Interference Effects in Coulomb and Nuclear Excitation

Coulomb excitation is based on the assumption that the interaction between the projectile and the target nucleus is purely electromagnetic. It is thus important to determine the safe bombarding energy below which contributions from nuclear-force interactions are negligible. This is of particular importance in Coulomb excitation experiments that are designed to determine quadrupole- and hexadecapole-moments of excited states which are based on relatively small second order effects. Some early experiments produced false results as a consequence of using too high bombarding energies.

A first experiment by the Pittsburgh group (ref. I.1, I.2) was carried out for the excitation of the first 2⁺ state in ¹¹⁴Cd, using α beams. The ratio of the inelastic to the elastic scattering cross section R_exp=dσ_inel/dσ_el was measured in the energy range between 8 and 16 MeV and for a scattering angle θ of 135°. The dots in figure I.1 represent the ratio R_exp/R_coul where R_coul is the theoretical value of dσ_inel/dσ_el , calculated under the assumption of pure Coulomb excitation. The figure shows that the ratio R_exp/R_coul drops below 1 at 10 MeV as a result of non-Coulombian effects that become important at this energy. The dip of the data below 1 is a result of the destructive interference between the effect of the repulsive Coulomb interaction and the attractive nuclear interaction. At higher bombarding energies the strong nuclear interaction dominates and the ratio R_exp/R_coul increases rapidly. The full and dashed lines in figure I.1 are the result of Distorted Wave Born Approximation (DWBA) calculations (ref. I.1, I.2) in which the effects of the nuclear interaction are represented by a so-called optical potential. This potential has an imaginary part, to simulate absorption. The full line is in excellent agreement with the experimental data. The calculation corresponding to the dashed line was done with a slightly different choice of parameters for the optical potential.

There is another interesting aspect to the interference effect beyond establishing "save" bombarding energies for Coulomb excitation experiments: From the measured quadrupole- and hexadecapole-moments it is possible, after making some reasonable assumptions, to extract deformation parameter β and β for the charge distribution. Previous experiments at bombarding energies far above the Coulomb barrier that were analyzed using coupled channel calculations based on deformed optical potentials, yielded nuclear deformation parameters &beta and β that were on average 22% smaller than the values of β and β (ref. I.3, I.4). This poses the intriguing question of whether this is indicative of a difference between the deformations of the charge density and the mass density. In view of the charge independence of the nuclear forces one would not expect a significant difference between the two deformations. In order to interpret such experiments, the quantum mechanical coupled channel code AROSA (ref. I.5) for Coulomb excitation was developed and expanded by the code INTE (ref. I.6) to include a deformed complex optical potential representing the nuclear interaction. First experiments by the Pittsburgh group were carried out in 1973 on ¹⁶⁸Er, ¹⁸⁴W, and ¹⁸⁶W (ref. I.7), using α-beams at energies between 12 and 19 MeV. The results are summarized in figure I.2. The circles in the first row show the ratio dσ_exp/dσ_Ce of the experimental elastic cross section divided by the elastic cross section calculated under the assumption of pure Coulomb excitation . The solid lines represent the result of theoretical calculations with the code INTE that include the optical potential. The next two rows show the experimental ratios (dσ_2⁺/dσ_elastic)_experiment divided by the same ratio calculated under the assumption of pure Coulomb excitation. The solid lines are again calculations including an optical potential and represent best fits to the data.

Figure I.3 demonstrates the sensitivity of coupled channels calculations to the relative size of charge- and optical potential deformation-parameters. In 1975 the Pittsburgh group in collaboration with a group from the Oak Ridge National Laboratory analyzed all experimental interference data available at that time. The results are shown in table I. The first column shows the isotopes, the second the deformation parameters &beta obtained from Coulomb excitation experiments at safe bombarding energies which agreed very well with the those obtained from the analysis of the interference data. Column three shows the nuclear deformation parameters &beta obtained from the analysis of the interference data and column four shows the averaged values of β obtained at high bombarding energies well above the Coulomb barrier i.e. 27.5, 30.0, 32.5 and 50 MeV (ref. I.3, I.4). It is interesting to note that the values of β and β agree with one another to within a few percent. Column 5 lists the difference (β - β )/β demonstrating that the values of &beta are consistently between 18 and 24% below the β values. This is, in view of the fact that the experimental uncertainties for the β₂ values are are of the order of 3% or less, a significant indication that the charge distribution is different from the mass distribution. There has however been some concern regarding the dependence of this difference on the (model) assumptions that must be made in extracting the deformation parameters β and β from the experimental data. In 1977 the mass distribution in ¹⁵⁴Sm was determined in a neutron scattering experiment (ref. I.9) resulting in β = 0.24, in good agreement with β = 0.25 in table I, thus supporting the difference between the shape of the charge and mass distribution. Such a difference could be attributed to the repulsive electromagnetic interaction between protons which prefers a distribution that is more elongated than that of the neutrons. An analogous investigation for &beta₄ values was not feasible because the experimental uncertainties for these values were 30% or higher.