In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is any interaction of a particle's spin with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron's atomic energy levels due to electromagnetic interaction between the electron's spin and the magnetic field generated by the electron's orbit around the nucleus. This is detectable as a splitting of spectral lines. 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. In the field of spintronics, spin-orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin-orbit interaction is one cause of magnetocrystalline anisotropy.
Spin–orbit interaction in atomic energy levels 
This section presents a relatively simple and quantitative description of the spin-orbit interaction for an electron bound to an atom, up to first order in perturbation theory, using some semiclassical electrodynamics and non-relativistic quantum mechanics. This gives results that agree reasonably well with observations. A more rigorous derivation of the same result would start with the Dirac equation, and achieving a more precise result would involve calculating small corrections from quantum electrodynamics.
Energy of a magnetic moment 
The energy of a magnetic moment in a magnetic field is given by:
Magnetic field 
We shall deal with the magnetic field first. Although in the rest frame of the nucleus, there is no magnetic field, there is one in the rest frame of the electron. Ignoring for now that this frame is not inertial, in SI units we end up with the equation
where v is the velocity of the electron and E the electric field it travels through. Now we know that E is radial so we can rewrite . Also we know that the momentum of the electron . Substituting this in and changing the order of the cross product gives:
Next, we express the electric field as the gradient of the electric potential . Here we make the central field approximation, that is, that the electrostatic potential is spherically symmetric, so is only a function of radius. This approximation is exact for hydrogen, and indeed hydrogen-like systems. Now we can say
where is the potential energy of the electron in the central field, and e is the elementary charge. Now we remember from classical mechanics that the angular momentum of a particle . Putting it all together we get
Magnetic Moment of the Electron 
where is the spin angular momentum vector, is the Bohr magneton and is the electron spin g-factor. Here, is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum.
The spin-orbit potential consists of two parts. The Larmor part is connected to the interaction of the magnetic moment of the electron with the magnetic field of the nucleus in the co-moving frame of the electron. The second contribution is related to Thomas precession.
Larmor interaction energy 
The Larmor interaction energy is
Substituting in this equation expressions for the magnetic moment and the magnetic field, one gets
Now, we have to take into account Thomas precession correction for the electron's curved trajectory.
Thomas interaction energy 
In 1926 Llewellyn Thomas relativistically recomputed the doublet separation in the fine structure of the atom. Thomas precession rate, , is related to the angular frequency of the orbital motion, , of a spinning particle as follows 
where is the Lorentz factor of moving particle. The Hamiltonian producing the spin precession is given by
To the first order in , we obtain
Total interaction energy 
The total spin-orbit potential in an external electrostatic potential takes the form
The net effect of Thomas precession is the reduction of the Larmor interaction energy by factor 1/2 which came to be known as the Thomas half.
Evaluating the energy shift 
Thanks to all the above approximations, we can now evaluate the detailed energy shift in this model. In particular, we wish to find a 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
Taking the dot product of this with itself, we get
(since L and S commute), and therefore
It can be shown that the five operators H0, J2, L2, S2, and Jz all commute with each other and with ΔH. Therefore, the basis we were looking for is the simultaneous eigenbasis of these five operators (i.e., the basis where all five are diagonal). Elements of this basis have the five quantum numbers: n (the "principal quantum number") j (the "total angular momentum quantum number"), l (the "orbital angular momentum quantum number"), s (the "spin quantum number"), and jz (the "z-component of total angular momentum").
To evaluate the energies, we note that
for hydrogenic wavefunctions (here is the Bohr radius divided by the nuclear charge Z); and
Final Energy Shift 
We can now say
Spin-orbit interaction in solids 
Crystalline solid (semiconductor, metal etc.) is characterized by its band structure. While on the overall scale (including the core levels) the spin-orbit interaction is still a small perturbation, it may play relatively more important role if we zoom in to bands close to the Fermi level (). The atomic interaction for example splits bands which would be otherwise degenerate and the particular form of this spin-orbit splitting (typically of the order of few to few hundred millielectronvolts) depends on the particular system. The bands of interest can be then described by various effective models, usually based on some perturbative approach. An example of how the atomic spin-orbit interaction influences the band structure of a crystal is explained in the article about Rashba interaction.
Examples of effective Hamiltonians 
Hole bands of a bulk (3D) zinc-blende semiconductor will be split by into heavy and light holes (which form a quadruplet in the -point of the Brillouin zone) and a split-off band ( doublet). Including two conduction bands ( doublet in the -point), the system is described by the effective eight-band model of Kohn and Luttinger. If only top of the valence band is of interest (for example when , Fermi level measured from the top of the valence band), the proper four-band effective model is
where are the Luttinger parameters (analogous to the single effective mass of a one-band model of electrons) and are angular momentum 3/2 matrices ( is the free electron mass). In combination with magnetization, this type of spin-orbit interaction will distort the electronic bands depending on the magnetization direction, thereby causing Magnetocrystalline anisotropy (a special type of Magnetic anisotropy). If the semiconductor moreover lacks the inversion symmetry, the hole bands will exhibit cubic Dresselhaus splitting. Within the four bands (light and heavy holes), the dominant term is
where the material parameter for GaAs (see pp. 72 in Winkler's book; the total Hamiltonian will be ). Two-dimensional electron gas in an asymmetric quantum well (or heterostructure) will feel the Rashba interaction. The appropriate two-band effective Hamiltonian is
where is the 2 × 2 identity matrix, the Pauli matrices and the electron effective mass. The spin-orbit part of the Hamiltonian, is parametrized by , sometimes called the Rashba parameter (its definition somewhat varies), which is related to the structure asymmetry.
References: Cardona and Winkler books (see below).
See also 
- L. H. Thomas, The motion of the spinning electron, Nature (London), 117, 514 (1926).
- L. Föppl and P. J. Daniell, Zur Kinematik des Born'schen starren Körpers, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 519 (1913).
- C. Møller, The Theory of Relativity, (Oxford at the Claredon Press, London, 1952).
- E. U. Condon and G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 0-521-09209-4.
- D. J. Griffiths (2004). Introduction to Quantum Mechanics (2nd edition). Prentice Hall.
- Landau, Lev; L. M. Lifshitz. "72. Fine structure of atomic levels". Quantum Mechanics: Non-Relativistic Theory, Volume 3.
- Yu, Peter Y.; Manuel Cardona. Fundamentals of Semiconductors.
- Winkler, Roland. Spin-Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems.