Zeeman Shifts

Goal

Compute the Zeeman shift.

In the presence of a magnetic field, an ion's levels will be shifted, leading to a loss of degeneracy in its various manifolds. For example, the \(4S_{1/2}\) manifold is two-fold degenerate because both \(m_F = \frac{1}{2}\) and \(m_F = -\frac{1}{2}\) have the same energy. However, when a magnetic field with magnitude \(B\) is applied, the degeneracy is broken: the \(m_F = \frac{1}{2}\) level will be shifted up and the \(m_F = -\frac{1}{2}\) will be shifted down.

In the weak field regime, this Zeeman shift is given by

\[ \Delta_B = m_F g_F \mu_B B \]

where \(\mu_B\) is the Bohr magneton (\(\approx 9.27 \times 10^{-24}\) J/T) and \(g_F\) is the total angular momentum F Lande g-factor. It can be written in terms of the spin-orbital angular momentum Lande g-factor, \(g_J\):

\[ g_F = g_J \frac{F(F+1) + J(J+1) - I(I+1)}{2F(F+1)} \]

where F, J, and I are the total, spin-orbital, and nuclear angular momentum quantum numbers, respectively. Finally,

\[ g_J = \frac{3}{2} + \frac{S(S+1) - L(L+1)}{2J(J+1)} \]

where \(S\) is the spin angular momentum quantum number.