volume of a crystal cell, ${\boldsymbol}g$ the reciprocal lattice vector, $S({\boldsymbol}g)$ the structure factor of the crystal, $m$ the electron mass and $e$ the electron charge.

In Fig. \[fig:ASF\] we present the results for gold, lead, uranium and xenon. Our analytical expressions for atomic scattering factors are comparable to the DHF calculation to within $25\%$.

As a realistic example, we also evaluated the relative difference for the emission intensity of parametric X-ray radiation (PXR) [@feranchuk_theoretical_1985; @baryshevsky2005parametric; @skoromnik_radical_2017] from relativistic electrons in a tungsten crystal. The intensity of PXR is proportional to the square of the absolute value of crystal polarizability. Therefore, the relative difference for the intensity of radiation with crystal polarizabilities from DHF calculation and from effective charge model is given by $(|\chi^{(0)}|^{2} -
|\chi_{\mathrm{DHF}}|^{2})/|\chi_{\mathrm{DHF}}|^{2}$. Consequently, for the relevant range of frequencies and Bragg’s angles the relative accuracy was $\sim 20\%$.

Highly charged ions {#sec:highly-charged-ions}
===================

The effective charge model is also suitable for the description of ions, since it does not require the charge of the nucleus to be equal to the number of electrons of an atom. Consequently, we fix the nuclear charge as $Z$ and use a state vector $|\lambda_{1},\ldots,\lambda_{N}\rangle$, where $N \neq Z$. As an 