---
title: Multipole Radiation and Selection Rules
module: Gamma Decay
moduleNumber: 7
lessonNumber: 1
order: 701
summary: >
  Gamma decay carries a nucleus from an excited state to a lower one by emitting
  a photon of definite angular momentum and parity. We correct the photon energy
  for nuclear recoil, expand the radiation field into electric and magnetic
  multipoles, and read off how the transition rate collapses with each increase
  in multipole order. The Weisskopf single-particle estimates set the scale, and
  angular-momentum and parity conservation fix which multipole dominates.
topics: [Gamma Decay]
draft: false
sources:
  - book: Krane
    ref: "Ch. 10 — Gamma Decay; §10.1 Energetics, §10.2 Classical Multipole Radiation, §10.3 Transition Rates, §10.4 Angular Momentum and Parity Selection Rules"
  - book: Wong
    ref: "Ch. 6 — Nuclear Collective Motion; §6-5 Electromagnetic Transitions"
---

Alpha and beta decay change the nuclear charge and mass; gamma decay changes
neither. A nucleus left in an excited state by a preceding decay or reaction
sheds the excitation energy as a photon, dropping from a level $E_i$ to a lower
level $E_f$ of the **same** nuclide. The photon is not emitted with arbitrary
properties: it carries away a definite angular momentum $L\hbar$ and a definite
parity, and those two quantum numbers, matched against the spins and parities of
the initial and final nuclear states, decide both which photons can appear and
how fast they do.[^krane-gamma]

## Recoil-corrected energetics

Write the level spacing as $\Delta E = E_i - E_f$. Momentum conservation forces
the nucleus to recoil against the emitted photon, so $\Delta E$ is split between
the photon energy $E_\gamma$ and the recoil kinetic energy $E_R$ of the daughter
nucleus of mass $M$. With photon momentum $p_\gamma = E_\gamma/c$ and a nucleus
initially at rest,

$$
\Delta E = E_\gamma + E_R, \qquad E_R = \frac{p_\gamma^2}{2M} = \frac{E_\gamma^2}{2Mc^2}.
$$

Solving the quadratic for $E_\gamma$ and expanding, since $E_\gamma \ll Mc^2$,

$$
E_\gamma = \Delta E\left(1 - \frac{\Delta E}{2Mc^2} + \cdots\right)
\approx \Delta E - \frac{(\Delta E)^2}{2Mc^2}.
$$

The recoil shift is minute. For a $1\ \mathrm{MeV}$ transition in a nucleus of
mass number $A \approx 100$, $Mc^2 \approx 9.3\times10^4\ \mathrm{MeV}$ and
$E_R \approx 5\ \mathrm{eV}$, a part in $2\times10^5$ of the photon energy. It is
negligible for level bookkeeping but not always physically negligible: the same
recoil is what destroys resonant absorption of nuclear gammas and motivates the
[Mössbauer effect](/nuclear-physics/gamma-decay/angular-correlations-mossbauer).

$$
% caption: A gamma transition drops the nucleus from level E_i to E_f; the level
% spacing Delta-E is shared between the emitted photon and a small nuclear recoil.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black!70, very thick] (0,3.4) -- (3.2,3.4);
  \node[black!70, anchor=west] at (3.3,3.4) {excited state};
  \draw[black!70, very thick] (0,0.6) -- (3.2,0.6);
  \node[black!70, anchor=west] at (3.3,0.6) {lower state};
  % transition arrow
  \draw[->, acc, very thick] (1.2,3.4) -- (1.2,0.6);
  \node[acc, anchor=east] at (1.1,2.0) {gamma photon};
  % level labels
  \node[black, anchor=east] at (-0.1,3.4) {$E_i$};
  \node[black, anchor=east] at (-0.1,0.6) {$E_f$};
  % recoil
  \draw[->, black] (2.5,0.6) -- (2.5,0.05);
  \node[black, anchor=west] at (2.55,0.25) {recoil};
\end{tikzpicture}
$$

## The classical multipole expansion

A localized time-varying charge and current distribution radiates a field that
separates into **multipole** components. Each component is labeled by an integer
$L \ge 1$ (the multipole order) and by whether the radiating source is the charge
density, giving **electric** multipoles $EL$, or the current and intrinsic
magnetization, giving **magnetic** multipoles $ML$. The order $L$ is the angular
momentum, in units of $\hbar$, that each emitted photon carries.

