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Sequential two-photon double ionization of the 4d shell in xenon

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Phys. B: At. Mol. Opt. Phys. 44 175602 (http://iopscience.iop.org/0953-4075/44/17/175602) View the table of contents for this issue, or go to the journal homepage for more

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JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

doi:10.1088/0953-4075/44/17/175602

J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 175602 (10pp)

Sequential two-photon double ionization of the 4d shell in xenon S Fritzsche1,2 , A N Grum-Grzhimailo3 , E V Gryzlova3 and N M Kabachnik3,4,5 1 2 3 4

Department of Physics, University of Oulu, PO Box 3000, Fin-90014, Finland GSI Helmholtzzentrum f¨ur Schwerionenforschung, D-64291 Darmstadt, Germany Institute of Nuclear Physics, Moscow State University, Moscow 119991, Russia European XFEL GmbH, Albert-Einstein-Ring 19, D-22607 Hamburg, Germany

E-mail: [email protected]

Received 3 April 2011, in final form 22 June 2011 Published 12 August 2011 Online at stacks.iop.org/JPhysB/44/175602 Abstract The sequential two-photon double ionization of 4d electrons of xenon in intense free-electron laser (FEL) radiation is theoretically investigated for photon energies of 70–200 eV. At these energies, which are available at the free-electron laser FLASH, the 4d photoionization dominates over the ionization of the valence shells, showing a giant resonance. The 4d vacancy produced by the first photon strongly couples to the (Auger) autoionization continuum with a lifetime of only ∼ 6 fs and thus the Auger decay competes with the second photoionization in a typical FEL pulse. The photoelectron angular distributions and angular correlation between emitted electrons are investigated as associated with the peculiar 4d character of the photoionization and Auger decay. (Some figures in this article are in colour only in the electronic version)

Among the noble gases, xenon is known for several peculiarities in spectroscopy and electron dynamics. As a mid-Z element, xenon has a rather large fine-structure splitting due to the strong spin–orbit interaction and is also strongly affected by electron correlations. For photon energies larger than about 70 eV, the photoionization of xenon is dominated by the ionization of the 4d shell with a giant resonance in the photoionization cross section due to virtual 4d→4f excitations [16]. This makes xenon especially interesting for studying the multi-electron dynamics in photon fields of increasing intensity, i.e. from a weak-field regime towards a strong-field regime. For xenon, moreover, an unusual strong multiple ionization has been observed in intense FEL radiation which led to charged states of Xe ions up to Xe21+ [17] and which is presumably connected with the existence of the giant resonance [18] (for alternative explanation and discussion see [19]). Therefore, the investigation of the sequential TPDI of xenon can shed further light on the role that giant resonances play in the multiple ionization of atoms in strong light fields. Furthermore, the 4d vacancy, created at the first step, quickly decays by emission of an Auger electron. The lifetime of about 6 fs [20] is comparable with the duration of the FEL pulse and, thus, the decay can occur during the FEL pulse.

1. Introduction The sequential two-photon double ionization (TPDI) is one of the simplest nonlinear processes in an extreme ultraviolet (XUV) region. Its experimental investigation became possible with the advent of free-electron lasers (FEL). In contrast to the direct (non-sequential) TPDI, where two electrons are emitted simultaneously and therefore share the total energy release, in the sequential TPDI, the absorption of the first photon leads to some well-defined intermediate state(s), which is subsequently ionized by the second photon from the same pulse. In this case, the energies of the two emitted electrons are fixed by the energies of the initial, intermediate and final states of the two-step photoionization process. The sequential TPDI is the dominant double-ionization process if the photon energy is larger than the ionization threshold of a singly ionized ion [1]. For the noble gas atoms heavier than He, it has been studied recently both in experiment [2–7] and in theory [8–15]. Until now, however, only the TPDI of the valence np subshells of the noble gases has been investigated. In this paper, we consider for the first time the TPDI of an inner d subshell, namely of the 4d subshell of xenon atoms. 5

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On the other hand, it was proved in a recent experiment on time-resolved ion spectrometry on xenon with FEL pulses of moderate intensity [21] that the 4d shell can be double ionized by two-photon absorption before a single vacancy is refilled by Auger decay. Such a competition between photoionization and Auger processes is a new feature in the sequential TPDI which does not occur in the photoionization of the np valence shell of the noble gas atoms. The existence of the two channels for the double ionization of xenon through (i) the sequential two-photon absorption and through (ii) the Auger decay of the singly ionized inner shell, may serve as a model system for more complicated sequences of photoabsorption and Auger processes which apparently determine the charged state distribution of the ions in multiphoton ionization in the hard x-ray regime [22]. Similar to our previous studies for light atoms [11–13, 15], we are interested not only in the cross sections but also in the angular distributions and angular correlations of emitted electrons. To calculate the required differential cross sections for the first- and second-step photoionization, we use the formalism based on the density matrix and statistical tensor description of the atomic (ionic) states [23], developed in our papers [11, 13, 15]. In the following, we consider the sequential TPDI of the 4d subshells of Xe in the photon energy range from 70 to 200 eV. In section 2, we give a short description of our theoretical approach and present the main expressions for the angular distributions of emitted electrons and their angular correlations. The Auger decay is included phenomenologically through the time evolution of the statistical tensors of the intermediate ionic states. The results of our calculations are presented and discussed in section 3. Finally, some conclusions are formulated in section 4. Atomic units are used throughout the paper unless otherwise indicated.

