Internal Conversion and Isomers

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.¹

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

where $B_n$ is the binding energy of the electron in its shell $n=K,L,M,\dots$. 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 of any accompanying decay. The vacancy left behind is filled by outer electrons, so conversion is followed by characteristic X-rays or Auger electrons.

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

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A nonrelativistic Born-approximation treatment reproduces the trends even though it is quantitatively superseded by tabulated relativistic calculations.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 $n$,

and for a magnetic multipole,

with ${\alpha }_{\mathrm{fs}}=e^2/4\pi {\epsilon }_0\hslash c\approx 1/137$ 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.

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,

An E0 transition therefore emits conversion electrons (and, above threshold, electron-positron pairs) but no gamma ray. Because the operator measures $⟨r^2⟩$, 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.

Internal pair formation

When the transition energy exceeds twice the electron rest energy, $\Delta E>2m_e{c}^2=1.022\mathrm{MeV}$, 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, 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.

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.

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.