Two facts fix the allowed values and follow directly from the transformation
properties of the fields.

- **No monopole radiation.** A photon has intrinsic spin $1$, so it removes at
  least one unit of angular momentum: $L = 0$ is forbidden. A $0 \to 0$
  transition therefore emits no photon at all and must proceed by the competing
  channels of the [next lesson](/nuclear-physics/gamma-decay/internal-conversion-isomers).
- **Parity of a multipole.** An electric $2^L$-pole field has parity
  $\pi(EL) = (-1)^L$; a magnetic $2^L$-pole field has parity
  $\pi(ML) = (-1)^{L+1}$. Electric and magnetic multipoles of the same order
  carry opposite parity.

The lowest orders name familiar patterns: $L=1$ is dipole radiation with its
single node and two-lobed intensity, $L=2$ is quadrupole radiation with a
four-lobed pattern. Higher $L$ spreads the emission into $2L$ lobes and pushes
more of the field's angular structure to large multipole index.

$$
% caption: Angular intensity patterns: electric or magnetic dipole radiation
% (L=1) forms two lobes with a node across the source axis, while quadrupole
% radiation (L=2) forms four lobes.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % dipole: two lobes meeting at a node on the source axis
  \begin{scope}[shift={(0,0)}]
    \draw[thick] (0,0.62) ellipse (0.46 and 0.62);
    \draw[thick] (0,-0.62) ellipse (0.46 and 0.62);
    \draw[black, dashed] (-1.3,0) -- (1.3,0);
    \fill[black!70] (0,0) circle (1.4pt);
    \node[black!70, anchor=north] at (0,-1.5) {dipole, L = 1};
  \end{scope}
  % quadrupole: four lobes at 45 degrees
  \begin{scope}[shift={(4.6,0)}]
    \foreach \a in {45,135,225,315}{
      \draw[thick, rotate=\a] (0.7,0) ellipse (0.62 and 0.32);
    }
    \fill[black!70] (0,0) circle (1.4pt);
    \node[black!70, anchor=north] at (0,-1.4) {quadrupole, L = 2};
  \end{scope}
\end{tikzpicture}
$$

## Transition rates

Quantizing the multipole field turns the classical radiated power into a photon
emission rate. For a transition emitting a photon of energy
$E_\gamma = \hbar\omega$, the rate for a $\sigma L$ multipole ($\sigma = E$ or
$M$) is

$$
\lambda(\sigma L) = \frac{2(L+1)}{\varepsilon_0\,\hbar\,L\,[(2L+1)!!]^2}
\left(\frac{E_\gamma}{\hbar c}\right)^{2L+1} B(\sigma L),
$$

where $(2L+1)!! = 1\cdot 3\cdot 5\cdots(2L+1)$ and $B(\sigma L)$ is the **reduced
transition probability**, the square of the nuclear matrix element of the
multipole operator between initial and final states. Two features control the
rate before any nuclear structure enters.

- **Energy dependence.** The factor $(E_\gamma/\hbar c)^{2L+1}$ makes the rate
  climb steeply with photon energy and with multipole order. Writing
  $k = E_\gamma/\hbar c$ and $R$ for the nuclear radius, each successive multipole
  brings an extra factor of order $(kR)^2$, and $kR \ll 1$ for nuclear gammas:
  at $E_\gamma = 1\ \mathrm{MeV}$ and $R \approx 6\ \mathrm{fm}$,
  $kR \approx 0.03$, so $(kR)^2 \approx 10^{-3}$.
- **Electric versus magnetic.** For the same order $L$, the magnetic matrix
  element is smaller than the electric one by roughly
  $(\hbar/M_p c R)^2 \sim 0.3\,A^{-2/3}$, so magnetic transitions run about two
  orders of magnitude slower than electric transitions of the same multipolarity.

