# Ionization of Xenon Clusters by a Hard X-ray Laser Pulse
## 3 Results and Discussions
### XFEL Pump-NIR Probe Results
The resonance frequency of an inhomogeneous electron system becomes lower than the Mie resonance frequency, where the electron density of the nanoplasma is assumed to be constant [73]. The resonance frequency tends to be a Mie frequency with the increase in the cluster size [73; 74]. In other words, the surface plasma resonance for the nanoplasma with an inhomogeneous electron system occurs at a lower electron density than a homogeneous one. The Mie-like resonance becomes much broader due to the motion in the fluctuation field of point-like ions [75]. This broadening must expand the range of electron density for causing the effective heating of nanoplasma. Therefore, if the electron density is inhomogeneous, effective heating occurs on the XFEL-ignited nanoplasma with an electron density lower than the critical electron density. Thus, we conclude that the gradual declines in the yield of multiply charged ions are attributed to the XFEL-ignited nanoplasma with a finite size having inhomogeneous electron density. Moreover, as mentioned above, the inhomogeneous electron density makes the resonance electron density lower than Mie resonance, which shifts the peaks of the multiply charged ions to a time delay later than what was expected. This finding is consistent with the experimental observations. A more detailed investigation using molecular dynamics simulations is necessary to quantitatively analyze the ion yield curves for the XFEL pump-NIR probe experiments because the molecular dynamics simulations address the real space dynamics of atoms, atomic ions,and electrons, including electron impact ionization, inelastic scattering, as well as electron-ion recombination processes. Moreover, the two-step molecular dynamics-hydrodynamic approach is expected to track the expansion of the quasi-neutral nanoplasma core in the longer picosecond timescale [76, 77]. The total energy absorption of the cluster irradiated by the XFEL and NIR pulses can be determined as the sum of the kinetic energies of the emitted electrons and ions, of the ionization potentials required to produce all ions, and of the energies of the photons emitted from electronically excited atoms. In the present study, the sum of kinetic energies of emitted ions and the ionization potentials required to produce all ions read as \(S_{\text{EI}}=\sum\limits_{\text{ion}}\int E_{\text{kin,ion}}Y_{\text{ion}}(E_{ \text{kin,ion}})dE_{\text{kin,ion}}\) and \(S_{\text{IP}}=\sum\limits_{\text{ion}}IP_{\text{ion}}\int Y_{\text{ion}}(E_{ \text{kin,ion}})dE_{\text{kin,ion}}\), respectively, where \(E_{\text{kin,ion}}\) is the kinetic energy of the ion, \(Y_{\text{ion}}(E_{\text{kin,ion}})\) is the ion yield with the kinetic energy of \(E_{\text{kin,ion}}\), and \(IP_{\text{ion}}\) is the ionization potential of the ion. \(IP_{\text{ion}}\) can be accurately evaluated from the atomic ionization potentials available in the literature [78]. Assuming that the ion system is in good thermal equilibrium with the electrons system, the total energy absorption is propotional to the sum of \(S_{\text{EI}}\) and \(S_{\text{IP}}\). Figure 7 shows the relative total energy absorption (\(S_{\text{EI}}+S_{\text{IP}}\)) of xenon clusters with an average size of 5000 atoms irradiated by the XFEL and NIR pulses as a function of the strength of the NIR-laser field, \(|E_{0}|\), where the peak fluence of XFEL pulse is \(\sim\)4.6 \(\upmu/\upmu\)m\({}^{2}\) and the time delay is 2 ps. The heating efficiency under this resonance condition is proportional to \(\
u^{-1}|E_{0}|^{2}\), where \(\
u\) is the electron-ion collision frequency [38]. The electron-ion collision frequency is proportional to \(|E_{0}|^{-3}\) according to the standard Coulomb formulas [79]. One can clearly see that the relative total energy absorption is proportional to \(|E_{0}|^{5}\) in Figure 7, which has also provided evidence that heating up the XFEL-ignited nanoplasma using an NIR-laser pulse can be ascribed to the the surface plasma resonance.