When the spectral lines are absorption lines, the effect is called Inverse Zeeman effect.
The presence of a magnetic field breaks the degeneracy, since it interacts in a different way with electrons with different quantum numbers, slightly modifying their energies. The result is that, where there were several configurations with the same energy, now there are different energies, which give rise to several very close spectral lines.
Without a magnetic field, configurations a, b and c have the same energy, as do d, e and f. The presence of a magnetic field splits the energy levels. A line produced by a transition from a, b or c to d, e or f now will be several lines between different combinations of a, b, c and d, e, f. Not all transitions will be possible, as regulated by the transition rules.
Since the distance between the Zeeman sub-levels is proportional to the magnetic field, this effect is used by astronomers to measure the magnetic field of the Sun and other stars.
There is also an anomalous Zeeman effect that appears on transitions where the net spin of the electrons is not 0, the number of Zeeman sub-levels being even instead of odd if there's an uneven number of electrons involved. It was called "anomalous" because the electron spin had not yet been discovered, and so there was no good explanation for it at the time that Zeeman observed the effect.
If the magnetic field strength is too high, the effect is no longer linear; at even higher field strength, electron coupling is disturbed and the spectral lines rearrange. This is called the Paschen-Back effect.
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 further on. Therefore,
where is the Bohr magneton, is the total electronic angular momentum, and is the 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 or (the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to the relativistic 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.
If the spin-orbit interaction dominates over the effect of the external magnetic field, and are not separately conserved, only the total angular momentum is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector . The (time-)"averaged" spin vector is then the projection of the spin onto the direction of :
and for the (time-)"averaged" orbital vector:
Using and squaring both sides, we get
Combining everything and taking , we obtain the magnetic potential energy of the atom in the applied external magnetic field,
where the quantity in square brackets is the Lande g-factor gJ of the atom ( and ) and is the z-component of the total angular momentum. For a single electron above filled shells .
In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S1/2 and 2P1/2 states into 2 levels each () and the 2P3/2 state into 4 levels (). The Lande g-factors for the three levels are:
Note in particular that the size of the energy splitting is different for the different orbitals, because the gJ values are different. On the left, fine structure splitting is depicted. This splitting occurs even in the absence of a magnetic field, as it is due to spin-orbit coupling. Depicted on the right is the additional Zeeman splitting, which occurs in the presence of magnetic fields.
When the magnetic-field perturbation significantly exceeds the spin-orbit interaction, one can safely assume . This allows the expectation values of and to be easily evaluated for a state :
The above may be read as implying that the LS-coupling is completely broken by the external field. The and are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the selection rule. The splitting is independent of the unperturbed energies and electronic configurations of the levels being considered.