Past exam of the particle and nuclear physics course of the University of Cambridge / 2013 / Paper 1 / 2 / a / Solution

A nuclear excited state has internal energy above its ground state, with levels labelled by spin and parity $J^{\pi }$ and, where useful, approximate isospin. In the shell model, nucleons occupy orbitals split by spin-orbit coupling; nuclear particle-hole excitation and broken pairs produce excited configurations. An orbital has parity $(-1{)}^l$, and an even-even paired ground state is commonly $0^{+}$. Residual interactions mix configurations, so shell spacings are not a complete spectrum.

Collective motion gives distinct patterns. A deformed nucleus can have a rotational band with $E_J-E_0={\hslash }^{2}J(J+1)/(2\mathcal{I})$, so a simple even-even $0^{+},2^{+},4^{+}$ band has $E_4/E_2=10/3$. Near spherical shape, a quadrupole vibration gives a one-phonon $2^{+}$ excitation; two such phonons can couple to $0^{+},2^{+},4^{+}$ near twice that energy in the harmonic limit. Anharmonicity and coupling to single-particle motion modify these ideal patterns.

De-excitation occurs by gamma emission, internal conversion, or particle emission if energetically open. photon multipolarity $\lambda \ge 1$ must satisfy the angular-momentum triangle rule. Electric multipoles change parity by $(-1{)}^{\lambda }$ and magnetic multipoles by $(-1{)}^{\lambda +1}$; a single photon cannot connect $0\to 0$. Large multipolarity or a small transition matrix element can make a long-lived nuclear isomer. Gamma energies, coincidence cascades, angular correlations and lifetimes constrain level schemes; inelastic scattering and transfer reactions supply complementary spin/configuration information. One observed line alone need not identify a unique level assignment.