Spin-orbit coupling

The spin-orbit coupling may be treated in a similar fashion as for one-electron atoms. The total angular momentum quantum number J = L + S will of course be used and proper eigenfunctions are those of L2, S2, J2 and Jz. We call these, which are the correct zero order functions,

The first order correction to the energy Es-o is then given by

J may take all integer values between L+S and |L-S| which means that the spin-orbit coupling splits a term into either 2L+1 or 2S+1 fine structure levels. Since the energy is independent of M, every level will have (2J+1)-fold degeneracy.

As for one-electron atoms, the spin-orbit coupling may be expressed in terms of the different angular momenta using

where A(L,S) is called the interval factor. The distance between adjacent fine structure levels J and J-1 is obtained as

This is the Landé interval rule.

Use the result from first order perturbation theory to determine the energy of the fine structure levels in 3P-state from a p2-configuration.

We have the following values of the quantum numbers: L=1, S=1, J=0,1,2

Inserting L and S gives the formula

with the following energies:

This result may be illustrated in a schematic extended energy level diagram for the p2-configuration, which incorporates the influence of spin-orbit coupling. It looks like this.

The spin-orbit splitting has been exaggerated in order to make a legible figure. Of course, spin-orbit splittings may be of this size for heavy atoms, but the perturbation treatment may then be less appropriate.

These energy levels are: E(3P0) = 0 eV; E(3P1) = 0,00203 eV; E(3P2) = 0,00539 eV;E(1D2) = 1,26386 eV; E(1S0) = 2,68406eV;

Last change: 21 February 2001