## Weisskopf single-particle estimates

$B(\sigma L)$ depends on the nuclear wavefunctions, but a useful benchmark comes
from assuming the transition is made by a single proton moving between two
shell-model orbitals in a uniform-density nucleus of radius
$R = R_0 A^{1/3}$ with $R_0 = 1.2\ \mathrm{fm}$. Evaluating the matrix element
with this **Weisskopf** ansatz gives closed-form single-particle rates. With
$E_\gamma$ in $\mathrm{MeV}$ and the rate in $\mathrm{s^{-1}}$, the lowest orders
are

$$
\begin{aligned}
\lambda(E1) &= 1.0\times10^{14}\,A^{2/3}\,E_\gamma^{3}, &
\lambda(M1) &= 3.1\times10^{13}\,E_\gamma^{3},\\[2pt]
\lambda(E2) &= 7.3\times10^{7}\,A^{4/3}\,E_\gamma^{5}, &
\lambda(M2) &= 2.2\times10^{7}\,A^{2/3}\,E_\gamma^{5},\\[2pt]
\lambda(E3) &= 34\,A^{2}\,E_\gamma^{7}, &
\lambda(M3) &= 10\,A^{4/3}\,E_\gamma^{7},\\[2pt]
\lambda(E4) &= 1.1\times10^{-5}\,A^{8/3}\,E_\gamma^{9}, &
\lambda(M4) &= 3.3\times10^{-6}\,A^{2}\,E_\gamma^{9}.
\end{aligned}
$$

Reading down a column, each increase of one unit in $L$ drops the rate by five to
six orders of magnitude for a $1\ \mathrm{MeV}$ transition in a medium-mass
nucleus. Reading across, the magnetic rate trails the electric rate of the same
order. The practical consequence is decisive: **the lowest multipole permitted by
the selection rules dominates**, and mixtures of two multipoles matter only when
the lower one is a magnetic transition competing with the electric transition one
order higher (the common $M1 + E2$ mixing).

$$
% caption: Weisskopf single-particle rates for a 1 MeV transition in a
% medium-mass nucleus fall by roughly five orders of magnitude per unit increase
% in multipole order; magnetic rates trail electric rates of the same order.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % axes
  \draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {multipole order L};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log rate};
  \foreach \x/\l in {1/1,2/2,3/3,4/4}{
    \node[black, anchor=north, font=\scriptsize] at (1.4*\x,0) {\l};
  }
  % electric series (descending, solid)
  \draw[acc, very thick] (1.4,4.3) -- (2.8,3.0) -- (4.2,1.7) -- (5.6,0.4);
  \foreach \x/\y in {1.4/4.3,2.8/3.0,4.2/1.7,5.6/0.4}{ \fill[acc] (\x,\y) circle (1.8pt); }
  \node[acc, anchor=west, font=\scriptsize] at (4.3,2.0) {electric};
  % magnetic series (descending, dashed, below)
  \draw[black, very thick, dashed] (1.4,3.6) -- (2.8,2.3) -- (4.2,1.0) -- (5.6,-0.3);
  \foreach \x/\y in {1.4/3.6,2.8/2.3,4.2/1.0}{ \fill[black] (\x,\y) circle (1.6pt); }
  \node[black, anchor=north east, font=\scriptsize] at (2.7,2.2) {magnetic};
\end{tikzpicture}
$$

Measured rates are quoted in Weisskopf units, the ratio of the observed rate to
this single-particle estimate. Electric quadrupole transitions in deformed nuclei
run at tens or hundreds of Weisskopf units, a direct measure of the collective
enhancement discussed with the
[collective model](/nuclear-physics/nuclear-models/collective-model-rotations-vibrations);
retarded transitions fall well below one Weisskopf unit and signal a structural
mismatch between the connected states.

## Angular-momentum and parity selection rules

The photon removes angular momentum $L$ and, through its parity, connects states
of definite relative parity. Conservation of angular momentum between an initial
state of spin $I_i$ and a final state of spin $I_f$ requires the vector sum
$\vec I_i = \vec I_f + \vec L$, which restricts the magnitude to

$$
|I_i - I_f| \le L \le I_i + I_f, \qquad L \ge 1.
$$

Parity fixes the electric-versus-magnetic character. If the two states have the
**same** parity, $\Delta\pi = $ no, the emitted multipole must have even parity:
even-order electric ($E2, E4, \ldots$) or odd-order magnetic ($M1, M3, \ldots$).
If the parities **differ**, $\Delta\pi = $ yes, the multipole must have odd
parity: odd-order electric ($E1, E3, \ldots$) or even-order magnetic
($M2, M4, \ldots$).

