Russell-Saunders or LS coupling

1. Russell-Saunders or LS coupling
The central field approximation, in which all the electrons are reckoned to move in a common spherically-symmetric field. The rest of their repulsion, the part which is not spherically symmetric, is called the residual electrostatic interaction. It is easy to write down the mathematical form of this residual electrostatic interaction. The precise complete Hamiltonian for the atom is:

…..(1)

while the approximate Hamiltonian used in the central field approximation was:

……(2)

Therefore the residual electrostatic interaction is:

.....(3)

The upshot is that the quantum numbers on which the energy of the whole atom depends are now ni; li; L; S, and this whole method of approximation is called LS coupling or Russell-Saunders coupling. The things which are coupled in this coupling are:
1. All the individual orbital angular momenta couple together to make ^L.
2. All the individual spin angular momenta couple together to make ^S.
2. Fine structure for many electrons
The central field approximation is the correct starting point for all atoms, and after that it have both the residual electrostatic interaction and the spin-orbit interaction:

……(4)

…..(5)

r-dependence into
There are three cases to consider:


……(6)
Where:

…….(7)



?

In this case, the interaction between the electrons is stronger than the spin-orbit interaction in each of them. example with a 2-electron atom:


Then, L and S couple to a total J.













This situation is called “LS-coupling”





?

In this case, the individual coupling between the electrons, via the spin-orbit interaction, is stronger than the electrostatic interaction between them. example with a 2-electron atom:
Then, j1 and j2 couple to a total J.












This situation is called “jj-coupling”.

For a filled shell :



Degenerescence in ML and MS
? (2L + 1)(2S + 1) degenerate states.

? “Landé’s interval rule”
This rule cam be used as a test of how well system can be described by LS-coupling.