Figure 1. Scheme of sequential TPDI of the 4d subshell in Xe including the Auger decay of the 4d vacancies. The numbers near the energy levels show their total angular momentum and energy (in eV) relative to the ground state of the Xe atom.

the two outgoing electrons from either the photoionization or autoionization of the atom. A radiative decay of the ions is also possible but much less likely for vacancies in the subvalence shells and will not be considered here. In figure 1, transitions (1)–(3) are schematically shown for a particular case of 4d photoionization of Xe. Only atomic (ionic) levels relevant to the discussion below are indicated. The first-step photoionization of the Xe 4d subshell creates an intermediate doublet 2 D of Xe+ which can be further ionized by the second photon to the states 4d−2 (for example 3 F ) of Xe2+ or can Auger decay to some of the low-lying levels of the Xe2+ ion. In the stepwise description of the sequential TPDI, the two photoionization processes (1) and (2) are considered as independent and connected only by the properties of the intermediate ionic state. In particular, interaction of the two emitted electrons is ignored which is a good approximation when the energies of electrons are notably different as is the case in all examples below. Processes (2) and (3) compete with each other, and their relative contribution to the electron spectrum depends on the photoionization cross section, the Auger lifetime and on the duration of the pulse. We note that processes (1)–(3) determine only part of the electron spectrum. Among other contributions there are, for example, the Auger decay of the doubly ionized atom which is produced in process (2) and the photoionization of the doubly ionized atom after the Auger decay (3). Thus, the spectrum is expected to be rather complex (see, for example, recent measurements for Ne [24]). However, the electrons produced in processes (1)–(3) can be sorted out by their energies and therefore studied separately. In the following, we shall be mainly concerned with the parameters that characterize the photoelectron emission (1) and (2), especially, the cross sections, angular distribution parameters, etc. A detailed description of the used formalism of the density matrix and statistical tensors in application to TPDI can be found elsewhere [11, 15]. Here, we describe only

2. Theory and computations 2.1. Theoretical description of the sequential TPDI The sequential TPDI of two inner-shell electrons of an atom may be described as a stepwise process. First, the target atom is ionized by a first photon and an inner-shell electron is emitted: γ1 + A(α0 J0 ) → A+ (αi Ji ) + e1 (l1 , j1 ).

(1)

The produced ionic state A+ (αi Ji ) can subsequently be further ionized either by a second photon from the same pulse γ2 + A+ (αi Ji ) → A2+ (αf Jf ) + e2 (l2 , j2 ) ,

(2)

or it decays under the emission of an Auger electron A+ (αi Ji ) → A2+ (αf Jf ) + eA (lA , jA ).

(3)

In the schematic notation of the two steps above, the J0 , Ji , Jf and Jf are the total angular momenta of the corresponding atomic or ionic states, α0 , αi , αf and αf refer to all other quantum numbers necessary for their unique specification and the lk , jk (k = 1, 2, A) characterize the orbital and total angular momenta of the partial waves of 2

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the main approximations used in the calculations and present the formulas necessary for the following discussion. For an initially unpolarized atom A(α0 J0 ) in its ground state |0 and a linearly polarized photon beam, the partial differential photoionization cross section for transition to the ionic state |i ≡ |αi Ji  due to single photoabsorption is given by the standard dipole expression  σ (0−i)  dσ (0−i,unpol) (θ ) = 1 + β2 P2 (cos θ ) . d 4π

parameter for photoelectrons from an ion aligned after first photoionization can be expressed as [11] (i−f ,unpol)

(i−f ,TPDI)

β2

4π 2 αω B(0, 0, 0), 2J0 + 1

β2

(i−f )

+ A20 (Ji ) a2 (i−f )

1 + A20 (Ji ) a0

(9)

in terms of the anisotropy parameter for an initially unpolarized (i−f ,unpol) and the alignment of the ion A20 (Ji ) times ion β2 some dynamical factors a0 and a2 describing the second-step (i−f ,TPDI) can similarly be photoionization. The parameter β4 written as

(4)

In this expression, P2 (cos θ ) is a second-order Legendre polynomial, θ is the polar angle of the electron emission with respect to the polarization direction, σ (0−i) is the corresponding partial photoionization cross section and (0−i,unpol) β2 denotes the asymmetry parameter that characterizes the photoelectron emission from an unpolarized atom. The differential cross section may also be written in terms of some dynamical B coefficients, B(k, ka , kb ), introduced in [11] which depend on the photoionization amplitudes. Using the B coefficients, the partial cross section is given by σ (0−i) =

=

(i−f )

(i−f ,TPDI)

β4

=

A20 (Ji ) a4

(i−f )

1 + A20 (Ji ) a0

.

(10)

Explicit expressions for the factors a0 , a2 and a4 in terms of the amplitudes of the second photoionization transition are given in [11]. In contrast to the second-step photoionization of the ion, the angular distribution of the emitted Auger electrons is given by an expression similar to (4) [23, 25]: WA (αf Jf )   (i−f  ,A) 1 + β2 P2 (cos θA ) (11) 4π where WA (αf Jf ) is the partial Auger decay rate to the final WA (θA ) =

(5)

where α is the fine-structure constant and ω is the photon frequency. In these notations, the asymmetry parameter becomes √ − 2 B(2, 0, 2) (0−i,unpol) β2 . (6) = B(0, 0, 0)

