Paschen-Back Effect

The Paschen-Back effect represents the high-field limit of the Zeeman effect. While the anomalous Zeeman effect describes the case where the external field is a small perturbation compared to the internal spin-orbit coupling, the Paschen-Back effect describes the opposite scenario. When the magnetic field is sufficiently strong, the interaction energy of the magnetic moments with the external field, \(\hat{H}_Z\), dominates over the spin-orbit interaction Hamiltonian, \(\hat{H}_{SO}\).

As a result, \(\vec{L}\) and \(\vec{S}\) are effectively decoupled. The ‘good’ quantum numbers are no longer J and \(m_J\), but rather \(m_L\) and \(m_S\), which describe the independent projections of orbital and spin angular momentum along the magnetic field axis. The first-order energy shift is then given by the sum of the independent interactions: \(\Delta E = (m_L + g_s m_S)\mu_B B\). With \(g_s \approx 2\), this leads to a splitting pattern that closely resembles the normal Zeeman triplet, although the fine-structure interaction, now treated as a small perturbation, causes each of these lines to have a small residual substructure. The transition from the anomalous Zeeman regime to the Paschen-Back regime is continuous and can be calculated using intermediate-field theories.