In the vector model of the atom, the angular momentum L and spin angular momentum vector precess around the total angular momentum J.
When the atom is placed in weak magnetic field B along z, the magnetic moment of atom associated with j ( μj) and j precess slowly around the magnetic field following quantum condition, i.e.
where mj takes ( 2j + 1) values from -j to +j. Thus due to the precession of J around B, the energy level is splitted into 2J + 1 levels called Zeeman levels.
Let us consider a system of the single valence electron, we have :
then, the magnetic field of an atom is given by :
The importance of g factor is that it gives directly the relative separations of Zeeman levels.
The expression for g-factor for multi electron under L – S coupling is same as above. The expression for total magnetic moment of atom now becomes :
Let us now, calculate magnetic interaction energy. From the last expression we’ve :
Thus, g determines the ratio of the total magnetic moment to the total angular momentum in states where the angular momentum is partly orbital and partly spin. (For S = 0 and so on
J=L,g=1;for L = 0 and so J = S , g = 2).
By Larmor’s theorem, the angular velocity of the precession of J around the field B is :
Thus, Anamolous Zeeman effect is explained with the help of the above expression.