  Proton-$^{133}$Cs elastic scattering at low momentum transfer is performed
using an in-ring reaction technique at the Cooler Storage Ring at the Heavy Ion
Research Facility in Lanzhou. Recoil protons from the elastic collisions
between the internal H$_2$-gas target and the circulating $^{133}$Cs ions at
199.4 MeV/u are detected by a silicon-strip detector. The matter radius of
$^{133}$Cs is deduced by describing the measured differential cross sections
using the Glauber model. Employing the adopted proton distribution radius, a
point-neutron radius of 4.86(21) fm for $^{133}$Cs is obtained. With the newly
determined neutron radius, the weak mixing angle sin$^2 \theta_W$ is
independently extracted to be 0.227(28) by fitting the coherent elastic
neutrino-nucleus scattering data. Our work limits the sin$^2 \theta_W$ value in
a range smaller than the ones proposed by the previous independent approaches,
and would play an important role in searching new physics via the high
precision CE$\nu$NS-CsI cross section data in the near future.
