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Predominance of the second order, two-step mechanism in the electron impact double ionization of Helium at intermediate impact energy

A Lahmam-Bennani 1,2 , E M Staicu Casagrande 1,2 , A Naja 1,2,5 , C Dal Cappello 3 and P Bolognesi 4

Introduction

The study of multiple ionization processes by charged particle impact is of considerable interest in many branches of physics, such as astrophysics or plasma physics, and also in other sciences such as life sciences where it is of prime importance for instance to understand the various mechanisms leading to energy deposition by radiation in matter. Electron impact double ionization (DI) is one of the simplest and the most fundamental of such processes. The advent of multi-parameter detection techniques, together with the progress in computational methods, has made it possible to perform complete experiments in which all kinematical parameters (vector momenta and energies) of all involved particles are determined. These are the so-called (e,3e) experiments, which have been successfully used during the last two decades to investigate in very fine details the ionization process on a variety of atomic targets, see and references therein. However, (e,3e) experiments remain difficult to perform because they necessitate the triple coincidence detection of the three final state electrons [1] (or alternatively the residual ion plus two electrons [15]). Besides, the corresponding cross section is inherently very small due to it's multiply (five-fold) differential character, consequently resulting in relatively modest statistics for the obtained data. It is therefore also of interest to consider (e,3-1e) experiments where, with respect to the (e,3e) case, one electron is not detected, hence only necessitating a double coincidence experiment similar to an (e,2e) single ionization (SI) but with energetics corresponding to DI. These experiments were first introduced by Lahmam-Bennani and coworkers (23)(24)(25)(26) and further developed in [27][28][29][30][31]. In all these works, it was clearly demonstrated that (e,3-1e) experiments provide a very sensitive mean to identify the various mechanisms responsible for electron impact DI and to gauge their relative importance.

Direct electron impact DI may proceed mostly via three routes [32][33]. (i) In the shakeoff mechanism (SO), the incident electron interacts only once with a single target electron which is ejected without further interaction with other target electrons. Subsequent ejection of a second electron occurs, due to the electronic relaxation caused by the sudden change in the distribution of potential seen by this electron. (ii) In the two-step mechanism termed TS1, the projectile electron interacts with one target electron which subsequently collides with a second electron leading to ejection of the pair. (iii) In the two-step mechanism termed TS2, the projectile electron successively collides with two target electrons, resulting in their double ejection. Both SO and TS1 involve only one single interaction of the projectile with the target, hence their denomination as first order or first Born mechanisms (in the Born series). In contrast, TS2 involves two successive projectile -target interactions and hence is a second order or second Born mechanism. The most obvious signature of the presence of these mechanisms is that first order mechanisms yield ejected electron(s) angular distributions which are symmetrical with respect to the momentum transfer vector direction, whereas this symmetry is broken in the case of second order, TS2 mechanism, see [34,35]. Lahmam-Bennani and coworkers [8,9,12] reported the observation of non-first order effects in their measurements of fully differential (e,3e) cross sections for DI of Helium. As one would expect [36], these effects were found to be small at 5.5 keV impact energy [8] while they become increasingly stronger as the energy is decreased to 1.1 keV [9] and to ~ 600 eV [12].

Similar non-first order effects were also reported in [18] for (e,3e) DI of He at 500 eV impact energy. In the low energy work of Lahmam-Bennani et al [12], the experimental angular distributions displayed large shifts of the forward peak from the momentum transfer direction, which could only partially be reproduced by second-Born calculations.

Similar large shifts were also observed in the (e,3-1e) experiments on DI of He and H 2 at ~ 600 eV [29]. A tentative description of this latter work on H 2 was developed by Mansouri et al [37] using a second-Born treatment (and correlated single center wavefunctions) which includes the double interaction between the incoming electron and the target. The results were able to produce only part of the shift of the experimental peaks. A more recent theoretical investigation using the external complex scaling method in spheroidal coordinates [38] also failed to describe the experimental findings for He as well as for H 2 . The authors concluded that the disagreement of theory with experiment is caused by the sequential TS2 mechanism of DI. On the other hand, a tentative description of the He (e,3-1e) results published in [29] was given by Götz et al [39,40] where the four-particle continuum was described in different approximations leading up to a 6C wavefunction which takes all two-body interactions into account as well as the interactions between each outgoing electron and the nucleus, and hence goes beyond the second Born approximation. However, the authors could not reproduce the strong asymmetry seen in the measured data, so that they came to implicitly question the experiments by writing [39]: <<we suggest that it is very important to establish the validity or otherwise of the asymmetry in the experimental data>>.

Given this situation and this questioning, we decided to reexamine the problem by taking advantage of the extended capabilities and sensitivity of our spectrometer gained by including to it new developments described in [41]. Our aim is to produce more extensive experimental data which would validate or invalidate the observations made in [29] and against which the theoretical predictions can be tested. To this end, we report here new (e,3-1e) experiments for DI of He performed at about the same impact energy of ~ 600 eV as in [29]. Besides, different energy sharings between the ejected electrons are investigated, in order to test the sensitivity of the results to this energy sharing, and to address the question whether features which appear very pronounced in the measured distributions and which are attributed to TS2 mechanism may strongly depend (or not) upon the difference in energy of the outgoing electrons. In the absence of elaborate non-first order theoretical calculations, the experimental data are compared with the results of a first-order theoretical model, namely the BBK model [42] which has proved to successfully describe experiments of this type in the few keV energy range [43]. Such comparison allows visualizing the strong shifts observed in the experimental data. Moreover, a qualitative discussion based on simple kinematical arguments is presented which shows that the observed large shifts can be related to the presence of TS2 mechanism which dominates over SO and TS1 mechanisms.

Experiment

The experimental set-up and experimental procedure used for the present work are described in detail in [41]. The main characteristic of the spectrometer is the unique combination of three high-efficiency, multi angle toroidal electrostatic energy analysers.

Briefly, an incident electron beam with energy E 0 collides with the gas jet formed at the collision centre. A coplanar geometry is used, where all electrons are observed in the collision plane defined by the incident and scattered momentum vectors, k 0 and k a , respectively. The

Results and discussion

Before discussing the He data, we start this Section by recalling that the experimental procedure and the good response of the spectrometer for measuring coincidence angular distributions were validated as described in [44,45]. Very briefly, the (e,2e) triple differential cross section (TDCS), d 3 new data are compared with the calculated results obtained using the very same model which has been used in [43] for DI of He, namely the first Born approximate BBK or 3C model where the final state is described by the product of three Coulomb waves [42,43]. Note that the experiments are obtained on a relative scale and have been normalized at the maximum of the forward lobe to the calculations. All the experimental as well as theoretical distributions of the 4DCS display the two-lobe structure well-known for the TDCS: a forward lobe pointing roughly in the momentum transfer direction (+K) and a backward lobe pointing in the opposite direction (these two directions are indicated by the vertical dotted lines in the figures). In this sense, we may find some resemblance between the experimental and theoretical distributions.

Also, the calculated backward-to-forward maximum intensity ratios are more or less in agreement with the measured ones, with the exception of the (E b : E c ) = (74 : 5) eV case (Fig.

1(c))

, where the theory predicts quite a small backward lobe. However, the most striking observations from both figures 1 and 2 are (i) the breaking of symmetry about ±K directions, (ii) the large shift in the angular position of the lobes, ~30° to 60°, and (iii) the existence of structures in these lobes. For all energy sharings considered here, the symmetry breaking and the large angular shifts are very much alike those reported in [29] which were obtained at (E b :

E c ) = (51 : 10) eV using an older version of our spectrometer. We think that this close resemblance between the 'old' results and the extended new data definitely confirms the validity of our previous results, and should put a final end to the question left open in [39] about this validity. To add one further argument to this statement, we recall that strong nonfirst Born effects were also reported in [12] and in [18], where fully differential (e,3e) cross sections were measured for He at E 0 ~ 600 eV and ~ 500 eV, respectively, that is under kinematics which are similar to the present ones. As stated in the Introduction, second Born calculations [12,37] could only partially account for the observed effects, likely because they include only discrete states as intermediate states and also use the closure approximation. It is not surprising that the strong non-first Born effects observed in (e,3e) cross section distributions are still significantly present in the (e,3-1e) 4DCS, in spite of the integration over the unobserved electron solid angle, dΩ c .

In terms of DI mechanisms, the features observed in Fig. 1 and 2 (i.e. breaking of symmetry, large angular shifts and existence of structures in the measured lobes) cannot be present in any first-order mechanism, namely the SO and TS1 which involve only one single interaction of the projectile with the target. Hence, they are clear signatures of non-first order processes such as TS2 which involves two successive target -projectile interactions. In the absence of second-or higher-order calculations (these are very difficult to perform when using the BBK model [START_REF] Ancarani | Electron and Photon Impact Ionisation and Related Topics[END_REF]) which could be compared with our data, we propose hereafter a discussion based on kinematical arguments built on our knowledge of SI (e,2e) processes.

This discussion is meant to give a qualitative interpretation of the observed structures as well as an estimate of the angular positions of the lobes, and hence to strengthen the conclusion that the measured strong non-first order effects are not due to an experimental artifact, but are physical effects due to the TS2 mechanism which reveals to play a preponderant role under the present kinematics. For the sake of brevity, we use the data of Fig. 1(c) at (E b : E c ) = (74 : 5) eV and E 0 = 658 eV for this discussion, but the ideas and the resulting conclusions hold for all the data presented in this work. The discussion is illustrated in figure 2.

