Spin-orbit interaction
Encyclopedia
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 nucleus's magnetic field. This is detectable as a splitting of spectral line
s. A similar effect, due to the relationship between angular momentum
and the strong nuclear force, occurs for proton
s and neutron
s 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 semiconductor
s and other materials are explored and put to useful work.
. 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
.
where μ is the magnetic moment
of the particle and B is the magnetic field
it experiences.
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
It is important to note at this point that B is a positive number multiplied by L, meaning that the magnetic field
is parallel to the orbit
al angular momentum
of the particle.
of the electron
is
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 interaction of
the magnetic moment of electron with magnetic field of nucleus in the co-moving frame of electron. The second contribution is related to Thomas precession
.
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.
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
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.
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 number
s: 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
where
For hydrogen, we can write the explicit result
For any hydrogen-like atom with Z protons
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
with its motion. The first and best known example of this is that spin-orbit interaction causes shifts in an electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
's atomic energy levels
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...
due to electromagnetic interaction between the electron's spin and the nucleus's magnetic field. This is detectable as a splitting of spectral line
Spectral line
A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from a deficiency or excess of photons in a narrow frequency range, compared with the nearby frequencies.- Types of line spectra :...
s. A similar effect, due to the relationship between angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
and the strong nuclear force, occurs for proton
Proton
The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....
s and neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
s moving inside the nucleus
Atomic nucleus
The nucleus is the very dense region consisting of protons and neutrons at the center of an atom. It was discovered in 1911, as a result of Ernest Rutherford's interpretation of the famous 1909 Rutherford experiment performed by Hans Geiger and Ernest Marsden, under the direction of Rutherford. The...
, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics
Spintronics
Spintronics , also known as magnetoelectronics, is an emerging technology that exploits both the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental electronic charge, in solid-state devices.An additional effect occurs when a spin-polarized current is...
, spin-orbit effects for electrons in semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
s and other materials are explored and put to useful work.
Spin–orbit interaction in atomic energy levels
Using some semiclassical electrodynamics and non-relativistic quantum mechanics, in this section we present a relatively simple and quantitative description of the spin-orbit interaction for an electron bound to an atom, up to first order in perturbation theoryPerturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
. This gives results that agree reasonably well with observations. A more rigorous derivation of the same result would start with the Dirac equation
Dirac equation
The Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928. It provided a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity, and...
, and achieving a more precise result would involve calculating small corrections from quantum electrodynamics
Quantum electrodynamics
Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...
.
Energy of a magnetic moment
The energy of a magnetic moment in a magnetic field is given by:where μ is the magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
of the particle and B is the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
it experiences.
Magnetic field
We shall deal with the magnetic fieldMagnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
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
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...
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
Electric potential
In classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
. Here we make the central field approximationCentral field approximation
In atomic physics, the central field approximation for many-electron atoms takes the combined electric fields of the nucleus and all the electrons acting on any of the electrons to be radial and to be the same for all the electrons in the atom...
, 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 energyPotential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...
of the electron in the central field, and e is the elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
. Now we remember from classical mechanics that the angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
of a particle
. Putting it all together we getIt is important to note at this point that B is a positive number multiplied by L, meaning that the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
is parallel to the orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
al angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
of the particle.
Magnetic Moment of the Electron
The magnetic momentMagnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
of the electron
Electron magnetic dipole moment
In atomic physics, the electron magnetic dipole moment is the magnetic moment of an electron caused by its intrinsic property of spin.-Magnetic moment of an electron:...
is
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 spinSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
, so the magnetic moment
Magnetic moment
The magnetic moment of a magnet is a quantity that determines the force that the magnet can exert on electric currents and the torque that a magnetic field will exert on it...
is antiparallel to the spin angular momentum.
The spin-orbit potential consists of two parts. The Larmor part is connected to interaction of
the magnetic moment of electron with magnetic field of nucleus in the co-moving frame of electron. The second contribution is related to Thomas precession
Thomas precession
In physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular...
.
Larmor interaction energy
The Larmor interaction energy isSubstituting in this equation expressions for the magnetic moment and the magnetic field, one gets
Now, we have to take into account Thomas precession
Thomas precession
In physics the Thomas precession, named after Llewellyn Thomas, is a special relativistic correction that applies to the spin of an elementary particle or the rotation of a macroscopic gyroscope and relates the angular velocity of the spin of a particle following a curvilinear orbit to the angular...
correction for the electron's curved trajectory.
Thomas interaction energy
In 1926 Llewellyn ThomasLlewellyn Thomas
Llewellyn Hilleth Thomas was a British physicist and applied mathematician. He is best known for his contributions to atomic physics, in particular:...
relativistically recomputed the doublet separation in the fine structure
Fine structure
In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to first order relativistic corrections.The gross structure of line spectra is the line spectra predicted by non-relativistic electrons with no spin. For a hydrogenic atom, the gross structure energy...
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 spinprecession
is given byTo the first order in
, we obtainTotal interaction energy
The total spin-orbit potential in an external electrostatic potential takes the formThe 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 operatorOperator (physics)
In physics, an operator is a function acting on the space of physical states. As a resultof its application on a physical state, another physical state is obtained, very often along withsome extra relevant information....
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 number
Quantum number
Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...
s: 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 radiusBohr radius
The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...
divided by the nuclear charge Z); and
Final Energy Shift
We can now saywhere
For hydrogen, we can write the explicit result
For any hydrogen-like atom with Z protons
See also
- Angular momentum couplingAngular momentum couplingIn quantum mechanics, the procedure of constructing eigenstates of total angular momentum out of eigenstates of separate angular momenta is called angular momentum coupling. For instance, the orbit and spin of a single particle can interact through spin-orbit interaction, in which case the...
- Stark effectStark effectThe Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to presence of an external static electric field. The amount of splitting and or shifting is called the Stark splitting or Stark shift. In general one distinguishes first- and second-order Stark effects...
- Zeeman effectZeeman effectThe Zeeman effect is the splitting of a spectral line into several components in the presence of a static magnetic field. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field...
- Kugel–Khomskii coupling
- Rashba effectRashba effectRashba effect — splitting by spin in solids by strong spin-orbit interaction. Named in honour of Emmanuel Rashba, co-worker of National Academy of Sciences of USSR who discovered it....