Fast calculator for X-ray emission due to Radiative Recombination and Radiative Electron Capture in relativistic heavy-ion atom collisions

Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx

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Fast calculator for X-ray emission due to Radiative Recombination and Radiative Electron Capture in relativistic heavy-ion atom collisions M.O. Herdrich a,b,c,⇑, G. Weber a,c, A. Gumberidze c, Z.W. Wu a,d, Th. Stöhlker a,b,c a

Helmholtz-Institute Jena, Fröbelstieg 3, 07743 Jena, Germany Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany c GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstraße 1, 64291 Darmstadt, Germany d Key Laboratory of Atomic and Molecular Physics & Functional Materials of Gansu Province, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, People’s Republic of China b

a r t i c l e

i n f o

Article history: Received 15 December 2016 Accepted 5 April 2017 Available online xxxx Keywords: Highly charged heavy ions GSI/FAIR Radiative Recombination Radiative Electron Capture ESR Characteristic X-ray lines

a b s t r a c t In experiments with highly charged, fast heavy ions the Radiative Recombination (RR) and Radiative Electron Capture (REC) processes have significant cross sections in an energy range of up to a few GeV=u. They are some of the most important charge changing processes in collisions of heavy ions with atoms and electrons, leading to the emission of a photon along with the formation of the ground and excited atomic states. Hence, for the understanding and planning of experiments, in particular for Xray spectroscopy studies, at accelerator ring facilities, such as FAIR, it is crucial to have a good knowledge of these cross sections and the associated radiation characteristics. In the frame of this work a fast calculator, named RECAL, for the RR and REC process is presented and its capabilities are demonstrated with the analysis of a recently conducted experiment at the Experimental Storage Ring (ESR) at the GSI Helmholtz Center for Heavy Ion Research in Darmstadt, Germany. A method is presented to determine unknown X-ray emission cross sections via normalization of the recorded spectra to REC cross sections calculated by RECAL. Ó 2017 Published by Elsevier B.V.

1. Introduction Experimental studies on accelerated and highly charged heavy ions at the GSI Helmholtz Center for Heavy Ion Research (GSI) in Darmstadt, Germany have proven to provide deep insights into atomic structures and interactions in the regime of extreme field strengths [1–4]. The Facility for Antiproton and Ion Research (FAIR), which is currently being built at the site of the GSI, will open up new possibilities for ion beam experiments covering virtually the full range of kinetic energies from near rest to several GeV=u for all charge states of projectiles ranging from hydrogen up to uranium [5]. At the CRYRING@ESR low-energy heavy ion interactions with electrons and atoms can be studied with projectile energies of a few MeV=u down to hundreds of keV=u [6]. In contrast, studies of heavy-ion atom collisions at the internal gas target of the HighEnergy Storage Ring (HESR) will feature heavy-ion beam energies up to several GeV=u [7]. For the planning and interpretation of future experiments covering this vast range of parameters, it is ⇑ Corresponding author at: Helmholtz-Institute Jena, Fröbelstieg 3, 07743 Jena, Germany. E-mail address: [email protected] (M.O. Herdrich).

crucial to have a good understanding of the most important interaction processes occurring during the collisions. Of particular interest are the Radiative Recombination (RR) and the closely related Radiative Electron Capture (REC) as they have significant cross sections in the whole regime of considered energies and may lead to the emission of distinct X-ray lines. The RR process describes the capture of a free electron into a vacancy of the projectile ion accompanied by the emission of a photon (i.e. the time-inversed photoelectric effect). The photon energy is determined by the sum of the kinetic energy of the incident electron and the binding energy of the final projectile state. Within the context of storage ring experiments the RR process mainly occurs in the electron cooler section where it leads to a charge change of the projectiles. As a consequence it may contribute significantly to the beam loss [8,9]. It also is important for the understanding of hot plasmas as they occur in astrophysics, in electron beam ion traps or in fusion experiments [10]. The REC process on the other hand describes the capture of a bound electron into the projectile ion accompanied by the emission of a photon. It can be observed in collisions between ion beams and dedicated targets. But it also occurs when stored ions collide with residual gas atoms in the beam line. Especially for heavy ions at relativistic speeds this

http://dx.doi.org/10.1016/j.nimb.2017.04.022 0168-583X/Ó 2017 Published by Elsevier B.V.

