# Ionization of Xenon Clusters by a Hard X-ray Laser Pulse
## 3 Results and Discussions
### XFEL Pump-NIR Probe Results
Here, we consider the electron density, \(n_{\rm e}\), inside a spherical nanoplasma to satisfy the surface plasma resonance condition. [PERSON] et al. described the surface plasma resonance effect for homogeneous nanoplasma [38], which can be mathematically tantamount to the Mie resonance. The heating efficiency of the surface plasma resonance depends on the surface plasma frequency, namely, the electron density inside a spherical nanoplasma. Surface plasma resonance is expected to occur when the surface plasma frequency, \(\Omega_{\rm pl}=\sqrt{\frac{e^{2}n_{\rm e}}{3e_{0}m_{\rm e}}}\), coincides with the probe-laser frequency, \(\omega_{\rm NIR}\). Here, \(m_{\rm e}\) is electron mass. The critical electron density, \(n_{\rm crit}\), has been defined to be \(\frac{e_{0}m_{\rm e}\omega_{\rm NIR}^{2}}{e^{2}}=\) 1.7 \(\times\) 10\({}^{21}\) cm\({}^{-3}\). At the resonance condition \(n_{\rm e}=3n_{\rm crit}=\) 5.1 \(\times\) 10\({}^{21}\) cm\({}^{-3}\), the heating efficiency in the cluster becomes enhanced. Again, \(\sim\)400 fast and \(\sim\)2900 slow electrons are emitted from the individual core-ionized atoms in the xenon cluster with a size of 5000 atoms irradiated by the 5.5 keV photons with the peak fluence of \(\sim\)4.6 \(\mu\)J/\(\mu\)m\({}^{2}\). Assuming that \(\sim\)400 fast electrons fly away from the ionized cluster system, the electron density inside the spherical nanoplasma is estimated to be \(\sim\)7.9 \(\times\) 10\({}^{21}\) cm\({}^{-3}\) by the volume of pristine Xe\({}_{5000}\) (3.7 \(\times\) 10\({}^{-19}\) cm\({}^{3}\)). We note that the electron density is expected to be greater than \(\sim\)7.9 \(\times\) 10\({}^{21}\) cm\({}^{-3}\) as a results of the additional collisional ionization events with the trapped slow electrons, even though some of them escape from the nanoplasma. When the electron density is much higher than 3\(n_{\rm crit}\), the probe-laser field inside the cluster is screened by the dense quasi-free electrons of the nanoplasma. Thus, the laser field inside the cluster becomes weaker than the field on the outside. For instance, \(n_{\rm e}\) is much greater than \(\sim\)7.9 \(\times\) 10\({}^{21}\) cm\({}^{-3}\) on the ionized clusters right after XFEL irradiation. Therefore, at \(\sim\)0 ps time-delay the heating up of the nanoplasma is hindered by the screening effect. The electron density gradually decreases with time via thermal electron emission [26] from, and the electron-ion recombination within, the nanoplasma [24; 37; 57], as well as the expansion of nanoplasma [68; 69; 70]. When the resonance condition, \(n_{\rm e}=3n_{\rm crit}=\) 5.1 \(\times\) 10\({}^{21}\) cm\({}^{-3}\), is fulfilled due to the evolution of the system on the picosecond timescale, the laser field can perforate into the XFEL-ignited nanoplasma and the heating efficiency becomes enhanced. Namely, the time-delay dependence features for the yields of the multiply charged ions must be attributed the time evolution of the expanding nanoplasma. Let us now discuss the time evolution of electron density in the quasi-neutral core of XFEL-ignited nanoplasma that is hydrodynamically expanding. Here, we simply assume that the expansion speed of the quasi-neutral core is \(\sim\)3.5 \(\times\) 10\({}^{3}\) m/s, which corresponds to the velocity of Xe\({}^{+}\) with the mean kinetic energy of \(\sim\)17 eV. The quasi-neutral core expands to its radius of 52 A at 200 fs time delay. Even if one excludes the reduction of quasi-free electrons due to the thermal electron emission and the electron-ion recombination, the electron density of a quasi-neutral core with a radius of 52 A decreases to \(\sim\)5.1 \(\times\) 10\({}^{21}\) cm\({}^{-3}\), reaching the critical electron density of 3\(n_{\rm crit}\).