# Ionization of Xenon Clusters by a Hard X-ray Laser Pulse
## 3 Results and Discussions  
### Electron Emission in Atomic Processes  
First of all, an outline of the ionization processes for individual atoms is shown as a premise of the discussion about the ionization dynamics of clusters. In the 5.5 keV X-rays, we can classify photo- and Auger electrons derived from xenon atoms into two groups based on their initial kinetic energy. We have defined slow and fast groups as the electrons whose kinetic energies are initially below 1.2 keV and above 2.4 keV, respectively.  
The deepest shell for photo-ionization of xenon atoms in the 5.5-keV X-rays is the \(L\)-shell, which is the most probable photo-ionization channel. Since the photo-absorption cross-section for xenon atoms at 5.5-keV of photon energy is 0.166 Mbarn [49] in the X-ray beam with the peak fluence of \(\sim\)4.6 \(\upmu\)/\(\upmu\)m\({}^{2}\) the photo-absorption probability is approximately 8.7%, which denotes that the probability of multi-photon absorption is rather small, although for a single atom. A xenon cluster with the size of 5000 atoms absorbs \(\sim\)430 photons, for example.  
The percentages of the photo-ionization probabilities for \(L_{1}\), \(L_{2}\), and \(L_{3}\) orbitals are 14%, 25%, and 45%, respectively [50]. The kinetic energies of electrons emitted from \(L_{1}\), \(L_{2}\), and \(L_{3}\) photo-ionization by the 5.5-keV photons are \(\sim\)47 eV, \(\sim\)400 eV, and \(\sim\)720 eV, respectively, which are slow electrons, as defined above. The other photo-ionization processes release fast electrons. Meanwhile, \(\sim\)43% of \(L_{1}\)- and \(\sim\)12% of \(L_{2}\)-vacancies produce slow electrons via Coster-Kronig transitions (LLN and LLO) [51; 52]. \(L\)-holes are predominantly filled by electrons from shallow orbits in another Auger process releasing a fast electron (\(\sim\)51% of \(L_{1}\), \(\sim\)80% of \(L_{2}\), \(\sim\)93% of \(L_{3}\) holes). In addition, the relaxation of \(M\)-, \(N\)-, and \(O\)-holes via Coster-Kronig transitions and Auger decays lead to the ejection of one slow electron. The mean charge state of xenon ions produced by 5.5 keV photons is \(\sim\)7.7 [53]. Finally, 6.8 slow and 0.9 fast electrons are ejected from individual atoms on average by the photo-ionization and subsequent Auger cascades.  
Collisional ionization is one of the dominant ionization events in a dense atomic environment, in which energetic free electrons interact with surrounding atoms to kick out a bound atomic electron. The secondary electrons produced by the collisional ionization event are typically categorized into the slow group.  
The mean free path, \(\lambda=(\sigma n)^{-1}\), is usually employed to explain the collisional ionization events in a medium, where \(\sigma\) is the cross-section of collisional ionization and \(n\) is the atomic density of the material. In a pristine xenon cluster, whose \(n\) is \(\sim\) 1.36 \(\times\) 10\({}^{28}\) m\({}^{-3}\)),\(\lambda\) becomes the shortest (\(\sim\)9.2 A) at \(\sim\)70 eV kinetic energy and monotonically longer with an increase in the kinetic energy [54].  
At the kinetic energy of 1.2 and 2.4 keV, which represent the upper limit of the slow electrons and the lower limit of the fast ones, respectively, their mean free paths are \(\sim\)39 and \(\sim\)82 A, respectively, which are shorter and longer than the radius of a 5000-atom cluster (44 A). In other words, the fast electrons do not create secondary electrons by the collisional ionization of xenon atoms in the cluster.