TS2 is a two-step process: in the first step (sketched in the top panel of Fig. 2 In the second step of TS2, the a*-electron plays the role of the incident projectile in a new (e,2e) ionization of the target, resulting in the pair of electrons effectively detected, that is the a-scattered one with (E a = 500 eV, θ a = -6°) and the fast b-ejected one with E b = 74 eV. Here, two scenarios may occur, depending whether the intermediate a*-electron is scattered under θ a* ~ + 12° or -12°. These scenarios are sketched in the middle and bottom panels of Fig. 2, using black and red colors, respectively. We show in the following that they might be responsible, respectively, for the forward and backward lobes observed in the data of Fig. 1.

(1) In scenario 1 (middle panel of Fig. 2), the a-electron is detected at θ a = -6°, hence at -18° from the a*-incident direction. The b-electron is essentially ejected from the target in a binary collision and appears in the corresponding θ K -direction, that is θ K ~ + 61° from the a*-incident direction (at the high energies involved here, the recoil intensity appearing about -K direction is quite small [52]). In other words, the b-electron is mostly ejected at θ b ~ + (61°+12°) = +73° from the primary incident beam direction. In the experimental data displayed in Fig. 1(c), the forward lobe is peaking at about 70°. On the one hand, this value is quite far from the momentum transfer direction θ K ~ + 34° where the firstorder SO and TS1 contributions should be at their maximum. But on the other hand, this value is quite close to + 73° derived from the above scenario 1. We thus conclude that the shift of the forward lobe in Fig. 1(c) is essentially due to the TS2 contribution, according to this scenario 1. SO and TS1 contributions might of course be also present, but their intensity is appreciably smaller than that of the TS2.

(2) In scenario 2 (bottom panel of Fig. 2), the a-electron is again detected at θ a = -6°, but now this corresponds to +6° from the intermediate a*-incident direction. The b-electron ejected from the target appears essentially in the corresponding θ K -direction, that is θ K ~ -40° from the a*-incident direction (here also, the recoil intensity appearing about -K direction is quite small). In other words, the b-electron is mostly ejected at θ b ~ -(40°+12°) = -52°, or alternatively at θ b ~ (360° -52°) = 308° from the primary incident beam direction. In the data displayed in Fig. 1(c), the backward lobe is peaking at ~ 310°.

On the one hand, this value is quite far from the -K direction, θ -K = 180° + 34° = 214°

where the first-order SO and TS1 contributions should be at their maximum. But on the other hand, this value is in excellent agreement with the value + 308° derived from scenario 2. We thus again conclude that the backward lobe position in Fig. 1(c) is essentially determined by the TS2 contribution, according to this scenario 2. Once more, SO and TS1 contributions are appreciably smaller than that of TS2.

This discussion and the resulting conclusions may be likewise repeated for all the data presented in figure 1. We may add one further observation: each of the experimental forward as well as backward lobes in Fig. 1 appears to be split in two sub-lobes. From the discussion above, the sub-lobes located at the largest angles can be attributed to the TS2 contribution, whereas the ones located at smallest angles are grossly peaking at +K or -K direction, and hence may be attributed to the first-order SO and/or TS1 contributions.

This splitting in two sub-lobes is most obvious in Fig. 1(b), but it can still be clearly noticed in all the data. We also note (at least for the backward sub-lobes) that the relative importance of the two sub-lobes strongly changes with the (E b : E c ) energy sharing, the maximum intensity of the second-order sub-lobe being respectively ~20% smaller, ~25% larger and much larger (2 to 3 times larger) than the maximum intensity of the first-order sublobe, when E b is increased from 17 to 37 and to 74 eV at constant E c = 5 eV (Fig. 1 (a) to(c)).

This means that for the kinematics considered here, the TS2 contribution becomes more and more preponderant with increasing E b and is dominant over the SO and TS1 ones. Such dominance confirms the observations made in previously published (e,3e) [12] and (e,3-1e) [29] experiments at about the same incident energy. This makes it clear that the introduction of beyond first-order terms of the Born series in any future theoretical work is necessary at the intermediate electron impact regime investigated here.

Conclusion

We have performed new relative (e,3-1e) experiments for double ionization of He at ~ 600 eV impact energy and for different energy-sharings (E b : E c ). Our data display large shifts of the forward and backward lobes in the ejected electron angular distributions, thus confirming and validating the observations made earlier in [29], and consequently removing the questioning put in [39] on these earlier results. We also present a qualitative analysis based on kinematical arguments which shows that under the present kinematics the two-step 2 (TS2) mechanism dominates over the SO and the TS1 and is mostly responsible for the structures and angular positions of the measured lobes.

Ackowledgments

A N and P B wish to acknowledge the Université de Paris-Sud 11 for an 'ATER position' and a 'Professeur invité position', respectively.

  Predominance of the second-order, two-step mechanism in the electron impact double ionization of helium at intermediate impact energy A Lahmam-Bennani, E M Staicu Casagrande, A Naja, C Dal Cappello, P Bolognesi To cite this version: A Lahmam-Bennani, E M Staicu Casagrande, A Naja, C Dal Cappello, P Bolognesi. Predominance of the second-order, two-step mechanism in the electron impact double ionization of helium at intermediate impact energy. Journal of Physics B: Atomic, Molecular and Optical Physics, 2010, 43 (10), pp.105201. 10.1088/0953-4075/43/10/105201. hal-00569788

  fast, forward-scattered electron (indexed 'a') is detected at the scattered energy E a = 500 eV and at two symmetrical scattering angles, θ a = + (6°± 3°) and -(6°± 3°) as set by input slits at the entrance to the electrostatic lenses associated with the a-toroidal analyser. Throughout this work, positive angles are counted clockwise starting from the incident electron beam direction. Among the two ejected electrons resulting from DI of the target, (labelled 'b' for the faster and 'c' for the slower), we choose to only detect the faster one, with energy E b , in coincidence with the a-scattered electron, hence an (e,3-1e) experiment. Of course, such distinction does not hold in the case of equal ejection energies, but we keep labelling 'b' the detected electron. These b-electrons are multi-angle analysed in a double toroidal analyser over the angular ranges θ b = 20° -160° and 200° -340°, where 0° is defined by the incident beam direction. In the off-line analysis, the total θ b -angular range is divided into sectors of width ∆θ b = 5°. Though the emission direction of the third, c-electron is unknown, its kinetic energy E c is known from the energy conservation E 0 -IP 2+ = E a +E b +E c , where IP 2+ is the double ionization potential. [Here, the translational energy of the target atom and the recoil energy imparted to the ion after the emission of the two electrons are neglected, due to the small electron to ion mass ratio]. The present experiments were performed at ejected electron energies corresponding either to an equal energy sharing of (E b : E c ) =(17 : 17) eV or an unequal sharing of (E b : E c ) = (17 : 5) eV ;(37 : 5) eV and (74 : 5) eV. The incident energy (E 0 ) is consequently adjusted to fulfil the energy conservation requirement for the Helium target under study, with IP 2+ = 79 eV. We recall that the previous work published in[29] was obtained with the same E a -value of 500 eV and (E b : E c ) = (51 : 10) eV, that is under similar enough kinematical conditions (though not exactly identical) to allow meaningful comparison.Given the above experimental parameters, the momentum transfer to the target, defined by K = k 0 -k a , varies from K = 0.89 au at E b + E c = 22 eV to K = 1.12 au at E b + E c = 79 eV, while, due to the spread in θ a -angle, the momentum transfer resolution amounts to ∆Κ ~ ± 0.2 au and the spread in the momentum transfer direction is ∆θ K ~ ±10•.

  σ /dE a dΩ a dΩ b , for single ionization of the He 1s orbital was measured at the energies E a and E b of the current DI experiments on He and H 2 , immediately before and after each individual (e,3-1e) run. The so-obtained He TDCS-distributions are in excellent agreement both with the experimental values as well as with the calculated convergent close coupling (CCC) results published in[46][47][48]. The CCC model is commonly known to produce very accurate TDCS results for electron impact single ionization of He when the residual ion is left in its ground state. Consequently, the invariance of the coincidence collection efficiency upon variation of the θ b -ejected electron angle is ensured. We now discuss the new DI data. The angular distributions of the (e,3-1e) four-fold differential cross sections (4DCS), d 4 σ /dE a dE b dΩ a dΩ b , for DI of He at (E b : E c ) = (17 : 5) eV ; (37 : 5) eV ; (74 : 5) eV and (17 : 17) eV are shown in figure 1(a) to 1(d), respectively. Our

  ) the slowest c-electron (E c = 5 eV) is ejected in an (e,2e)-like process where the corresponding scattered electron (called 'a*') has an energy E a* = 658-25-5 = 628 eV (the He SI potential is taken here to be 25 eV) and appears with the highest probability at the Compton scattering angle, ±θ a* , corresponding to the Bethe ridge condition[START_REF] Inokuti | [END_REF]51] : E 0 -E a* = E 0 sin 2 (θ a* ). This corresponds to θ a* ~ ± 12°. The ± sign stands for the fact that the intermediate, scattered a*-electron may appear on both sides of the incident beam direction, as shown in figure2by the black and red colors, respectively. (Positive angles are here counted clockwise starting from the incident direction). The associated c-electron in this (e,2e) process is most likely ejected in the corresponding momentum transfer direction, that is θ K* ~ ± 78° with respect to the incident direction. Note that neither the a*-nor the c-electrons are detected in the (e,3-1e) experiments.

