Weak field zeeman effect, why don't we still use degenerate perturbation theory?

[alemsalem]

the spin orbit coupling removes the degeneracy but not completely, should we still use the degenerate perturbation theory. is it because of relativistic corrections?

Thanks!

 

[alemsalem]

Since nobody is answering I will say what I think the answer is although I'm not very sure, please correct me if I'm wrong.
the spin orbit Hamiltonian is only degenerate up to first order in perturbation theory, higher order ( or exact) energy eigenvalues will not give a non-degenerate spectrum.

I'm not sure about this answer because then the magnetic field energy splitting has to be smaller than the SECOND order splitting due to Spin orbit hamiltonian.

I hope I'm making sense.