| $\lvert I_i - I_f\rvert$ | $\Delta\pi = $ no (same parity) | $\Delta\pi = $ yes (opposite parity) |
|:---:|:---:|:---:|
| $0$ (with $I_i = I_f \neq 0$) | $M1,\ E2$ | $E1,\ M2$ |
| $1$ | $M1,\ E2$ | $E1,\ M2$ |
| $2$ | $E2,\ M3$ | $M2,\ E3$ |
| $3$ | $M3,\ E4$ | $E3,\ M4$ |
| $4$ | $E4,\ M5$ | $M4,\ E5$ |

Only the two lowest allowed multipoles are listed; higher orders are permitted
but are slower by the Weisskopf factors and never compete. The rule that the
lowest permitted multipole dominates makes the observed multipolarity a direct
readout of the spin and parity change: measuring that a transition is pure $E2$
fixes $\Delta I = 2$ with no parity change, while a pure $E1$ fixes $\Delta I \le 1$
with a parity change.

> **Worked example (Assigning multipolarities in a level scheme).**
> A nucleus has states $2^+$ at $1.33\ \mathrm{MeV}$, $2^+$ at $0.83\ \mathrm{MeV}$,
> and a $0^+$ ground state. The $1.33 \to 0$ transition connects $2^+$ to $0^+$:
> $\Delta I = 2$ with no parity change, so the lowest allowed multipole is $E2$.
> The $1.33 \to 0.83$ transition connects $2^+$ to $2^+$: here $\Delta I$ can be
> $0, 1,$ or $2$ with no parity change, so $M1$ and $E2$ are both allowed and the
> transition is an $M1 + E2$ mixture. The $0.83 \to 0$ transition is again
> $2^+ \to 0^+$, pure $E2$. A $0^+ \to 0^+$ transition, were one present, would be
> forbidden to all multipoles and could de-excite only by internal conversion.

$$
% caption: A level scheme with each transition labeled by its dominant
% multipolarity; the 2-plus to 2-plus link is a mixed M1 plus E2, the others are
% pure E2, and a 0-plus to 0-plus link would emit no photon.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % levels
  \draw[black!70, very thick] (0,4.0) -- (3.6,4.0);
  \node[black, anchor=west] at (3.7,4.0) {$2^+$ upper};
  \draw[black!70, very thick] (0,2.3) -- (3.6,2.3);
  \node[black, anchor=west] at (3.7,2.3) {$2^+$ lower};
  \draw[black!70, very thick] (0,0.4) -- (3.6,0.4);
  \node[black, anchor=west] at (3.7,0.4) {$0^+$ ground};
  % transitions
  \draw[->, black, very thick] (0.7,4.0) -- (0.7,0.4);
  \node[black, anchor=east, font=\scriptsize] at (0.6,2.2) {E2};
  \draw[->, acc, very thick] (1.8,4.0) -- (1.8,2.3);
  \node[acc, anchor=west, font=\scriptsize] at (1.9,3.15) {M1 + E2};
  \draw[->, black, very thick] (2.8,2.3) -- (2.8,0.4);
  \node[black, anchor=west, font=\scriptsize] at (2.9,1.35) {E2};
\end{tikzpicture}
$$

The selection rules are strict, but they only forbid; they do not by themselves
say how fast an allowed transition runs. That rate, set by the Weisskopf estimate
and the nuclear matrix element, is what makes some excited states decay in
$10^{-12}\ \mathrm{s}$ and others survive as long-lived
[isomers](/nuclear-physics/gamma-decay/internal-conversion-isomers). When the
only permitted multipole is high-order and the transition energy is low, the
Weisskopf rate can fall so far that internal conversion takes over the
de-excitation entirely.

[^krane-gamma]: **Krane**, _Introductory Nuclear Physics_, Ch. 10, §10.1–§10.4: recoil-corrected energetics, the electric and magnetic multipole expansion, the transition-rate formula, the Weisskopf single-particle estimates (Table 10.2), and the angular-momentum and parity selection rules. The collective enhancement of $E2$ rates in Weisskopf units is developed in **Wong**, _Introductory Nuclear Physics_, §6-5. Measured level energies and multipole assignments are tabulated by the National Nuclear Data Center, [nndc.bnl.gov](https://www.nndc.bnl.gov/).