Editing Angular momentum coupling (section)

==Spin–orbit coupling==
{{Main article|Spin–orbit coupling}}

The behavior of [[atoms]] and smaller [[Subatomic particle|particles]] is well described by the theory of [[quantum mechanics]], in which each particle has an intrinsic angular momentum called [[Spin (physics)|spin]] and specific configurations (of e.g. 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 couple 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 (physics)|coupling]], of the particle [[Spin (physics)|spin]] and the [[orbital motion (quantum)|orbital motion]] of this particle, e.g. the [[electron]] spin and its motion around an [[atom]]ic [[atomic nucleus|nucleus]]. One of its effects is to separate the energy of internal states of the atom, e.g. 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 [[solid-state physics]], the spin coupling with the orbital motion can lead to splitting of [[Electronic band structure|energy bands]] due to [[Dresselhaus effect|Dresselhaus]] or [[Rashba effect|Rashba]] effects.

In the [[macroscopic]] world of [[astrodynamics|orbital mechanics]], the term ''spin–orbit coupling'' is sometimes used in the same sense as [[orbital resonance|spin–orbit resonance]].

===LS coupling===
[[File:LS coupling (corrected).png|thumb|250x250px|Illustration of L–S coupling. Total angular momentum '''J''' is green, orbital '''L''' is blue, and spin '''S''' is red.]]
In light atoms (generally ''Z''&nbsp;≤&nbsp;30<ref>[http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/Electronic_Spectroscopy/The_atomic_spectrum/Atomic_Term_Symbols/The_Russell_Saunders_Coupling_Scheme The Russell Saunders Coupling Scheme] R. J. Lancashire, UCDavis ChemWiki (accessed 26 Dec.2015)</ref>), electron spins '''s'''<sub>''i''</sub> interact among themselves so they combine to form a total spin angular momentum '''S'''. The same happens with orbital angular momenta '''ℓ'''<sub>''i''</sub>, forming a total orbital angular momentum '''L'''. The interaction between the quantum numbers '''L''' and '''S''' is called '''Russell&ndash;Saunders coupling''' (after [[Henry Norris Russell]] and [[Frederick Albert Saunders|Frederick Saunders]]<!-- (1875-1963) -->) or '''LS coupling'''. Then '''S''' and '''L''' couple together and form a total angular momentum '''J''':<ref>{{cite book|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles|edition=2nd|author=R. Resnick, R. Eisberg|publisher=John Wiley & Sons|year=1985|page=[https://archive.org/details/quantumphysicsof00eisb/page/281 281]|isbn=978-0-471-87373-0|url=https://archive.org/details/quantumphysicsof00eisb/page/281}}</ref><ref>{{cite book|title = Physics of Atoms and Molecules|url = https://archive.org/details/physicsatomsmole00bran_159|url-access = limited|author=B.H. Bransden, C.J.Joachain|publisher=Longman|year=1983|pages=[https://archive.org/details/physicsatomsmole00bran_159/page/n346 339]–341|isbn=0-582-44401-2}}</ref>

:<math>\mathbf J = \mathbf L + \mathbf S, \, </math>

where '''L''' and '''S''' are the totals:

: <math>\mathbf L = \sum_i \boldsymbol{\ell}_i, \ \mathbf S = \sum_i \mathbf{s}_i. \, </math>

This is an approximation which is good as long as any external magnetic fields are weak.  In larger magnetic fields, these two momenta decouple, giving rise to a different splitting pattern in the energy levels (the [[Paschen–Back effect]]), and the size of LS coupling term becomes small.<ref>{{cite book|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles|edition=2nd|author=R. Resnick, R. Eisberg|publisher=John Wiley & Sons|year=1985|isbn=978-0-471-87373-0|url=https://archive.org/details/quantumphysicsof00eisb}}</ref>

For an extensive example on how LS-coupling is practically applied, see the article on [[term symbol]]s.

===jj coupling===
In heavier atoms the situation is different. In atoms with bigger nuclear charges, spin–orbit interactions are frequently as large as or larger than spin–spin interactions or orbit–orbit interactions. In this situation, each orbital angular momentum '''ℓ'''<sub>''i''</sub> tends to combine with the corresponding individual spin angular momentum '''s'''<sub>''i''</sub>, originating an individual total angular momentum '''j'''<sub>''i''</sub>. These then couple up to form the total angular momentum '''J'''
:<math>\mathbf J = \sum_i \mathbf j_i = \sum_i (\boldsymbol{\ell}_i + \mathbf{s}_i).</math> 
This description, facilitating calculation of this kind of interaction, is known as ''jj coupling''.