Selection rule

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In physics, especially in the context of quantum mechanics, a selection rule is a condition constraining the physical properties of the initial system and the final system that is necessary for a process to occur with a nonzero probability.

See also: angular momentum coupling

In many cases, a transition involves the emission of radiation, that is, a photon is emitted. In general, electric (charge) radiation or magnetic (current, magnetic moment radiation) can be classified into multipoles Eλ (electric) or Mλ (magnetic) of order 2^λ, e.g. E1, E2, E3 for electric dipole, quadrupole or octupole. The radiation field will be a sum of the multipole contributions; however, usually one or two multipoles dominate.

The emitted particle carries away an angular momentum λ, which for the photon must be at least 1, since it is a vector particle (i.e., it has J^P = 1⁻). Thus there is no E0 (electric monopoles) or M0 (magnetic monopoles) radiation (the latter is forbidden because magnetic monopoles do not seem to exist).

Since the total angular momentum has to be conserved during the transition, we have that

$\mathbf J_{\mathrm{i}} = \mathbf{J}_{\mathrm{f}} + \boldsymbol{\lambda}$

where $\Vert \boldsymbol{\lambda} \Vert = \sqrt{\lambda(\lambda + 1)} \, \hbar$ , and its z-projection is given by $\lambda_z = \mu \, \hbar.$ The corresponding quantum numbers λ, μ must satisfy

$| J_{\mathrm{i}} - J_{\mathrm{f}} | \le \lambda \le J_{\mathrm{i}} + J_{\mathrm{f}}$

and

$\mu = M_{\mbox{i}} - M_{\mbox{f}}\,.$

Parity is also preserved. For electric multipole transitions

$\pi(\mathrm{E}\lambda) = \pi_{\mathrm{i}} \pi_{\mathrm{f}} = (-1)^{\lambda}\,$

while for magnetic multipoles

$\pi(\mathrm{M}\lambda) = \pi_{\mathrm{i}} \pi_{\mathrm{f}} = (-1)^{\lambda+1}\,.$

Thus, parity does not change for E-even or M-odd multipoles, while it changes for E-odd or M-even multipoles.

These considerations generate different sets of transitions rules depending on the multipole order and type. The expression forbidden transitions is often used; 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. These are the so-called forbidden transitions. The transition rate decreases by a factor of approximately 10⁻³ from one multipole to the next one, so the lowest multipole transitions are most likely to occur.

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.

[edit] Summary table

		Electric dipole (E1)	Magnetic dipole (M1)	Electric quadrupole (E2)	Magnetic quadrupole (M2)	Electric octupole (E3)	Magnetic octupole (M3)
Rigorous rules	(1)	$\begin{matrix} \Delta J = 0, \pm 1 \\ (J = 0 \not \leftrightarrow 0)\end{matrix}$		$\begin{matrix} \Delta J = 0, \pm 1, \pm 2 \\ (J = 0 \not \leftrightarrow 0, 1;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2}\end{matrix})\end{matrix}$		$\begin{matrix}\Delta J = 0, \pm1, \pm2, \pm 3 \\ (0 \not \leftrightarrow 0, 1, 2;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2} \end{matrix}, \begin{matrix}{3 \over 2}\end{matrix};\ 1 \not \leftrightarrow 1) \end{matrix}$
	(2)	$\Delta M_J = 0, \pm 1$		$\Delta M_J = 0, \pm 1, \pm2$		$\Delta M_J = 0, \pm 1, \pm2, \pm 3$
	(3)	$\pi_{\mathrm{f}} = -\pi_{\mathrm{i}}\,$	$\pi_{\mathrm{f}} = \pi_{\mathrm{i}}\,$		$\pi_{\mathrm{f}} = -\pi_{\mathrm{i}}\,$		$\pi_{\mathrm{f}} = \pi_{\mathrm{i}}\,$
LS coupling	(4)	One electron jump Δl = ±1	No electron jump Δl = 0, Δn = 0	None or one electron jump Δl = 0, ±2	One electron jump Δl = ±1	One electron jump Δl = ±1, ±3	One electron jump Δl = 0, ±2
LS coupling	(5)	If ΔS = 0 $\begin{matrix}\Delta L = 0, \pm 1 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}$	If ΔS = 0 $\Delta L = 0\,$	If ΔS = 0 $\begin{matrix}\Delta L = 0, \pm 1, \pm 2 \\ (L = 0 \not \leftrightarrow 0, 1)\end{matrix}$		If ΔS = 0 $\begin{matrix}\Delta L = 0, \pm 1, \pm 2, \pm 3 \\ (L=0 \not \leftrightarrow 0, 1, 2;\ 1 \not \leftrightarrow 1)\end{matrix}$
Inter- mediate coupling	(6)	If ΔS = ±1 $\Delta L = 0, \pm 1, \pm 2\,$		If ΔS = ±1 $\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2, \pm 3 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}$	If ΔS = ±1 $\begin{matrix}\Delta L = 0, \pm 1 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}$	If ΔS = ±1 $\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2, \pm 3, \pm 4 \\ (L = 0 \not \leftrightarrow 0, 1)\end{matrix}$	If ΔS = ±1 $\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}$