---
title: Internal Conversion and Isomers
module: Gamma Decay
moduleNumber: 7
lessonNumber: 2
order: 702
summary: >
  A nucleus can shed excitation energy without emitting a photon by handing it
  directly to an atomic electron. We define the internal-conversion coefficient,
  trace its growth with atomic number, multipole order, and decreasing energy,
  and treat the electron-only E0 transitions and internal pair formation. When
  the lowest allowed multipole is high and the energy low, the gamma rate falls
  so far that the excited state survives as a metastable isomer.
topics: [Gamma Decay]
draft: false
sources:
  - book: Krane
    ref: "Ch. 10 — Gamma Decay; §10.6 Internal Conversion, §10.7 Lifetimes for Gamma Emission"
  - book: Wong
    ref: "Ch. 6 — Nuclear Collective Motion; §6-5 Electromagnetic Transitions"
---

Gamma emission is not the only way an excited nucleus reaches its ground state.
The same electromagnetic multipole field that would radiate a photon can instead
couple directly to a bound atomic electron and eject it. This **internal
conversion** competes with gamma emission for every transition, dominates when
gamma emission is slow, and provides the only de-excitation route at all for
$0 \to 0$ transitions, which no photon can carry. The physics that makes some
transitions slow is also what produces long-lived isomeric states.[^krane-ic]

## Internal conversion

The nuclear multipole field extends over the atomic electron cloud. For an
inner-shell electron whose wavefunction overlaps the nucleus, the interaction can
transfer the full transition energy $\Delta E$ to the electron, which leaves the
atom with kinetic energy

$$
T_e = \Delta E - B_n,
$$

where $B_n$ is the binding energy of the electron in its shell $n = K, L, M,
\ldots$. This is a one-step process, not the emission of a photon that is later
photo-absorbed by the same atom: the coupling is direct, and it occurs even at
energies where no real photon channel is open. Because the shell binding energies
are discrete, the ejected **conversion electrons** appear as sharp lines in the
electron spectrum, one per subshell, superimposed on the continuous
[beta spectrum](/nuclear-physics/beta-decay/beta-decay-energetics-neutrino) of any
accompanying decay. The vacancy left behind is filled by outer electrons, so
conversion is followed by characteristic X-rays or Auger electrons.

$$
% caption: An excited nucleus de-excites either by emitting a gamma photon or by
% transferring the energy directly to a K-shell electron, which is ejected with
% kinetic energy equal to the transition energy minus its binding energy.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  % nucleus
  \draw[thick] (0,0) circle (0.42);
  \node[black, anchor=north] at (0,-0.55) {nucleus};
  % electron shells
  \draw[black] (0,0) circle (1.15);
  \draw[black] (0,0) circle (1.75);
  \node[black, anchor=south, font=\scriptsize] at (0,1.15) {K};
  \node[black, anchor=south, font=\scriptsize] at (0,1.75) {L};
  % gamma channel
  \draw[->, black, very thick] (0.5,0.4) -- (2.6,1.8);
  \node[black, anchor=west, font=\scriptsize] at (2.6,1.8) {gamma photon};
  % conversion channel: K electron ejected
  \fill[acc] (1.15,0) circle (2.2pt);
  \draw[->, acc, very thick] (1.3,-0.1) -- (2.9,-1.1);
  \node[acc, anchor=west, font=\scriptsize] at (2.9,-1.1) {conversion electron};
\end{tikzpicture}
$$

The two channels are independent, so their rates add. The total de-excitation
rate is $\lambda_t = \lambda_\gamma + \lambda_e$, and the branching between them
is measured by the **internal-conversion coefficient**

$$
\alpha = \frac{\lambda_e}{\lambda_\gamma},
\qquad \lambda_t = \lambda_\gamma(1 + \alpha).
$$

The coefficient is a sum over the shells that can convert,
$\alpha = \alpha_K + \alpha_L + \alpha_M + \cdots$, with the $K$ shell usually
largest because its electrons have the greatest amplitude at the nucleus (until
the transition energy drops below the $K$ binding energy, which closes that
subshell). The coefficient can range from far below unity for fast high-energy
transitions to $10^3$ or more for slow, low-energy, high-multipole transitions in
heavy elements.

