Everything about Spin-spin Coupling totally explained
In
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 it's useful to couple the spin and orbit angular momentum of the particle. Or two charged particles, each with a well-defined angular momentum, may interact by
Coulomb forces, in which case coupling of the two one-particle angular momenta to a total angular momentum is a useful step in the solution of the two-particle
Schrödinger equation.
In both cases the separate angular momenta are no longer
constants of motion, but the sum of the two angular momenta usually still is. Angular momentum coupling in atoms is of importance in atomic
spectroscopy. Angular momentum coupling of
electron spins is of importance in
quantum chemistry. Also in the nuclear
shell model angular momentum coupling is ubiquitous.
» spin-orbit coupling in
astronomy reflects the general law of
conservation of angular momentum, which holds for celestial systems as well. In simple cases, the direction of the
angular momentum vector is neglected, and the spin-orbit coupling is the ratio between the frequency with which a
planet or other
celestial body spins about its own axis to that with which it orbits another body. This is more commonly known as
orbital resonance. Often, the underlying physical effects are
tidal forces.
General theory and detailed origin
Angular momentum is a property of a physical system that's a
constant of motion (is time-independent and well-defined) in two situations: (i) The system experiences a spherical symmetric potential field. (ii) The system moves (in quantum mechanical sense) in isotropic space. In both cases the angular momentum operator commutes with the Hamiltonian of the system. By Heisenberg's uncertainty relation this means that the angular momentum can assume a sharp value simultaneously with the energy (eigenvalue of the Hamiltonian).
An example of the first situation is an atom whose electrons only feel the Coulomb field of its nucleus. If we ignore the electron-electron interaction (and other small interactions such as spin-orbit coupling), the
orbital angular momentum l of each electron commutes with the total Hamiltonian. In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherical symmetric electron-nucleus interactions. The individual electron angular momenta
l(i) commute with this Hamiltonian. That is, they're conserved properties of this approximate model of the atom.
An example of the second situation is a
rigid rotor moving in field-free space. A rigid rotor has a well-defined, time-independent, angular momentum.
These two situations originate in classical mechanics. The third kind of conserved angular momentum, associated with
spin, doesn't have a classical counterpart. However, all rules of angular momentum coupling apply to spin as well.
In general the conservation of angular momentum implies full rotational symmetry
(described by the groups
SO(3) and
SU(2)) and, conversely, spherical symmetry implies conservation of angular momentum. If two or more physical systems have conserved angular momenta, it can be useful to add these momenta to a total angular momentum of the combined system—a conserved property of the total system.
The building of eigenstates of the total conserved angular momentum from the angular momentum eigenstates of the individual subsystems is referred to as
angular momentum coupling.
Application of angular momentum coupling is useful when there's an interaction between subsystems that, without interaction, would have conserved angular momentum. By the very interaction the spherical symmetry of the subsystems is broken, but the angular momentum of the total system remains a constant of motion. Use of the latter fact is helpful in the solution of the Schrödinger equation.
As an example we consider two electrons, 1 and 2, in an atom (say the helium atom). If there's no electron-electron interaction, but only electron nucleus interaction, the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. Both operators,
l(1) and
l(2), are conserved.
However, if we switch on the electron-electron interaction depending on the distance
d(1,2) between the electrons, then only a simultaneous
and equal rotation of the two electrons will leave
d(1,2) invariant. In such a case neither
l(1) nor
l(2) is a constant of motion but
L =
l(1) +
l(2)
is. Given eigenstates of
l(1) and
l(2), the construction of eigenstates of
L (which still is conserved) is the
coupling of the angular momenta of electron 1 and 2.
In
quantum mechanics, coupling also exists between angular momenta belonging to different
Hilbert spaces of a single object,
for example its
spin and its orbital
angular momentum.