(i−f  ,A)

state αf Jf and β2 = α2 A20 (Ji ), with α2 being the anisotropy parameter determined by the Auger decay matrix elements [25]. If both photoelectrons are detected in coincidence, the angular correlation function can be written as [15]  CKk1 k2 W TPDI (θ1 , φ1 ; θ2 , φ2 ) = (π αω)2 (2J0 + 1)−1

Note that expression (4) is valid for a single-photon absorption. If this transition is the first step of a TPDI process, the angular distribution of the first electron differs from (4) and will be discussed below. After the ionization, the photoion is usually aligned along the polarization vector of the incoming light. For a well-isolated ionic state |i, the degree of the alignment is given by the reduced statistical tensor (alignment parameter) A20 (Ji ) = ρ20 (αi Ji )/ρ00 (αi Ji ), where the statistical tensors ρk0 (αi Ji ) are determined by the photoionization amplitudes [23]. The alignment parameter can also be expressed in terms of B coefficients: √ − 2 B(0, 2, 2) . (7) A20 (Ji ) = B(0, 0, 0)

Kk1 k2

×4π {Yk1 (θ1 , φ1 ) ⊗ Yk2 (θ2 , φ2 )}K0 ,

(12)

where {Yk1 (ϑ1 , ϕ1 ) ⊗ Yk2 (ϑ2 , ϕ2 )}K0 are bipolar spherical harmonics [26]. The coefficients CKk1 k2 may be presented as    Z(k1 , k2 , kγ1 , kγ2 , ki ) kγ1 0, kγ2 0 | K0 CKk1 k2 = kγ × 1 k2

k i k γ1 k γ2

k γ2 k1

K γ γ ρ ρ , k i k γ1 0 k γ2 0

(13)

where we use the standard notations for the 6j symbol, γ (kγ1 0, kγ2 0 | K0) is a Clebsch–Gordan coefficient, ρkγ 0 (i = i 1, 2) are the statistical tensors of the photons and  Z(k1 , k2 , kγ1 , kγ2 , ki ) = h(αx Jx , αx Jx )kˆ γ1 kˆ 1−1

If the pulse of the radiation is intense enough, the produced aligned ion can be further ionized by a photon from  the same pulse. This second-step transition |i → |f  ≡ αf Jf gives rise to the angular distribution of the second photoelectron:

αx αx Jx Jx

¯ i , k2 , kγ2 ). ×B(k1 , ki , kγ1 )B(k (14) √   2a + 1 and h(αx Jx , αx Jx ) denotes a Here, aˆ ≡ depolarization factor which will be discussed below. ¯ i , k 2 , k γ2 ) The dynamical coefficients B(k1 , ki , kγ1 ) and B(k characterize the first- and second-step photoionization processes, respectively, and can be expressed in terms of the corresponding dipole matrix elements. Detailed expressions for these dynamical coefficients can be found in [11]. If the general expression for the two-electron correlation function (12) is integrated over the emission angles of the

W TPDI (αf Jf ) W TPDI (θ2 ) = 0 4π   (i−f, TPDI) (i−f, TPDI) P2 (cos θ2 ) + β4 P4 (cos θ2 ) . (8) × 1 + β2 In this case, the term proportional to P4 (cos θ2 ) appears in the angular distribution because two dipole photons are absorbed by the atom, providing each one unit of angular momentum to the system. Let us note, moreover, that the anisotropy 3

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first electron, expression (8) for the angular distribution of the second-step electron in non-coincidence measurements is readily obtained. On the other hand, integration of expression (12) over the emission angles of the second electron gives an expression for the angular distribution of the first-step electron: W0  1 + β2(0−i,TPDI) P2 (cos θ1 ) W TPDI (θ1 ) = 4π  + β4(0−i,TPDI) P4 (cos θ1 ) , (15)

contribution of the intermediate levels. The non-diagonal matrix elements characterize the coherent contribution due to the interference of the intermediate states. From the analysis in [15], it follows that the interference can affect the angular distribution in particular for two different physical scenarios. First, when the involved levels overlap, i.e. J J   ωJ J  . Then the non-diagonal elements are considerable even for long incoherent pulses. Such a case is realized, for example, in the excitation and decay of several core-excited resonances [27–29]. A second scenario, in which the non-diagonal elements are important, is when the exciting pulse is coherent and so short that its bandwidth is larger than the energy difference ωJ J  .

which is analogous to the second-step electron angular distribution. While this looks perhaps surprising at the first glance, it follows from the fact that despite the integration over the angular coordinates, the second electron is still supposed to be emitted, independent of whether its direction and the final state of the photoion remain unobserved. If neither the energies of the second electron nor of the photoion are recorded, the average has to be taken over all final states of the TPDI process, and this will typically reduce the magnitude of the β4(0−i,TPDI) parameter, but will not turn it to zero. Formally, the appearance of the fourth-order Legendre polynomial is associated with the absorption of two dipole photons in the TPDI process (see the discussion in [11]). Concluding this section we note that only a single asymmetry parameter β2 occurs in the photoemission and Auger emission for the absorption of a single photon (cf equations (4) and (10)), while the angular distribution of the first and second photoelectrons (equations (15) and (8)) is determined by the two parameters β2 and β4 in TPDI . This type of angular distribution with second- and fourth-order Legendre polynomials is characteristic for the electron emission in two photon processes and is independent of whether the other electron is detected in some particular experiment.

