This article contains my notes about the giant resonances in nuclear physics. The information is mainly from Giant Resonances: Fundamental High-Frequency Modes of Nuclear Excitation by M. N. Harakeh. Please see that book for more details!

The pygmy resonance is a phenomenon observed within the excitation of isovector giant dipole resonances (IVGDR) in light nuclei with a large neutron skin, such as radioactive nuclei. In light nuclei (\(A \leq 40\)), the IVGDR strength distribution is significantly fragmented compared to heavier nuclei (\(A \geq 90\)). This fragmentation arises from configuration splitting, where single-particle transitions (\(1p \to sd\) and \(sd \to fp\)) result in subgroups of transitions that do not fully mix due to the residual interaction. As a result, the photo-absorption spectrum appears as a broad distribution rather than a single peak.

Adding neutrons to light nuclei strongly affects the E1 strength distribution. For example, in \(^{12}\mathrm{C}\) and \(^{16}\mathrm{O}\), the addition of neutrons drastically alters the relative strengths of photo-neutron and photo-proton cross sections, particularly enhancing strength in the \(10-20\ \mathrm{MeV}\) energy range while maintaining a relatively stable overall absorption cross-section. This phenomenon is particularly evident in \(^{18}\mathrm{O}\), where a high-resolution experiment revealed fine structures in the \(9-13\ \mathrm{MeV}\) range. The fragmentation of strength in light nuclei also results in the presence of low-energy strength, often associated with pygmy resonances.

In sd-shell nuclei, the IVGDR is fragmented due to the splitting of \(1p\)-\(1h\) transitions into distinct energy regions. For nuclei like \(^{12}\mathrm{C}\) and \(^{16}\mathrm{O}\), only transitions to higher energy regions (\(B\)-type transitions) occur. However, with increasing neutron number, lower energy transitions (\(A\)-type transitions) become more prominent, culminating in a smoother strength distribution, as observed in \(^{40}\mathrm{Ca}\).

In heavier nuclei (\(A \geq 40\)), the coupling of \(1p\)-\(1h\) states to \(2p\)-\(2h\) states significantly dampens fine structure, leading to a more uniform strength distribution. However, in light sd-shell nuclei, such damping is less pronounced, and coupling to the continuum contributes to broadening and intrinsic widths in the range of a few MeV. The presence of isospin splitting further contributes to the fragmentation of IVGDR strength in non-self-conjugate nuclei (\(N \neq Z\)).

The scissors mode is an orbital magnetic dipole (M1) excitation observed in deformed nuclei, first discovered in the strongly deformed nucleus \( ^{156} \mathrm{Gd} \) through inelastic electron scattering. This mode is characterized by an angular oscillation of the axially deformed distributions of protons and neutrons around an axis perpendicular to their symmetry axes. It exhibits a predominantly orbital nature, as evidenced by its weak excitation in inelastic proton scattering. The mean excitation energy of the mode is approximately \( E_x \approx 66 \delta A^{-1/3} \mathrm{MeV} \), where \( \delta \) is the nuclear deformation parameter, and its total M1 strength, typically around \( 3 \mu_N^2 \) for mid-shell rare earth nuclei, is proportional to \( \delta^2 \).

A sum rule derived by Lo Iudice and Richter describes the total M1 strength \( B(\mathrm{M}1) \), incorporating factors like the deformation parameter \( \delta \), nucleon numbers \( Z \) and \( N \), and their respective orbital gyromagnetic factors \( g_p \) and \( g_n \): $$ B(\mathrm{M} 1) \uparrow \approx 0.0042 \frac{4 N Z}{A^2} \omega_{s c} A^{5 / 3}\left(g_p-g_n\right)^2 \delta^2 \mu_N^2. $$ This formula aligns well with experimental data, especially for well-deformed nuclei. The scissors mode has also been observed in actinide nuclei, showing a similar \( \delta^2 \)-dependent behaviour as in rare earth nuclei.