Fine Structure Spectra: Understanding Electron Intrinsic Spin Effects

[zengodspeed1]

I understand that the intrinsic spin of an electron causes discrete differences in energy for transitions, due to the interaction of the magnetic dipole moment of the electron. Also, that this in turn creates the fine structure spectra.
But what I am currently struggling to picture is why this occurs at specific frequencies. Which properties of electronic configuration determine the energy difference?

Thanks

 

[Charles Link]

I believe the fine structure is due to an $\vec{L} \cdot \vec{S}$ term in the Hamiltonian. With a little algebra, this can be converted to $\vec{J}'s$, and the $\vec{J}$ levels get quantized in the form $J_z=m \hbar$, where $m$ is an integer or half integer. Perhaps others can give a better answer, but I think this explains most of it.

 

[DrClaude]

I believe the fine structure is due to an $\vec{L} \cdot \vec{S}$ term in the Hamiltonian.

Yes. You can picture it as the interaction of the magnetic moment due to orbital motion interacting with the spin magnetic moment. The relative orientation of the two will determine the energy shift, and since these orientations are quantized, the shift in energy is discrete.