that the zeroth-order approximation correctly reproduces all qualitative features of the corresponding DHF result and contrary to the relativistic TFD model provides density oscillations and a correct asymptotic behavior for large values of $r$. In addition, we point out that the relativistic TFD model [@dolg_relativistic_2012; @nakajima_douglaskrollhess_2012; @marini_relativistic_1981; @waber_relativistic_1975], unlike its nonrelativistic counterpart [@landau1981quantum], does not have a universal dependence on the charge of the nucleus. For this reason, in order to obtain the electronic density, the relativistic TFD equation needs to be repeatedly solved numerically, which is a nontrivial procedure. (See Appendix \[sec:solut-relat-tf\]).

Atomic scattering factors {#sec:atom-scatt-fact}
-------------------------

Another observable characteristics that can be extracted from the effective charge model are the atomic scattering factors. According to their definition [@hau2012high], they are expressed by the Fourier transforms of electronic density [@prince_international_2006]: $$\begin{aligned}
  f({\boldsymbol}{q}) = \int \rho({\boldsymbol}{r})e^{i {\boldsymbol}{q} \cdot {\boldsymbol}{r}}
  d{\boldsymbol}{r}, \label{eq:15}\end{aligned}$$ which in our approach can be calculated analytically (see Appendix \[sec:appendix-a.-details\]).

The atomic scattering factors are very important for crystallography and X-ray physics, since the crystal polarizability $\chi$ as the function of X-ray frequency $\omega_{\mathrm{r}}$, can be evaluated by employing the relation [^1] [@ahmadi_parametric_2013; @baryshevsky2005parametric] $$\begin{aligned}
  \chi({\boldsymbol}g, \omega_{\mathrm{r}}) = \frac{4 \pi S({\boldsymbol}g)}{\Omega_{0}
  \omega_{\mathrm{r}}^{2}}
  \left(-\frac{e^{2}}{m}f({\boldsymbol}g)\right), \label{eq:17}\end{aligned}$$ where $\Omega_0$ is the 