Multiple Sequential Ionization of Valence n = 4 Shell of Krypton by Intense Femtosecond XUV Pulses

: Sequential photoionization of krypton by intense extreme ultraviolet femtosecond pulses is studied theoretically for the photon energies below the 3 d excitation threshold. This regime with energetically forbidden Auger decay is characterized by special features, such as time scaling of the level population. The model is based on the solution of rate equations with photoionization cross sections of krypton in different charge and multiplet states determined using R-matrix calculations. Predictions of the ion yields and photoelectron spectra for various photon ﬂuence are presented and discussed.


Introduction
Multiple ionization of atoms by intense pulses generated by free-electron laser (FEL) operating in the extreme ultraviolet (XUV) has been observed since the first experiments at the Free-electron LASer in Hamburg (FLASH) [1]. Such studies are of great importance to benchmark theoretical models for the description of simple non-linear process in the XUV. Two regimes can be distinguished in the multiple ionization process: (i) The photon energy of the FEL is high enough to eject an electron from an inner shell. Within a few femtoseconds the hole is filled by an electron originating from an outer shell, through a single or cascaded Auger decay mechanism. The ultrafast dynamics of the Auger decay competes with the absorption by the target ion of another photon from the same femtosecond FEL pulse. Therefore, ionization of the target often proceeds through a chain of consecutive photoionization and Auger decay events [2]. Usually, the ion yields of the different charge states are measured in the experiment as a function of the FEL pulse parameters and they are compared with the corresponding predictions of theoretical models [3,4]. So far, only a limited number of electronic spectra in this regime have been reported in the literature [2,5]. (ii) The photon energy is not enough for creating a hole in an inner shell and therefore, the Auger process is energetically forbidden, if multiphoton ionisation is neglected. The atom is then ionized only by sequential absorption of photons by valence electrons as far as it is energetically allowed. Sequential photoionization in this regime was observed in noble gases [6][7][8][9][10] and the experimental results were compared with theoretical predictions [11][12][13][14]. For these processes, in addition to the ion yield and photoelectron spectra, also the angular distribution and even angular correlation [6,12,15,16] of two emitted electrons were studied both experimentally and theoretically. The important role of autoionizing resonances was also investigated [10,17,18].
To the best of our knowledge, no studies investigating multiple ionization beyond triple charged ions in the (ii) regime has been reported so far, except a general theoretical formulation for the photoelectron angular distributions in [19].
The current study belongs to the class (ii). Our main purpose is to analyse theoretically and make predictions for sequential ionization of Kr at photon energies in the interval 50-80 eV, i.e., below the excitation threshold of the 3d-hole (91.2 eV [20]). In this energy interval the 3d-hole is not produced and the Auger decay is excluded. In the next section we describe the process and outline a theoretical approach for modelling the interaction based on the solution of a system of rate equations. In Section 3, we present the results focusing on the time eVolution of the atomic Kr target under the FEL pulse and on the resulting electron spectra. In Section 4 a method for calculating the photoionization cross sections required for the rate equations is described. Examples of the ionization cross sections between electronic multiplet of different charge states of Kr are presented. The last section contains our conclusions.