  Figure 2

Caption to figures


Krzysztof Gawryluk

Miross Law Brewczyk

Mariusz Gajda

Kazimierz

Kazimierz Rzażewski

Free expansion of a

come

Since the first experimental realization of Bose-Einstein condensation in dilute atomic gases [1] the measurement techniques based on time-of-flight expansion became a powerful method to study ultracold atomic systems. In fact, this kind of measurement was used to prove the existence of a condensate. Starting from an axially (cigar-or disc-shaped) symmetric atomic cloud it happened after its release from a trap that the ratio of the axial and radial both condensate and thermal cloud widths systematically change during the expansion. Eventually the anisotropy inversion for a condensate was observed which was a crucial distinction from the behavior of a thermal cloud. The thermal part, in agreement with the classical Maxwell distribution of velocities eventually takes a spherical shape. For a small condensate (like the very first rubidium condensate consisting of some 2000 atoms only) the anisotropy inversion is just a direct manifestation of the Heisenberg uncertainty principle -more spatial squeeze -higher momenta. For larger samples similar inversion is a result of the interaction energy stored anisotropically in the trapped condensate. It is worth adding that the free expansion technique was also used for degenerate fermionic gases, for instance to probe the superfluidity of strongly interacting atomic Fermi mixtures [2] or to measure the p-wave Feshbach resonances for fermionic atoms [3].

The main purpose of this work is to investigate an influence of a thermal cloud on the dynamics of an expanding condensate. If such an impact exists another question is whether it is restricted only to times just after the release or is it continued over the whole expansion time. Finally, it would be interesting to know how the influence during the expansion compares to the influence while the system is confined.

To investigate the mutual interaction between the condensed and thermal components during the expansion we employ the classical field approximation in a version described in Ref. [4]. So, we start with the N -particle Hamiltonian written in terms of the field operator Ψ(r, t) satisfying the bosonic commutation relations. Assuming the usual contact interaction potential for colliding atoms the Hamiltonian takes the form:

H = d 3 r Ψ+ (r, t) - 2 2m ∇ 2 + V tr (r, t) Ψ(r, t) + g 2 d 3 r Ψ+ (r, t) Ψ+ (r, t) Ψ(r, t) Ψ(r, t) , (1)
where the interaction strength g = 4π 2 a/m and a is the s-wave scattering length. The trapping potential V tr (r, t) is time-dependent and is switched off instantaneously to trigger the expansion. The main equation of the classical field approximation reads:

i ∂ ∂t Ψ(r, t) = - 2 2m ∇ 2 + V tr (r, t) Ψ(r, t) +g Ψ * (r, t)Ψ(r, t)Ψ(r, t) ( 2 )
and is just the Heisenberg equation of motion for the field operator stripped of its operator character. The complex wave function Ψ(r, t) which we call the classical field describes both condensed and noncondensed atoms. The use of the classical field instead of the field operator is justified when only macroscopically occupied modes are taken into consideration. This reasoning remains in analogy with the treatment of an intense light beam which although consisting of single photons may be described by the electric and magnetic fields. An important question how to get out of the classical field the information on a condensate and a thermal cloud is resolved by using the Penrose and Onsager definition of Bose-Einstein condensation [5] and by taking into consideration a measurement process. Since any detector has a limited spatial and temporal resolutions a complicated (both in space and time) behavior of the high energy classical field is smoothed out during the measurement. Therefore, the quantity which is physically important is a time and/or space averaged one-particle density matrix. According to the Penrose and Onsager definition the condensate wave function is an eigenvector corresponding to the dominant eigenvalue of a one-particle coarse-grained (i.e., averaged over time and/or space) density matrix. We closely follow the experiment and realize the averaging as a column integration along one of the radial directions. The physically important (averaged along y Confidential: not for distribution. Submitted to IOP Publishing for peer review 30 March 2010 direction) one-particle density matrix is given by ρ(x, z, x , z ; t) = dy Ψ(x, y, z, t) Ψ * (x , y, z , t) (3) and the splitting procedure requires the diagonalization of (3). This kind of averaging was already used to investigate a decay of multiply charged vortices [6]. The splitting procedure is then summarized as:

ρ = k N k ϕ k (x, z, t) ϕ * k (x , z , t) ( 4 )
ψ 0 (x, z, t) = N 0 ϕ 0 (x, z, t) ( 5 ) ρ T (x, z, t) = ρ(x, z, x, z; t) -|ψ 0 (x, z, t)| 2 . (6)
Here, ϕ k are the macroscopically occupied modes, N 0 is the dominant eigenvalue, ψ 0 is the condensate wave function, and ρ T is the density of thermal cloud.

Having introduced the classical field approximation we now describe our numerical procedure. First, we find the classical field corresponding to the 87 Rb Bose gas (with a scattering length a = 5.82 nm) at equilibrium confined in a harmonic trap with frequencies ω ⊥ ≡ ω x,y = 2π × 137.4 Hz and ω z = 2π ×12.6 Hz. Details on how to obtain an equilibrium state for a given number of atoms and at particular temperature are explained in Ref. [7]. Here, we just briefly outline our procedure. First, we solve iteratively the equations of the self-consistent Hartree-Fock method [START_REF] Pethick | Bose-Einstein Condensation in Dilute Gases[END_REF] and obtain in this way the densities of the condensed, n c (r), and the thermal, n th (r), fractions corresponding to the system with the total number of N atoms being at the equilibrium at a given temperature T . Next, we built an initial classical field:

Ψ(r, t = 0) = n c (r) + n th (r) e iϕ(r) , (7)
and randomize both the phase ϕ(r) and the density n th (r) in such a way that the total energy per atom in the classical field equals the corresponding energy in the Hartree-Fock model. The presence of the phase factor in the second term in ( 7) is necessary, otherwise the classical field suffers from a lack of kinetic energy in comparison with the Hartree-Fock method. Then we let the field thermalize by evolving it according to the Eq. ( 2). However, the Eq. ( 2) is solved on a grid with a particular value of the spatial step for which the condensate fraction N 0 /N does not change in time. The optimal spatial step ∆x is approximately related to temperature by p 2 max /2m ≈ k B T , where p max = π/∆x. Since we intend to investigate the influence of thermal atoms on the condensate expansion we generate numerically various equilibrium states, all having the same number of condensed atoms (N 0 = 30000 or N 0 = 90000). Next, we suddenly turn off the trapping potential and let the atomic cloud to expand. Technically speaking, we solve the Eq. ( 2) on a larger grid (but having the same spatial step) and without any trap. We monitor the momentum distribution (i.e., the Fourier transform of the classical field) during the expansion and find that it changes for the first few milliseconds only. In other words, a few milliseconds is required to convert fully the interaction energy into the kinetic energy. Afterwards, the classical field evolves freely and can be found with the help of the propagator of the free Schrödinger equation. Finally, at a desired time the splitting of the classical field into the condensed and noncondensed components is performed. We have also investigated the case of switching off the trapping potential in a finite time of a few hundreds of microseconds as it is done in the experiment, but the procedure just outlined showed no differences in the results. In Fig. 1 we plot the radial and axial densities of an expanding atomic cloud at 22 ms. Initially the condensate is a cigar-shaped like a trap. Its aspect ratio in the Thomas-Fermi approximation is given by R ⊥ /R z = ω z /ω ⊥ [START_REF] Pethick | Bose-Einstein Condensation in Dilute Gases[END_REF] which equals approximately 1/10. After 22 ms, as can be seen in Fig. 1, the initial anisotropy is, actually, inverted. The radial size gets larger than the axial one. This, of course, is not true for the thermal cloud in which case the final density becomes spherical.

Fig. 2 reveals some technical details related to the read-out procedure. After the splitting of the classical field is concluded and the two-dimensional condensate density is known, this density is fitted by a twodimensional inverted parabola (since according to the large number of atoms in the condensate the Thomas- Fermi approximation is valid). The fit is performed based on the least squares method. In fact, fits depend on the realization and, as will be shown later, the aspect ratio is a quantity which is most sensitive to the realization.

The main result of this work is presented in Fig. 3. It shows the radial and axial condensate widths after 22 ms of free expansion as a function of a condensate fraction. There are two sets of data included in this figure. The first one (squares and circles) depicts the behavior of the system with the same number of condensed atoms (N 0 = 90000) independently of the condensate fraction (i.e., temperature). The second set (stars and triangles) represents data for the systems with N 0 = 30000. Horizontal lines are the radial and axial sizes of 9 × 10 4 and 3×10 4 atoms condensate calculated within the expansion model formulated by Castin and Dum [9]. This model describes the free expansion of a pure condensate within the Thomas-Fermi limit. Our data suggest that thermal atoms somehow temper the expansion of the condensate.

There are three experimental papers discussing the temperature effects having influence on the ballistic expansion of a condensate [10][11][12]. All of these papers claim that the behavior of the condensed cloud measured in terms of its size during the expansion depends on the temperature of the system before the expansion. So, the quantitative comparison between the numerical calculations and the experiment is possible.

