# Spin-orbit coupling in a mercury atom

1. Dec 14, 2007

### juzbe

1. The problem statement, all variables and given/known data

The problem is to determine which has a more dominant effect on the energy of a given state in mercury, spin-orbit interaction or the Zeeman effect, when the applied magnetic field B is about 2T.

2. Relevant equations

As long as the spin-orbit interaction is the dominant effect, I can calculate the Zeeman energy from
$$E_{Z}=\mu_{B}\g_{J}Bm_{J}$$,
but I'm at a loss trying to figure out whether this is a lot or very little compared to the spin-orbit interaction.

3. The attempt at a solution

The first-order spin-orbit correction to the energy of a hydrogen level is given by

$$E^{1}_{so}=\frac{E_{n}^{2}}{2mc^{2}}\left(\frac{n[j(j+1)-l(l+1)-s(s+1)]}{l(l+1/2)(l+1)}\right)$$

How does this generalize to atoms with more than 1 electron? Can I just substitute for j, l, and s the atomic quantum numbers J, L, and S, and get a rough estimate? and if so, what do I substitute for $$E_{n)$$? (I understand that for the exact answer I'd have to take into account the two electron's interaction with each other's orbitals plus some exchage terms as well, but all I really need to know is if the two figures are in the same ballpark...)