How Does a Strong Magnetic Field Affect LS-Coupling in Atoms?

[jonas_nilsson]

A question concerning atoms in magnetic fields:

Why does the LS-coupling break down in a strong magnetic field (Paschen-Back effect)? I have until now only gotten this stated as a fact, but if anyone could give a few arguments it would be appreciated. Why aren't the individual components of $$\mu_s$$ and $$\mu_l$$ quantised in weak fields?

 

[Physics Monkey]

You have essentialy two limiting cases that define a set of good quantum numbers. In the case where the spin-orbit interaction is much stronger than Zeeman interaction, the good quantum numbers of the system are $$J,\,M_J, \,L,\, S$$. In the opposite limit, where the Zeeman intereaction is much stronger than spin-orbit interaction, the good quantum numbers are $$M_L,\,M_S,\, L,\, S$$ but not $$J$$. Why? The Zeeman interaction isn't proportional to $$J_z$$ because of the different g-factors associated with spin and orbital angular momentum which translates into the statement $$[J^2 , H_{Zeeman} ] \neq 0$$. When doing perturbation theory in a weak field (spin-orbit splittings much bigger than zeeman splittings), you can label the primary splittings with the spin-orbit quantum numbers and then treat the Zeeman effect as a perturbation on these states, but when doing perturbation theory in a strong magnetic field (zeeman splittings much bigger than spin-orbit splittings), you label the primary splittings using the Zeeman quantum numbers. Here is a good little site that has some nice diragrams: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/paschen.html

Hope this helps.