Coulomb excitation of unstable nuclei at intermediate energies

We investigate the Coulomb excitation of low-lying states of unstable nuclei in intermediate energy collisions ($E_{lab}\sim10-500$ MeV/nucleon). It is shown that the cross sections for the $E1$ and $E2$ transitions are larger at lower energies, much less than 10 MeV/nucleon. Retardation effects and Coulomb distortion are found to be both relevant for energies as low as 10 MeV/nucleon and as high as 500 MeV/nucleon. Implications for studies at radioactive beam facilities are discussed.

participating nuclei stay outside the range of the nuclear strong force, the excitation cross section can be expressed in terms of the same multipole matrix elements that characterize excited-state gamma-ray decay, or the reduced transition probabilities, B(πλ; J i → J f ).
Hence, Coulomb excitation amplitudes are strongly coupled with valuable nuclear structure information. Therefore, this mechanism has been used for many years to study the electromagnetic properties of low-lying nuclear states [1].
Coulomb excitation cross sections are large if the adiabacity parameter satisfies the con- where a 0 is half the distance of closest approach in a head-on collision for a projectile velocity v, and E x = ω f i is the excitation energy. This adiabatic cut-off limits the possible excitation energies below 1-2 MeV in sub-barrier collisions. A possible way to overcome this limitation, and to excite high-lying states, is to use higher projectile energies. In this case, the closest approach distance, at which the nuclei still interact only electromagnetically, is of order of the sum of the nuclear radii, R = R P + R T , where P refers to the projectile and T to the target. For very high energies one has also to take into account the Lorentz contraction of the interaction time by means of the Lorentz factor γ = (1 − v 2 /c 2 ) −1/2 , with c being the speed of light. For such collisions the adiabacity condition, Eq. (1), becomes From this relation one obtains that for bombarding energies around and above 100 MeV/nucleon, states with energy up to 10-20 MeV can be readily excited [3].
An appropriate description of Coulomb excitation at intermediate energies (E lab = non-relativistic Coulomb excitation formalism described in ref. [1], nor the relativistic one formulated in refs. [3,4] are appropriate. This is discussed in details in ref. [2] where it is shown that the correct values of the Coulomb excitation cross sections differ by up to 30-40% when compared to the non-relativistic and relativistic treatments used to calculate experimental observables (cross sections, gamma-ray angular distributions, etc.).
We follow the formalism of ref. [2] to calculate cross sections for Coulomb excitation from energies varying from 10 to 500 MeV/nucleon. These are the energies where most radioactive beam facilities are or will be operating around the world. The calculated cross sections will be of useful guide for future experiments. We also compare the accurate calculations with those obtained by using simple analytical formulas and test the regime of their validity.
The cross sections for the transition J i → J f in the projectile are calculated using the where π = E or M stands for the electric or magnetic multipolarity, and are the reduced transition probabilities. In these equations, ǫ = 1/ sin(Θ/2), with Θ being the deflection angle, a 0 = Z P Z T e 2 /m 0 v 2 and a = a 0 /γ. The complex functions S(πλ, µ) are integrals along Coulomb trajectories corrected for retardation. Their calculation and how they relate to the non-relativistic and relativistic theories are described in details in ref. [2].
Here we will introduce another comparison tool for the total cross section, which is obtained by integration of eq. 3 over scattering angles. The code COULINT [2] was used to calculate the orbital integrals S(πλ, µ) and the cross sections of eq. 3 (for more details, see ref. [2]).
Using the theory described in ref. [4], it is easy to show that approximate values of the cross sections for E1, E2, and M1 transitions can be obtained by means of the relations where K n are the modified Bessel functions of the second order, as a function of ξ given by eq. 2, with R corrected for recoil by the modification R → R + πa/2 [3].
Here we will only consider the excitation of the lowest lying states in light and medium heavy nuclei. For nuclear masses A < 20, the TUNL nuclear data evaluation web site was of great help [5]. The electromagnetic transition rates at the TUNL database are given in Weisskopf units and are transformed to the appropriate B(πλ, comparison, a few medium mass nuclei, as well as a few stable nuclei, were included in the calculation. Other data were taken from refs. [6,7,8,9]. The value of B(E2) for 16 C based on this formula is at least one order of magnitude larger than what is observed experimentally in a Coulomb dissociation experiment [9]. The anomalously strong hindrance of the 16 C transition is not well explained theoretically. This is just an example of the power of Coulomb excitation as a tool to access the new physics inherent of poorly known rare nuclear species.
Another example is the strong E1 transition in 11 Be. 11 Be is an archetype of a halo nucleus and exhibits the fastest known dipole transition between bound states in nuclei. The which is still a matter of investigation [12,13,14]. It is thus seems clear that predictions based on traditional nuclear structure and reaction theory often yields results in disagree-ment with experimental data. In spite of that, when proper corrections are accounted for (e.g. channel-coupling, nuclear excitation, relativistic corrections), Coulomb excitation of radioactive beams is a powerful complementary tool to investigate electromagnetic properties of nuclei far from the stability line.
In Table 1  We should stress that a full theoretical treatment of relativistic dynamics of strong and electromagnetic interactions in many-body systems is very difficult and still does not exist [15].

