarrow g_{s_\text{eff}}=g_s + \delta g_s$ can then be inserted to the formulae above to have a more precise estimation.