  Proton elastic scattering off Sn isotopes at $E_p=295$ MeV in the mass number
range of $A=$116--124 was investigated using calculation employing relativistic
impulse approximation (RIA) with theoretical densities obtained for the Sn
isotopes from relativistic Hartree-Bogoliubov (RHB) and nonrelativistic Skyrme
Hartree-Fock-Bogoliubov (SHFB) calculations of spherical nuclei. In the RIA
calculations, a modified version of the Murdock and Horowitz model that
includes a density dependence in the effective nucleon-nucleon ($NN$)
interaction was used. A calculation using the theoretical density obtained from
a relativistic calculation employing the DD-ME2 interaction successfully
reproduced the experimental data for $^{122}$Sn$(p,p)$, but it overestimated
the $^{116}$Sn$(p,p)$ and $^{118}$Sn$(p,p)$ cross sections at backward angles.
Isotopic analyses of the $(p,p)$ reactions combined with nuclear structure
properties were performed based on reaction calculations that used a model
density modified from the DD-ME2 density to optimize the neutron density of the
Sn isotopes by fitting the isotopic cross section ratios. The resulting
optimized density reproduced the experimental $(p,p)$ data for the series of Sn
isotopes from $^{116}$Sn to $^{124}$Sn. The neutron root-mean-square(rms) radii
and the skin thickness of the Sn isotopes obtained in the present analysis
exhibited smooth $A$ dependences in the range of $A=$116--124, which are
consistent with the theoretical predictions obtained using the DD-ME2
interaction but seem to contradict the experimental results determined from the
$(p,p)$ data. In a detailed analysis of the surface neutron density probed by
proton elastic scattering, a signal of the shell effect at $N=66$ in Sn
isotopes was found.
