Solving coupled equations by iteration for heavy ion multiple Coulomb excitation

The set of coupled linear second-order differential equations which has to be solved for quantum-mechanical calculations of inelastic scattering processes with multiple excitation can be rewritten as an equivalent set of coupled first-order integral equations. When Airy functions are used as piecewise analytic reference solutions, it makes it possible to evaluate analytically the integrals that arise in the set of integral equations. This set can be solved iteratively with a considerable reduction of computation time in cases of heavy ion scattering, when compared to quantum-mechanical coupled-channel calculations of the conventional type. The efficiency of two iteration schemes, an inward-outward and a perturbative one, has been investigated for some test cases dealing with multiple Coulomb excitation of ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$ by Kr and Pb. It turns out that, for heavy ion scattering, only the inward-outward iteration scheme has a practical importance. Finally, the excitation probabilities for ${}_{}{}^{238}{}_{}{}^{}\mathrm{U}$, Coulomb excited by 385 MeV Kr up to $I=24\mathrm{}\hslash$, are shown for a reduced $E2\mathrm{}$ transition matrix element of 3.5 eb and they are compared with the excitation probabilities calculated according to the semiclassical theory.