SpinOrbit

Spin-orbit coupling, often abbreviated as SOC, is a quantum mechanical interaction between a particle’s spin and its orbital motion in an electromagnetic field, arising from relativity. In atoms, SOC is the interaction between the electron’s orbital angular momentum L and its spin S, leading to a coupling term in the Hamiltonian.

Origin of SOC can be described in the electron’s rest frame, where the Coulomb field of the nucleus is perceived as a magnetic field that couples to the electron’s magnetic moment. The resulting effective Hamiltonian is H_so = (1/2m^2 c^2) (1/r dV/dr) L·S, or more generally H_so = ξ(r) L·S. This interaction produces fine-structure splittings between levels with the same principal quantum number n but different total angular momentum j = l ± s.

In solids, SOC enters the band structure as a momentum-dependent coupling and can cause spin splitting in crystals lacking inversion symmetry (Rashba, Dresselhaus) or give rise to spin-momentum locking in topological insulators. It is a central ingredient in spintronics and can influence transport phenomena such as the spin Hall effect and various magneto-transport phenomena.

In nuclear physics, a spin-orbit term is included in shell-model potentials to reproduce observed magic numbers, where nucleon j = l ± s energy splitting organizes shells. The strength of SOC generally increases with atomic number, being more pronounced in heavy elements. Its presence affects spectroscopy, chemical behavior, and material properties.