  The expression for the radial moments $\left\langle r^{n}\right\rangle_{c}$
of the nuclear charge density has been discussed under the plane wave Born
approximation (PWBA) method recently, which is significant to investigate the
nuclear surface thickness and neutron distribution radius. In this paper, we
extend the studies of extracting second-order moment $\left\langle
r^{2}\right\rangle_{c}$ and fourth-order moment $\left\langle
r^{4}\right\rangle_{c}$ from the Coulomb form factors $|F_{C}(q)|^2$ by the
distorted wave Born approximation (DWBA) at the small momentum transfer $ q $
region. Based on the relativistic mean-field (RMF) calculations, the DWBA form
factors $F_{C}^{DW}(q)$ are expanded into $ q^4 $, where the corresponding
charge distributions are corrected by the contributions of neutron and
spin-orbit densities. In the small $ q $ region, it is found that the
experimental $|F_{C}(q)|^2$ can be well reproduced by considering the
contributions of the $\left\langle r^{4}\right\rangle_{c}$ at the small $ q $
region. Through further analyzing the second-order and fourth-order expansion
coefficients of the $F_{C}^{DW}(q)$, the relationship between the expansion
coefficients and proton number $ Z $ is obtained. By the relationship, we
extract the $\left\langle r^{2}\right\rangle_{c}$ and $\left\langle
r^{4}\right\rangle_{c}$ from the limited experimental data of form factors at
the small $ q $ region. Within the permissible range of error, the extracted
$\left\langle r^{n}\right\rangle_{c}$ are consistent with the experimental data
in this paper.