Please cite this article in press as: M.O. Herdrich et al., Fast calculator for X-ray emission due to Radiative Recombination and Radiative Electron Capture in relativistic heavy-ion atom collisions, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.04.022

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strongly impacts the lifetime of the stored ion beams [11] and therefore is important to characterize rigorously. If the field strength induced by the bypassing projectile is big compared to the initial binding potential of the target electrons and also the ions have a high velocity compared to the bound electron momenta, it is valid to treat the bound electrons as being quasifree. In the socalled Impulse Approximation (IA) the REC cross sections can be estimated by convoluting the RR predictions with the momentum distribution of the bound target electron prior to the collision [12]. The total cross section for recombination into the K-shell (K-RR) can be estimated conveniently up to beam energies of roughly 1 GeV=u using the non-relativistic Stobbe-equation. [12]. The calculation of the angular distribution and the polarization of the emitted photons, however, requires a fully relativistic approach and therefore can be numerically very expensive. Existing data sets created by Ichihara and Eichler [13] are sparse and, moreover, do not feature predictions about the polarization of the RR radiation. To improve this situation, in the frame of this work a database was created to enable a fast calculation of RR/REC radiation characteristics within a large parameter range. This calculator program, in the following referred to as RECAL, yields results by fast interpolation on a dense, precalculated data grid. Theoretical methods developed and implemented by Fritzsche and Surzhykov were used to generate the entries of this database as described in chapter 2. A comparison between the interpolated data and time consuming, rigorous calculations shows very good agreement as presented in chapter 3. Furthermore, an evaluation of experimental data obtained at the internal gas target of the ESR storage ring at GSI was performed to demonstrate the usefulness of the database.

2. Methods The computation of REC angular differential cross section and angular-dependent linear polarization data with the RECAL calculator is performed in two separate steps. RR calculations The theoretical treatment that was used to calculate the entries for the RR radiation process database was provided by Fritzsche and Surzhykov [14,15]. This method describes the RR process using the density matrix approach. Both, the differential cross section and the degree of linear polarization can be extracted from the finale-state matrix using the so called detector operator and the density operator itself. This method requires that the initially free electron states are expressed as an infinite sum over all eigenstates of the ion. For practical reasons, the expansion is approximated by taking a finite amount of partial waves j into account. Surzhykov kindly provided a script which implements the whole procedure for the recombination of electrons into initially bare ions for arbitrary capture orbitals, atomic numbers and kinetic energies. As shown in previous publications presenting the first steps towards the RR/REC calculator [16], for the calculation of accurate RR data good knowledge of the minimum number of partial wave orders that need to be included is crucial. To be more specific, too few partial wave orders lead to a distortion of the angular differential data, that is particularly expressed for forward emission angles. On the other hand, taking more orders into account makes the calculations slower as computation time scales with a factor of approximately j3 . Finally, if contributions of too many partial waves are being summed over, numerical artifacts start adding up and strong oscillations of the data become immanent. That is why in the frame of this work a method was applied that searches for the optimal j for each set of parameters (i.e. collision energy, projectile atomic number and capture orbital). The algorithm scans over a range of j-parameters and calculates the maximum relative deviation between the data sets for two consecutive numbers of j. A convergence is found when all points stay

within a predefined error margin a ¼ 1%. This is illustrated in Fig. 1 for the case of U92þ with a kinetic energy of 500 MeV=u for the angular differential cross section of the Radiative Recombination into the initially bare K-shell. To increase the efficiency of the algorithm, before starting the evaluation of RR cross sections and polarization values over a large range of parameters, an estimation for the optimal j is prepared for each collision system. For that a prescan over a coarse parameters grid is performed. The optimal number of partial wave orders can be estimated by interpolating between the prescanned data points. Finally, the interpolated j-values are used as a guess value for the optimum search of the remaining calculations. REC calculations REC radiation characteristics can easily be generated utilizing the RR database. According to the Impulse Approximation the bound target electron is approximated by a free electron that has a momentum distribution matching the one of the bound electron. The orbital-wise momentum distributions are taken from tables of Biggs et al. [17], who computed the data using Hartree–Fock wave functions. To fold the target electron momentum distribution with the RR cross sections that depend on the relative momentum between the target electron and the projectile system, first, the corresponding kinetic energy of the electron in the projectile system has to be determined. For that a relativistic transformation of the electron momentum parallel to the beam axis is performed [18]. Initial electron momenta perpendicular to the beam axis can be neglected, because they are small compared to the high projectile velocities. For each electron energy in the projectile system the corresponding RR data sets are extracted from the database and a weighted summation is performed incorporating the momentum distribution. If the target atoms are populated by more than one electron, the calculations are performed for each target orbital individually and finally all spectra are superposed taking the number of electrons per orbital into account. 3. Results RR Database For the K- and L-RR a fine grid of data sets spanning the collision energy and atomic number has been generated. The calculations took about half a year on a standard desktop PC. Available energies range from 1 MeV=u up to 1 GeV=u for 15 elements