2.3. Computational procedure All computations have been made by following similar lines as in our previous works [11, 12] for neon, argon and krypton. In most cases, both multiconfiguration Hartree–Fock (MCHF) and Dirac–Fock (MCDF) wavefunction expansions have been utilized to describe the initial, intermediate and final states of the photoion. All calculations were performed in the velocity gauge, which gives better results for the first ionization step when theory and experiment are compared for the 4d cross section and angular distribution of photoelectrons. In the MCHF computations, the atomic structure calculations for both discrete and continuum spectra were performed with the program package of Froese Fischer et al [30]. The ground Xe state was described by fully selfconsistent Hartree–Fock calculation by accounting for the 4d10 5s2 (5p6 + 5p4 5d2 ) 1 S configuration mixing with all intermediate terms. The 5p4 5d2 configuration with the 5d pseudo-orbitals, being optimized on the 5p6 state, accounts for the majority of the electron correlations in the initial state [31]. The ionic state after the first-step ionization is similarly described by the mixture of the 4d9 5s2 (5p6 + 5p4 5d2 ) 2 D configurations with the orbitals frozen from the groundstate calculation. For the final state of the first step, the frozen-core Hartree–Fock [Xe+ (4d−1 ) + Ep/f] 1 P continuum wavefunctions were used. Experimental ionization thresholds were taken for the calculations. The second-step ionization was calculated rather similar to the first step. In particular, the fully self-consistent configuration mixture 4d9 5s2 (5p6 + 5p4 5d2 ) 2 D was used to describe the initial state. The chosen final double-charged ionic state was obtained by the 4d8 5s2 (5p6 + 5p4 5d2 ) 3 F mixing with all orbitals frozen from the single-charged-ion calculations, except the 5d pseudo-orbital, which was re-optimized by taking into account all possible intermediate terms of the 5p4 5d2 configuration. The doublecharged ionic 3 F core was frozen to find the term-dependent Hartree–Fock continuum electron functions [Xe2+ (4d−2 ) 3 F + Ep/f] 2 P,2 D,2 F, which were finally used to obtain the secondstep dipole ionization amplitudes. The nonorthogonality of the 5d pseudo-orbitals in the initial and final states was taken into account. In MCDF computations, the relativistic analogue to the above MCHF reference configurations was used to represent the ground and the 4d hole states of xenon. For these

2.2. Depolarization factor for the 4d hole states The depolarization factor h(αx Jx , αx Jx ) is determined by the evolution of the intermediate states in the interval between the absorption of the first and second photons. Consequently, it depends on both the properties of the intermediate ionic states and those of the incoming photon pulse. Since in TPDI experiments one has no information about the particular moments in which the first and second absorptions occur in course of the pulse propagation, one should average over all possible intervals, taking into account the possible decay of the states. Thus,  +∞   dt1 I (t1 ) h(αJ, α J ) = N −∞  ∞ dt2 exp[(iωJ J  − J J  )(t2 − t1 )] I (t2 ), (16) × t1

where I (t) is the intensity of the photon pulse, J J  = (J + J  )/2 and ωJ J  = EJ − EJ  with J and EJ being the width and the energy of the intermediate state |J , respectively, while N is a normalization factor which is chosen to be equal to unity for pulses much shorter than the inverse-energy splitting of the intermediate states. A detailed analysis of the depolarization factor can be found in [15]. The value h(αJ, α  J  ) is a matrix whose diagonal matrix elements determine the incoherent 4

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Table 1. Excitation energies (in eV) of the 4d−1 and 4d−2 hole states with regard to the 1 S0 ground state of neutral xenon. Only those levels are shown which are relevant for the following discussion. Excitation energy, eV Atom/ion

Level

This work

Other calculations

Experiment

Neutral Xe Xe+

5p6 4d9 4d9 4d8 4d8 4d8

0 67.32 69.29 151.2 153.0 153.0

0 66.45a 68.47a 152.0c 153.4c 153.4c

0 67.55b 69.54b 151.5d 153.2d 153.2d

Xe2+

1

S0 D5/2 2 D3/2 3 F4 3 F3 3 F2 2

a

Kutzner et al [36], b King et al [37], c Hansen [38], d Svensson et al [39].

calculations, wavefunctions have been generated by means of the GRASP code [32] and utilized within the RATIP program [33] in order to evaluate the required dipole amplitudes for the 4d photoionization of xenon. Apart from 5p → 5d double excitations with regard to the 4dm 5s2 5p6 (m = 10, 9, 8) reference configuration for the initial, intermediate and final states of the 4d inner-shell photoionization, which are included in the above MCHF case, we also explored the role of (double) excitations into the 4f, 6s and 6p shells. While these virtual excitations helped to improve the ionization thresholds of the 4d−1 and 4d−2 hole states, only excitations into the 5d shell were taken into account eventually by means of the 4dm 5s2 (5p6 + 5p4 5d2 ) configurations for evaluating the dipole amplitudes. The accuracy achieved in the computations of the ionization thresholds is demonstrated by table 1. The calculated values deviate from the experimental ones by 0.2–0.3 eV. For an open d shell of the photoion, unfortunately, the size of the relativistic wavefunction expansions increases so rapidly [34] that no attempt could be made to incorporate further excitations into the construction of the wavefunctions, nor that the convergence of the cross sections and rates could be monitored in detail. To calculate the dipole amplitudes, approximate scattering states of the final system ‘photoion + emitted electron’ need to be constructed with well-defined total angular momenta and parities. In this construction, the continuum spinors were generated within a spherical but level-dependent potential to properly account for the final state of the photoion of the corresponding step in the photoionization; this so-called optimal-level scheme includes the exchange interaction of the emitted electron with the bound-state electron density and has been described in further detail in [35].