Process Description
Since the Auger decay is energetically forbidden, the main processes, which we consider are photoionization from either the 4s or the 4p energy levels with emission of a photoelectron e ph or radiative transition from the 4p to the 4s level with emission of a fluorescence photon γ f l Sequential ionization of the valence shells of Kr and its ions is schematically shown in Figure 1, where ionization paths between the ionic configurations from neutral Kr to the triply charged ion Kr 3+ are indicated. "Horizontal" radiation transitions occur between energy levels of a same ion. In fact each indicated level includes all possible multiplet states of the configuration, which were all taken into account in the calculations. For photon energies equal or lower than 50.85 eV, ionization from the Kr 3+ 4s k 4p m electron configurations is energetically forbidden. For energies between 50.85 eV and 80 eV ionization channels of Kr 3+ 4s k 4p m are allowed, but we neglect them (see below). Thus, we consider the sequential ionization up to triple, three-photon ionization and the scheme in Figure 1 is restricted to transitions up to Kr 3+ . We do not consider the fine-structure levels of the multiplet states and consider summations over the fine-structure levels. This approach implies that the spectral width of the FEL pulse and resolution of the electron detector are comparable with the fine-structure splitting of the ions: 4s 2 4p 5 2 P o (0.65 eV), 4s 2 4p 4 3 P (0.66 eV), 4s 1 4p 5 3 P o (0.69 eV), 4s 2 4p 4 2 D (0.20 eV), 4s 2 4p 3 2 P o (0.39 eV), 4s 1 4p 4 4 P (0.66 eV), 4s 1 4p 4 2 D (0.11 eV), 4s 1 4p 4 2 P (0.34 eV). More details on the transitions between the multiplet states in the first three ionization steps from Kr to Kr 3+ are presented in Table 1. Scheme of transitions in sequential multiphoton ionization of Kr. Solid red arrows-photoionization, dashed blue arrows-radiative transitions. Only configurations are shown without multiplet splitting. The horizontal dash-dotted line indicates the limit for the sequential three-photon triple ionization (see text). Table 1. List of transitions in sequential three-photon ionization of Kr. Columns and lines correspond to initial and final states of the transitions within the LS-coupling scheme, respectively. Capital letters denote photoelectron lines for further convenience. Numbers after the letters present experimental ionization thresholds [21] for the corresponding transitions in eV, averaged over a multiplet. Transitions not marked by a capital letter are weak and their contribution to the photoelectron spectra is negligible, although they are included in Equations (3).  To follow the dynamics of the state populations, we apply a method of solving rate equations extensively used in the description of sequential ionization of atoms by X-ray FEL pulses (for example [22][23][24][25][26][27][28][29]). For femtosecond pulses the radiative 4p → 4s transitions can be neglected and the temporal dynamics is dominated by photoionization. The rate equations for the level populations then take the form (we use atomic units until otherwise indicated) where N i (t) is the population of level i, σ i→j is the photoionization cross section from level i of Kr n+ to level j of Kr (n+1)+ , and j(t) is the intensity of the incident radiation, which varies with time according to the pulse shape. M is the number of states, which are accounted for in treating the temporal dynamics of the sequential ionization. In our case M = 17 according to Table 1. In the set (3) we do not include shake-up and one-photon double ionization channels [30], but account for the shake-up, as well as autoionizing resonances in the calculation of the photoionization cross sections σ i→j . Details of the cross section calculations are presented in Section 4. The discrete levels can influence the process through two-photon resonance single ionization [31,32]. Photon energies above 50 eV, are above the 4s and 4p ionization thresholds in Kr and Kr + and therefore the two-photon resonance ionization via their discrete states is not possible. However, for energies around 50 eV, the channel of one-photon ionization of the 4s electron from Kr 2+ is closed (see Table 1, lines [13][14][15][16][17]. Therefore, the two-photon resonance ionization might occur in the latter case. Nevertheless, we neglect this process, because it can proceed for Kr 2+ at photon energies around 50 eV only via high Rydberg states with small excitation and ionization probability. For the photon energies higher than ∼58 eV all the 4s ionization channels for Kr 2+ open and the two photon resonance ionization channels disappear. We assume a temporal Gaussian distribution of the incident photon flux density (number of photons per surface per time): where t p is related to the full width at half maximum (FWHM) of the pulse, FWHM = 2 √ ln 2 t p . The photon flux is related to the fluence F, i.e., to the integral number of photons per 1 Å 2 in the entire pulse, as Equation (5) is obtained by considering the integral of (4) over time and transforming from atomic units.
For a fixed fluence, Equations (3) are invariant under changes of the time scale, t → at, where a is a constant. This scaling feature breaks down when additional terms not proportional to j(t), such as fluorescence and Auger decays, are added to the right side of (3). Thus, the calculations for a fixed fluence can be performed for a generic pulse duration and then scaled. Our case differs in this respect substantially from previous calculations, where the competing Auger decay had to be included and each pulse duration had to be calculated individually. In this paper we fix the pulse duration to t p = 18 fs (FWHM = 30 fs), which is comparable to the typical pulse duration obtained at the seeded FEL FERMI [33,34].
In order to predict the electron spectrum generated by the pulse, we calculate the probability P ij (F) (which depends on the fluence F) that an ion (atom) in a state i is ionized into the ion in a state j over the entire pulse and build up the function, where ε is the kinetic energy of the photoelectron at the photon energy ω, I ij is the threshold of ionization of the state i to the state j (binding energy) and Γ represents the resolution of the electron detector. We assumed Γ = 0.42 eV (corresponding to the resolution FWHM = 0.7 eV), in order to leave the fine structure of levels unresolved, as explained above.

