Nuclear Isomers

Metastable Nuclear States

A nuclear isomer is a metastable excited state of an atomic nucleus that has a measurably long half-life before decaying to a lower energy state. The term was coined by Frederick Soddy in 1917. Unlike chemical isomers, nuclear isomers differ only in the excitation energy and angular momentum of the same nucleus.

A state is considered isomeric (denoted by a superscript "m", e.g., $^{99m}$Tc) when its half-life is long enough to be measured — typically $t_{1/2} > 10^{-9}$ s. The longevity arises from large differences in spin between the isomeric state and available lower-lying states, making electromagnetic transitions highly suppressed.

The isomeric transition (IT) proceeds by gamma-ray emission or internal conversion, with no change in $Z$ or $A$.

Selection Rules for Gamma Transitions

The angular momentum and parity carried away by a photon of multipolarity $L$ determine the selection rules. For a transition between states $J_i^{\pi_i} \to J_f^{\pi_f}$:

The parity selection rules distinguish electric (E) and magnetic (M) transitions:

Higher multipolarities are strongly suppressed. The dominant transition is always the lowest allowed multipole. Mixed transitions (e.g., M1+E2) occur when multiple multipolarities are allowed.

Weisskopf Single-Particle Estimates

Victor Weisskopf derived estimates for transition rates assuming a single nucleon makes the transition within a uniform-density nucleus of radius $R = r_0 A^{1/3}$. These serve as reference values — actual rates are quoted as Weisskopf units (W.u.):

The key feature is the steep energy dependence: $E_\gamma^{2L+1}$. For high multipolarities and low energies, transition rates become extremely small, producing isomeric states. A ratio $T(EL)/T(E(L+1)) \sim 10^7$ per unit increase in $L$ at 1 MeV.

Internal Conversion

Internal conversion (IC) is an alternative de-excitation process where the nuclear electromagnetic field directly ejects an atomic electron (usually from the K-shell) instead of emitting a gamma ray. The ejected electron has kinetic energy:

where $B_e$ is the electron binding energy. The internal conversion coefficient (ICC) is defined as:

The total transition rate is then $T_{\text{total}} = T_\gamma(1 + \alpha)$. The ICC depends strongly on multipolarity, atomic number, and transition energy:

IC is dominant for heavy nuclei ($Z \gg 1$), low transition energies, and high multipolarities — precisely the conditions that create isomeric states. IC allows transitions that are strictly forbidden for gamma emission, such as $0^+ \to 0^+$electric monopole (E0) transitions.

Islands of Isomerism

Isomeric states cluster in specific regions of the nuclear chart called "islands of isomerism." These occur where the nuclear shell structure produces states with very different spins close in energy. The main islands are:

Transition Probabilities: B(EL) and B(ML)

The reduced transition probability $B(\sigma L)$ contains all the nuclear structure information and is independent of the transition energy:

where $\sigma = E$ or $M$. The Weisskopf unit provides a natural scale:

Collective transitions (e.g., rotational E2) can have $B(E2) \sim 100$ W.u., while single-particle transitions are $\sim 1$ W.u. Isomeric transitions are typically strongly retarded: $B(\sigma L) \ll 1$ W.u.

Simulation: Weisskopf Transition Rates

This simulation computes and plots the Weisskopf single-particle estimates for electromagnetic transition rates as a function of gamma-ray energy, showing the dramatic suppression at high multipolarities that creates nuclear isomers.

Fortran Simulation: Internal Conversion

This Fortran program calculates approximate internal conversion coefficients and demonstrates why K-isomers in deformed nuclei achieve such extraordinary half-lives through the K-selection rule.