Since our numerical parameters were taken in a way to match the parameters of experiment of Ref. [11] we start with this paper. In that experiment the authors make effort to keep constant the number of condensed atoms while expanding the atomic samples at various temperatures. They found the increase of radial and axial condensates lengths when the temperature gets higher (see Fig. 4 in [11]). The measured widths are larger than the corresponding Castin-Dum values. So, the authors claim that their experiment is performed in the non-Thomas-Fermi regime. An increase of radial and axial widths with temperature is explained by an assumption that at equilibrium in a trap the thermal atoms exert a force on condensed atoms towards the center of a trap thus compressing the condensate cloud. This compression results in a faster expansion in all directions after the trap is released. However, our calculations within the classical field approximation show that, actually, no compression of a condensate cloud occurs in a trap. We stress that this statement is true also within the self-consistent Hartree-Fock model [START_REF] Pethick | Bose-Einstein Condensation in Dilute Gases[END_REF]. In Fig. 4 we plot axial and radial condensate densities obtained by solving selfconsistently the equations of the Hartree-Fock model for a particular number of condensed atoms (N 0 = 90000) but at various temperatures (solid lines which correspond to the condensate fractions 0.15, 0.30, and 0.60). We also added to the figure the densities of a pure condensate consisting of 90000 atoms calculated by solving the Gross-Pitaevskii equation in imaginary time (dotted line) as well as by the Thomas-Fermi formula (dashed line). Clearly, Fig. 4 shows no compression due to the presence of a thermal cloud.

A qualitative difference between numerical results and the experimental data of Ref. [11] comparison with other experimental works. For example, in Ref. [10] a deviation from ballistic expansion is also reported. In Fig. 5a of that paper the authors plot the aspect ratio of the condensed component after 22.3 ms of free expansion as a function of reduced temperature. It is clear from this figure that for higher temperatures the results are different than the Castin-Dum limit [START_REF] Castin | [END_REF] (the aspect ratio shows deviation from the Castin-Dum values also in [11]). Therefore, we look separately at the axial and radial sizes of the expanding condensate in the case of Orsay experiment. The results are presented in Fig. 5. Here, the experimental data are compared with the Castin-Dum values. This figure clearly shows that the experimental data stay close to the Castin-Dum values. In axial direction the thermal cloud seems to temper the expansion of a condensate whereas the interplay between the condensate and the thermal component in radial direction gets more complicated. Differences are on the level of a few percent similarly to what we obtain from our numerics (although for a different trap geometry) and in opposite to what is reported in paper [11].

In other experimental work, Ref. [12], the authors find finite temperature correction to the Thomas-Fermi approximation as a function of the condensate fraction by measuring the ratio R5 /N 0 , where R is the condensate radius defined as R = (R 2 ⊥ R z ) 1/3 and N 0 is the number of condensed atoms. There is an agreement with Castin-Dum predictions [START_REF] Castin | [END_REF] for low temperatures, however, when the temperature gets higher the ratio R5 /N 0 departs from the Castin-Dum value getting larger (see Fig. 4 of [12]). The authors explain this behavior by us- ing a combination of a modified Hartree-Fock model to describe the condensed and thermal fractions in a trap and an expansion model formulated by Castin and Dum [START_REF] Castin | [END_REF]. They conclude that the influence of the thermal cloud on the condensate during the expansion is negligible which seems to be in opposition to what is claimed in Refs. [10] and [11]. It contradicts also our findings. Therefore, we decided to compare all experiments and our numerical results on a graph where we plot R/N 0 (actually, normalized to the value given by the Castin-Dum approach to make the comparison feasible) as a function of the condensate fraction. In Castin-Dum formulation one has

R5 (t)/N 0 = 15 a ā4 (λ 2 ⊥ (t) λ z (t)) 5/3 (8)
where

λ ⊥ (t) = 1 + t 2 λ z (t) = 1 + β 2 (t arctan t -ln 1 + t 2 ) , (9)
β = ω z /ω ⊥ and time t is expressed in units of 1/ω ⊥ whereas ā is an oscillatory unit length calculated based on the geometric mean of all angular frequencies. Fig. pening during the expansion of condensate and thermal cloud.

Finally, in Fig. 7 we plot the aspect ratio for a condensate for various condensate fractions. The aspect ratio seems to be a quantity which is most sensitive to the interplay between the thermal cloud and the condensate. For low temperatures the aspect ratio approaches the Castin-Dum value whereas for higher temperatures it is getting larger. Moreover, we find that the aspect ratio is very sensitive to the initial conditions fulfilled by the classical field. Different realizations lead to different aspect ratios what is marked by error bars in Fig. 7. FIG. 7: Aspect ratio (R ⊥ /Rz) at 22 ms defined as the ratio of radial and axial condensate widths for various condensate fractions. Two sets of data are presented corresponding to the number of condensed atoms N0 = 90000 (solid line) and N0 = 30000 (dashed line). Error bars depict the dependence on the realization. Horizontal dashed (thin) line is the aspect ratio obtained within the Thomas-Fermi approximation.

In conclusion, we have studied the expansion of the Bose-Einstein condensate in the presence of thermal atoms. Using the classical field approximation we have shown that thermal atoms change the dynamics of a condensate in a way that both radial and axial condensate widths get smaller in comparison with the case when there is no thermal cloud. It results in a change of condensate aspect ratio which becomes bigger for smaller condensate fraction (i.e., larger thermal cloud). While all papers agree that the thermal cloud does play a role in the expansion of the condensate, the details remain unclear. The three experimental papers are not mutually in agreement and also our results do not coincide with some measurements. Clearly more work is needed to clarify this somewhat confusing situation.

FIG. 1 :

 1 FIG.1: Axial (upper frame) and radial (lower frame) cuts of the total (thick solid line), condensate (dashed line), and thermal (thin solid line) densities as obtained by splitting the free expanding classical field at 22 ms as described in the text. The condensate fraction is equal to 0.3 and the number of condensed atoms N0 = 90000. The oscillatory unit of length is defined based on the axial trap frequency: q mωz and equals 3.0 µm.

6 FIG. 2 :

 62 FIG.2: Condensate axial (left frame, dashed line) and radial (right frame, dashed line) densities for two single realizations (upper and lower frames, respectively) at 22 ms. Condensate density is extracted from the classical field by the splitting procedure and is fitted to two-dimensional inverse parabola. Black lines show axial (left frame) and radial (right frame) cuts of such a fit. The parameters are the same as in Fig.1.

FIG. 3 :

 3 FIG.3:(Color online) Radial (squares and stars) and axial (circles and triangles) condensate radii as a function of the condensate fraction after 22 ms of ballistic expansion. Two sets of points marked by squares and circles correspond to the systems with the number of condensed atoms N0 = 90000 whereas two other sets (marked by stars and triangles) represent the systems with N0 = 30000 condensed atoms. Solid lines are shown to guide the eye. Horizontal dashed lines are the Thomas-Fermi values for the radial lengths (two upper lines, the upper line for N0 = 90000 and the lower line for N0 = 30000) and the axial lengths (two lower lines, the upper line for N0 = 90000 and the lower line for N0 = 30000). Note that both radial and axial widths get shorter in comparison with the size of a pure condensate of the same number of atoms.

ZFIG. 4 :

 4 Fig.4in[11]). The measured widths are larger than the corresponding Castin-Dum values. So, the authors claim that their experiment is performed in the non-Thomas-Fermi regime. An increase of radial and axial widths with temperature is explained by an assumption that at equilibrium in a trap the thermal atoms exert a force on condensed atoms towards the center of a trap thus compressing the condensate cloud. This compression results in a faster expansion in all directions after the trap is released. However, our calculations within the classical field approximation show that, actually, no compression of a condensate cloud occurs in a trap. We stress that this statement is true also within the self-consistent Hartree-Fock model[START_REF] Pethick | Bose-Einstein Condensation in Dilute Gases[END_REF]. In Fig.4we plot axial and radial condensate densities obtained by solving selfconsistently the equations of the Hartree-Fock model for a particular number of condensed atoms (N 0 = 90000) but at various temperatures (solid lines which correspond to the condensate fractions 0.15, 0.30, and 0.60). We also added to the figure the densities of a pure condensate consisting of 90000 atoms calculated by solving the Gross-Pitaevskii equation in imaginary time (dotted line) as well as by the Thomas-Fermi formula (dashed line). Clearly, Fig.4shows no compression due to the presence of a thermal cloud.A qualitative difference between numerical results and the experimental data of Ref.[11] motivated us to make a

FIG. 5 :

 5 FIG.5: Radial (upper panel) and axial (lower panel) widths as a function of reduced temperature. Experimental data (points with error bars) correspond to the experiment of Ref.[10] whereas numerical results (solid lines) are calculated based on the Castin-Dum approximation.

  Color online) Ratio R/N0 (normalized to the value obtained from the Castin-Dum approximation) as a function of the condensate fraction. The mean size of the condensate is defined as R = (R 2 ⊥ Rz) 1/3 . Four sets of data are presented in the figure according to the numerical calculations (solid and open squares), the results of Ref.[10] (stars), Ref.[12] (diamonds), and Ref.[11] (triangles).

