* Spin-orbit coupling* is the interaction between a particle’s spin angular momentum and orbital angular momentum. An electron orbiting around the nucleus ‘sees’ the nucleus circling it, just like a person on earth perceives the sun circling the earth as the latter orbits around the sun.

This apparent nuclear orbit creates a magnetic field that exerts a torque on the electron’s spin magnetic dipole moment , resulting in an additional term of (where ) in the multi-electron Hamiltonian. To derive this term, we consider a 1-electron atom.

Let be the orbital angular momentum of the electron and be the proton’s current loop, which generates a magnetic field of magnitude given by the Biot-Savart law. Since , we have

Substitute eq259 in , we have, or , where and are unit vectors. Since and point in the same direction, . Multiplying both sides of by , we have , which we substitute in eq65 (where is the spin analogue of eq61) to give . Substitute eq164 and in ,

For a 1-electron atom, and so

Eq260 can be written in terms of the Larmor frequency of the electron. From eq149, . So, . Swapping with the Thomas precession rate , we obtain the correction term of . The total spin-orbit Hamiltonian is

For a multi-electron atom,