The Lande' gfactor is a geometric factor which arises in the evaluation of the magnetic interaction which gives the Zeeman effect. The magnetic interaction energy
which is continuous in the classical case takes on the quantum form
which is like a vector operation based on the vector model of angular momentum

The problem with evaluating this scalar product is that L and S continually change in direction as shown in the vector model. The strategy for dealing with this problem is to use the direction of the total angular momentum J as a coordinate axis and obtain the projection of each of the vectors in that direction. This is done by taking the scalar product of each vector with a unit vector in the J direction.
These vector relationships must be evaluated and expressed in terms of quantum numbers in order to evaluate the energy shifts. Carrying out the scalar products above leads to
