positive($+$) and negative ($-$) helicities. As the electromagnetic interaction is not sensitive to the above difference, the asymmetry becomes dependent of the weak interaction between the electron and the target. Moreover, we know from the Standard Model that the neutral Z-boson couples more strongly to the neutron than to the proton. Those reasonings were then used in [@Don] to first propose a clean way to determine the neutron distribution in nuclei. If we consider elastic scattering on an even-even target nucleus, the asymmetry can be written in the form:

$$\mathcal{A}=\frac{Gq^2}{2\pi\alpha\sqrt{2}}a[\beta_{V}^{p}+\beta_{V}^{n}\frac{\rho_{n}(q)}{\rho_{p}(q)}].\label{ass}$$

In the above expression, G, $\alpha$, $a$ and $\beta_{V}^{p,n}$ are Standard Model coupling constants as defined in [@Don], $q$ is the transferred momentum by the electron to the nucleus and,

$$\rho_{n(p)}(q)=\int~d^{3}r~j_{0}(qr)\rho_{n(p)}(\mathbf {r}),$$

$\rho_{n(p)}(\mathbf {r})$ being the neutron (proton) distribution in configuration space and $j_{0}$ the spherical Bessel function of order zero. It is then clear that a small $q$ measurement of the asymmetry gives the neutron radius of the distribution once the proton radius is well known. The proton and neutron mean-square radius are defined as

$$R_{i}^{2}=\frac{\int~d^{3}r r^2 \rho_{i}(\mathbf {r})}{\int~d^{3}r
 \rho_{i}(\mathbf {r})}, \quad i=p,n.$$

The neutron skin thickness is defined as $$\theta=R_n-R_p. \label{skin}$$

In the PREX experiment mentioned in the 