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The explanation of anamolous Zeeman effect is explained by taking spin of electron. 

In the vector model of 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 ╬╝jand j precess slowly around magnetic field following quantum condition, i.e.
where mj takes ( 2j + 1) values from -j to +j.
Thus due to precession of J around Bthe energy level is splitted into  2J + 1 levels called Zeeman levels.
Let us consider a system of single valence electron, we have :
then, magnetic field of atom is given by :

Using Cosine law :

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 precession of around the field B is :

Thus, Anamolous Zeeman effect is explained with the help of above expression.