**Weak Field (Zeeman Effect)**

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:

Thus,

Using and squaring both sides, we get

and: 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 LandÃ© g-factor g_{J} of the atom ( and ) and is the z-component of the total angular momentum. For a single electron above filled shells and, the LandÃ© g-factor can be simplified into:

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