A procedure to connect a model-independent phase-shift analysis with the solution of the inverse quantum scattering problem has been developed and applied to experimental differential cross sections of ${}_{}{}^{12}{}_{}{}^{}\mathrm{C}$${+}^{12}$C elastic scattering in the energy range ${\mathit{E}}_{\mathrm{c}.\mathrm{m}.\mathrm{}}$=8–12 MeV. The minimization of the error square function ${\mathrm{\chi }}^2$ is performed with respect to the spectral coefficients involved in the inverse procedure. Input quantities are measured differential cross sections; output results are complex potentials. The real part of the potentials, so obtained, is characterized by a pronounced minimum value of -(7–14) MeV at relative distances in the range 2.4–3 fm and by a Coulomb barrier of height 6–7 MeV in the outer region around r≊8–9 fm. In addition a second minimum, very shallow or vanishing at some incident energies, is found to exist in the region 5–6 fm. The imaginary part of the potential exhibits positive maxima in those regions of radial distances where the real part has minimum values indicating a possible feedback effect of flux to the elastic channel. The overall energy dependence of the potentials shows a shape transition resulting in diminishing the outer potential minimum between ${\mathit{E}}_{\mathrm{c}.\mathrm{m}.\mathrm{}}$ of 9 and 12 MeV. The inverted (real) potentials yield phase shifts of π/2 in those partial waves where resonances are known to exist. The procedure is tested by recalculating differential cross sections from the inverted energy-dependent potentials with the result that consistent agreement with the experimental input data is found.

• Received 27 September 1993

DOI:https://doi.org/10.1103/PhysRevC.49.2608