Figure 2. 4d partial photoionization cross section as a function of the photon energy. Computations within the MCHF model (dashed line) and MCDF model (solid line; summed over the 2 D3/2 and 2 D5/2 photolines) are compared with the experimental data from Becker et al [41] (dots) and K¨ammerling et al [42] (triangles).

(0−i,unpol)

Figure 3. Comparison of the β2 anisotropy parameters of the 5p6 1 S0 →4d9 2 D photoline as a function of the photon energy from MCHF (dashed line) and MCDF (solid line) calculations, averaged over the two 2 D3/2 and 2 D5/2 fine-structure levels, and with the experimental data by Southworth et al [43].

two (MCHF and MCDF) model computations are displayed together with the selected experimental results for the single photoionization cross section and asymmetry parameter (0−i,unpol) , respectively. β2 We note that the fine structure is neither resolved for the intermediate nor for the final states in typical experiments. Therefore, only the total (summed) cross section and averaged angular distribution parameters are measured and can be compared to the corresponding summed (averaged) theoretical values. On the other hand, the two spin–orbit components of the 4d−1 2 D term are well separated by 1.98 eV in Xe+ [20, 45] and do not overlap with each other. We may therefore consider

3. Results and discussions 3.1. Photoionization of the 4d subshell in Xe The ionization of the 4d subshell in the Xe atom has been studied in the past both experimentally and theoretically (see [40] and references therein). We use it as a testground in order to choose optimal approximations for the wavefunctions involved. The quality of our calculations can be seen in figures 2 and 3, where the results from the 5

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(i−f ,TPDI)

Figure 4. Alignment of the 2 D3/2 level as a function of the photon energy, calculated within MCHF (dashed line) and MCDF (solid line) approaches. The experimental data are from K¨ammerling et al [42] and Snell et al [44].

Figure 5. Angular distribution parameters β2(2) ≡ β2 and (i−f ,TPDI) averaged over the intermediate and final terms for β4(2) ≡ β4 the 4d 9 2 D3/2,5/2 →4d 8 3 F2,3,4 photoline in the TPDI of xenon as a function of the photon energy.

theoretically the generation and ionization of the intermediate states separately and sum the corresponding contributions. Figures 2 and 3 demonstrate good agreement between the two calculations and the experimental data. Our results are very close to those by Kutzner et al [36] (not shown). The good agreement with experiment in the region of the giant resonance shows that the chosen set of configurations describes well the important electron correlations responsible for the formation of this resonance. The giant resonance in the cross section is reflected in a characteristic strong variation of the anisotropy parameter (see figure 3) in this energy region. As discussed above, both the angular distribution of the TPDI photoelectrons in non-coincidence experiments and their angular correlation function in coincidence measurements depend on the alignment of the ion in the intermediate state. In figure 4, the calculated alignment parameter of the Xe(4d−1 ) ion is shown. Earlier this parameter was calculated by several authors [46–48], and their results are very close to ours. The experimental values of the alignment parameter as derived from the angular distributions of the Auger electrons [42, 44] are also displayed in figure 4. The agreement with the theory is satisfactory. As was shown in paper [47], the alignment parameter is rather sensitive to intershell correlations as well as to relaxation effects. Typically, however, the alignment of photoinduced states is most sensitive in the region of its maximum (modulus) which is close to the Cooper minimum, where measurements are hindered by the low cross section.

(not shown) also give a cross-section maximum in this energy region; however, a direct comparison with experiment is not possible since the initial state in the experiment [49] is Xe+ (5p−1 ), while we supposed a Xe+ (4d−1 ) hole state in the calculations. The influence of the giant resonance is clearly seen in the energy dependence of the asymmetry parameter of photoelectrons (see figure 5) which exhibit a characteristic strong variation in this energy region (compare with figure 3). In figure 5, we show the calculated asymmetry parameters which characterize the angular distribution of the second-step photoelectrons (8) for the transition 4d 9 2 D3/2,5/2 →4d 8 3 F2,3,4 . We have chosen this particular transition to the 3 F term of the Xe2+ ion because this term is expected to be strongly populated in TPDI of the 4d subshell and is energetically well separated from other terms of the 4d−2 hole-state configuration. It should be therefore easily identified in forthcoming experiments. To facilitate the comparison with measurements of typical resolution in present-day FEL experiments, the displayed parameters are averaged over the fine structure of the intermediate and final terms. Although the energy dependence of the calculated parameters is similar in both models, the curves are shifted with (i−f,TPDI) respect to each other. Note that the β4 parameter which characterizes the deviation from the standard dipole angular distribution is small in both models. A similar result was obtained earlier for the np subshells of the noble gases [11, 12].

3.2. Photoelectron angular distribution in the second-step photoionization

3.3. Modification of the first-step angular distribution due to a second photoionization

The second photon in sequential TPDI is absorbed by the ion Xe+ . The experimental data, which are available for the photoionization of Xe+ [49], also show a strong and broad maximum (giant resonance) at the photon energy of 85– 115 eV similar to that in the Xe atom. Our calculations

As was discussed in section 2, the angular distribution of the first photoelectron in TPDI differs from the standard form for the absorption of single dipole photons by the term containing the β4(0−i,TPDI) parameter and the fourth Legendre polynomial. In figure 6, as an example, we present 6

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(b)

(c)

Figure 8. Polar plot of the second-step photoelectron angular distribution if the first-step electron is observed under the angle θ1 = 0◦ , 45◦ and 90◦ (φ1 = 0). Here the z-axis (photon polarization direction) is vertical, the arrow shows the direction of the x-axis. The calculations have been performed within the MCHF approach for the TPDI transition 4d10 → 4d9 2 D→ 4d8 3 F averaged over the fine structure of the intermediate and final terms.