		2
		1.8
		1.6
	R R CD	1.2 1.4
		1
		0.8
		0	0.2	0.4	0.6	0.8	1
				CONDENSATE FRACTION
	FIG. 6: (
							6 shows that our results (solid squares for N 0 = 90000
							and open squares for N 0 = 30000) stay in a quite good
							agreement with Orsay experimental data.
							Certainly, further experimental and theoretical effort
							is required to gain more insight to what, indeed, is hap-

Acknowledgments

We are grateful to all three experimental groups (Orsay, Toruń, and São Carlos) for helpful discussions and for providing us with their raw data. The authors acknowledge support by Polish Government research funds for 2009-2011. Some of the results have been obtained using computers at the Interdisciplinary Centre for Mathematical and Computational Modeling of Warsaw University.


S Sheinerman

P Lablanquie

F Penent

Y Hikosaka

T Kaneyasu

E Shigemasa

K Ito

PCI effects in Argon 2p double Auger decay probed by multielectron coincidence methods

Keywords: numbers: 32, 70, Jz, 32, 80, Hd

Electron correlations in three electron emission associated with innershell photoionization are investigated for the first time through a combined theoretical and experimental approach. Namely, different kinds of Post Collisional Interactions (PCI) occurring in the decay of Argon 2p holes by emission of two Auger electrons are isolated experimentally with a powerful coincidence technique and are modeled with an eikonal approach. It is shown that PCI distortion of the photoelectron line shape is stronger than in the single Auger paths and depends critically on the process: in direct Double Auger decays where two Auger electrons are emitted simultaneously, direct interaction of the 2p photoelectron with the two Auger electrons is demonstrated; in cascade Double Auger processes, the PCI distortion depends only slightly on the last electron emitted in the cascade. This perturbation depends on and reveals the lifetime of the intermediate Ar 2+ state.

Introduction

An inner atomic vacancy created by photoionization can decay by emission of radiation or by Auger decay. For L shells, the Auger decay is generally more probable and can occur with emission of one electron (single Auger decay) or of a few electrons (multiple Auger decay). The probability of multiple Auger decay is usually lower than the probability of single Auger decay but can be high enough to allow experimental observations (see e.g. Aberg 1975). The first evidence of a double Auger (DA) process was reported by [START_REF] Carlson | [END_REF]Krause (1965, 1966). Recently, progress in coincidence measurements opened the path to a deeper search of the double Auger decays (Penent et al 2005). DA implies that the filling of the inner shell vacancy by an outer electron can cause the ejection of two electrons. This process is in its turn divided into the direct double Auger decay (DDA) when the two electrons are emitted simultaneously and cascade double Auger decay (CDA) when the electrons emission occurs in two steps through the creation and decay of an intermediate quasi-stationary state. Viefhaus et al 2004 were the first to demonstrate directly the existence of the DDA path.

Recently, coincidence measurements based on the use of magnetic bottle time of flight spectrometers (Eland 2009) have opened the possibility to obtain the complete state selected triple photoionization (TPI) continua over a wide range of kinetic energies (Hikosaka et al 2009). The application of this method to the multi-electron decays of an Ar 2p inner shell reveals the following (Lablanquie et al 2007, Palaudoux et al 2008, Hikosaka et al 2009, 2009A): i) One can distinguish the electrons emitted due to the decay of the 2p 3/2 subshell or of the 2p 1/2 one. ii) The observed electron lines can be unambiguously filtered according to the final state (e.g. configuration ...3s 2 3p 3 with the different terms ( 4 S, 2 D, 2 P ).) iii) One can separate the electron lines with respect to the decay process of the 2p innershell: emission of one, two and up to three Auger electrons. iv) The position and shapes of the photoelectron lines which are associated with the DA decay differ for the DDA and CDA processes. v) The line shapes ascribed to the DA processes are strongly distorted by post collision interaction (PCI) effects.

PCI is known as a special kind of electron correlation associated with the interaction between the charged particles of a resonant process, through the creation and decay of an intermediate quasi stationary state. In the case of inner shell photoionization, PCI reduces to the interaction of the emitted photoelectron with the Auger electrons and with the ion field which varies during the Auger decay. PCI in single Auger decay following inner shell photoionization is quite well documented both experimentally and theoretically (see for example the reviews of Kuchiev andSheinerman 1989, Schmidt 1992).

PCI in the DA processes has been much less studied. There are a number of experimental investigations on the PCI influence on the threshold photoelectron yield in the vicinity of inner vacancies: Kr 1s -1 , Kr 2p -1 , Kr 3p -1 , Ar 1s -1 , Ar 2p -1 , Xe 4p -1 , Xe 4d -1 (Hayaishi et al 1988,1990,1994, Kjeldsen et al 1996, Hayaishi et al 1999,2000,2002, Matsui et al 2002). These experiments present coincidences between threshold electrons, which are the more affected by PCI effect, and the residual ions. PCI in DA processes was also studied after 4d ionization in Xenon by the method of threshold electron /Auger electron coincidences (Lablanquie et al 2001(Lablanquie et al , 2002)). Finally only a few studies presented a theoretical analysis of PCI associated with few Auger electron decay: a classical model to describe the release of threshold electrons in the DA process (Kjeldsen et al 1996), a classical model of cascade Auger decay (Koike 1994), and a quantum mechanical eikonal approach to the DDA and CDA processes (Sheinerman 1994(Sheinerman , 1998)).

In this study, we reveal and investigate the PCI effects in DA processes, using Ar 2p innershell ionization as an example. For the first time we present a systematic investigation of photoelectron spectra measured in coincidence with two Auger electrons. The PCI distortion of the photoelectron and Auger lines is shown to be important for DDA and CDA processes. Our measurements are supported by calculations based on an eikonal approach to PCI. The results of these calculations are in good agreement with the experimental data thus demonstrating the adequacy of the theoretical model to the phenomenon considered. The analysis of the experimental and theoretical results shows that the photoelectron line shape depends both on the variation of the ionic field and on the interaction with the Auger electrons. Moreover, since the line shapes for DDA and CDA processes are slightly different, their analysis allows us to extract properties of the associatted DA decays.

The paper is organized as follows. Section 2 presents a brief description of our experimental method and shows 2 dimensional maps of the coincidences for double Auger decays. Section 3 presents the theoretical approach for calculation of the cross section of the DA processes. In section 4 we present the results of our measurements with calculation and the analysis of these data. The atomic system of units |e| = m e = = 1 is used throughout.

Experiment

Two different experimental setups have been used here: a magnetic bottle time of flight analyzer for powerful coincidence detection and a very high resolution hemispherical analyzer for the spectroscopic characterization of the Ar 2+ states involved in the cascade double Auger decay.

The non coincident high resolution spectra were obtained on the brand new undulator beamline BL6U at the UVSOR storage ring in the Institute for Molecular Science in Okazaki, Japan. BL6U is equipped with a variable included angle grazing incidence monochromator with a varied line spacing plane grating, without entrance slit. We used a high performance hemispherical electron energy analyzer MBS-A1, developed by the MB Scientific AB company. It is equipped with a gas cell and a CCD camera and reaches a theoretical 1.3 meV (FWHM) resolution when operated with a pass energy of 2 eV and a slit opening of 0.2 mm. The energy scale was calibrated with the strongest Ar 2p DA cascade peak (Lablanquie et al 2007), set at 2.19 ± 0.01eV. We checked carefully this value with the magnetic bottle analyzer, by measuring simultaneously He photoelectron lines, in order to avoid possible shifts due to contact potentials. Slow drift of the peak positions occurred with time and were minimized by short scans of less than a few minutes.

The main part of the results were acquired with a magnetic bottle time of flight analyzer. This kind of apparatus has been extensively described before (Eland et al 2003, Penent et al 2005, Hikosaka et al 2006, Eland 2009) and will only be briefly introduced here. Ionization occurs close (∼ 1 mm) to a strong (∼ 0.5 T) permanent magnet in a zone where the strong divergent magnetic field creates a magnetic mirror. Almost all released electrons are guided along the magnetic field lines through a long (2.5 m) solenoid (∼ 10 -3 T) to the detector. A multi-hit time-to-digital converter then records the arrival times of each electron with respect to the light bunch and stores them in on-line computers for further analysis. Time to energy calibration was performed by measuring photolines associated with He + (N=1) and He + (N=2) formation at known photon energies. The experiments were performed at the Photon Factory and used light from the undulator beam line BL-16B (Shigemasa et al 1998). The photon bandwidth was set at 30 meV. Single bunch operation of the storage ring provided 200 ps-wide light pulses every 624 ns. It follows that time of flight of isolated electrons could only be measured modulo the 624 ns inter light pulse period. It was however not needed to use a mechanical chopper (Ito et al 2009) to increase this period in this favourable case: it is possible to retrieve absolute electron time of flights in coincidence events where a 2p photoelectron and all associated Auger electrons are recorded, because one of the Auger electrons is then necessarily a fast one, with a flight time shorter than 624 ns. Typical count rate in coincidence experiments was 3000 Hz, which allows us to neglect random coincidences between electrons produced by subsequent light pulses.

