On the Analysis of Intermediate-Energy Coulomb Excitation Experiments

In a recent publication (Bertulani et al., PLB 650 (2007) 233 and arXiv:0704.0060v2) the validity of analysis methods used for intermediate-energy Coulomb excitation experiments was called into question. Applying a refined theory large corrections of results in the literature seemed needed. We show that this is not the case and that the large deviations observed are due to the use of the wrong experimental parameters. We furthermore show that an approximate expression derived by Bertulani et al. is in fact equivalent to the theory of Winther and Alder (NPA 319 (1979) 518), an analysis method often used in the literature.


Introduction
Coulomb excitation (CE) of radioactive heavy-ion beams on high-Z targets at intermediate to relativistic beam energies (E/A > 30 MeV) is an important and now widely used method to study the structure of nuclei furthest from stability and a wealth of experimental information was obtained in laboratories worldwide [2,3,4,5,6] since the method was first applied in 1995 [2]. Even at these high beam energies it is possible to suppress the influence of the strong interaction in the excitation process, by rejecting events with small impact parameter, and to extract largely model independent nuclear structure information.
Experimentally, usually the de-excitation γ ray yields after CE are measured and from these the excitation cross sections are deduced. It is important to note that the deflection angle of the outgoing scattered particle is restricted either by the experimental setup [2,3,7,8] or during the off-line analysis of the data, which are recorded event-by-event [9]. The so imposed maximum scattering angle θ max corresponds to a certain minimum impact parameter b min which is chosen to exceed the sum of the radii of projectile and target by several femtometer, ensuring the dominance of the Coulomb interaction in the excitation process. The angle integrated CE cross section depends of course strongly on the chosen maximum scattering angle θ max , which is therefore an important parameter in the analysis. For further details review articles [10,11,12] or the original references should be consulted.
In order to extract the relevant quantities, namely the reduced transition probabilities B(πλ) and in particular B(E2; 0 + gs → 2 + 1 ) values, various analysis techniques are used: • distorted-wave theory (using the ECIS code [13]) used e.g. in [2,14,15,16], • the semi-classical theory of Winther and Alder [17] used e.g. in [3,9,18], and • the virtual photon method [19] used e.g. in [20]. 1 More than one method has been applied to certain nuclei and consistent results were obtained, see e.g. [2,14,21,22]. Recently, a compilation of data obtained on stable nuclides was published and a comparison of the B(E2; 0 + gs → 2 + 1 ) values obtained by various methods was given [23]. It was concluded that high-energy CE is at least as precise and accurate as so-called standard methods such as direct lifetime measurements, low-energy "safe" CE, and electron scattering.
Recently, however, a new approach was proposed [24] and previously published results were called into question in [1] claiming that either only non-relativistic, or only partially relativistic, expressions are used or the effect of the orbit of the nuclei, due to their Coulomb repulsion, is not fully accounted for. Using their new model, changes in the calculated cross sections of up to 30% were reported with respect to results from the literature.
We would like to dispute that assertion and show that the previously published results [3,25,9] are indeed correct and consistent.

