Spin-orbit coupling and jj scheme

The hamiltonian representing spin-orbit interaction is

We can treat as a perturbation to . If we're not dealing with other perturbation of the same order ( which means spin-orbit corrections are the leading ones), we simply have to find a basis which is able to diagonalize and simultaneously. That basis is , where represent the total angular momentum of each electron. We can pass from the previous basis to the new one through Clebsch-Gordan coefficients:

The mathematical need for this basis represents the physical fact that when spin-orbit interaction becomes strong (in heavy atoms, ), each orbital angular momentum tends to couple with its own spin:. The then combines to give the total (jj coupling)

Individual spin orbitals are now labeled with the new good quantum numbers:

The effect of spin-orbit coupling consists in a partial removal of degeneracy according to value:

For example level, characterized by , is split in .

If we want to fill level with electrons we simply have to add an apix representing the occupation number. After placing all the electrons in the jj levels compute the possible values of the total J:

When we take into account non-spherical corrections on these states is no longer a good quantum number and we need to label states with . Degeneracy of configurations with same is removed:

Each configuration with different value of has now different energy. The lowest energy configuration is given by Hund's third rule [1]

Remark
  • can have a maximum of electrons
  • If a level , which can contain electrons is split in the equality must hold. For example level is split in and we have
 
  1. For a detailed discussion about how to find atomic terms read JJ scheme
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