Data
Projectile Target E lab πλ B(πλ) θ range E x σ exp σ th σ app  In figure 1 we show a comparison between the experimental data and our calculations.
We notice that the cross sections calculated with help of eq. 5 are not much different than those calculated with eq. 3. They are systematically lower, up to 10%, than the exact calculation following eq. 3. As we discuss below, this is not always the case, specially for the excitation of high-lying states. In fact, this is a good check of eq. 3, which is done in a very different way than the analytical calculations of eq. 5. But as we will see below, this agreement is not always the case, specially when one includes small impact parameters for which the sensitivity to the relativistic corrections is higher (see ref. [2]). The dashed curve in figure 1 is a guide to the eye. It helps to see that the experimental cross sections are on average larger than the calculated ones, either with eq. 3 (open circles), or with eq. 5 (open triangles).    [12,14] can give rise to large high-order effects [21]. The interpretation of data could be distorted as in the case of Coulomb dissociation of 8 B at low energy [24], which was completely misinterpreted in terms of first-order calculations. In some situations, when higher-order effects are relevant, the effect of the nuclear breakup cannot be neglected either [22,23]. Thus, the choice of the incident energy would depend on the experimental conditions. Identification of gamma-rays from de-excitation using Doppler shift techniques are often more advantageous at higher energies. Moreover, except for few cases (e.g. 11   the E2 ( fig. 2c). The reason is that the E2 Coulomb field ("tidal field") is very sensitive to the details of the collision dynamics at low energies. These conclusions can be deceiving since even for the E1 and M1 cases the approximations in eq. 5 may strongly differ from the exact calculations if the excitation energy is large (see discussion in ref. [2]). This is shown in figure 2d, where we plot the Coulomb excitation cross section of the E x = 13.09 MeV state in 16 O. In this case, the cross sections based on eq. 5 is a factor of 10 smaller than the exact calculation at 10 MeV/nucleon. At 100 MeV/nucleon this difference drops to 10%, which still needs to be considered with care.
In summary, in this article we have used the formalism of ref.
[2] to predict the cross sections for Coulomb excitation of several light projectiles with electromagnetic transitions found in the literature, listed in the TUNL database [5], and for a few other selected cases.
These estimates will be useful for planing Coulomb excitation experiments at present and future heavy ion facilities. It is evident that the inclusion of relativistic effects combined with Coulomb distortion are of the utmost relevance. The cross section inferred by using non-relativistic or pure relativistic treatments can be wrong by up to 30% even at 100 MeV/nucleon, as shown here and in ref. [2]. Finally, the use of Coulomb excitation to produce nuclei in high-lying states is an important tool to study particle emission processes.
For example, the excitation of 18 Ne and its subsequent decay by two-proton emission is a process of large theoretical and experimental interest. Experimental work in this direction is in progress [25].