  Background: Precise measurements of atomic transitions affected by
electron-nucleus hyperfine interactions offer sensitivity to explore basic
properties of the atomic nucleus and study fundamental symmetries, including
the search for new physics beyond the Standard Model of particle physics. In
particular, such measurements, augmented by atomic and nuclear calculations,
will enable extraction of the higher-order radial moments of the charge density
distribution.
  Purpose: The nuclear charge density is composed of the proton point
distribution folded with the nucleonic charge distributions. The latter induce
subtle relativistic corrections due to the coupling of nucleon magnetic moments
with the nuclear spin-orbit density. We assess the precision of nuclear charge
density calculations by studying the behavior of relativistic corrections.
Special attention has been paid to the magnetic spin-orbit density associated
with the local variations of the spin-orbit current.
  Methods: The calculations are performed in the framework of self-consistent
mean-field theory using Skyrme energy density functionals and density-dependent
pairing force. We used the general expression for the spin-orbit form factor
that is valid for spherical and deformed nuclei.
  Results:
  We studied the impact of the spin-orbit densities on the charge radii and the
fourth radial moments $\rf$ of spherical and deformed nuclei. The corresponding
corrections to charge radii show strong shell fluctuations and sometimes are
quite appreciable when aiming at high-precision predictions.
  Conclusions: To establish reliable constraints on the existence of new forces
from isotope shift measurements, precise calculations of nuclear charge
densities are needed. The proper inclusion of the spin-orbit charge density is
essential when aiming at extraction of tiny effects.
