Spin–orbit coupling

Generally, an atomic electron possesses both spin and orbital angular momentum. They may be coupled by spin–orbit interaction to create a total electronic angular momentum j , with resultant magnetic moment

m = γ j ....... (1)
It is conventional to use lower-case letters , s, j to denote the the angular momentum quantum numbers of a single electron. Upper-case letters L, S, J are reserved for the multielectron atoms and ions discussed in the next chapter. The numbers may be integral or half-integral, except for  and L, which can only be integral. Bold symbols represent angular momentum vectors, which
have units of h.

From the electron’s geocentric point of view, the nucleus revolves around it with speed v. The motion is equivalent to a current loop I_n = Zev/2πr, which creates a magnetic field μ₀I_n/2r  at the centre. The spin–orbit interaction is due to this magnetic field, B_so = μ₀Zev/4πr², acting on the intrinsic magnetic moment of the electron. The electron’s magnetic moments associated with  and s are oppositely aligned. The interaction energy ε_so = −μ_BB_so can be written approximately in terms of the Bohr magneton and the Bohr radius, since r ≃ a₀/Z for an inner electron and r = na₀ for an outer electron, and mevr ≈ h. In the former case,

.......(2)

The Z variation means that the spin–orbit interaction, while weak for light elements becomes much more important for heavy elements and especially for inner shells. The associated magnetic field is of order 10 T for boron or carbon. The correct version of the spin–orbit interaction, resulting from a relativistic calculation. The expression (2) is modified by a factor 2. The interaction for a single electron is represented by the spin–orbit Hamiltonian

.......(3)

where λ is the spin–orbit coupling energy. ˆl and ˆs are dimensionless operators the h2 has been absorbed into λ, thus giving it dimensions of energy.