Zeeman Effect - Theoretical Presentation

Theoretical Presentation

The total Hamiltonian of an atom in a magnetic field is

where is the unperturbed Hamiltonian of the atom, and is perturbation due to the magnetic field:

where is the magnetic moment of the atom. The magnetic moment consists of the electronic and nuclear parts, however, the latter is many orders of magnitude smaller and will be neglected here. Therefore,

where is the Bohr magneton, is the total electronic angular momentum, and is the Landé g-factor. The operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum and the spin angular momentum, with each multiplied by the appropriate gyromagnetic ratio:

where and (the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to Quantum Electrodynamics effects). In the case of the LS coupling, one can sum over all electrons in the atom:

where and are the total orbital momentum and spin of the atom, and averaging is done over a state with a given value of the total angular momentum.

If the interaction term is small (less than the fine structure), it can be treated as a perturbation; this is the Zeeman effect proper. In the Paschen-Back effect, described below, exceeds the LS coupling significantly (but is still small compared to ). In ultrastrong magnetic fields, the magnetic-field interaction may exceed, in which case the atom can no longer exist in its normal meaning, and one talks about Landau levels instead. There are, of course, intermediate cases which are more complex than these limit cases.