Cross sections
In [1] Bertulani et al. studied the CE cross sections at intermediate energies (10-500 MeV/u) using their model, published earlier in [24], which aims to correctly describe the CE process at all beam energies. In addition, since the orbital integrals appearing in the expressions for the excitation amplitudes have to be determined numerically, an approximate expression for the CE cross section was derived, called σ app or σ app in [1], which is a simple expression containing modified Bessel functions. Using these expressions and using the numerically determined orbital integrals it was found that the calculated excitation cross sections for a given excitation strength are smaller by 10-30% in comparison to values from the literature [3,9,25] and consequently that the published B(E1)↑ and B(E2)↑ values are in fact too small by 10-30%.
Since this would be a substantial alteration we tried to reproduce (i) our original results in [3,9,25] and (ii) the results of Ref. [1] using Eq. (5) of [1]. Our findings are listed in table 1 and are displayed in figure 1. The number shown in column 1 of the table and plotted on the abscissa in the figure corresponds to the "Data set" as given in [1]. Contrary to Bertulani et al. we essentially found exact agreement between the published cross sections (circles in the figure) and the results of our own calculation using Eq. (5) of [1] (filled diamond in the figure). However, the results reported in [1] (triangles) deviate significantly from those.
In order to understand the differences we performed a calculation using the beam energies and scattering angles listed in table 1 of [1]. We would like to point out that at least for data sets number 7-12 (original publications [3,9]) the given values correspond to the incident beam energy and maximum laboratory scattering angle. If we assume, however, that the listed beam energy is the average beam energy in the mid-plane of the CE target 2 and the angle is the scattering angle in the center-of-mass system, we can in all cases, except for 11 Be, reproduce the values given in [1] (see open diamonds in figure 1).
We therefore conclude that mentioned discrepancies are due to an interchange 2 To determine the effective (theoretical) cross section σ the beam energy dependent cross section σ(E b ) should be averaged over the target thickness d t where E b (x) is the beam energy at target depth x. Due to the almost linear dependence of the cross section on beam energy within the small energy loss in the target it is sufficient to use the beam energy in the center of the target in the calculation. [1] Eq. 5 (*) [1] Eq. 5 Exp.  table 1). The open diamond shows the result of our own application of Eq. (5) of [1] using the input parameters from the rows without a "*", i.e. the ones used in [1]. The triangle shows the result published in [1].
of center-of-mass and laboratory angles and the use of the incident beam energy instead of the mid-target beam energy.
We furthermore found that the expressions given for σ app in [1] are in fact analytically identical to the one of Winther and Alder [17] (see Appendix). Since for all experimental cross sections listed in table 1 and shown in figure  1 the semi-classical theory of Winther and Alder was employed during the analysis of the data, the σ app results must be identical to the original results σ exp provided the proper input parameters are used. This conclusion is in fact supported by an earlier work of Bertulani et al. [24] where it is shown in figure 3 (curve RR) that for the case of an E2 excitation the cross section determined within the Winther and Alder theory deviates by less than 5% from the exact treatment for beam energies above 30 MeV/u. We would also like to note that the analysis of CE data using a simple maximum scattering angle is not applicable for all experiments in the literature. For instance for the analysis of the CE of 32 Mg [2] the use of a sharp cutoff θ max is meaningless ("Data set" 6 in [1]). Here the particle detection efficiency depended (smoothly) on the scattering angle with maximum values of about 80%. In this case the differential cross section must first be multiplied by the  Table 1 Coulomb excitation cross sections are shown in the three columns on the right-hand side with σ exp being the experimental CE cross section, σ app the cross section from reference [1] and σ ′ app the cross section calculated by us using Eq. (5) of [1]. The B(Eλ)↑ values are listed in units of e 2 fm 2λ . Values in bold were taken verbatim from [1] and the "Data set" number # corresponds to the one used in [1]. In the rows starting with "*" the correct center-of-mass scattering angles θ cm max and mid-target (m-t) beam energies E m−t beam are listed; an agreement of σ exp and σ ′ app is evident for all cases. The rows not starting with "*" list the values as given in [1] and σ ′ app is calculated by us using these values. In most cases σ app and σ ′ app agree, but σ exp and σ app do not. experimental particle detection efficiency before integrating over the scattering angle. This can only be done when the efficiency curve is known, which is shown in figure 2 (lower panel) in [2].
We furthermore found the following errors in [1]: • There is a misprint in Eq. (5). For the E2 excitation cross section the power of E γ should be 2 (E 2 γ ) and not 3 (E 3 γ ). • The wrong citation was used for 38,40,42 S and 44,46 Ar. It should be Scheit et al. [3] and not Chromik et al. [20]. • An M1 excitation was considered for 17 Ne, but the excitation process proceeds through an E2 transition with only a very weak M1 component. The decay, though, is an almost pure M1 transition.
• The deviation of the cross sections published in [1] (triangle in figure 1) for 11 Be ("Data set" 3) seems to be due to a forgotten factor of e 2 = α c ≈ 197/137 ≈ 1.44 in [1].

Conclusion
We conclude that the theory presented in [1,24] does not represent a significant improvement over analysis methods used so far. In fact, a good approximation [1] to the full theory is equivalent to the theory of Winther and Alder [17], which is widely used in the analysis of CE data at high beam energies. We furthermore conclude that the large differences of up to 30% reported in [1] are likely solely due to an interchange of center-of-mass and laboratory angles and the use of incident beam energy instead of the mid-target beam energy in the calculation of the CE cross sections.
We nevertheless agree with the authors of [1] that a refined theory which includes relativistic effects and properly treats the orbit of the projectile is needed and should be applied in the future, when high-precision results become available at next-generation radioactive nuclear beam facilities. The currently available data will be hardly affected due to rather large statistical and systematical uncertainties.

Acknowledgement
This work was supported by the US National Science Foundation under grant PHY-0606007.