6.2.3 Magnetic Dipole Moments: Schmidt Lines

 

 

In the previous posting, we derived the magnetic dipole moment for the single particle state as follows:

(Link: 2024.04.11 - [Nuclear Physics/From Nucleons to Nucleus] - 6.1.6 Electromagnetic Multipole Moments )

\begin{eqnarray} \mu_{\rm sp} = \mu_N \frac{(1 - (-1)^{l+j+\frac{1}{2}}(2 j+ 1))}{4(j+1)} \left[g_s - g_l \left( 2 + (-1)^{l+j+\frac{1}{2}} (2j+1) \right) \right]. \end{eqnarray}

 

Using the above formula, for $j= l \pm 1/2$, one can evaluate the dipole magnetic moment.

 

For $j= l + \frac{1}{2}$,  

\begin{eqnarray} \mu_{\rm sp} &=& \mu_N \frac{(1 - (-1)^{j-\frac{1}{2}+j+\frac{1}{2}}(2 j+ 1))}{4(j+1)} \left[g_s - g_l \left( 2 + (-1)^{j-\frac{1}{2}+j+\frac{1}{2}} (2j+1) \right) \right] \\[12pt] &=& \mu_N \frac{(1 - (-1)^{2j}(2 j+ 1))}{4(j+1)} \left[g_s - g_l \left( 2 + (-1)^{2j} (2j+1) \right) \right] \\[12pt] &=& \mu_N \frac{(2(j+ 1)}{4(j+1)} \left[g_s - g_l \left( - 2j + 1 \right) \right] = \mu_N \left( \frac{g_s - g_l}{2} + g_l j \right). \end{eqnarray}

 

For $j= l - 1/2$, 

\begin{eqnarray} \mu_{\rm sp} &=& \mu_N \frac{(1 - (-1)^{j+\frac{1}{2}+j+\frac{1}{2}}(2 j+ 1))}{4(j+1)} \left[g_s - g_l \left( 2 + (-1)^{j+\frac{1}{2}+j+\frac{1}{2}} (2j+1) \right) \right] \\[12pt] &=& \mu_N \frac{(1 - (-1)^{2j+1}(2 j+ 1))}{4(j+1)} \left[g_s - g_l \left( 2 + (-1)^{2j+1} (2j+1) \right) \right] \\[12pt] &=& \mu_N \frac{-2j}{4(j+1)} \left[g_s - g_l \left(2j+3\right) \right] = \mu_N \frac{-j}{2(j+1)} \left[g_s - g_l \left(2j+2 + 1\right) \right] \\[12pt] &=& \mu_N \left[ \left( \frac{-j}{2(j+1)} \right) g_s - \left( \frac{-j}{2(j+1)} \right) g_l \left( 2j+2 \right) - g_l \left( \frac{-j}{2(j+1)} \right) \right] \\[12pt] &=& \mu_N \left[ j g_l - \left( g_s - g_l \right) \left( \frac{j}{2(j+1)} \right) \right]. \end{eqnarray}

As shown in both results, two equations depend only on $j$. Therefore, two equations can be plotted as functions of $j$, whose plots with bare $g$ factor are called Schmidt lines.

 

As shown in the previous posting (2024.04.29 - [Nuclear Physics/From Nucleons to Nucleus] - 6.2.2 Examples: Transitions in One-Hole Nuclei 15N and 15O), the one-particle or one-hole nuclei formula is precisely the same as the $\mu_{\rm sp}$. So, the experimental dipole moments of such nuclei are also close to the Schmidt values. 

 

For other nuclei, experimental dipole moments are found to lie between the Schmidt lines. To reproduce experimental results, one can introduce effective $g$ factors reflecting many-body effects.