Auger Spectra and Different Ionic Charges Following 3s, 3p and 3d Sub-Shells Photoionization of Kr Atoms

The decay of inner-shell vacancy in an atom through radiative and non-radiative transitions leads to final charged ions. The de-excitation decay of 3s, 3p and 3d vacancies in Kr atoms are calculated using Monte-Carlo simulation method. The vacancy cascade pathway resulted from the de-excitation decay of deep core hole in 3s subshell in Kr atoms is discussed. The generation of spectator vacancies during the vacancy cascade development gives rise to Auger satellite spectra. The last transitions of the de-excitation decay of 3s, 3p and 3d holes lead to specific charged ions. Dirac-Fock-Slater wave functions are adapted to calculate radiative and non-radiative transition probabilities. The intensity of Kr^{4+} ions are high for 3s hole state, whereas Kr^{3+} and Kr^{2+} ions have highest intensities for 3p and 3d hole states, respectively. The present results of ion charge state distributions agree well with the experimental data.


Introduction
The relaxation of inner-shell ionized atom via successive Auger, Coster-Kronig and radiative transitions leads to production of highly charged ions. In the course of de-excitation decay pathway, multiple vacancies are generated after each Auger transition. The distribution of generated vacancies does not dependent on the initial ionization process. The Auger cascades are accompanied with emission of photon spectra or electron spectra for each pathway branch. The spectra are conditioned by the transition rates and transition energies of multi-vacancy states. The generation of vacancies in the course of de-excitation cascade is accompanied by characteristic energy shifts in the electronic levels. The influence of the additional vacancies during the cascades may close some low-energy Auger channels (forbidden energies). Understanding the influence of the spectator vacancies on Auger transitions, gives more detailed information about the vacancy cascade development. The overlapping spectra emitted from parallel branches of the de-excitation cascades lead to different low-energy highly charged ions. These low-energy ionic charges are important in the field of astrophysical plasma [1]. Study of ion charge state distributions following inner-shell ionization of atoms provides information to estimate the characteristic relaxation time constants and thermal equilibrium of ion gas that stored in trap [2,3,4]. The ion charge state distributions following inner-shell ionized rare gas atoms are measured using both energy ionization from x-ray tube [5,6], and synchrotron radiation [7,8,9,10,11]. Tamenori et al. [12] measured the branching ratio of multiply charged ions formed through photoionization of Kr 3d, 3p and 3s subshells using a coincidence technique. The calculations of ion charge state distributions resulted after core-hole creation of atom are carried out by many researchers [13,14,15,16,17,18,19,20,21,22].
The Auger transition rates and transition energies in multi-ionized neon atom are measured using heavy-ion and electron bombardment [23] and calculated using Hartree-Fock-Slater wave functions [24]. Larkins [25] investigated the influence of multi-vacancy states on the electron binding energies and transition rates in K, L, and M subshells in the calculation of ion charge state distributions. Auger and x-ray spectra formed during vacancy cascades are calculated using Monte-Carlo method [26,27]. Cooper et al. [28] measured and calculated the L 23 MM Auger spectra of argon excited at energies between those of K-and L-thresholds and affected by L 1 L 23 M intermediate transition. The L 23 MM Auger spectra of argon emitted after photoexcitation are measured using broad-band synchrotron radiation of energies largely lying above the K-shell ionization potential [29].
In the present work, the different ionic charges followed 3s, 3p and 3d subshells photoionization of Kr are calculated using Monte Carlo simulation technique. The calculation is considered an extension to the previous work carried out by Abdullah et al. [21], in which ion charge state distributions yield after 1s, 2s and 2p vacancy creation in Kr atoms are obtained. The deexcitation decay branches of parallel Coster-Kronig and Auger transitions followed 3s, 3p and 3d hole production are discussed. The Auger transition rates and electron shake off are obtained using Dirac-Fock-Slater wave functions. The present results are compared with available theoretical [16] and experimental values [12].

