# Ionization of Xenon Clusters by a Hard X-ray Laser Pulse
## 3 Results and Discussions
### Ion TOF Spectrum of Xenon Clusters Irradiated with XFEL Pulses
Here, we simply assume that the slow electrons stay in the ionized cluster while the fast ones fly away from it. When a spherical object with the radius of 44 A has the charge of +400, the electrostatic potential on its surface is 130 eV. As mentioned in Section 3.2, a number of the slow electrons escape from the charged clusters, and thus clusters' electrostatic potentials may become greater than 130 eV with the increasing charge. Again, the electrostatic potential on the surface of charged cluster provides the upper limit of the available electron temperature. Plasma electrons with the expected temperature cause the hydrodynamic expansion of nanoplasma that begins with surface ions. A fraction of outermost ions with high kinetic energy can expand fast enough to avoid electron capture. Now we consider the ions with high kinetic energies that have been ejected from the vicinity of the charged cluster surface and assume that the XFEL-ignignited nanoplasma has been constructed with a quasi-neutral core and a charged thin spherical shell [24; 61; 62; 63]. In the charged thin spherical shell, the ions are expected to be accelerated by their own Coulomb repulsive forces. The potential energy of a singly charged ion at cluster's surface is \(U_{\text{Coul}}=\dfrac{e^{2}}{4\pi e_{0}}\dfrac{N_{\text{eff}}}{R_{0}}\), taking into account the screening of ions by trapping potential [64]. Here, we consider that the charges have spread into the system already and define \(e\) is the elementary charge, \(e_{0}\) as the dielectric constant of the vacuum, \(N_{\text{eff}}\) as the effective number of unscreened charges on the cluster, and \(R_{0}\) as the radius of the cluster. The \(N_{\text{eff}}\) contributing to the electrostatic potential is approximated by the difference between the total number of ion charges and the number of trapped electrons: \(N_{\text{eff}}=Q\). When the cluster size is 5000 atoms, the \(R_{0}\) of the xenon cluster is 44 A. When the ions are singly charged in the spherical shell, their mean kinetic energy corresponds to the energy stored per ion of the cluster's surface as \(E_{\text{kin}}=\left(\dfrac{1}{2}\dfrac{e^{2}}{4\pi e_{0}}\dfrac{N_{\text{eff} }^{2}}{R_{0}}\right)/N_{\text{eff}}=\dfrac{1}{2}U_{\text{Coul}}\)[64]. We extended the use of this formula for the mean kinetic energy of the highly charged ions: \(E_{\text{kin,Xe}^{++}}=\dfrac{1}{2}qU_{\text{Coul}}\). When \(Q\) is 400, the mean kinetic energies of Xe\({}^{2+}\) and Xe\({}^{3+}\) are \(\sim\)130 and \(\sim\)200 eV, respectively. The estimated mean kinetic energy of Xe\({}^{3+}\) agrees with the present observation (see Figure 2b); on the other hand, the estimated mean kinetic energy of Xe\({}^{2+}\) is slightly higher than the observed one. We note that the experimental results have been volume integrated [25; 65], increasing the contribution from the lower fluence region of the beam focus. As will be discussed in the following Section 3.4, the low-fluence XFELs reduce \(Q\) of the clusters and then provide ions with lower charges rather than higher ones. The low-fluence region of the XFEL beam's focus may be the main contributor to the creation of the Xe\({}^{2+}\) ions, and thus the observed kinetic energy becomes lower than the expected energy.