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Selection rule

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.

In many cases, a transition involves the emision 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 2λ, 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.

This 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 JP = 1). Thus there is no E0 (electric monopoles) or M0 (magnetic monopoles) radiation (the latter is also forbidden because magnetic monopoles do not exist).

Since the total angular momentum has to be conseved 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 $\boldsymbol{\lambda}_z = \mu \, \hbar.$ For the corresponding quantum numbers λ, μ it must hold:

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

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. Usually, the expression forbidden rules is used; it does not mean that they cannot occur, it means electric-dipole forbidden. These transitions are perfectly possible, they just occur at a slower rate. If the probability for a E1 transition is different from zero, then the transition is called permitted; if it is zero, then M1, E2, etc. can still produce radiation, with very low probability but still with non negligible intensity. Those are the so-called forbidden transitions. There is an approximate drop of 10−3 in a the transition probability from one multipole to the next one, so the lowest multipoles are the most probable.

Semi-forbidden transitions are electric dipole (E1) transitions for which the selection rule that the spin does not change is violated.

Summary table

 Electric dipole (E1) Magnetic dipole (M1) Electric quadrupole (E2) Magnetic quadrupole (M2) Electric octupole (E3) Magnetic octupole (M3) $\begin{matrix} \Delta J = 0, \pm 1 \\ (J = 0 \not \leftrightarrow 0)\end{matrix}$ $\begin{matrix} \Delta J = 0, \pm 1, \pm 2 \\ (J = 0 \not \leftrightarrow 0, 1;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2}\end{matrix})\end{matrix}$ $\begin{matrix}\Delta J = 0, \pm1, \pm2, \pm 3 \\ (0 \not \leftrightarrow 0, 1, 2;\ \begin{matrix}{1 \over 2}\end{matrix} \not \leftrightarrow \begin{matrix}{1 \over 2} \end{matrix}, \begin{matrix}{3 \over 2}\end{matrix};\ 1 \not \leftrightarrow 1) \end{matrix}$ $\Delta M_J = 0, \pm 1$ $\Delta M_J = 0, \pm 1, \pm2$ $\Delta M_J = 0, \pm 1, \pm2, \pm 3$ $\pi_{\mathrm{f}} = -\pi_{\mathrm{i}}\,$ $\pi_{\mathrm{f}} = \pi_{\mathrm{i}}\,$ $\pi_{\mathrm{f}} = -\pi_{\mathrm{i}}\,$ $\pi_{\mathrm{f}} = \pi_{\mathrm{i}}\,$ One electron jumpΔl = ±1 No electron jumpΔl = 0,Δn = 0 None or one electron jumpΔl = 0, ±2 One electron jumpΔl = ±1 One electron jumpΔl = ±1, ±3 One electron jumpΔl = 0, ±2 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$\begin{matrix}\Delta L = 0, \pm 1, \pm 2, \pm 3 \\ (L=0 \not \leftrightarrow 0, 1, 2;\ 1 \not \leftrightarrow 1)\end{matrix}$ 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$\begin{matrix}\Delta L = 0, \pm 1, \\ \pm 2 \\ (L = 0 \not \leftrightarrow 0)\end{matrix}$