Multipole Radiation and Selection Rules

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 $L\hslash$ 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.¹

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,

Solving the quadratic for $E_{\gamma }$ and expanding, since $E_{\gamma }\ll M{c}^2$,

The recoil shift is minute. For a $1\mathrm{MeV}$ transition in a nucleus of mass number $A\approx 100$, $M{c}^2\approx 9.3\times 10^4\mathrm{MeV}$ and $E_R\approx 5\mathrm{eV}$, a part in $2\times 10^5$ 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.

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 $\hslash$, 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.
• 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.

Transition rates

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

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 $(E_{\gamma }/\hslash c{)}^{2L+1}$ makes the rate climb steeply with photon energy and with multipole order. Writing $k=E_{\gamma }/\hslash c$ 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 $(\hslash /M_{p}cR{)}^2\sim 0.3A^{-2/3}$, 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

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).

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; 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

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 ($E2,E4,\dots$) or odd-order magnetic ($M1,M3,\dots$). If the parities differ, $\Delta \pi =$ yes, the multipole must have odd parity: odd-order electric ($E1,E3,\dots$) or even-order magnetic ($M2,M4,\dots$).

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.

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. 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.