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In the previous sections, we have discussed that electron as a constituting element of an atom possesses orbital as well as spin angular momentum. Thus the total angular momentum of an atom will be the vector sum of the orbital and spin angular momenta of its electrons. The determination of total angular momentum of an atom depends on the process of coupling of the orbital and spin angular momenta. The two methods of coupling has been developed and described in following sections :

Russel-Saunder’s or LS Coupling

In the LS coupling, the orbital angular momentum vectors and the spin angular momentum vectors of the optical electrons of an atom coupled strongly separately. The total angular momentum of the atom is thus the vector sum of the resultant orbital momentum vector L and spin angular momentum S. Mathematically this scheme can be represented as

J = | L – S|, |L – S| + 1 …. |L + S|                                … (2)

 j-j Coupling

This coupling scheme is applicable for heavy atoms in which the interaction between the orbital and spin momenta of a single electron interaction among either is much stronger than the orbital or spin momenta of different electrons.

We know that the resultant angular momentum of an electron J = l + S. The vector sum of total angular momentum of all the individual electrons of atom will give total J values i.e.,

J = ∑ji                                                                  … (3)