Parametrization of energy sharing distributions in direct double photoionization of He

We present experimental results on the characteristic sharing of available excess energy, ranging from 11–221 eV, between two electrons in single-photon direct double ionization of He. An effective parametrization of the sharing distributions is presented along with an empirical model that describes the complete shape of the distribution based on a single experimentally determinable parameter. The measured total energy sharing distributions are separated into two distributions representing the shake-off and knock-out parts by simulating the sharing distribution curves expected from a pure wave collapse after a sudden removal of the primary electron. In this way, empirical knock-out distributions are extracted and both the shake-off and knock-out distributions are parametrized. These results suggest a simple method that can be applied to other atomic and molecular systems to experimentally study important aspects of the direct double ionization process.

overview of the possible relaxation network is shown in Fig. 1. In addition to the KO and SO mechanisms, recent experimental studies have identified a third ionization mechanism called the quasifree mechanism 6,7 . The mechanism was predicted by Amusia et al. 8 in 1975 and results in the two electrons being emitted back to back with similar energy. The mechanism relates to the non-dipole part of the interaction and its contribution to the direct double ionization cross section is therefore small (~1% at a photon energy of 800 eV) 6 . It is therefore assumed negligible for the energy range considered in this study. For the rest of this work, we will focus primarily on the two paths in Fig. 1 that leads to double escape. A total, fully differential transition amplitude for double escape was defined by Pattard et al. 2 in terms of a sum of two independent transition amplitudes representing KO and SO, By taking the modulus squared of Eq. 1 and integrating over the angles, one gets the total energy sharing distribution, S tot as f i f i tot , SO 2 , KO 2 int where C int describes the interference between the two mechanisms and ε the kinetic energy of an electron. Based on the definition of SO and KO, proposed by Pattard et al. 2 , and the assumption of negligible interference (C int = 0), as suggested for He 4 , we can write this distribution as a mixture of two independent distributions representing the energy sharing distribution of the KO and SO mechanism, respectively. Strong angular differences in how the two mechanisms populate the same energy eigenstates is thought to explain why the role of interference appears to be negligible, which has gained further support by angularly resolved experiments 9,10 . The SO mechanism depends, by definition 2 , only on correlations in the initial state wave function, while the KO mechanism takes all post-absorption electron-electron interaction into account. Hence, dynamical information from the energy distribution during the post-absorption process is exclusively found in S KO . Experimental data with information reflecting the dynamics on time-scales considered in direct double photoionization is difficult to obtain. Experimental benchmarks for S KO are thus interesting for indirect testing of the validity of different dynamical models for direct double photoionization.

