**2. Basics of Low-Energy Coulomb Excitation**

Coulomb excitation is an inelastic scattering process, in which the two colliding nuclei are excited via a mutually-generated, time-dependent electromagnetic field. If the distance between the collision partners is sufficiently large, the short-range nuclear interaction has a negligible influence on the excitation process, which is governed solely by the well-known electromagnetic interaction. This condition can be quantified using the Cline's safe distance criterion [7], appropriate for heavy nuclei, which states that if the distance of the closest approach between the surfaces of the collision partners exceeds 5 fm, contributions from the nuclear interaction to the observed excitation cross sections are below 0.5%.

The excitation cross sections depend on electromagnetic matrix elements coupling the low-lying states in the nucleus of interest, including diagonal *E*2 matrix elements related to spectroscopic quadrupole moments. The decay of Coulomb-excited states is governed by the same set of electromagnetic matrix elements, although the influence of specific matrix elements on the excitation and decay processes may be very different as illustrated by Figure 1. Namely, low-energy Coulomb excitation favours the population of collective states through *E*2 and *E*3 transitions, while other multipolarities typically have a small impact on the measured cross sections (see [8] for further details). The *M*1 and *E*1 multipolarities, however, remain important in the de-excitation process. The quantities measured in low-energy Coulomb-excitation experiments are, most commonly, *γ*-ray yields in coincidence with at least one of the collision partners. It is, however, also possible to measure Coulomb-excitation cross sections by detecting only scattered particles or only *γ* rays.

**Figure 1.** Low-lying level scheme of a fictitious even–even nucleus outlining dominant excitation (**left**) and de-excitation (**right**) patterns in low-energy Coulomb excitation. The transitions are labelled with the corresponding matrix elements. The inset on the left depicts the magnetic substates *m* of the 2<sup>+</sup> <sup>1</sup> state and illustrates the reorientation effect. Some allowed transitions are neglected for simplicity.

While Coulomb-excitation cross sections can be calculated using a full quantummechanical treatment, a semi-classical approach is typically employed to overcome difficulties arising from the long-range of the Coulomb interaction and complex level schemes of the colliding nuclei. In this approach, introduced by K. Alder and A. Winther [9], the relative motion of collision partners is described using classical equations, and the quantal treatment is limited to the excitation process. The validity of this procedure, which provides a significant simplification of the calculations without a relevant loss of accuracy, stems from the fact that the interaction in the Coulomb-excitation process is dominated by the Rutherford term. For the semi-classical approximation to be valid, the de Broglie wavelength associated with the projectile must be small compared to the distance of closest approach, and the energy transferred in the excitation process must be small

compared with the total kinetic energy in the centre-of-mass reference system. These two conditions are well satisfied in low-energy Coulomb-excitation experiments involving heavy ions, but when light nuclei are involved (i.e., protons, deuterons, *α* particles), a full quantum-mechanical analysis is required.