Results and Discussion
Our main results are presented in Figures 2 and 3 for the population of the different ionic states and Figure 4 for the photoelectron spectra.

Time Evolution of Population
In Figure 2, we present results obtained with 50 eV (left column) and 80 eV (right column) photon energies and fluences F = 100 ph/Å 2 (solid lines) and F = 400 ph/Å 2 (dash-dotted lines). We summed up over the populations of different terms within one configuration.
The population of the ionic states Kr + , Kr 2+ and Kr 3+ is presented as a function of time in the first, second and third rows, respectively. Different colors correspond to the configurations without 4s-hole (red), with one 4s-hole (blue) and double 4s-hole (green). With the value of t p = 18 fs the above fluences correspond to intensities between 10 15 and 10 16 W/cm 2 . As explained above, the curves in Figure 2 remain the same for fixed values of F, after appropriate scaling of the pulse duration and of the timescale. In particular, the final populations of the levels at the end of the pulses are invariant for changes of the pulse duration (for a fixed fluence).
For the photon energy of 50 eV, ionization of the valence 4p shell dominates; the configurations Kr n+ 4s 2 4p −n (red curves in Figure 2a,c,e) are quickly populated, reaching maximum of the population at slightly different times, depending on the ionic charge and the fluence. As expected, the population maxima are reached at later instants for increasing ionic charge states and decreasing pulse fluences. At the end of the pulse, the populations become constants, because the Auger decay of the holes is energetically forbidden and the fluorescence occur on much longer time scale. The sum of all values presented in Figure 2a,c,e, as well as in Figure 2b,d,f corresponding to the same conditions (time, fluence, photon energy) equals unity.
For the fluence of F = 400 ph/Å 2 , the majority of the krypton atoms, ∼90.5%, are found after the pulse in the triply charged ionic state in its ground configuration Kr 3+ 4s 2 4p 3 (Figure 2e). Configurations of ions with one 4s-hole are populated on the level of ∼5.5%, while the amount of ions Kr 2+ and Kr 3+ with double 4s-holes is negligible. This result is explained by the very small 4s ionization cross section in the region of the Cooper minimum around 45 eV (see Section 4). The concentration of neutral atoms at the end of the pulse is also negligible (Figure 2a). The number of singly and doubly charged ions first increases with time, but then it drops down (Figure 2a,c), because the fluence is high enough to further ionize them to Kr 3+ within the pulse.
For the smaller fluence of F = 100 ph/Å 2 , already 17% of atoms are left as neutrals (Figure 2a For the photon energy of 80 eV, the final concentration of the neutral atoms is much higher (compare Figure 2a,b) than for the photon energy of 50 eV, because the 4p ionization cross section rapidly decreases with the photon energy in interval from 50 eV to 80 eV. At the same time, the production of ions with the 4s-vacancy is much more efficient, because the 4s ionization cross section increases in this energy interval, which is just above the corresponding Cooper minimum. At F = 400 ph/Å 2 the number of the Kr ions after the pulse in each charge state with a 4s-hole exceeds the number of the ions with only 4p ionized electrons. Furthermore, the number of Kr ions with the double 4s-hole reaches 4.5% of all the target atoms.
Finally, in Figure 3 we present the overall ionic yields at fluences F = 40 ph/Å 2 , F = 100 ph/Å 2 and F = 400 ph/Å 2 and photon energies ω = 50 eV and ω = 80 eV. The population of the different ionic states strongly depends on the photon energy and fluence. Figure 3a,b,d show a typical situation for multiple ionization in a regime far from saturation, when the ions with higher charge have smaller yields. Although ionization of Kr 3+ is energetically possible for 80 eV photons, Figure 3b,d show that at the fluences of F = 40 ph/Å 2 and F = 100 ph/Å 2 the contribution of higher charge states is negligible. The opposite behavior is shown in Figure 3e, in which nearly all ions are in highest allowed charged state, because a further ionization step is energetically forbidden for the photon energy of 50 eV. In this case the intensity of the radiation and the cross section of the 4p-ionization are high enough to promote three 4p electrons into the continuum during the pulse. More uniform distribution in Figure 3c,f show an intermediate regime, caused by an interplay between energy dependence of the ionization cross sections and fluence. Note that the last column in Figure 3f actually presents the sum yield (∼25%) of ions with charges three and higher. Although we cut the treatment at Kr 3+ , the photoelectron spectra presented below, are not influenced by this fact in the considered interval of the electron energies.