## How the coefficient scales

A nonrelativistic Born-approximation treatment reproduces the trends even though
it is quantitatively superseded by tabulated relativistic calculations. For an
electric multipole converting in the shell of principal quantum number $n$,

$$
\alpha(EL) \approx \frac{Z^3}{n^3}\,\alpha_{\mathrm{fs}}^{4}\,
\frac{L}{L+1}\left(\frac{2m_e c^2}{E_\gamma}\right)^{L+5/2},
$$

and for a magnetic multipole,

$$
\alpha(ML) \approx \frac{Z^3}{n^3}\,\alpha_{\mathrm{fs}}^{4}
\left(\frac{2m_e c^2}{E_\gamma}\right)^{L+3/2},
$$

with $\alpha_{\mathrm{fs}} = e^2/4\pi\varepsilon_0\hbar c \approx 1/137$ the
fine-structure constant. Four dependences follow.

- **Atomic number.** The $Z^3$ factor makes conversion negligible in light nuclei
  and dominant in heavy ones. It reflects the electron density at the nucleus,
  which grows with $Z$.
- **Transition energy.** The rate falls steeply as $E_\gamma$ increases, opposite
  to the gamma rate, because a more energetic transition matches the electron
  wavefunction less well.
- **Multipole order.** Higher $L$ raises the exponent, so conversion grows with
  multipolarity exactly where gamma emission is being suppressed.
- **Electric versus magnetic.** The magnetic exponent $L + 3/2$ is smaller than
  the electric $L + 5/2$, so magnetic transitions convert relatively more
  strongly at a given energy.

$$
% caption: Internal-conversion coefficients rise as the transition energy falls
% and rise with multipole order; on log axes each multipole is a descending line,
% steeper for higher order.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {log transition energy};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log alpha};
  % E1 (shallow)
  \draw[black, very thick] (0.4,3.4) -- (5.8,1.1);
  \node[black, anchor=south, font=\scriptsize] at (5.0,1.55) {E1};
  % E2 (steeper)
  \draw[black, very thick] (0.4,4.2) -- (5.4,0.4);
  \node[black, anchor=south, font=\scriptsize] at (4.4,1.3) {E2};
  % E3 (steepest, dashed)
  \draw[black, very thick, dashed] (0.4,4.4) -- (4.6,0.2);
  \node[black, anchor=east, font=\scriptsize] at (2.6,2.2) {E3};
\end{tikzpicture}
$$

Because the coefficients depend on $L$ and on the electric-versus-magnetic
character in a calculated, $Z$-dependent way, measuring $\alpha$ (or ratios such
as $\alpha_K/\alpha_L$) against the tabulated values determines the
multipolarity of a transition. Conversion-electron spectroscopy thereby fixes the
spin and parity change even for transitions too weak in gamma emission to analyze
by other means.

## E0 transitions

The selection rules forbid any photon in a $0 \to 0$ transition, since a photon
must carry $L \ge 1$. When two states of spin zero and the same parity are
connected, gamma emission is impossible, yet the states are not stable against
de-excitation: internal conversion proceeds through the **monopole** operator,
whose matrix element is set by the difference in the mean-square charge radius
between the two states,

$$
\rho(E0) \propto \langle f | \textstyle\sum_k r_k^2 | i \rangle.
$$

An E0 transition therefore emits conversion electrons (and, above threshold,
electron-positron pairs) but no gamma ray. Because the operator measures
$\langle r^2 \rangle$, the E0 strength is a sensitive probe of a change in nuclear
shape between the two $0^+$ states, the signature of shape coexistence. The
$6.05\ \mathrm{MeV}$ excited $0^+$ state of $^{16}\mathrm{O}$ decays to the $0^+$
ground state purely by E0, entirely through pair formation and conversion.

$$
% caption: A 0-plus to 0-plus (E0) transition cannot emit a photon; it
% de-excites only by ejecting an atomic electron or, above 1.022 MeV, by creating
% an electron-positron pair.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black!70, very thick] (0,3.2) -- (3.0,3.2);
  \node[black, anchor=west] at (3.1,3.2) {$0^+$ excited};
  \draw[black!70, very thick] (0,0.5) -- (3.0,0.5);
  \node[black, anchor=west] at (3.1,0.5) {$0^+$ ground};
  % no gamma: crossed-out
  \draw[black, dashed] (0.8,3.2) -- (0.8,0.5);
  \draw[black, thick] (0.55,2.1) -- (1.05,1.6);
  \draw[black, thick] (0.55,1.6) -- (1.05,2.1);
  \node[black, anchor=east, font=\scriptsize] at (0.5,1.85) {no gamma};
  % conversion electron
  \draw[->, acc, very thick] (1.9,3.2) -- (1.9,0.5);
  \node[acc, anchor=west, font=\scriptsize] at (2.0,1.85) {electron or pair};
\end{tikzpicture}
$$

## Internal pair formation

When the transition energy exceeds twice the electron rest energy,
$\Delta E > 2 m_e c^2 = 1.022\ \mathrm{MeV}$, the multipole field can create a real
electron-positron pair, which shares the energy $\Delta E - 1.022\ \mathrm{MeV}$
as kinetic energy. This **internal pair formation** is a third de-excitation
channel, again competing with gamma emission and internal conversion. Its
coefficient grows with transition energy, opposite to the trend for internal
conversion, so pair formation is the electron-emitting channel that survives at
high energy. For $0 \to 0$ transitions above threshold it is often the dominant
route, as in the $^{16}\mathrm{O}$ example.

