# Spin-orbit coupling in itinerant electron systems

Itinerant electron systems are, e.g., metals. The electrons are not anymore localised at the atomic nucleus, but they behave as Bloch waves, which have a non-neglecting probability of travelling somewhere in the crystal.

My understanding of spin-orbit coupling: In the rest frame of the electron, the atomic nucleus orbits the electron producing at the position of the electron a magnetic field $B$ which couples to the spin of the electron.

As a general equation the spin-orbit Hamiltonian is given by: $\hat H_{so}=\alpha \vec L\cdot \vec S$, where $\alpha$ is proportional to $\frac 1r \frac{\partial U(\vec r)}{\partial r} \hat e_r$. The potential $U$ is approximately radial.

What I do not get, is how the given equation can be true for itinerant electron systems. Firstly, the atomic nucleus can not orbit the electron, since the electron has (only) a very small probability to be there (to be localised). Secondly, I do not understand the variable $r$ in context of itinerant electron systems. It should be the orbital distance electron-atomic nucleus.

Is this equation still valid for itinerant electron systems?

Thanks in advance for any help.

• I don't really buy the claim that the form you've given for spin-orbit coupling is used for lattice potentials (I'd want to see a reference to be sure), and since you don't fully specify $\alpha$ it is hard to judge, but in any case I don't see why you find the $r$ dependence troublesome, since $L$ itself is only defined w.r.t. a fixed origin. – Emilio Pisanty Nov 29 '17 at 19:00