Photoelectron Spectra
The photoelectron spectrum contains much more information on the pathways of the sequential ionization than the ion yield, because it gives information on the relative population of the intermediate states of the process (see Figure 1 and Table 1). This information becomes more detailed by improving the energy resolution of the electron detector and decreasing the pulse spectral bandwidth.
The generated photoelectron spectra for three photon energies, 50 eV, 65 eV, and 80 eV are displayed in Figure 4a-c, respectively.
We considered the modifications of the spectra in a broad range of photon fluences. Figure 4a-c show, as examples, spectra for the two fluences, F = 100 ph/Å 2 and F = 400 ph/Å 2 . As follows from Table 1, lines A and E originate from single ionization of Kr and are produced by absorption of one photon. Lines B, C, D, and F are due to one-photon ionization of Kr + and, therefore, in the sequential two-photon double ionization of Kr. Lines G, H, J, L-R are produced in the sequential three-photon triple ionization of Kr. Lines I and K represent an overlap between a few lines from the sequential twoand three-photon ionization. In Figure 4a-c the lines are concentrated in certain groups: three groups for the photon energy of 50 eV and four groups for the photon energies 65 eV and 80 eV. The three groups at ω = 50 eV (Figure 4a) correspond to the line from the 4p-shell ionization of neutral Kr (electron energy 36 eV), lines from ionization of mostly Kr + 4s 2 4p 5 (electron energies 21-26 eV) and from ionization of mostly Kr 2+ 4s 2 4p 4 (electron energies 8-14 eV). The fourth group of lines appears at the photon energies 65 eV and 80 eV (Figure 4b,c), when ionization of the 4s-electron from Kr 2+ 4s 2 4p 4 and Kr 2+ 4s 1 4p 5 is opened. Note that some of the lines in different panels of Figure 4 are not indicated because of negligible contributions to the spectra.
Comparison of Figure 4a,b shows that the intensity of the photoelectron lines drops down for increasing ionic charges at the photon energy of 65 eV faster than at 50 eV, for a fixed fluence. This is caused by the decrease of the 4p-subshell ionization cross-section with increasing photon energy in this range (see Section 4). For higher photon energies the role of the multiple ionization increases (Figure 4c), due to the increase of the ionization cross-section of the 4s-subshell after the corresponding Cooper minimum (see Section 4, Figure 5b). The change of the relative contribution of 4s-and 4p-ionization channels leads to substantial modifications of the photoelectron spectrum. For example, lines E and I with large contribution from the 4s-ionization, are not observed at ω = 50 eV, but dominate in their group at ω = 80 eV. This is opposite to neon, where ionization from the subvalence 2s shell modifies noticeably the photoelectron spectrum at all photon energies within the (ii) regime [10,15]. We expect that the interference Cooper minimum in the 3s-ionization of Ar around ω = 40 eV [35,36], leads to modifications of the spectra as function of the photon energy similar to the present Kr case. Figure 6 shows the fluence dependence of the intensities of some selected spectral lines. The curves clearly indicate the one-, two-and three-photon origin of the spectral features A and EF, I and BCD, and G, respectively. The saturation starts to show up at fluences above 100 ph/Å 2 , progressing from the first (one-photon) to the third (three-photon) steps of the sequential ionization process.  [38] (dash-dotted curve), configuration interaction (CI) approximation from [39] (short-dashed curve), relativistic random phase approximation (RRPA) from [40] (dash-dot-dot curve). Experimental data from [41] (squares), [42] (crosses), and [43] (open circles).

Method of the Cross Section Calculations
Photoionization cross sections of multiplet states 4s k 4p m 2S+1 L for variously charged Kr ions were calculated by the B-spline R-matrix approach [44], which fully takes into account non-orthogonality of the electron functions before and after the ionization. For all the three steps, the basis wave functions of each of the initial nine states listed in Table 1 were obtained in independent Hartee-Fock calculations in the LS-coupling approximation with variation of 4s and 4p orbitals and the 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 Ar-like core frozen from the self-consistent calculations for the ground state of Kr. The final ionic states (numbers 2-17 in Table 1) were similarly calculated in independent Hartree-Fock calculations with variable 4s and 4p electron orbitals. For each step, the corresponding set of final ionic states were included in the R-matrix expansion, giving the total angular momentum L tot and spin S tot of the system (final ion + electron) satisfying the dipole selection rules. Overall 19 R-matrix runs were needed for the reactions of the type where n = 0, 1, 2 and are the orbital angular momentum of the photoelectron. Each of the reactions (7) describes a few ionization channels with one of the fixed nine initial states from Table 1 and fixed values of L tot S tot , but all possible from 17 final Kr (n+1)+ states and . The ionization cross section to a particular final ionic state is obtained by summation over all corresponding channels with different sets of , L tot , S tot . Figure 5 presents photoionization cross sections of the 4p (Figure 5a) and 4s (Figure 5b) electrons from Kr 4s 2 4p 6 1 S, respectively, i.e., for the first step of the sequential ionization. These cross sections can be compared with experiment and other calculations [35,[37][38][39][40][41][42][43]45,46]. Here and below we use results in the velocity gauge which better agrees with experiment for neutral Kr.