Figure 6. ‘Non-dipole’ angular distribution parameter β4(1) ≡ β4(0−i,TPDI) for the first step 5p6 1 S0 →4d 9 2 D3/2,5/2 photolines, if a second photoionization is supposed to occur to the term 3 F, as a function of the photon energy. MCHF calculations for the 4d 9 2 D3/2 (dash-dotted line) and 4d 9 2 D5/2 (short-dashed line) photolines are compared with the corresponding MCDF computations (long-dashed line and solid line, respectively).

more detailed information about the photoionization process. In figures 7 and 8, for example, we show the theoretically expected angular correlation and distribution functions for two realistic set-ups. The results are obtained within the MCHF approximation; similar results were also found in the MCDF approach which we do not present for brevity. In figure 7, the 3D plots of the angular correlation function (12) for TPDI of Xe are shown for the transition 4d10 → 4d9 Ji → 4d8 3 FJf as a function of the polar angles θ1 and θ2 of the two electrons at φ1 = φ2 = 0. The photon energy is chosen to be 100 eV which corresponds to the maximum in the photoionization cross section of the 4d shell (see figure 2). The corresponding photoelectron energies are E1 =∼ 31 eV and E2 =∼ 48 eV, and are quite different from each other so that post-collision interaction between the two electrons can be ignored. Different panels show the results for individual transitions for particular Ji and Jf

the calculated β4(0−i,TPDI) parameter for the TPDI transitions through 4d9 2 D3/2 and 4d9 2 D5/2 intermediate levels of Xe+ to the final 3 F term of Xe2+ (averaged over the fine structure of the term). Interestingly, the β4(0−i,TPDI) parameter in this case is not small and might be measured. However, in order to extract the β4(0−i,TPDI) parameter experimentally, the first-step photoelectron should be detected in coincidence either with the second-step photoelectron (possibly in 4π geometry) or with some proper fluorescence photon to assure that the TPDI transition leads indeed to the 3 F2,3,4 final states. 3.4. Angular correlations between two emitted electrons When two electrons are detected in coincidence, the angular correlations between them can be measured [7] which provide

Figure 7. The 3D plots of the angular correlation function for the TPDI partial transitions 4d10 → 4d9 2 DJi → 4d8 3 FJf in Xe as a function of θ1 and θ2 (φ1 = φ2 = 0). The photon energy is 100 eV. The calculations have been performed within an MCHF approach. 7

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fine structure components6 . The angular correlation patterns strongly depend on the particular intermediate and final states that are involved in the TPDI process. In a real experiment it is sometimes convenient to fix the emission angle of one of the photoelectrons and to measure the angular distribution of the second one. As an example figure 8 displays the angular distribution of the second electron when the first electron is detected at three different angles. When the first electron is detected along the photon polarization vector, the angular distribution of the second one must be axially symmetric, which is indeed the case for figure 8(a). When the angle of the first emission relative to the polarization direction increases, then the angular distribution becomes strongly axial asymmetric, see figure 8(b). As we have shown previously [15] that for p subshell photoionization, the axial symmetry preserves also when one of the electrons is emitted perpendicular to the polarization vector. This should not be the case for d subshell photoionization. Indeed, one can see in figure 8(c) that the angular distribution of the second electron is clearly asymmetric (compare to figure 6 of [15]).

1

h(J,J)

0.8 0.6 0.4 0.2 0 0

10

20 30 40 Pulse duration (fs)

50

Figure 9. The factor h(αJ, αJ ), which gives the ratio of probability of TPDI to the sum of probabilities for TPDI and the Auger decay, as a function of the Gaussian pulse duration (FWHM).

cross sections or the angular distributions and correlations in accordance with the above intuitive expectation. With increasing pulse duration, the factor h(αJ, αJ ) diminishes. For very long pulses, much longer than 1/ J , the averaged probability of TPDI becomes very small and tends to zero. In this case, the doubly charged ions will be produced predominantly by the Auger decay (3), and not by the second ionization (2). Thus the most interesting situation for studying the influence of the Auger decay on the TPDI is when τ J ∼ 1, where τ is the lifetime of the Auger state. Since the lifetime of the Xe 4d vacancy is about 6 fs, the TPDI with the production of the double 4d hole will effectively occur with the FEL pulses of a few fs duration. We note that the above discussion is valid when the TPDI is the only process that competes with the Auger decay. This is correct for moderate photon intensities, for which higher order ionization of the intermediate ions can be ignored. For sufficiently high intensities, other channels of multiple ionization of the ions should be taken into account. Thus the problem becomes much more complex. In the considered formalism additional channels can be included by introducing an induced width of the ionic state. However, a more detailed discussion of this problem is beyond the scope of this paper. Concluding this section we note, that in the considered case of TPDI in the 4d subshell of Xe, excitation by femtosecond pulses does not induce coherence of the intermediate ionic states. However, if the pulse is extremely short, about 100 as or even shorter, the spectral bandwidth becomes larger than the spin–orbit splitting of the 4d vacancy and both components of the doublet are excited coherently. In this case the non-diagonal elements of the factor h(αx Jx , αx Jx ) are of the same value as the diagonal ones, which means that interference effects play a considerable role. The angular correlation functions of the type discussed in the previous section will be notably modified due to this interference.