The key point for the success of the present experiment is its ability to define precisely the conditions in which PCI is observed, thanks to the coincidence techniques. Fig 1 a) and 1 b) demonstrate first the ability to resolve the final states reached respectively in double and triple ionization events. The histogram of the sum of the kinetic energy of two electrons detected in coincidence in Fig 1 a) reveals the Ar 2+ final state. Most of the events correspond to the creation and Auger decay of a 2p 1/2 or a 2p 3/2 innershell hole, along with a minor valence double ionization contribution. Similarly, the final Ar 3+ states populated by triple ionization are obtained in Fig. 1 b) through the histogram of the sum of the kinetic energies of three electrons detected in coincidence, and they originate mainly from the decay of 2p holes by emission of two Auger electrons. The resolution, slightly improved compared to our previous study (Lablanquie et al 2007), allows us to distinguish but not isolate fully the Ar 3+ (3p -3 ) 2S+1 L final states. A clear separation between direct and cascade Double Auger decays is not always possible. For instance, the two processes overlap heavily in the case of the 4d decay in Xenon (Penent et al 2005), but the 2p decay in Argon offers a favorable system, where distinction is possible (Viefhaus et al 2004, Lablanquie et al 2007). This is demonstrated in We selected in this study two cascade processes characterized by the emission of a 8.5 eV electron for the first one, and or of a 2.19 eV electron for the second. As for the DDA path, we selected also two zones, with emission of an Auger electron in the intervals (30 -70) eV and (4.9 -5.1) eV. The experimental coincidence spectra presented in

Theoretical approach to PCI effects in DA processes

Treatment of PCI associated with inner shell photoionization implies taking into account the interaction of the charged particles in the intermediate and final states of the process. It means that the photoelectron interacts first with the ionic field the charge of which varies from +1 to +3 during the DA decay of the inner vacancy. Then this photoelectron interacts with the two Auger electrons which are emitted in the Auger decay. For descriptions of these effects we will use a quantum mechanical eikonal approach which has been used successfully for the case of single Auger decay (Sheinerman et al 2006, Palaudoux et al 2008).

This approach implies that the main contribution to the interaction between the escaping particles occurs at large distances when R ≡ V ph /Γ 1. Here V ph is the photoelectron velocity and Γ is the width of the inner vacancy. In this case we can consider the kinetic energy of the relative motion of the emitted particles to be significantly larger than the potential energy of their interaction. This assumption is valid if the following condition is fulfilled (Kuchiev and Sheinerman 1986): E ph Γ 2/3 •(E 0 /2) 1/3 where E 0 is the atomic unit of energy and E ph is the photoelectron energy. Here, the widths values of Γ(2p 1/2 ) = Γ(2p 3/2 ) = 118 meV (Jurvansuu et al 2001) ensure that this condition is fulfilled rather well for a photon energy in excess over threshold of E ph > 2 eV and quite reasonable for 2 eV > E ph > 0.7 eV. Therefore we can use the eikonal approach to analyse our data at least for the energies E ph > 0.7 eV.

PCI in direct double Auger decay (DDA)

This decay occurs when the two Auger electrons are emitted simultaneously. The equation of the reaction has the form:

ω + Ar → e 1 + Ar + * (2p -1 1/2,3/2 ) → → e 1 + e 1A (E 2 ) + e 2A (E 3 ) + Ar 3+ ((3p -3 ) 4 S, 2 D, 2 P ) .
(1)

The PCI theory for the DDA processes on the base of the eikonal approach was developed in (Sheinerman 1998). According to this theory the PCI-distorted cross section of DDA process when all three electrons are registered in coincidence is given by :

d 4 σ dE 2 dE 3 dΩ 3 = dΩ 1 dΩ 2 σ 0 (ω, E (0) 1 , Ω 1 )× × Γ d (E (0) 2 , E (0) 3 , Ω 2 , Ω 3 ) 2 π [(ε 2 + ε 3 ) 2 + Γ 2 /4] • π ξ sinh(π ξ) exp 2 ξ tan -1 2 (ε 2 + ε 3 ) Γ . (2)
Here

E (0) 1 , E (0) 2 , E (0)
3 are unshifted values of energies of photoelectron, first and second Auger electrons, which are ejected in the directions characterized respectively by the solid angles Ω 1 , Ω 2 , Ω 3 . The energy variables ε 2 , ε 3 denote the energies of the first and second Auger electrons measured from their unshifted values:

ε 2 = E 2 -E (0) 2 , ε 3 = E 3 -E (0)
3 . σ 0 is the inner shell photoionization cross section and Γ d is the partial width of the double Auger decay. It should be noted that the lineshape is determined only by the inner vacancy width Γ and by the kinematics of the process which is included in the dimensionless parameter ξ:

ξ = -2 V 1 + 1 |V 1 -V 2 | + 1 |V 1 -V 3 | , (3)
with V 1 , V 2 , V 3 being the velocities of the escaping electrons.

PCI in cascade double Auger decay (CDA)

This decay occurs when the two Auger electrons are emitted sequentially. It means that the CDA occurs through the creation and decay of an excited state of the Ar 2+ * ion and that the second Auger electron is emitted due to the decay of this intermediate quasistationary Ar 2+ * state:

ω + Ar → e 1 + Ar + * (2p -1 1/2,3/2 ) → e 1 + e 1A (E 2 ) + Ar 2+ * → → e 1 + e 1A (E 2 ) + e 2A (E 3 ) + Ar 3+ ((3p -3 ) 4 S, 2 D, 2 P ) . ( 4
)
The theory for the CDA processes based on the eikonal approach was developed in (Sheinerman 1994). According to this theory the PCI distorted cross section of CDA process is given by :

d 4 σ dE 2 dE 3 dΩ 2 = σ 0 (ω, E (0)
1 ) Γ

(1)

d (E (0)
2 ) Γ

(2)

d (E (0) 3 ) 4π 2 [(E j -E i -E 2 -E 3 ) 2 + Γ 2 1 /4] [(E 3 -E (0) 3 ) 2 + Γ 2 2 /4] × dΩ 1 dΩ 3 |N ξ | 2 exp 2 ξ 1 tan -1 2 (E j -E i -E 2 -E 3 ) Γ 1 × exp -2 (ξ 2 + ξ 3 ) tan -1 2 (E 3 -E (0) 3 ) Γ 2 × 2 F 1 1 + i ξ 1 , -i ξ 2 , 2 + i (ξ 1 + ξ 3 ), 1 - E 3 -E (0) 3 + iΓ 2 /2 E 2 + E 3 + E i -E j + iΓ 1 /2 2 .
(5)

Here 2 F 1 is the hypergeometric function symbol and E i , Γ 1 are the energy and the width of the inner vacancy; Γ 2 is the width of the intermediate quasistationary state of the Ar 2+ * ion; Γ

(1)

d and Γ

(2)

d are the partial Auger decay widths of the initial and intermediate vacancies; E j is the energy of final ionic state. To simplify we assume the factors Γ

(1) d , Γ

(2) d , σ 0 do not depend on the direction of the electron ejection.

The dimensionless parameters ξ i are defined as:

ξ 1 = -1 V 1 + 1 |V 1 -V 2 | ; ξ 2 = -1 V 2 + 1 |V 2 -V 3 | ; ξ 3 = -1 V 1 + 1 |V 1 -V 3 | . ( 6
)
And the normalization factor |N ξ | 2 is given by

|N ξ | 2 = π 2 ξ 1 ξ 2 sinh(π ξ 1 ) sinh(π ξ 2 ) • sinh[π (ξ 1 + ξ 2 )] sinh[π (ξ 1 + ξ 2 + ξ 3 )] × (ξ 1 + ξ 2 + ξ 3 ) [1 + (ξ 1 + ξ 2 + ξ 3 ) 2 ] (ξ 1 + ξ 2 ) [1 + (ξ 1 + ξ 2 ) 2 ] . (7)
It is important to note that in this case the cross section is determined by both the widths of the initial inner vacancy Γ 1 and of the intermediate doubly charged excited state Γ 2 .

The equations ( 2) and ( 5) present the cross sections of the processes when all three electrons are registered in coincidence and have precisely define energies E 1 , E 2 , E 3 . If we want to describe the real experimental conditions when the photoelectron yield is measured and the energy of one of the Auger electrons, for example e 2A , belongs to some definite interval ∆E = E max -E min , then these cross sections have to be integrated over the energy E 3 :

σ(E 1 , ∆E) = Emax E min d 4 σ dE 2 dE 3 dΩ 3 dE 3 . (8)
Of course, carrying out of such an integration we have to take into account the energy conservation law:

E 1 + E 2 + E 3 = ω -E(Ar 3+ (3p -3 )) , ( 9
)
where ω is the energy of the incident photon.

In the same way we can obtain the line shape of the second Auger electron in the CDA process carrying out an integration of the cross section (5) over the energies of the photoelectron E 1 :

σ(E 3 , ∆E) = Emax E min d 4 σ dE 2 dE 3 dΩ 3 dE 1 . ( 10
)

Results of measurements and calculation

To analyze the PCI distortion in DA spectra we have selected two energies of the incident photon ω = 251.63 eV and ω = 260.63 eV that corresponds to excess energy of 3 eV and 12 eV above the 2p 3/2 threshold.

PCI distortion in direct double Auger decay (DDA)

One of our aims is to investigate the contribution of direct interaction between the photoelectron and Auger electrons to the PCI distortion. Hence we have chosen two different intervals for the Auger electrons which are emitted during the DDA, ∆E = 30-70 eV and ∆E = 4.9-5.1 eV. As discussed above, analysis of the coincidence spectra shows that the Auger electrons of these energies are ejected purely in DDA processes. The energies of the second Auger electron are in this case approximately 95 -135 eV and 160 eV respectively. The corresponding measured and calculated spectra of the coincident photoelectron line shape are presented in the figures 3a (for ω 1 = 251.63 eV) and 3b (for ω 2 = 260.63 eV). In each figure the experimental results are shown as error bars, while calculated data are represented by solid lines. A blue color is used when one of the Auger electron energies belongs to the interval (∆E = 30 -70 eV) while red refers to the second group (∆E = 4.9 -5.1 eV). For comparison, the PCI distortion associated with the decay of 2p holes by emission of a single Auger electron and populating Ar 2+ (3p -2 ) final states are represented in black.

