  A method is presented to extract the neutron-skin thickness of atomic nuclei
from the second and fourth moments of the electric charge distribution. We show
that the value of the proton fourth moment must be independently known in order
to estimate the neutron skin thickness experimentally. To overcome this
problem, we propose the use of a strong linear correlation among the second and
fourth moments of the proton distribution as calculated with several energy
density functionals of common use. We take special care in estimating the
errors associated with the different contributions to the neutron radius and
show, for the first time, the analytic expressions for the spin-orbit
contribution to the charge fourth moments of neutrons and protons. To reduce
the uncertainty on the extraction of the neutron radius, two neighboring
even-even isotopes are used. Nevertheless, the error on the fourth moment of
the proton distribution, even if determined or assumed with large accuracy,
dominates and prevents the present method from being applied for a sound
determination of the neutron skin thickness.