Ionization cross sections for the second step into different multiplet states of the doubly charged ion are shown in Figure 7: ionization from Kr + 4s 2 4p 5 2 P o and from Kr + 4s4p 6 2 S are displayed in Figure 7a,b, respectively. Figure 8 is related to the third step and shows ionization cross sections from six multiplet states of Kr 2+ to different multiplet states of Kr 3+ . The transitions correspond to those indicated in Table 1.  Table 1 are presented. The blue lines in (a) correspond to ionization of the 4p electron into different terms of the Kr 2+ 4s 2 4p 4 configuration. The green dash-dotted line in (b) corresponds to producing of the double-hole state Kr 2+ 4s 0 4p 6 . Identical red lines in the both panels show transitions to the same final states from Kr + 4s 2 4p 5 (panel (a)) and Kr + 4s4p 6 (panel (b)), respectively. Ionization cross section of a subvalence 4s shell in Kr shows a deep interference Cooper minimum around the photon energy of 45 eV [35] (Figure 5b). The position and the depth of the minimum is very sensitive to the model. In our study the particular value of the 4s ionization cross section in the minimum is not so important except the fact that it is at least two orders of magnitude less than the 4p ionization cross section. The Cooper minimum occurs in the 4s ionization from the Kr + 4s 1 4p 6 state (Figure 7b, green line), but looks moving under the threshold in ionization from the Kr + 4s 2 4p 5 state (Figure 7a, red lines). In the ionization of the doubly charged ion, the appearance of the Cooper minimum strongly depends on the initial and final ion configurations (Figure 8).
There are series of Rydberg autoionizing states with configurations 4s n 4p m ( 2S f +1 L f )np in the energy region under consideration. The series appear in the ionization of 4p shell at photon energy below the 4s ionization threshold for all terms and configurations with only one exception, 4s 0 4p m . Besides, the Rydberg series appear in the ionization of 4s shell between the split multiplet thresholds, for example between 4s4p 5 3 P o and 4s4p 5 1 P o . For the sake of clarity, in Figures 7 and 8, we show only sample resonance structures on the upper curves, obtained in our R-matrix calculations, cutting other curves in the near-threshold region in other cases. Although our calculations automatically accounts for the autoionizing resonances, a careful analysis showed that with the currently available spectral width of intense XUV pulses and the electron detector resolution, it is hardly possible to observe resonance effects in integrated observables such as ionic or electron yields. In order to reveal these resonances, more detailed measurements like photoelectron angular distribution [10,47] or angular correlation functions [18] are needed.

Conclusions
Time eVolution and photoelectron spectra in sequential photoionization of atomic krypton by intense femtosecond XUV pulses at the photon energies from 50 eV to 80 eV are studied theoretically. In this regime the Auger decay channel is closed and the ionization proceeds through photoemission of 4s and 4p electrons. Within our model, the results are applicable to pulses with arbitrary duration within the femtosecond domain, due to the time scaling of the rate equations. Lines from single, double and triple ionization processes are predicted in the photoelectron spectra. The intensity of the lines is proportional to the corresponding power of the fluence up to the saturation fluence of about 10 2 ph/Å 2 . The Cooper minimum in the 4s ionization cross section influences the time eVolution of the target and eVolution of the photoelectron spectrum with the photon energy. Nevertheless, the population of states with the 4s-hole after the pulse can reach tens percents. The present study is a natural step before turning to higher photon energies, when the 3d excitation threshold is opened and the Auger decay makes the dynamics more complex.
Author Contributions: The authors have contributed to the manuscript equally. All authors took part in discussions and preparation of the manuscript. All authors have read and agreed to the published version of the manuscript.