  Measurements of atomic transitions in different isotopes offer key
information on the nuclear charge radius. The anticipated high-precision
experimental techniques, augmented by atomic calculations, will soon enable
extraction of the higher-order radial moments of the charge density
distribution. To assess the value of such measurements for nuclear structure
research, we study the information content of the fourth radial moment $\rf$ by
means of nuclear density functional theory and a multiple correlation analysis.
We show that $\rf$ can be directly related to the surface thickness of nuclear
density, a fundamental property of the atomic nucleus that is difficult to
obtain for radioactive systems. Precise knowledge of these radial moments is
essential to establish reliable constraints on the existence of new forces from
precision isotope shift measurements .
