2.2.1. Multi-Step Excitation and Relative Signs

To understand the importance of multi-step excitation, it is useful to consider the population of two excited states , *I<sup>π</sup>* = 0<sup>+</sup> <sup>2</sup> , 4<sup>+</sup> <sup>1</sup> , in an even–even nucleus (see Figure 1). As Coulomb excitation via an *E*0 transition is strictly forbidden, two-step excitation is the only way to populate the 0<sup>+</sup> <sup>2</sup> state. The 4<sup>+</sup> <sup>1</sup> state can be Coulomb-excited in two ways: directly from the ground state, via an *E*4 excitation, or with an *E*2 two-step excitation through the first excited state. Since the probability of Coulomb-exciting a given state through an *E*4 transition is much smaller than through the *E*2 excitation [8], the two-step excitation is typically dominant. Consequently, by measuring the intensities of the 4<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> , 0<sup>+</sup> <sup>2</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> 1 *γ*-ray transitions with respect to the 2<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> <sup>1</sup> decay, and relating them to excitation cross sections, it is possible to extract the *B*(*E*2; 4<sup>+</sup> <sup>1</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> ) and *<sup>B</sup>*(*E*2; 0<sup>+</sup> <sup>2</sup> <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>1</sup> ) values.

In some cases, single-step and multi-step excitations are comparable in magnitude; an example is the 2<sup>+</sup> <sup>2</sup> state in an even–even nucleus (see Figure 1). This state can be populated by a direct *E*2 transition from the ground state and by a two-step excitation through the first excited state. The total excitation probability for the 2<sup>+</sup> <sup>2</sup> state can be written as:

$$P(\mathbf{0}^+\_{\mathbf{g},\mathbf{s}.} \to \mathbf{2}^+\_2) = |a^{(1)}(\mathbf{0}^+\_{\mathbf{g},\mathbf{s}.} \to \mathbf{2}^+\_2) + a^{(2)}(\mathbf{0}^+\_{\mathbf{g},\mathbf{s}.} \to \mathbf{2}^+\_1 \to \mathbf{2}^+\_2)|^2,\tag{1}$$

where *a*(1), *a*(2) are first-order and second-order excitation amplitudes. Consequently, *P*(0<sup>+</sup> g.s. <sup>→</sup> <sup>2</sup><sup>+</sup> <sup>2</sup> ) includes a term related to one-step excitation (2<sup>+</sup> <sup>2</sup> ||*E*2||0<sup>+</sup> g.s.2), one related to two-step excitation (2<sup>+</sup> <sup>2</sup> ||*E*2||2<sup>+</sup> <sup>1</sup> 22<sup>+</sup> <sup>1</sup> ||*E*2||0<sup>+</sup> g.s.2) and the interference term

$$
\langle \mathfrak{L}\_2^+ || E 2 || 0\_{\mathfrak{g}, \text{s.}}^+ \rangle \langle \mathfrak{L}\_2^+ || E 2 || 2\_1^+ \rangle \langle \mathfrak{L}\_1^+ || E 2 || 0\_{\mathfrak{g}, \text{s.}}^+ \rangle. \tag{2}
$$

In this last term, at variance with all the others, the matrix elements are not squared. As the total Coulomb-excitation cross section will be different for a negative (destructive) and a positive (constructive) interference term, its sign becomes an observable.

More complex interference terms can influence the Coulomb-excitation cross sections if states are populated through several excitation patterns involving multiple intermediate states. As such terms include non-squared matrix elements, their appearance leads to the experimental sensitivity to relative signs of transitional matrix elements. A sign convention should be adopted to ensure consistent analysis and facilitate a comparison with model predictions. Usually, signs of all in-band transitional *E*2 matrix elements are assumed to be positive, and, for each band head, a positive sign is imposed for one of the transitions linking it with a state in the ground-state band. The signs of all remaining matrix elements can be determined relative to those.

The probability of exciting a state via a process involving two or more steps can be comparable to that of one-step excitation, depending, for instance, on the magnitude of the involved matrix elements. Multi-step excitation is enhanced for larger scattering angles and masses of the collision partners. Experiments aiming at extracting reduced transition probabilities between the ground state and an excited state are typically performed in conditions reducing multi-step excitations, by limiting the scattering angle in the forward direction and selecting a light collision partner. In contrast, if the relative signs of transitional matrix elements and reduced transition probabilities between excited states are the objective of the experiment, the detection of scattered particles at backward angles and the use of a heavy collision partner is preferable.