Results
By detecting the kinetic energy of both ejected electrons in a direct double ionization event, one can estimate S tot experimentally. Having an experimental estimate of S tot and an analytical formula for S SO at hand, it is straightforward to extract S KO . To arrive at a formula for S SO , we rely on the same procedure as Schneider et al. 3,4 . We assume that one electron has been removed suddenly, and that the residual electron can be described with a hydrogenic ns-wavefunction, ψ(r), with effective charge Z eff . We let the wavefunction of the residual electron collapse into a free electron Coulomb wavefunction, ν, where ν corresponds to an s-wave. From conservation of energy, www.nature.com/scientificreports www.nature.com/scientificreports/ which means that the kinetic energy of either the primary, ε 1 or secondary electron, ε 2 , immediately defines the energy of the other. As the electrons are indistinguishable, we symmetrize a SO about ε = E/2. Integrating over the angles of the shake-off electron, the final form for the energy sharing distribution of the SO mechanism reads We estimated S tot experimentally for several excess energies in the range between 11-221 eV, using a coincidence detection technique described in the methods section. The coincidence detection setup allows identification of events with a particular total kinetic energy of two electrons originating from the same ionization event. The total kinetic energy of the two electrons can be used to identify the double ionization potential (DIP) 1 2 corresponding to a particular final state that the process led to. Such identification can be seen in the 2D-histogram in Fig. 2. This figure shows how a particular DIP correlates with two individual electron kinetic energies. The graph on the top of this histogram represents the projected intensity onto the x-axis and the graph on the left side represents the projection onto the y-axis. We obtain the experimental shape for S tot by forming a graph similar to the one shown on the y-axis in Fig. 2 but only for the electrons involved in the formation of He 2+ . To extract good estimates of the distributions it is crucial to not only maximize the signal-to-noise level but to remove the influence of the noise profile in each bin forming S tot . To ensure this, noise estimates were subtracted for each selected DIP region. The intensity and shape of the noise was estimated by taking slices of the same size adjacent to the final state in the 2D-map. The subtracted intensity map was integrated over the final state region to form S tot . By normalizing S SO so that KO tot S O after having normalized S tot to a proper probability distribution, such that In this way, P SO (E) and P KO (E) = 1 − P SO (E) correspond the conditional probabilities of SO and KO given double ionization, respectively. We define S SO according to Eq. 6 with Z eff = Z SO = 1.49, as suggested by Schneider et al. 4 . An example of how the experimental estimate of S tot is used to extract an estimate of S KO is shown in Fig. 3, and a sample of measured energy sharing distributions is shown and compared with previous theoretical and experimental distributions in Fig. 4.
At low excess energies, the two electrons share the energy in a nearly uniform way and S tot is nearly flat. However, at higher excess energies S tot takes on a ∪-shaped form with edges that become sharper as the excess energy increases. This is a well known behaviour 3,4,9,11-14 but has not been parametrized or modelled www.nature.com/scientificreports www.nature.com/scientificreports/ systematically over a large range of excess energies. The shape of the distributions can be well reproduced using the model function    Fig. 5 show the optimized f-values for each experiment and distribution. The results indicate a linear trend with excess energy for all three distributions. This suggests that we can set f(E) = −kE and simplify the model to where k is the estimated slope of the linear trend. The model depends now on a single experimentally determinable parameter, and can be related to the differential equation is an odd function that ensures that the derivative changes sign at ε = E/2. The formula in Eq. 12 is obtained by solving Eq. 13 after choosing g(ε) as a first order polynomial but the odd function g(ε) could in principle take on other forms. The same fit procedure was performed for the theoretical distributions presented by Schneider et al. 3,4 and the estimated fit parameters of these distributions suggest a linear trend with excess energy as well. The fitted values for the k-parameters are summarized in Table 1. At ε = E/2, the model reduces to an exponential function that decays as E increases. In principle, this means that the shapes can be readily determined simply by fitting an exponential function to the ratio between the maximum and minimum value of a distribution for a few excess energies. It is clear from Fig. 5 that the SO, KO and total distributions all change gradually from flat to ∪-shaped as the excess energy increases. The S SO distribution is always more ∪-shaped and sharper at the edges than the S KO distribution.

Discussion
Energy exchange by scattering becomes less likely when the two electrons have a large difference in their momenta. Hence, it becomes more likely to observe a secondary electron that has received just enough kinetic energy to be ionized when the primary electron is fast. This explains the ∪-formation of the KO distributions and its dependence on excess energy. The model for the SO distributions is constructed by the sum of a decaying function, centred at ε = 0, and the same function but reflected and centred at ε = E (see Eq. 6). The derivatives of these two partial functions, and hence the shape of the SO distribution, is defined by the effective shake charge used in Eq. 4. The photon probes the correlated two-electron wave function of the initial state and essentially filters on the primary electron positions that are most likely to absorb at a given frequency. This filtering of the initial state wave function defines the effective screening felt by the shake electron. Hence, the effective shake charge may depend on energy, and to treat it as constant may not hold true over a large range of excess energies. The significance of a varying effective shake charge is an interesting aspect but is difficult to probe with this technique, and such analysis would require clean SO-distributions recorded over a long range of photon energies.
However, the gradual ∪-shape formation of the SO distributions is still expected when assuming constant shapes for the two partial functions in Eq. 6 (i.e. keeping the effective charge fixed). The two functions will overlap more strongly at the symmetry point, ε = E/2, when the excess energy is low, and effectively flatten the SO www.nature.com/scientificreports www.nature.com/scientificreports/ distribution. In contrast, as the excess energy increases, the overlap at the symmetry point reduces, and a more pronounced ∪-shape is formed. The effect can be seen for the two partial functions at two arbitrary excess energies in Fig. 6a,b, and the resulting SO distributions for the two excess energies are compared in Fig. 6c. Figure 6 also illustrates the interesting transition region where the discrete and continuum parts of the spectrum meet. As far as we know, this transition region has not yet been studied in single-photon double ionization of He, in terms of the two ionization mechanisms. Studying the transition region could potentially give further insights into the interplay between the transition channels leading to satellite states (KU and SU) and the channels leading to direct double ionization (KO and SO). Referring to the smooth merger of the Rydberg series and the continuum in single-photon absorption cross section of hydrogen, one might expect a similar merging of the SU Rydberg series and the SO continuum for high excess energies. If so, there might even be an observable enhancement of the higher n s-states due to a continued overlap of the two partial distributions into the discrete part of the spectrum. This is, however, beyond the scope of the present study.

