  In this work, we study the odd-even effect in binding energies and charge
radii, and systematic behavior of differential radii, to identify the
underlying components of the effective nuclear interaction. We apply nuclear
density functional theory using a family of Fayans and Skyrme energy density
functionals fitted to similar datasets but using different optimization
protocols. We inspect various correlations between differential charge radii,
odd-even staggering in energies and radii, and nuclear matter properties.
Detailed analysis is carried out for medium-mass and heavy semi-magic nuclei
with a particular focus on the Ca chain. By making the surface and pairing
terms dependent on density gradients, the Fayans functional offers the superb
simultaneous description of odd-even staggering effects in energies and charge
radii. Conversely, when the data on differential radii are added to the pool of
fit-observables, the coupling constants determining the strengths of the
gradient terms of Fayans functional are increased by orders of magnitude. The
Skyrme functional optimized in this work with the generalized Fayans pairing
term offers results of similar quality. We quantify these findings by
performing correlation analysis based on the statistical linear regression
technique. The nuclear matter parameters characterizing Fayans and Skyrme
functionals optimized to similar datasets are fairly close.
