Spin–orbit interaction

A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus.

This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two effects: the apparent magnetic field seen from the electron perspective due to special relativity and the magnetic moment of the electron associated with its intrinsic spin due to quantum mechanics.

A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model.

This section presents a relatively simple and quantitative description of the spin–orbit interaction for an electron bound to a hydrogen-like atom, up to first order in perturbation theory, using some semiclassical electrodynamics and non-relativistic quantum mechanics.

A rigorous calculation of the same result would use relativistic quantum mechanics, using the Dirac equation, and would include many-body interactions.

Achieving an even more precise result would involve calculating small corrections from quantum electrodynamics.

Although in the rest frame of the nucleus, there is no magnetic field acting on the electron, there is one in the rest frame of the electron (see classical electromagnetism and special relativity).

Here we make the central field approximation, that is, that the electrostatic potential is spherically symmetric, so is only a function of radius.

is the potential energy of the electron in the central field, and e is the elementary charge.

Now we have to take into account Thomas precession correction for the electron's curved trajectory.

In particular, we wish to find a new basis that diagonalizes both H0 (the non-perturbed Hamiltonian) and ΔH.

To find out what basis this is, we first define the total angular momentum operator

For the exact relativistic result, see the solutions to the Dirac equation for a hydrogen-like atom.

[4] However the rest frame calculation is sometimes avoided, because one has to account for hidden momentum.

The bands of interest can be then described by various effective models, usually based on some perturbative approach.

In this case, a (2J + 1)-fold degenerated primary multiplet split by an external CEF can be treated as the basic contribution to the analysis of such systems' properties.

The energies and eigenfunctions of the discrete fine electronic structure are obtained by diagonalization of the (2J + 1)-dimensional matrix.

The fine electronic structure can be directly detected by many different spectroscopic methods, including the inelastic neutron scattering (INS) experiments.

The energies and eigenfunctions of the discrete fine electronic structure (for the lowest term) are obtained by diagonalization of the (2L + 1)(2S + 1)-dimensional matrix.

can be found from direct diagonalization of Hamiltonian matrix containing crystal field and spin–orbit interactions.

This technique is based on the equivalent operator theory[11] defined as the CEF widened by thermodynamic and analytical calculations defined as the supplement of the CEF theory by including thermodynamic and analytical calculations.

, Fermi level measured from the top of the valence band), the proper four-band effective model is

are the Luttinger parameters (analogous to the single effective mass of a one-band model of electrons) and

If the semiconductor moreover lacks the inversion symmetry, the hole bands will exhibit cubic Dresselhaus splitting.

Two-dimensional electron gas in an asymmetric quantum well (or heterostructure) will feel the Rashba interaction.

, sometimes called the Rashba parameter (its definition somewhat varies), which is related to the structure asymmetry.

In cubic crystals, it has a symmetry of a vector and acquires a meaning of a spin–orbit contribution

between the conduction and heavy hole bands, Yafet derived the equation[13][14]

While in ESR the coupling is obtained via the magnetic part of the EM wave with the electron magnetic moment, the ESDR is the coupling of the electric part with the spin and motion of the electrons.

This mechanism has been proposed for controlling the spin of electrons in quantum dots and other mesoscopic systems.