Editing Selection rule (section)

=== Angular momentum ===
{{See also|angular momentum coupling}}

In general, electric (charge) radiation or magnetic (current, magnetic moment) radiation can be classified into [[multipole moments|multipoles]] E{{mvar|λ}} (electric) or M{{mvar|λ}} (magnetic) of order 2<sup>{{mvar|λ}}</sup>, e.g., E1 for electric [[dipole]], E2 for [[quadrupole]], or E3 for octupole. In transitions where the change in angular momentum between the initial and final states makes several multipole radiations possible, usually the lowest-order multipoles are overwhelmingly more likely, and dominate the transition.<ref>{{cite book |last=Softley |first=T. P. |title=Atomic Spectra |publisher=[[Oxford University Press]] |location=Oxford, UK |date=1994 |isbn=0-19-855688-8}}</ref>

The emitted particle carries away angular momentum, with quantum number {{mvar|λ}}, which for the photon must be at least 1, since it is a vector particle (i.e., it has [[total angular momentum quantum number|{{mvar|J}}]]<sup>[[Parity (physics)|{{mvar|P}}]]</sup> = 1<sup>−</sup>). Thus, there is no radiation from  E0 (electric monopoles) or  M0  ([[magnetic monopole]]s, which do not seem to exist).

Since the total angular momentum has to be conserved during the transition, we have that
: <math>\mathbf J_\text{i} = \mathbf{J}_\text{f} + \boldsymbol{\lambda},</math>
where <math>\|\boldsymbol{\lambda}\| = \sqrt{\lambda(\lambda + 1)} \, \hbar,</math> and its {{mvar|z}} projection is given by <math>\lambda_z = \mu \hbar;</math> and where <math>\mathbf J_\text{i}</math> and <math>\mathbf J_\text{f}</math> are, respectively, the initial and final angular momenta of the atom. 
The corresponding quantum numbers {{mvar|λ}} and {{mvar|μ}} ({{mvar|z}}-axis angular momentum) must satisfy
: <math>|J_\text{i} - J_\text{f}| \le \lambda \le J_\text{i} + J_\text{f}</math>
and
: <math>\mu = M_\text{i} - M_\text{f}.</math>

Parity is also preserved. For electric multipole transitions 
: <math>\pi(\mathrm{E}\lambda) = \pi_\text{i} \pi_\text{f} = (-1)^{\lambda},</math>
while for magnetic multipoles
: <math>\pi(\mathrm{M}\lambda) = \pi_\text{i} \pi_\text{f} = (-1)^{\lambda+1}.</math>
Thus, parity does not change for E-even or M-odd multipoles, while it changes for E-odd or M-even multipoles.

{{anchor|anchor_forbidden_trans}}These considerations generate different sets of transitions rules depending on the multipole order and type. The expression ''[[forbidden transition]]s'' is often used, but this does not mean that these transitions ''cannot'' occur, only that they are ''electric-dipole-forbidden''. These transitions are perfectly possible; they merely occur at a lower rate. If the rate for an E1 transition is non-zero, the transition is said to be permitted; if it is zero, then M1, E2, etc. transitions can still produce radiation, albeit with much lower transitions rates. The transition rate decreases by a factor of about 1000 from one multipole to the next one, so the lowest multipole transitions are most likely to occur.<ref>{{cite book |last1=Condon |first1=E. V. |last2=Shortley |first2=G. H. |orig-year=1935 |title=The Theory of Atomic Spectra |url=https://archive.org/details/in.ernet.dli.2015.212979 |publisher=Cambridge University Press |isbn=0-521-09209-4<!-- reprint --> |year=1999<!-- reprint -->}}</ref>

Semi-forbidden transitions (resulting in so-called intercombination lines) are electric dipole (E1) transitions for which the selection rule that the spin does not change is violated. This is a result of the failure of [[LS coupling]].

