  Isospin-violating dark matter (IVDM) provides a possible mechanism to
ameliorate the tension among recent direct detection experiments. For IVDM, we
demonstrate that the results of direct detection experiments based on
neutron-rich target nuclei may depend strongly on the density dependence of the
symmetry energy which is presently largely unknown and controls the neutron
skin thickness that reflects the relative difference of neutron and proton form
factors in the neutron-rich nuclei. In particular, using the neutron and proton
form factors obtained from Skyrme-Hartree-Fock calculations by varying the
symmetry energy within the uncertainty region set by the latest
model-independent measurement of the neutron skin thickness of $^{208}$Pb from
PREX experiment at JLab, we find that, for IVDM with neutron-to-proton coupling
ratio fixed to $f_n/f_p=-0.7$, the form factor effect may enhance the
sensitivity of Xe-based detectors (e.g., XENON100 and LUX) to the DM-proton
cross section by a factor of $3$ in the DM mass region constrained by
CMDS-II(Si) and even by more than an order of magnitude for heavy DM with mass
larger than $80$ GeV, compared with the results using the empirical Helm form
factor. Our results further indicate that the form factor effect can
significantly modify the recoil spectrum of Xe-based detectors for heavy IVDM
with $f_n/f_p=-0.7$.
