Effect of screening by external charges on the atomic orbitals and photoinduced processes within the Hartree-Fock-Slater atom

X-ray free-electron lasers (XFELs) are a promising tool for the structural determination of macro- and biomolecules, using coherent diffractive imaging. During imaging, the intense XFEL pulses also efficiently ionize the molecules, so it is important to estimate how the charged environment within the molecule modifies atomic properties, in comparison to the case of an isolated atom. Here, we apply the xatom toolkit to obtain predictions on the modified ionization thresholds and rates of some photoinduced processes in carbon. The Hartree-Fock-Slater model is extended to include the electron screening and ion correlation effects, induced by external charges. With this extended model, we obtain predictions on modifications of orbital energies, photoabsorption cross sections, Auger decay rates, fluorescence emission rates, and atomic scattering factors as a function of the density and temperature of the surrounding charges. Our results have implications for the studies of dynamics within XFEL irradiated samples, in particular for those dedicated to coherent diffraction imaging.

Figure 1

Ion-ion radial distribution function for liquid carbon with a density of $2.0\mathrm{g}/{\mathrm{cm}}^3$. Comparison between the MD calculation by Wang et al. [59] (crosses) and the fitting formula by Broyles et al. [57, 58] (solid line). The fitted parameters are $a=7.0$, $b=1.2$ Å${}^{-1}$, $c=5.4$ Å${}^{-1}$, and $d=6.1$.

Figure 2

Ion-ion radial distribution function for liquid carbon with a density of $4.4\mathrm{g}/{\mathrm{cm}}^3$. Comparison between the MD calculation by Wang et al. [59] (crosses) and the fitting formula by Broyles et al. [57, 58] (solid line). The fitted parameters are $a=1.2$, $b=0.16$ Å${}^{-1}$, $c=7.2$ Å${}^{-1}$, and $d=8.4$.

Figure 3

Energy shifts $\Delta E$ between the unscreened HFS model and the Debye-screened HFS model for carbon as a function of the number of the electrons bound on carbon, $Z_b$. The calculations were performed for several Debye screening lengths (${\lambda }_D=1.72,3.14,5.0,10.0$ $a_0$). Shifts for the various orbitals 1$s$, 2$s$, and 2$p$ are shown.

Figure 4

Energy shifts $\Delta E$ between the unscreened HFS model and the ion-sphere screened HFS model for carbon as a function of the number of the electrons bound on carbon, $Z_b$. The external ion density is $2.0$ g/cm${}^3$. External ions have charge ${\overline{Z}}_i=1$ (dashed lines) and ${\overline{Z}}_i=2$ (dotted lines), respectively.

Figure 5

Energy shifts $\Delta E$ between the unscreened HFS model and the ion-sphere screened HFS model for carbon as a function of the number of the electrons bound on carbon, $Z_b$. The external ion density is $4.4$ g/cm${}^3$. External ions have charge ${\overline{Z}}_i=1$ (dashed lines) and ${\overline{Z}}_i=2$ (dotted lines), respectively.

Figure 6

Orbital energy for the 1$s$(red lines) and 2$s$ (blue lines) shells for C${}^{3+}$ in the ground state as a function of the average charge of external ions, ${\overline{Z}}_i$. The solid line shows the result of the unscreened HFS model. The dashed and the dotted lines show the ion-sphere screened results for the carbon densities of $2.0$ g/cm${}^3$ and $4.4$ g/cm${}^3$, respectively.

Figure 7

Comparison of 1$s$, 2$s$, and 2$p$ radial wave functions of neutral carbon for the unscreened, Debye-screened, and ion-sphere models. For the Debye-screened model, ${\lambda }_D=5a_0$ is used, and for the ion-sphere model, the ion density of 2.0 g/cm${}^3$ and ${\overline{Z}}_i=2$ are used.

Figure 8

Total photoabsorption cross section for carbon in the ground state calculated with xatom as a function of the photon energy for (i) the unscreened HFS model (black solid line), (ii) the Debye-screened HFS model (${\lambda }_D=5.0$ $a_0$, black double-dot-dashed line; ${\lambda }_D=10$ $a_0$, black dot-dashed line), and (iii) the ion-sphere screened HFS model (density of $2.0$ g/cm${}^3$, ${\overline{Z}}_i=1,2$, red dashed line and red dotted line).

Figure 9

Subshell photoabsorption cross section for carbon in the ground state for different shells calculated with xatom for the unscreened HFS model (solid lines) and for the Debye-screened HFS model (${\lambda }_D=5.0$ $a_0$, dashed lines) as a function of the photon energy.

Figure 10

Atomic scattering factor of carbon for different electronic configurations calculated as a function of the momentum transfer. We show the unscreened HFS case (black solid line), the Debye-screened model for ${\lambda }_D=1.72$ $a_0$ (black dot-dashed line), and the ion-sphere screened HFS model with the density of $2.0\mathrm{g}/{\mathrm{cm}}^3$, ${\overline{Z}}_i=2$ (red dashed line).