3.5. Influence of the Auger decay of the 4d vacancy on TPDI As was mentioned, the Auger decay of the intermediate 4d hole state competes with the second photoionization in sequential TPDI. Obviously, if the Auger lifetime is very short, much shorter than the pulse duration, all produced vacancies will decay before the second ionization occurs. For another extreme case of a very long lifetime, the Auger decay does not affect the TPDI and all doubly charged ions will be produced by the photoionization process. In our model, the Auger lifetime (width) enters the formalism only through the depolarization factor h(αx Jx , αx Jx ). Let us consider it for the 4d photoionization of Xe. The spin–orbit splitting of the 4d doublet (1.98 eV [45] ) is larger than the natural width of the 4d hole states (∼ 0.1 eV [20]) and the fieldinduced width due to the presently available FEL intensities (∼ 0.03 eV [15]). Thus J J   ωJ J  and the intermediate levels do not overlap. Moreover, the spectral width of the incoming photon beam is also smaller than the spin–orbit splitting provided that the pulse duration is larger than ≈ 300 as. Under these circumstances, one can ignore the nondiagonal terms of h(αJ, α  J  ) with J = J  compared with the diagonal ones. Then expression (16) simplifies to  +∞ h(αJ, α  J  ) = Nδα,α δJ,J  dt1 I (t1 ) −∞  ∞ dt2 exp[−J (t2 − t1 )] I (t2 ), (17) × t1

which actually describes the influence of the Auger decay on the probability of the second photoabsorption. The dimensionless factor h(αJ, αJ ), calculated for the case of the Xe 4d shell ionization by the Gaussian XUV pulse is shown in figure 9 as a function of the pulse duration (FWHM). If the pulse is short, much shorter than 1/ J , the factor h(αJ, αJ ) → 1 and the Auger decay does not influence either 6 The absolute values of the cross section (in atomic units) can be obtained by multiplying the values in the figure by 4π 3 ω2 α 2 .

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[7] Kurka M et al 2009 J. Phys. B: At. Mol. Opt. Phys. 42 141002 [8] Kheifets A S 2007 J. Phys. B: At. Mol. Opt. Phys. 40 F313 [9] Kheifets A S 2009 J. Phys. B: At. Mol. Opt. Phys. 42 134016 [10] Hamonou L, van der Hart H W, Dunseath K M and Terao-Dunseath M 2008 J. Phys. B: At. Mol. Opt. Phys. 41 015603 [11] Fritzsche S, Grum-Grzhimailo A N, Gryzlova E V and Kabachnik N M 2008 J. Phys. B: At. Mol. Opt. Phys. 41 165601 [12] Fritzsche S, Grum-Grzhimailo A N, Gryzlova E V and Kabachnik N M 2009 J. Phys. B: At. Mol. Opt. Phys. 42 145602 [13] Grum-Grzhimailo A N, Gryzlova E V, Strakhova S I, Kabachnik N M and Fritzsche S 2009 J. Phys: Conf. Ser. 194 012004 [14] Gryzlova E V, Grum-Grzhimailo A N, Kabachnik N M and Fritzsche S 2009 Uzhhorod Univ. Sci. Herald, Ser. Phys. 24 73 [15] Gryzlova E V, Grum-Grzhimailo A N, Fritzsche S and Kabachnik N M 2010 J. Phys. B: At. Mol. Opt. Phys. 43 225602 [16] Amusia M Ya and Connerade J-P 2000 Rep. Prog. Phys. 63 41 [17] Sorokin A A, Bobashev S V, Feigl T, Tiedtke K, Wabnitz H and Richter M 2007 Phys. Rev. Lett. 99 213002 [18] Richter M, Amusia M Ya, Bobashev S V, Feigl T, Jurani´c P N, Martins M, Sorokin A A and Tiedtke K 2009 Phys. Rev. Lett. 102 163002 [19] Makris M G, Lambropoulos P and Miheliˇc A 2009 Phys. Rev. Lett. 102 033002 [20] Jurvansuu M, Kivima¨aki A and Aksela S 2001 Phys. Rev. A 64 012502 [21] Krikunova M et al 2009 New J. Phys. 11 123019 [22] Young L et al 2010 Nature 466 56 [23] Balashov V V, Grum-Grzhimailo A N and Kabachnik N M 2000 Polarization and Correlation Phenomena in Atomic Collisions. A Practical Theory Course (New York: Kluwer Plenum) [24] Rouz´ee et al 2011 Phys. Rev. A 83 031401 [25] Kabachnik N M, Fritzsche S, Grum-Grzhimailo A N, Meyer M and Ueda K 2007 Phys. Rep. 451 155 [26] Varshalovich D A, Moskalev A N and Khersonskii V K 1988 Quantum Theory of Angular Momentum (Singapore: World Scientific) [27] Ueda K, Shimizu Y, Kabachnik N M, Sazhina I P, Wehlitz R, Becker U, Kitajima M and Tanaka H 1999 J. Phys. B: At. Mol. Opt. Phys. 32 L291 [28] Ueda K, Shimizu Y, Chiba H, Kitajima M, Tanaka H, Fritzsche S and Kabachnik N M 2001 J. Phys. B: At. Mol. Opt. Phys. 34 107 [29] Kitajima M et al 2001 J. Phys. B: At. Mol. Opt. Phys. 34 3829 [30] Froese Fischer C, Brage T and J¨onsson P 1997 Computational Atomic Structure: An MCHF Approach (Bristol: Institute of Physics Publishing) [31] Swanson J R and Armstrong L Jr 1977 Phys. Rev. A 15 661 [32] Parpia F A, Froese Fischer C and Grant I P 1996 Comput. Phys. Commun. 94 249 [33] Fritzsche S 2001 J. Electron Spectrosc. Relat. Phenom. 114–116 1155 [34] Fritzsche S 2002 Phys. Scr. T 100 37 [35] Fritzsche S, Fricke B and Sepp W-D 1992 Phys. Rev. A 45 1465 [36] Kutzner M, Radojevic V and Kelly H P 1989 Phys. Rev. A 40 5052 [37] King G C, Tronc M, Read F H and Bradford 1977 J. Phys. B: At. Mol. Phys. 10 2479 [38] Hansen J E 1996 Private communication, cited in S Br¨uhl, PhD Dissertation University of Hamburg, p 119 [39] Svensson S, Mårtenson N, Basilier E, Malmquist P, Gelius U and Siegbahn K 1976 Phys. Scr. 14 141