First we can see two peaks in the spectra which are associated with the photoelectrons emitted from the 2p 1/2 or 2p 3/2 subshells. The line shapes of these peaks are shifted and distorted by the PCI. The PCI distortion is stronger for smaller energies of the photoelectron in agreement with the general behavior of PCI. It is seen that, generally, the PCI distortion for DDA processes (blue and red curves) is larger than for the single Auger process (black curves) because of the larger variation of the ionic field (on +2 |e| units). However it appears that the red and black curves almost coincide in Fig 3b ): this is when the photoelectron is faster than one of two Auger electrons (of 5 eV) emitted in the DDA decay. In this case and in a classical picture, the photoelectron is never overtaken by the slow Auger electron, and always feels a +1 central charge, as in the single Auger decay.

It is clear that the distortion of the photoelectrons lines associated with Auger electrons of ∆E = 30 -70 eV are always stronger than for Auger electrons of ∆E = 4.9 -5.1 eV. It can be easily understood if we take into account both the interaction of the photoelectron with the ion field and with the slower Auger electron (the interaction of the photoelectron with the fast Auger electron of > 100 eV energy can be neglected). For the photoelectron energies considered here the main contribution comes from the interaction with the ion field. The weaker interaction with the slower Auger electron has an opposite sign and its contribution is larger for the smaller energies of the Auger electron (see the equation ( 3)). As a result, the difference between the photoelectron line shapes associated with Auger electrons of different energies shows clearly the important role of direct interaction between photo and Auger electrons.

We can note quite a good agreement between the measured and calculated spectra. Note that all calculated and experimental curves are normalized at their maximum. For the comparison we have convolved the calculated curves with Gaussian of FWHM = 50 meV for ω 1 and of FWHM = 150 meV for ω 2 to model the experimental resolution. We used the experimental ratio of 0.6 : 1 for the intensities of the 2p 1/2 and 2p 3/2 peaks, as extracted from our measurements at ω 2 = 260.63 eV. Note that this ratio apparently increases at ω 1 = 251.63 eV but no attempt was made to adjust its value in the calculation as it does not improve significantly agreement with experiment for photoelectron energies of less than 1 eV (see below). The energy scale of the calculated curves has been shifted by +140 meV for ω 1 and by +90 meV for ω 2 to match experiment. This discrepancy could arise from the uncertainties in electron and photon energy calibration and in the values of the 2p binding energies that we took from the literature (King et al 1977). Disagreement between calculated and measured data for photoelectron energies of less than 1 eV is expected because the validity of the eikonal approach breaks down for small photon excess energies, and our model does not intend to describe this region precisely. However one notes that the trends of the PCI distortion in this region are also correctly described by our model.

As a preliminary conclusion on DDA processes, we can say that: 1) our coincidence method allows us to reveal the notable contribution to PCI from the interaction between the photoelectron and Auger electrons that strongly depends on the Auger electron(s) energies; 2) the eikonal theory for PCI describes adequately the direct double Auger processes.

PCI distortion in cascade double Auger decay (CDA)

For the analysis of PCI effects in CDA processes we have selected the two strongest cascade paths characterized respectively by the emission of Auger electrons of 2.19 and 8.5 eV. These electrons correspond to the second Auger electron emitted in the CDA path. More precisely they are associated with excited Ar 2+ * intermediate states of 86.31 eV (A1) and 96.94 eV (A2) binding energy which decay respectively towards the Ar +3 (3p -3 4 S) and Ar +3 (3p -3 2 P ) final states (Lablanquie et al 2007).

Characterization of the Ar 2+ intermediate state involved in the cascade double

Auger decay Figure 4 shows high resolution electron spectra of the second-step Auger electron of the two CDA processes that we have selected. They were obtained with the MBS high resolution spectrometer at a photon energy of 270 eV, in order to minimize the PCI distortion of the peaks. Fig 4a shows the 2.19 eV peak, measured with a 2 eV pass energy. Analysis of the peak width in a series of similar measurements reveals a lifetime broadening of 3.7 ± 0.4 meV, taking into account the theoretical 1.3 meV resolution of the analyzer and the 2.9 meV Doppler contributions.

Lifetime broadening of the 8.5 eV peak in Fig 4b) is obviously larger. A 20 eV pass energy was used here and is enough to reveal the doublet structure of the peak, associated with the decay of the A2 Ar 2+ * intermediate state to the 1/2 and 3/2 spin orbit components of the Ar 3+ (3p -3 ( 2 P )) final state. The fit represented in the Fig 4b) uses the literature value of 22 meV for this splitting [START_REF] Ralchenko | NIST Atomic Spectra Database (version 3.1.5)[END_REF] and takes into account the broadening due to the 13 meV theoretical resolution of the analyzer plus the 5.7 meV Doppler contribution. We deduce a 35 ± 5 meV broadening associated with the lifetime of the A2 Ar 2+ * intermediate state. The same procedure as above has been adopted for the comparison of the calculated and experimental results (convolution and shift of the theoretical curves). The main result is that red and black curves are almost exactly the same: PCI distortion is almost the same in single Auger decays and in cascade Double Auger decays. It is easily understood if we take into account that the interaction between the photo-and the second Auger electron occurs at large distances where this interaction is small and contributes only a little to PCI. The main contribution to PCI comes from the interaction of the photoelectron with the ion field and this interaction is the same in the single Auger and in the cascade Double Auger process before the second Auger electron emission.

PCI distortion of the photoelectron lineshape in cascade double

However a closer inspection reveals that if the PCI distortion in single and cascade double Auger paths is the same, within experimental resolution, when we consider path A1 and a 2.19 eV Auger electron in Fig 5, it differs slightly for path A2 in Fig 6 . More precisely, a slight difference is visible in Fig 6a) when the photoelectron is slower than the 8.5 eV Auger electron emitted in the last step of the A2 cascade. The experiment reveals that the smaller the photoelectron energy, the larger the difference. This observation implies that the photoelectron can can be influenced by a second Auger electron if it has a lower kinetic energy than this Auger electron. But to reveal this effect another condition is required: that the lifetime of the Ar 2+ * intermediate state be short enough.

That is why we detect this effect for the A2 intermediate state of 19 fs lifetime (35 meV lifetime broadening) in Fig 6a ) but not for the A1 state of 180 fs lifetime (3.7 meV lifetime broadening) in Fig 5a). (note that these times have to be compared with the lifetime of the 2p hole itself, which is 5.6 fs as deduced from its 118 meV lifetime broadening (Jurvansuu et al 2001).

The calculations reproduce exactly this behaviour, when introducing the lifetime of the Ar 2+ * intermediate. Fig 5 shows that even with a Γ 2 = 15 meV lifetime broadening, no effect is predicted for path A1. Using a 40 meV lifetime broadening for path A2 reproduces correctly the slight difference of PCI distortion for the red curves around 3 eV in Fig 6a). Agreement is less good for photoelectron energies smaller than 1 eV, as expected in this zone where eikonal approach is not valid, but the trend is correctly reproduced.

Finally this effect should be detected also in the kinetic energy distribution of the second Auger electron emitted in the last step of the cascade path. As expected we did not observe, within experimental error bars, any change in the 2.19 eV line shape associated with the decay of A1 state. In the experiment with the high resolution MBS spectrometer, we observed a distortion and an increase of the apparent kinetic energy of the 8.5 eV peak when decreasing the excess energy above the 2p thresholds. However the coincidence experiments, in spite of their lower resolution (120 meV), reveal more clearly the effect thanks to the ability to isolate contributions from the 2p 3/2 and 2p 1/2 decays. The result is displayed in Fig 7, where blue, black and red curves correspond respectively to coincidences of the second step Auger line with a ∼ 12 eV 2p 3/2 photoelectron (from Fig 6b), a ∼ 3 eV 2p 3/2 photoelectron and a ∼ 0.9 eV 2p 1/2 photoelectron (from Fig 6a). A shift of the order of a ∼ 100 meV is observed experimentally. The calculations of the second Auger electron distribution (solid curves) was performed by using formula (10) and (5); we use a value 40 meV for the width Γ 2 of the A2 intermediate state. Calculated curves were convolved with a FWHM = 120 meV Gaussian to take into account the experimental resolution and were all shifted by -50 meV to match experimental observations. We note a rather good agreement between the measured and calculated curves for photoelectrons energies of ∼ 12 eV and ∼ 3 eV.

Hence the PCI distortion of the line shapes of both the photoelectrons and the Auger electrons has been revealed clearly by our investigation of the CDA processes. Note that our approach allows us to distinguish and even select the contribution of different kinds of electron interactions in the total PCI distortion.