## Isomers

Most excited nuclear states de-excite in $10^{-9}\ \mathrm{s}$ or faster. A state
whose lifetime is long enough to be measured directly, conventionally longer than
about $10^{-9}\ \mathrm{s}$ and sometimes seconds, hours, or years, is a
**metastable** state or **isomer**, written with an $m$ superscript
($^{99m}\mathrm{Tc}$). The long lifetime traces to the Weisskopf rate: from the
[transition-rate estimates](/nuclear-physics/gamma-decay/gamma-multipole-radiation),
the rate collapses when the lowest allowed multipole is high (a large spin
difference $\Delta I$ between the isomer and the states below it) and the
transition energy is low. Both conditions together can lower the rate by ten or
more orders of magnitude.

Isomers cluster in the **islands of isomerism**, the regions of $N$ or $Z$ just
below the magic numbers $50, 82,$ and $126$. There the shell model places a
high-spin orbital ($g_{9/2}, h_{11/2}, i_{13/2}$) next to low-spin orbitals, so
the lowest states of a nucleus differ in spin by several units while lying close
in energy, exactly the combination that stalls gamma decay.

> **Worked example (The $^{99m}\mathrm{Tc}$ isomer).**
> Technetium-99m is a metastable state at $142.7\ \mathrm{keV}$ with spin-parity
> $1/2^-$, decaying to the $9/2^+$ ground state of $^{99}\mathrm{Tc}$. The direct
> transition would require $\Delta I = 4$ with a parity change, hence an $M4$
> multipole, whose Weisskopf rate at this low energy is extremely small. The
> result is a half-life of about $6.0\ \mathrm{h}$. Because it emits a clean
> $140.5\ \mathrm{keV}$ photon (through an intermediate level) with little
> accompanying particle radiation, and because its parent $^{99}\mathrm{Mo}$ is
> easily produced and eluted, $^{99m}\mathrm{Tc}$ is the most widely used isotope
> in diagnostic nuclear medicine.

The hindrance grows with the spin change: plotting the isomer half-life against
$\Delta I$ shows the Weisskopf suppression directly, with each additional unit of
angular momentum extending the lifetime by orders of magnitude. The longest-lived
isomers, such as the $9^-$ state of $^{180m}\mathrm{Ta}$, are effectively stable.

$$
% caption: Isomeric half-life rises steeply with the spin change of the
% de-exciting transition; each additional unit of angular momentum forces a
% higher multipole and lengthens the lifetime by orders of magnitude.
\begin{tikzpicture}[>=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[right, black!70] {spin change};
  \draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {log half-life};
  \foreach \x/\l in {1/2,2/3,3/4,4/5}{
    \node[black, anchor=north, font=\scriptsize] at (1.4*\x,0) {\l};
  }
  % rising trend of points
  \draw[acc, very thick] (1.4,0.7) -- (2.8,1.9) -- (4.2,3.1) -- (5.6,4.2);
  \foreach \x/\y in {1.4/0.7,2.8/1.9,4.2/3.1,5.6/4.2}{ \fill[black] (\x,\y) circle (1.9pt); }
  \node[black, anchor=north west, font=\scriptsize] at (4.3,3.1) {M4 isomers};
\end{tikzpicture}
$$

Internal conversion, E0 decay, pair formation, and isomerism are all consequences
of the same electromagnetic multipole coupling that governs gamma emission. What
these channels change is only how the energy leaves the nucleus, not the level
scheme itself. The complementary tool that uses the emitted radiation to read the
level scheme back out is the angular correlation between successive gammas, taken
up in the [next lesson](/nuclear-physics/gamma-decay/angular-correlations-mossbauer).

[^krane-ic]: **Krane**, _Introductory Nuclear Physics_, Ch. 10, §10.6 (Internal Conversion) and §10.7 (Lifetimes for Gamma Emission): the conversion coefficient and its $Z$, energy, and multipole dependences, the E0 monopole transition, internal pair formation, and the origin of isomeric states in the islands of isomerism below the magic numbers. Isomer half-lives, spins, and the $^{99m}\mathrm{Tc}$ decay scheme are tabulated by the National Nuclear Data Center, [nndc.bnl.gov](https://www.nndc.bnl.gov/).