4. Conclusions We have theoretically investigated the sequential TPDI of the 4d subshell of Xe in the photon energy range 70–200 eV which includes the photoabsorption maximum (giant resonance). The main peculiarity of this particular process is a competition between the Auger decay of the 4d vacancy, produced in the first ionization, and the second ionization by the photons from the same pulse. We have shown that the relative effectiveness of the two competing processes in producing doubly charged ions depends on the duration of the XUV pulse. For short pulses t < 1/  the TPDI process dominates, for long pulses t > 1/  the Auger process dominates. Thus by changing the pulse duration one can control the path of the Xe2+ production. The most interesting interval of the pulse durations is several femtoseconds, comparable with the lifetime of the 4d vacancy (6 fs). We have analysed also the angular distributions of the first and second electrons in the sequential TPDI as well as the angular correlations between the two emitted electrons. A strong variation of the asymmetry parameter β2 , characteristic for the giant resonance region, is predicted not only for the first-step ionization of the Xe atom but also for the secondstep ionization of the Xe+ ion. In TPDI of the 4d subshell in Xe, the angular distribution of the first-step photoelectrons displays a rather large ‘non-dipole’ term proportional to the fourth Legendre polynomial. This effect predicted by theory [11] is difficult to observe (see [7]) since usually this term is small. An angle-resolved measurement of the sequential TPDI in Xe (4d) could help to confirm this prediction.

Acknowledgments The authors are grateful to Kiyoshi Ueda for his interest in this work and stimulating discussions. SF acknowledges the support by the FiDiPro programme of the Finnish Academy and by the DFG (project no FR 1251/13). EVG, ANGG and NMK acknowledge the support by the Russian Foundation for Basic Research (RFBR) under grant 09-02-00516. NMK acknowledges the hospitality of the European XFEL GmbH. This work is part of the project ‘Correlation and Polarization Phenomena in Ionization of Dilute Species by XUV and X-ray Radiation’ in the framework of the German–Russian collaboration ‘Development and Use of Accelerator-Based Photon Sources’.

References [1] Bachau H and Lambropoulos P 1991 Phys. Rev. A 44 R9 [2] Braune M, Reink¨oster A, Viefhaus J, Lohmann B and Becker U 2007 Int. Conf. on Photonic, Electronic and Atomic Collisions (ICPEAC) (Freiburg, Germany) Book of Abstracts, Fr034 [3] Moshammer R et al 2007 Phys. Rev. Lett. 98 203001 [4] Rudenko A et al 2008 Phys. Rev. Lett. 101 073003 [5] Jiang Y H et al 2009 J. Phys. B: At. Mol. Opt. Phys. 42 134012 [6] Fukuzawa H et al 2010 J. Phys. B: At. Mol. Opt. Phys. 43 111001 9

J. Phys. B: At. Mol. Opt. Phys. 44 (2011) 175602

S Fritzsche et al

[40] Becker U and Shirley D A (ed) 1996 VUV and Soft X-Ray Photoionization (New York: Plenum) p 135 [41] Becker U, Szostak D, Kerkhoff H G, Kupsch M, Langer B, Wehlitz R, Yagishita A and Hayaishi T 1989 Phys. Rev. A 39 3902 [42] K¨ammerling B, Kossmann H and Schmidt V 1989 J. Phys. B: At. Mol. Opt. Phys. 22 841 [43] Southworth S H, Kobrin P H, Truesdale C M, Lindle D, Owaki S and Shirley D A 1981 Phys. Rev. A 24 2257 [44] Snell G, Kukk E, Langer B and Berrah N 2000 Phys. Rev. A 61 042709

[45] Ralchenko Yu, Kramida A E and Reader J 2008 and NIST ASD Team NIST Atomic Spectra Database (version 5.1.4) http://physics.nist.gov/asd3 [46] Berezhko E G, Kabachnik N M and Rostovsky V S 1978 J. Phys. B: At. Mol. Phys. 11 1749 [47] Berezhko E G, Ivanov V K and Kabachnik N M 1978 Phys. Lett. A 66 474 [48] Kleiman U and Lohmann B 2003 J. Electron Spectrosc. Relat. Phenom. 131–2 29 [49] Andersen P, Andersen T, Folkmann F, Ivanov V K, Kjeldsen H and West J B 2001 J. Phys. B: At. Mol. Opt. Phys. 34 2009

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