Semiclassical approach for nuclear Coulomb excitation

Nuclear Coulomb excitation is often calculated using a semiclassical (SC) approach, where the projectile follows classical trajectories and excites the target nucleus through a time-dependent Coulomb interaction. While the validity of the SC approach has been well established for electric quadrupole ($E2$) transitions, its accuracy for higher-order multipole transitions remains insufficiently benchmarked. In this paper, we compare Coulomb excitation cross sections for higher-order multipole transitions calculated using the SC approach with those obtained through a quantum mechanical (QM) approach, where the projectile is described by wave functions. For $E2$ transitions, the excitation cross sections from both approaches are of the same order of magnitude, consistent with existing validations. However, for higher-order multipole transitions, the SC approach yields significantly higher cross sections, deviating possibly by orders of magnitude from the QM results. This discrepancy underscores the necessity of the QM approach for accurate calculations of the Coulomb excitation cross sections. The failure of the SC approach is explained through using the Wentzel-Kramers-Brillouin approximation.