Electric moments and charge deformation in even rare-earth nuclei

We present measurements of $E2\mathrm{}$ and $E4\mathrm{}$ transition matrix elements, and related ${\beta }_2$ and ${\beta }_4$ deformations, for eight even rare earth nuclei, by exploiting the effect of electric quadrupole and electric hexadecapole excitation on the multiple Coulomb excitation of states in their ground state bands. The Coulomb excitation probabilities for the $0^{+}$, $2^{+}$, and $4^{+}$ states in the ground state bands of the even rare earth nuclei ${}_{}{}^{152,154}{}_{}{}^{}\mathrm{Sm}$, ${}_{}{}^{158,160}{}_{}{}^{}\mathrm{Gd}$, ${}_{}{}^{164}{}_{}{}^{}\mathrm{Dy}$, ${}_{}{}^{166,168}{}_{}{}^{}\mathrm{Er}$, and ${}_{}{}^{174}{}_{}{}^{}\mathrm{Yb}$ have been measured with ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ projectiles at several incident energies below the respective Coulomb barriers by direct detection, at backward angles, of elastically and inelastically scattered particles. The direct detection of scattered ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ particles was employed in order to eliminate the uncertainties involved in the detection of the deexcitation $\gamma$ rays, and to facilitate the determination of $M(E2;0^{+}\to 2^{+})$ to about the $\frac{1}{2}$% accuracy required to obtain a meaningful measurement of $M(E4;0^{+}\to 4^{+})$. Coulomb-nuclear interference effects in elastic and inelastic scattering were investigated, and it was demonstrated that Coulomb nuclear interference occurs at a much lower projectile energy than predicted by the simple classical formula for the Coulomb barrier, and manifests itself at different incident energies for elastic and inelastic scattering. The data on excitation probabilities were analyzed both with a semiclassical Coulomb excitation calculation, corrected for quantal effects in second order perturbation theory, and with a full quantal calculation, to determine the reduced electric transition matrix elements $M(E2;0^{+}\to 2^{+})$ and $M(E4;0^{+}\to 4^{+})$. Significant differences were found between the perturbation theory quantal treatment and the full quantal calculations. Static quadrupole and hexadecapole moments of the charge distribution were extracted from the measured transition matrix elements within the context of the rotational model. Using an axially symmetric deformed Fermi shape to parametrize the charge distribution, charge deformation parameters ${\beta }_2^C$ and ${\beta }_4^C$ deduced from the static moments, were found to follow the same general trends as the potential deformation parameters measured in nuclear scattering experiments well above the Coulomb barrier. For some nuclei the Coulomb excitation ${\beta }_{\lambda }^C$ values show what may be significant deviations from the nuclear potential deformations parameters. However, it is emphasized that, presently, the significance of such a comparison as well as the significance of a comparison with deformation parameters determined with other electromagnetic probes, is subject to the considerations introduced by the model dependence of each measurement, and diluted by the ambiguous correspondence between the quantities being measured in the different experiments.