Conclusion

Our paper presents for the first time a detailed investigation of the the PCI interactions between a positive ion and three electrons (1 photoelectron and two Auger electrons) in different conditions. Both our experimental coincidence method and our theoretical eikonal approach allow us to reveal reliably the PCI distortion of the photoelectron and the Auger electron line peaks and to show the dependence of this distortion on the energies of the emitted electrons in the DDA and CDA processes. In the case of the CDA decay, the analysis of the second Auger electron line shape allows us to estimate the widths of the intermediate quasistationary Ar 2+ * state. The quite good agreement between calculated and measured line shapes show that the eikonal theory of PCI describes adequately the direct and cascade double Auger decays, it also suggests that PCI distortion can be used to judge if a particular Auger electron is released in a direct or cascade Double Auger process. The methods presented here can be extended to the investigation of inner shells in other atomic or molecular systems. A further step will be to test the dependence of the PCI distortions upon the relative emission angles of the photoelectron and Auger electrons. The present success of our model suggests that it should also predict those dependences reliably. On the experimental front, angleresolved 3 electron coincidences are unfortunately beyond the present experimental capabilities.

Acknowledgments

We are grateful to the Photon Factory staff for the efficient and stable operation of the PF ring. We warmly thank professor John Eland for many discussions and for his constant support. Financial supports from JSPS and CNRS (PICS program no. 3407) are acknowledged. This work was performed with the approval of the Photon Factory Advisory Committee (proposal no. 2004G210). states by the relation : ω = E B (Ar 3+ ) + E1 + E2 + E3. Uncertainty in the energy scale are evaluated to be of the order of 0.3 eV.

Figure Captions

Figure 2: a) Two dimensional coincidence map showing the energy correlations between three electrons detected in coincidence. The sum of the energies of the three electrons is imposed to lie between 160 and 170 eV to ensure that a final Ar 3+ (3p -3 ) is reached. Coincidence counts are represented as a function of two electrons of the triplet only, as the energy of the third electron is determined by the relation : Figure 3: a) PCI distortion of the Ar 2p photoelectron lines associated with a direct Double Auger decay populating Ar 3+ (3p -3 ) final states. Photon energy was ω = 251.63 eV. The kinetic energy of one of the Auger electron has been chosen in the (30-70) eV range (blue) or in the (4.9 -5.1) eV one (red). It is compared with the PCI distortion in a single Auger decay forming Ar 2+ (3p -2 ) final states (in black). Dots with error bars (two standard deviations from the counting statistics) represent experimental results. Lines give the prediction from our eikonal model, after a +140 meV shift of the energy scale and their convolution with a Gaussian of FWHM = 50 meV to account for the experimental resolution. b) Same but obtained at a photon energy ω = 260.63 eV, 12 eV above the 2p 3/2 level. Here, the shift of the calculated line shapes is +90 meV and convolution is done with a FWHM = 150 meV Gaussian.

ω = E B (Ar 3+ ) + E1 + E2 + E3.
Figure 4: Non coincident high resolution electron spectra obtained with the MBS electrostatic analyzer at a photon energy of 270 eV. a) was obtained with a 2 eV pass energy and is fitted by the sum of two 5.2 meV FWHM Gaussians (red line). The component on the high energy side of the main peak is attributed to the contribution of a weak CDA path mediated by an Ar 2+ intermediate different from A1. b) was obtained with a 20 eV pass energy and is fitted by two Voigt profiles (in blue, with their sum represented by the red curve), corresponding to the decay of the A2 intermediate state to the Ar 3+ (3p -3 ( 2 P 3/2 )) and ( 2 P 1/2 ) final states. ) eV filter on the Auger electron energy selects the Ar 2+ * excited state of 96.9 eV binding energy decaying to Ar 3+ (3p -3 ( 2 P )).

Figure 7: PCI shift of the 8.5 eV Auger electron emitted in the A2 Cascade Double Auger path. The data comes from events where three electrons have been detected in coincidence and are associated to the formation of a final Ar 3+ (3p -3 ) state. Blue, black and red curves correspond respectively to coincidences of the second step Auger electron with a ∼ 12 eV 2p 3/2 a ∼ 3 eV 2p 3/2 photoelectron and a ∼ 0.9 eV 2p 1/2 photoelectron, see text.

  photoelectron lines, in order to avoid possible shifts due to contact potentials. Slow drift of the peak positions occurred with time and were minimized by short scans of less than a few minutes.The main part of the results were acquired with a magnetic bottle time of flight analyzer. This kind of apparatus has been extensively described before(Eland et al 2003, Penent et al 2005, Hikosaka et al 2006, Eland 2009) and will only be briefly introduced here. Ionization occurs close (∼ 1 mm) to a strong (∼ 0.5 T) permanent magnet in a zone where the strong divergent magnetic field creates a magnetic mirror. Almost all released electrons are guided along the magnetic field lines through a long (2.5 m) solenoid (∼ 10 -3 T) to the detector. A multi-hit time-to-digital converter then records the arrival times of each electron with respect to the light bunch and stores them in on-line computers for further analysis. Time to energy calibration was performed by measuring photolines associated with He + (N=1) and He + (N=2) formation at known photon energies. The experiments were performed at the Photon Factory and used light from the undulator beam line BL-16B(Shigemasa et al 1998). The photon bandwidth was set at 30 meV. Single bunch operation of the storage ring provided 200 ps-wide light pulses every 624 ns. It follows that time of flight of isolated electrons could only be measured modulo the 624 ns inter light pulse period. It was however not needed to use a mechanical chopper(Ito et al 2009) to increase this period in this favourable case: it is possible to retrieve absolute electron time of flights in coincidence events where a 2p photoelectron and all associated Auger electrons are recorded, because one of the Auger electrons is then necessarily a fast one, with a flight time shorter than 624 ns. Typical count rate in coincidence experiments was 3000 Hz, which allows us to neglect random coincidences between electrons produced by subsequent light pulses.The key point for the success of the present experiment is its ability to define precisely the conditions in which PCI is observed, thanks to the coincidence techniques. Fig1 a) and 1 b) demonstrate first the ability to resolve the final states reached respectively in double and triple ionization events. The histogram of the sum of the kinetic energy of two electrons detected in coincidence in Fig1 a) reveals the Ar 2+ final state. Most of the events correspond to the creation and Auger decay of a 2p 1/2 or a 2p 3/2 innershell hole, along with a minor valence double ionization contribution. Similarly, the final Ar 3+ states populated by triple ionization are obtained in Fig.1 b) through the histogram of the sum of the kinetic energies of three electrons detected in coincidence, and they originate mainly from the decay of 2p holes by emission of two Auger electrons. The resolution, slightly improved compared to our previous study(Lablanquie et al 2007), allows us to distinguish but not isolate fully the Ar 3+ (3p -3 ) 2S+1 L final states. A clear separation between direct and cascade Double Auger decays is not always possible. For instance, the two processes overlap heavily in the case of the 4d decay in Xenon(Penent et al 2005), but the 2p decay in Argon offers a favorable system, where distinction is possible(Viefhaus et al 2004, Lablanquie et al 2007). This is demonstrated in Fig 2, which presents a 2 dimensional coincidence map of the 2p DA decay measured at the same ω = 251.63 eV photon energy as for Fig 1 a) and b). Only events where

  Fig 3, 5 and 6 display the 2p photoelectron line shapes associated with the emission of such electrons, they are obtained by projection on the x axis of the horizontal bands in Fig 2 a) associated with these Auger electrons.

  Auger decay (CDA) Fig 5 and 6 display in red the distortion of the photoelectron lineshape in CDA paths mediated respectively through the A1 and A2 Ar 2+ * intermediate states, which emit a 2.19 or 8.5 eV secondary electron upon their decay. For comparison we also give in black and blue the same curves as in Fig 3, to show the PCI distortion in a single Auger decay (black) and in a direct path (blue) where the two emitted Auger electrons are rather fast (around 50 ± 20 eV and 115 ± 20 eV).

Figure 1 :

 1 Figure 1: a) Ar 2+ states populated by photons of ω = 251.63 eV. The histogram was obtained from the energy sum of all two-electron pairs detected in coincidence.It is converted to binding energy of the final Ar 2+ states by the relation : ω = E B (Ar 2+ ) + E1 + E2. The discretisation step is 50 meV. Accumulation time was 4 hours and the number of electrons detected was kept constant at 3000/s by adjusting the width of the monochromator exit slit, in order to compensate for the decrease with time of the photon flux. b) Ar 3+ states populated by photons of ω = 251.63 eV. In the same way as for a), the histogram was obtained from the energy sum of all three electrons detected in coincidence. It is converted in binding energy of the final Ar 3+

  Intensity is encoded in color with a log scale. Green rectangles visualize the energy regions analyzed in Fig 3, 5 and 6. b) displays the projection of the coincidence map onto the ordinate axis. It gives an image of the energy distribution of one of the electrons emitted by ω = 251.63 eV photons when producing Ar 3+ (3p -3 ) states.

Figure 5 :

 5 a) In red: PCI distortion of the Ar 2p photoelectron lines associated with the cascade Double Auger decay. Photon energy was ω = 251.63 eV. A filter on the kinetic energy of the second Auger electron has been imposed, (2.1 -2.3) eV, so that we select the Ar 2+ * excited state of 86.31 eV binding energy decaying to Ar 3+ (3p -3 ( 4 S)). It is compared with the PCI distortion in single Auger (black) and double Auger decays (blue) from figure3. Other conventions are the same as in Figure3. b) Same but obtained at a photon energy ω = 260.63 eV .

Figure 6 :

 6 Same as for Fig 5, but the (8.2 -8.7

  FIGURE 1