==== Summary table ====
<math>J = L + S</math> is the total angular momentum, 
<math>L</math> is the [[azimuthal quantum number]], 
<math>S</math> is the [[spin quantum number]], and
<math>M_J</math> is the [[Total angular momentum quantum number|secondary total angular momentum quantum number]].
Which transitions are allowed is based on the [[hydrogen-like atom]]. The symbol <math>\not\leftrightarrow</math> is used to indicate a forbidden transition.
{| class="wikitable" style="text-align:center"
|-
! colspan=2 | Allowed transitions
! Electric dipole (E1)
! Magnetic dipole (M1)
! Electric quadrupole (E2)
! Magnetic quadrupole (M2)
! Electric octupole (E3)
! Magnetic octupole (M3)
|-
! rowspan=3 | Rigorous rules 	
! (1)
| colspan=2 | <math>\begin{matrix} \Delta J = 0, \pm 1 \\ (J = 0 \not \leftrightarrow 0)\end{matrix}</math>
| colspan=2 | <math>\begin{matrix} \Delta J = 0, \pm 1, \pm 2 \\ (J = 0 \not \leftrightarrow 0, 1;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2}\end{matrix})\end{matrix}</math>
| colspan=2 | <math>\begin{matrix}\Delta J = 0, \pm1, \pm2, \pm 3 \\ (0 \not \leftrightarrow 0, 1, 2;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2} \end{matrix}, \begin{matrix}{3 \over 2}\end{matrix};\ 1 \not \leftrightarrow 1) \end{matrix}</math>
|-
! (2)
| colspan=2 | <math>\Delta M_J = 0, \pm 1 \  (M_J = 0 \not \leftrightarrow 0</math> if <math>\Delta J=0)</math>
| colspan=2 | <math>\Delta M_J = 0, \pm 1, \pm2</math>
| colspan=2 | <math>\Delta M_J = 0, \pm 1, \pm2, \pm 3</math>
|-
! (3)
| <math>\pi_\text{f} = -\pi_\text{i}</math>
| colspan=2 | <math>\pi_\text{f} = \pi_\text{i}</math>	
| colspan=2 | <math>\pi_\text{f} = -\pi_\text{i}</math>
| <math>\pi_\text{f} = \pi_\text{i}</math>
|-
! rowspan=2 | LS coupling
! (4)
| One-electron jump<br/><math>\Delta L = \pm 1</math>
| No electron jump<br/><math>\Delta L = 0</math>,<br><math>\Delta n = 0</math>
| None or one-electron jump<br/><math>\Delta L = 0, \pm 2</math>
| One-electron jump<br/><math>\Delta L = \pm 1</math>
| One-electron jump<br/><math>\Delta L = \pm 1, \pm 3</math>
| One-electron jump<br/><math>\Delta L = 0, \pm 2</math>
|-	
! (5)
| If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math>
| If <math>\Delta S = 0</math>:<br/><math>\Delta L = 0\,</math>
| colspan=2 | If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \pm 2 \\ (L = 0 \not \leftrightarrow 0, 1)\end{matrix}</math>
| colspan=2 | If <math>\Delta S = 0</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \pm 2, \pm 3 \\ (L=0 \not \leftrightarrow 0, 1, 2;\ 1 \not \leftrightarrow 1)\end{matrix}</math>
|-
! Intermediate coupling
! (6)
| colspan=2 | If <math>\Delta S = \pm 1</math>:<br/><math>\Delta L = 0, \pm 1, \pm 2\,</math>
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2, 
\pm 3 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math>
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math> 	
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2, \pm 3, \pm 4 \\ (L = 0 \not \leftrightarrow 0, 1)\end{matrix}</math> 
| If <math>\Delta S = \pm 1</math>:<br/><math>\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}</math> 
|}
In [[hyperfine structure]], the total angular momentum of the atom is <math>F = I + J,</math> where <math>I</math> is the [[Quantum number#Nuclear angular momentum quantum numbers|nuclear spin angular momentum]] and <math>J</math> is the total angular momentum of the electron(s). Since <math>F = I + J</math> has a similar mathematical form as <math>J = L + S,</math> it obeys a selection rule table similar to the table above.