Method of calculations
The highly excited atoms resulted after electron emission from inner-shell will rapidly relax back to a lower energy state by radiative or non-radiative process. In case of radiative transition, the vacancy transfers to a higher shell under emission of characteristic x-rays: where the core hole in the target atom A is produced by photoionization (hν), (A 1+ ) ⋆ is the atom in highly excited state and e p is the primary emitted electron. If the vacancy is filled via Auger transition, two vacancies in the higher shells of atom will be created and the transition is given by: where A 2+ is doubly ionized atom.
In an Auger transition, one electron falls from a higher level to fill the initial vacancy and another electron is ejected into continuum. The kinetic energy of emitted Auger electron can be estimated from the binding energies of the various levels involved in the transition: where E i is the binding energy of initial state, E j is the binding energy of electron filling the hole and E k is the binding energy of the electron leaving the atom.
The simulation of the cascade branches is based on the selection of all possible radiative and non-radiative branching ratios that may fill the inner-shell vacancies in atoms. The radiative branching ratio is defined as the probability that the vacancy in an atom is filled through x-ray transitions (photon emission), while the non-radiative branching ratio is defined as the vacancy filled through Auger and Coster-Kronig processes. The radiative and non-radiative branching ratio are given by: For fluorescence yield: and for Auger yield: where i is the initial configuration decaying into the final configuration f . Γ is the sum of partial radiative widths Γ R if and non-radiative width Γ A if which is given by: The calculations of radiative transition rates are performed for singly ionized atoms using multiconfiguration-Dirac-Fock (MCDF) wave functions [30]. The non-radiative transition rates and electron shake off processes are computed using Dirac-Fock-Slater (DFS) wave functions [31]. The Monte-Carlo simulation technique is described in detail by Abdullah et al. [21], in which the electron shake off process is considered. In radiative transitions case, the inner-shell vacancy will transfer to higher shell under emission of characteristic x-rays without changing the number of vacancies thus a new position of vacancy is created. If the vacancy is filled via Auger process, two vacancies in the higher shells of atom will be generated. The de-excitation decays are repeated until all vacancies reach the outermost shell and no transitions are possible. The successive Auger cascades lead to the production of specified low-energy charged ions. The Auger cascades correspond to transitions with emission of photon spectra or electron spectra for each pathway branches. In the course of cascade development, Auger spectra arise from spectator vacancies left by preceding Auger transitions and by electron shake off process accompanying the Auger transitions. Fig. 1 shows a schematical diagram for all main de-excitation decay pathway resulted after 3s vacancy creation in Kr atom. The solid arrows indicate the selected Auger channels following inner-shell ionization, while dotted ones show the Coster-Kronig channels. The solid line indicates the radiative transition; while dotted lines indicate the electron shake off processes occuring due to the change of atomic potential after primary ionization or after Auger and Coster-Kronig transitions. Electron shake off process resulted from the initial ionization or deexcitation decay produces additional vacancies in higher shells leading to an increase in the number of vacancies. Auger and Coster-Kronig transitions are indicated outside the brackets, while the spectator vacancy configurations are indicated inside the brackets. Each branch of the de-excitation leads to an ion of a specific charge as shown in the diagram. Considering of electron shake off process in the calculation gives a final charge state of ions which agree well with the experimental data. Fig. 2 shows the charge state of ions resulted after 3s vacancy creation in Kr atoms. The present results are compared with other theoretical values from Kochur et al. [16] and experimental data from Tamenori et al. [12]. The relaxation of ionized atom occurs through radiative and/or non-radiative transitions. At 3s hole state in Kr atoms, the probability of radiative transitions is lower than the probability of non-radiative transitions. The radiative branching ratios will occur in 1 % of the cases as given by the fluorescence yield. In the remaining 99 % of the  Kr 5+ and Kr 6+ ions are formed from the de-excitation via parallel Auger and Coster-Kronig channels and electron shake off process. The intensities of the Kr 5+ and Kr 6+ ions are 21 % and less than 4 %, respectively. It is found that the creation of additional N vacancies during the de-excitation decay of 3s hole closes the M 23 M 45 M 45 super Coster-Kronig channel, i.e. the super Coster-Kronig transition energy is forbidden. The intensity of Kr 4+ and Kr 5+ ions is high while a Kr 1+ , Kr 2+ , Kr 3+ and Kr 6+ ions is low. The most probable de-excitation pathway of a 3s shell vacancy is via Coster-Kronig transitions, which will create M 45 vacancies with additional N sub-shell vacancies. So each spectator vacancy configuration will give rise to an M 45 NN satellite spectrum. The present results of ion charge state distributions after 3s vacancy production agree well with the experimental data [12].

Results and discussions
The ion charge state distributions produced after 3p hole creation in Kr are shown in Fig. 3. The de-excitation decay of 3p hole gives rise to different ionic charges Kr q , q = 1, . . . , 5. The  In the 3p hole state, the ionization of the 3p electrons increases the intensity of triply charged ions Kr 3+ and decreases the quadruply charged ions Kr 4+ in comparison with the corresponding charged ions that produced after 3s vacancy. The formation of Kr 1+ is caused by radiative transition with ionizing electron from N subshells. The intensity of Kr 1+ ions after 3p shell ionization is less than 1 %. The decay of 3p vacancy via normal M 23 NN Auger transition leads to the formation of doubly charged ions Kr 2+ with an intensity of 3 %. The formation of Kr 3+ ions occurs from the decay of 3p hole via initial Coster-Kronig channel and subsequent Auger transition. The triply charged ions Kr 3+ followed after the 3p ionization has highest intensity (66 %). The Kr 4+ ions with intensity of 25 % are formed via Coster-Kronig, Auger transitions and electron shake-off from N shell. The probability of electron shake off from N shell after 3p ionization is about 7 %. The intensity of Kr 5+ ions yields from vacancy cascade followed 3p hole production is 5 %. In general, the de-excitation decay of 3p hole of Kr leads to dominant production of Kr 3+ and Kr 4+ charged ions and lower production of Kr 2+ and Kr 5+ ions, respectively.
The percentage fraction of the ion charge state distributions yields from de-excitation decay of 3d hole in Kr are shown in Fig. 4. The highest intensity Kr 2+ ions formed from de-excitation decays of 3d photoionization. The de-excitation decays of 3d vacancy yields Kr 2+ with higher intensity and Kr 3+ ions with lower intensity in comparison with the corresponding results from 3p vacancy. The intensities of doubly and triply charged ions are 66 % and 35 %, respectively. It is important to mention that the present results of ion charge state distributions produced after 3s, 3p and 3d vacancies production in Kr atoms agree well with the experimental data from Tamenori et al. [12].

Conclusions
The core hole creation followed by successive Auger, Coaster-Kronig transitions and electron shake off leads to different ionic charge. Monte Carlo simulation method is performed to calculate the highly charged ions after 3s, 3p and 3d vacancies creation in Kr atom. The Auger and Coster-Kronig branching ratio and electron shake off process are calculated using Dirac-Fock-Slater wavefunctions. The cascade of decay pathway branches followed 3s, 3p and 3d holes are discussed. The Kr 4+ ions have highest intensity in the 3s shell ionization, while Kr 3+ and Kr 2+ ions have the highest intensity in the 3p and 3d shell ionization, respectively. The present results are compared with other theoretical values [16] and experimental [12] and are found to agree well with the experimental values.