Reiterating slightly differently the above: one expands the
quantum states of composed systems (
for example made of subunits like two
hydrogen atoms or two
electrons) in
basis sets which are made of
direct products of
quantum states which in turn describe the subsystems individually. We assume that the states of the subsystems can be chosen as eigenstates of their angular momentum operators (and of their component along any arbitrary
z axis). The subsystems are therefore correctly described by a set of
l,
m quantum numbers (see
angular momentum for details). When there's interaction between the subsystems, the total Hamiltonian contains terms that don't commute with the angular operators acting on the subsystems only. However, these terms
do commute with the
total angular momentum operator. Sometimes one refers to the non-commuting interaction terms in the Hamiltonian as
angular momentum coupling terms, because they necessitate the angular momentum coupling.
Footnote
Spin-orbit coupling
The behavior of
atoms and smaller
particles is well described by the theory of
quantum mechanics, in which each particle has an intrinsic angular momentum called
spin and specific configurations (of for example electrons in an atom) are described by a set of
quantum numbers. Collections of particles also have angular momenta and corresponding quantum numbers, and under different circumstances the angular momenta of the parts add in different ways to form the angular momentum of the whole. Angular momentum coupling is a category including some of the ways that subatomic particles can interact with each other.
In
atomic physics,
spin-orbit coupling also known as
spin-pairing describes a weak magnetic interaction, or
coupling, of the particle
spin and the
orbital motion of this particle, for example the
electron spin and its motion around an
atomic
nucleus. One of its effects is to separate the energy of internal states of the atom, for example spin-aligned and spin-antialigned that would otherwise be identical in energy. This interaction is responsible for many of the details of atomic structure.
In the
macroscopic world of
orbital mechanics, the term
spin-orbit coupling is sometimes used in the same sense as
spin-orbital resonance.
LS coupling
In light atoms (generally Z<30), electron spins
si interact among themselves so they combine to form a total spin angular momentum
S. The same happens with orbital angular momenta
li, forming a single orbital angular momentum
L. The interaction between the quantum numbers
L and
S is called
Russell-Saunders coupling or
LS coupling. Then
S and
L add together and form a total angular momentum
J:
» where
This description, facilitating calculation of this kind of interaction, is known as
jj coupling.
Spin-spin coupling
See also: J-coupling and Dipolar coupling in NMR spectroscopy
Spin-spin coupling is the coupling of the intrinsic angular momentum (
spin) of different particles.
Such coupling between pairs of nuclear spins is an important feature of
nuclear magnetic resonance (NMR) spectroscopy as it can
provide detailed information about the structure and conformation of molecules.
Spin-spin coupling between nuclear spin and electronic spin is responsible for
hyperfine structure in atomic spectra.
Term symbols
Term symbols are used to represent the states and spectral transitions of atoms, they're found from coupling of angular momenta mentioned above. When the state of an atom has been specified with a term symbol, the allowed transitions can be found through
selection rules by considering which transitions would conserve
angular momentum. A
photon has spin 1, and when there's a transition with emission or absorption of a photon the atom will need to change state to conserve angular momentum. The term symbol selection rules are. ΔS=0, ΔL=0,±1, Δl=±1, ΔJ=0,±1
Relativistic effects
In very heavy atoms, relativistic shifting of the energies of the electron energy levels accentuates spin-orbit coupling effect. Thus, for example, uranium molecular orbital diagrams must directly incorporate relativistic symbols when considering interactions with other atoms.
Nuclear coupling
In atomic nuclei, the spin-orbit interaction is much stronger than for atomic electrons, and is incorporated directly into the nuclear shell model. In addition, unlike atomic-electron term symbols, the lowest energy state isn't L - S, but rather, l + s. All nuclear levels whose l value (orbital angular momentum) is greater than zero are thus split in the shell model to create states designated by l + s and l - s. Due to the nature of the
shell model, which assumes an average potential rather than a central Coulombic potential, the nucleons that go into the l + s and l - s nuclear states are considered degenerate within each orbital (for example The 2p3/2 contains four nucleons, all of the same energy. Higher in energy is the 2p1/2 which contains two equal-energy nucleons).
Further Information
Get more info on 'Spin-spin Coupling'.
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