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Explaination of Anomalous Zeeman effect

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

Anomalous Zeeman Effect

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

Using Cosine law :

Anomalous Zeeman Effect
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 :

Anomalous Zeeman Effect
Let us now,  calculate magnetic interaction energy. From the last expression we’ve :
Anomalous Zeeman Effect
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 around the field B is :
Anomalous Zeeman Effect
Thus, Anamolous Zeeman effect is explained with the help of the above expression.
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