Inverse methods and nuclear radii

In considering spherically symmetric three-dimensional systems, inverse methods are applied to the nuclear bound-state problem. While retaining only the self-interactions of the (occupied) bound-state levels, an analytical solution is obtained for the potential. The simplest possible approximation to it corresponding to a single fictitious bound state is used to evaluate (root mean square) radii. Combining this formula with the well-known $A^{\frac{1}{3}}$ dependence of the nuclear radii, a new formula is obtained containing the collective binding energy effect and the one of the saturation of nuclear forces. For absolute and relative radii (of isotopes of Sn, Xe, Nd, Dy, Yb, Os, Hg, Pb, and Pu), the results compare favorably with experiment. In spite of the crude approximations made, this approach yields the typical curvature of the plot of the experimental relative radii as a function of the mass number. The extreme simplicity of the formula recommends its use for global discussions or predictions. Yet, for a correct description of the finer details it is necessary to account explicitly for shell effects and deformations.