A more accurate approach is to take into account that the operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum formula_10 and the spin angular momentum formula_11, with each multiplied by the appropriate gyromagnetic ratio:  where formula_13 and formula_14 (the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to the effects of quantum electrodynamics). In the case of the LS coupling, one can sum over all electrons in the atom:  where formula_16 and formula_17 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 formula_18 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, formula_18 exceeds the LS coupling significantly (but is still small compared to formula_20). In ultra-strong magnetic fields, the magnetic-field interaction may exceed formula_2, in which case the atom can no longer exist in its normal meaning, and one talks about Landau levels instead. There are intermediate cases which are more complex than these limit cases. If the spin-orbit interaction dominates over the effect of the external magnetic field, formula_22 and formula_23 are not separately conserved, only the total angular momentum formula_24 is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector formula_25. The (time-)"averaged" spin vector is then the projection of the spin onto the direction of formula_25:  and for the (time-)"averaged" orbital vector:  Thus,  Using formula_30 and squaring both sides, we get  and: using formula_32 and squaring both sides, we get  Combining everything and taking formula_34, we obtain the magnetic potential energy of the atom in the applied external magnetic field,  where the quantity in square brackets is the Landé g-factor gJ of the atom (formula_36 and formula_37) and formula_38 is the z-component of the total angular momentum. For a single electron above filled shells formula_39 and formula_40, the Landé g-factor can be simplified into:  Taking formula_42 to be the perturbation, the Zeeman correction to the energy is  The Lyman-alpha transition in hydrogen in the presence of the spin–orbit interaction involves the transitions  In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S1/2 and 2P1/2 levels into 2 states each (formula_46) and the 2P3/2 level into 4 states (formula_47).