Angular Momenta and Magnetic Dipole Moments — Question 7

Question 7

A Lande factor connects weak-field splitting to coupled angular momentum

For $g_L=1$ and $g_s=2$, use $g_J=1+[j(j+1)+s(s+1)-l(l+1)]/[2j(j+1)]$. A p electron has $l=1,s=1/2$. In a weak field, $\delta E=g_J\mu_BBm_j$.

1. Calculate the two Lande factors.

2. Find the ratio of adjacent Zeeman spacings in the $j=3/2$ and $j=1/2$ multiplets.

Question 7 - Solution

A Lande factor connects weak-field splitting to coupled angular momentum

(a) Method and calculation

$$g_{3/2}=1+\frac{15/4+3/4-2}{2(15/4)}=\boxed{4/3}.$$$$g_{1/2}=1+\frac{3/4+3/4-2}{2(3/4)}=\boxed{2/3}.$$

(b) Interpretation and further calculation

$$\Delta E_j=g_J\mu_BB\quad(\Delta m_j=1).$$$$\boxed{\Delta E_{3/2}/\Delta E_{1/2}=(4/3)/(2/3)=2}.$$

Physical check

The field must be weak relative to the spin-orbit splitting so that j remains a useful label. These factors are effective projections of orbital and spin moments onto total J.