= 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 $E_J-E_0=\hbar^2J(J+1)/(2\mathcal I)$, 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 $\lambda\ge1$ 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 $0\to0$. 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.
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