Spin-Orbit Coupling

an interaction of particles that depends on the values and mutual orientations of the particles’ orbital and spin angular momenta and that leads to the fine-structure splitting of the system’s energy levels. Spin-orbit coupling is a relativistic effect. Formally, it is obtained if the energy of particles moving rapidly in an external field is found with an accuracy of v²/c², where v is the speed of the particle and c is the speed of light.

A simple physical interpretation of spin-orbit coupling can be obtained by considering, for example, the motion of an electron in a hydrogen atom. We shall use here a semiclassical model wherein the electron is viewed as moving around the nucleus along some “orbit.” The electron has an intrinsic angular momentum, or spin, which is responsible for the existence of the electron’s spin magnetic moment. The electric charge of the nucleus generates a Coulomb electric field that should affect the spin magnetic moment of the electron moving along its orbit. This situation can be easily understood if we imagine the frame of reference where the electron is at rest—that is, the frame of reference moves with the electron. For this stationary electron, the nucleus will appear as moving, and, like any moving charge, it will produce a magnetic field H. The magnetic field will affect the magnetic moment H. of the electron. The contribution thereby made to the electron energy depends on the orientation of µ, and H and is equal to – µH = – µ._HH. Since the projection µ,_H of the magnetic moment µ, on the vector of the field H can assume two values (±ℏ/2, where ℏ is Planck’s constant), the spin-orbit coupling leads to the splitting of energy levels in the hydrogen atom and in hydrogen-like atoms into two close-lying sublevéis (a doublet structure). For atoms with more than one electron, a complicated multiplet splitting of energy levels occurs. It may be noted that the atoms of alkali metals, whose total electron spin is ℏ/2, also have levels of doublet structure.

Spin-orbit coupling is also found for neutral particles, such as neutrons, that have both orbital and spin angular momenta. The spin-orbit coupling of nucleons (protons and neutrons) in atomic nuclei is of great importance, since its contribution to the total interaction energy reaches 10 percent.