conclusion
We have demonstrated a method to parametrize the energy sharing distribution of two electrons ionized in single-photon double ionization of He. The method relies on a single parameter that can be readily obtained by comparing the minimum and maximum value of the distribution curve. The parameter sets the shape of the distribution and how it depends on the excess energy. The complete experimental distributions were used to indirectly extract the KO distributions by using a theoretical SO model. This method requires no additional angular information for testing the difference of KO and SO. In using theoretical SO distributions, experimental KO distributions were analyzed and parametrized in a similar manner as the total sharing distributions. A parametrization and full characterization of the SO and KO mechanisms was achieved separately for excess energies between 11-221 eV. The separation revealed that there is a significant difference in how the ∪-shape evolves for the two ionization mechanisms. This way of separating the KO distribution could potentially serve as a relatively straightforward way for experimentalists to form benchmark distributions for testing dynamic correlations in single-photon double ionization events. Furthermore, this way of separating the two contributions can help in probing the importance of interference between KO and SO at intermediate excess energies. Finally, a proper energy sharing characterisation could simplify analyses of doubly ionized states, formed by a mixture of direct double ionization and single ionization followed by autoionization. A proper subtraction of the continuous direct double photoionization curve from such mixture spectra would greatly simplify the analysis of the interesting interference phenomena between discrete and continuous states in the continuum.

Methods
The experiments were carried out at beam line U49/2-PGM-2 of the BESSY-II storage ring in Berlin, Germany. The beam line allows selecting photon energies over a large energy range (~80-2000 eV) with high precision and accuracy. The photon energy sets the excess energy as E = hν − DIP, where DIP is the double ionization potential. The storage ring was operated in single bunch mode at approximately 1.25 MHz repetition rate. To minimize the risk of detecting slow and fast electrons originating from different light pulses, the repetition rate was reduced to about 78 kHz by a synchronized mechanical chopper 15 . A repetition rate of 78 kHz corresponds to a detection window for electron flight-times up to about 12 μs, which is long enough to extract even the slowest electrons. Typical count rates for these experiments were about a few hundred electron detections per second, and the accumulation time for each excess energy about one or a few hours. The magnetic bottle spectrometer that was used 16 www.nature.com/scientificreports www.nature.com/scientificreports/ collects essentially all electrons emitted from an ionization event by using a strong, divergent magnetic field in the interaction volume. The magnetic field is created by a ~1 T permanent neodymium-iron magnet and the field geometry is shaped by a conical soft iron pole piece. The electrons are guided towards a 2.2 m long flight tube held at ~10 −6 mbar pressure. The flight tube is surrounded by a solenoid producing a weak axial magnetic field that couples to the strong field. The coupling of the two magnetic fields leads to a "bottle-neck" shape of the resulting magnetic field lines. The solenoidal field guides the electrons on spiral trajectory towards a micro-channel plate detector located at the end of the tube. The collection-detection efficiency of the spectrometer is about 55%, but most importantly effectively constant over the measured range of kinetic energies as explicitly shown in the works of Hult Roos et al. 17,18 , which were carried out with the same spectrometer during the same experimental campaigns as the current work. This also implies that the transmission function of the spectrometer is essentially constant over the kinetic energy range of concern for the present work. Furthermore, the collection-detection efficiency mentioned implies that it is equally likely to miss an electron pair over the whole distribution and should hence not introduce any intensity biases. However, the MCP detector has a short dead time, and two electrons of the same or very nearly the same kinetic energies are thus not detectable. This can be observed as a clear intensity drop at the center of each recorded energy sharing distribution. There is a potential risk of enhanced intensity close to the drop at the center of the distribution if the two electron signals recorded in coincidence are close enough in time to effectively overlap each other. This could lead to an enhanced signal voltage, which could make it more probable to trigger and detect an electron pair of similar (but not equal) kinetic energies. If present, this effect should primarily affect the high excess energies. The inverse relationship between flight-times and kinetic energies shifts the center-point of the sharing distributions shifts toward shorter time-of-flights, which would make an overlap in the time-domain extend over a larger region in the energy domain. The overall risk of detection biases appears to be small in the current experimental data and the corresponding effect on the analysis negligible.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.