  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. Such
measurements impose higher precision requirements on a theoretical description.
  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.
  Methods: The calculations are performed 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 various correction terms on the charge
radii, fourth radial moments, diffraction radii, and surface thickness of
spherical and deformed nuclei. The spin-orbit corrections to charge radial
moments and surface thickness show strong shell fluctuations which impact
high-precision predictions of isotopic shifts.
  Conclusions: To establish reliable constraints on the existence of new forces
from isotope shift measurements,precise calculations of nuclear charge
densities of deformed nuclei are needed. The proper inclusion of the spin-orbit
charge density and other correction terms is essential when aiming at
extraction of subtle effects which become particularly visible in isotopic
trends.