Fig. 1. The graph shows the convergence behavior of the differential RR cross section for various values of the maximum order of partial waves j taken into account. For U92þ at 500 MeV=u beam energy the K-RR was calculated with maximum partial wave orders between j ¼ 2 and the optimum value j ¼ 20. With increasing j the cross sections (dashed lines) converge towards the final result (bold solid line).

Please cite this article in press as: M.O. Herdrich et al., Fast calculator for X-ray emission due to Radiative Recombination and Radiative Electron Capture in relativistic heavy-ion atom collisions, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.04.022

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between hydrogen and uranium. All calculations were performed for recombination into initially bare ions only. A recombination into an ionic system where electrons are already present can be approximated by using an effective nuclear charge for the projectile, while ignoring electron–electron interactions. The latter is justified when the binding energy of the captured electron is much larger than the contribution by electron–electron interactions. Generating angular differential cross section and linear polarization data for the M-RR was more challenging, because the typical required number of partial wave orders that had to be included is higher than for the capture into the K- and L-shell of the projectile. For some cases convergence could not be reached, as j approached the numerical limit of the current code. Therefore, the database does currently not contain a grid for the M-RR as dense as for the recombination into the lower orbitals. In particular, the capture into the 3d3 and 3d5 substate could not yet be 2

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included into the database. However, the cross section for the recombination into orbitals with high orbital angular-momentum is in general much smaller than for the s and p states. Tabulated cross sections that were calculated rigorously using the transition matrix elements of the RR process [13] were used to validate the angular differential cross sections generated by RECAL. A visualization of this comparison is presented in Fig. 2. Overall, the point-wise relative deviation of our results from the rigorous data was found to be well below 1% for all entries of the database. Compared to the results generated by a previous version of RECAL [16] this is a significant improvement. Note that due to the lack of reference data sets, the RECAL output for the linear polarization of RR radiation could not be validated in the same way. Demonstration of RECAL X-ray spectra from a heavy-ion atom collision experiment at the internal gas target of the Experiment Storage Ring (ESR) at GSI were used as a test case for the application of the RECAL program. U91þ projectile ions with a beam energy of 400 MeV=u were overlapped with a H2 gas-jet target. Several Ge (i) hard X-ray detectors were placed around the target chamber at observation angles of 90 ; 120 and 150 with respect to the beam axis. Furthermore, particle counters were installed in the first bending section after the target area to detect ions with devi-

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ating trajectories resulting from a charge change that occurred during the ion-target interaction. The signal of these particle detectors was used in coincidence and anti-coincidence mode with the semiconductor detectors to select photon signals resulting from electron capture processes. A detailed description of the setup and the experimental procedure can be found in [19]. To demonstrate the usability of the RR/REC calculator developed within the frame of this work, REC cross section data yielded by RECAL was utilized to obtain an estimation of the emission cross sections of other features in the spectrum, namely the K a1 and K a2 lines. As these transitions are arising mainly from REC into excited states of the projectile, a correct estimation of the K a cross sections should in good approximation be consistent with REC predictions for capture into the n > 1 orbitals. Fig. 3a exemplarily shows the X-ray spectrum recorded by the Ge(i) detector placed under 120 in coincidence with the particle detector for down-charged projectiles. The spectrum contains two significant features. When an electron is captured into an ion orbital above the ground state of the system, an excited electronic state is created. Almost instantaneously this state will relax into a lower energy state by emitting a characteristic X-ray photon. The energy of this photon is determined by the energy separation of the involved states. Thus, besides the REC radiation several characteristic transition lines (K a ; K b ) are present in the spectra. A term diagram of He-like uranium including the transitions leading to the emission of K a radiation is presented in Fig. 3b. Emission cross sections of characteristic transitions in few-electron ions are in general not trivial to compute, as they do not result from direct population of the excited states alone. Complex decay cascades from higher orbitals into the contributing excited states have to be taken into account as well [20]. Moreover, the population of excited states can be caused by different electron capture processes. However, in the present case the use of a high collision energy together with a low-Z target rules out significant contributions by non-radiative electron capture as well as resonant capture processes like dielectronic recombination. Thus, the population of the excited projectile states is almost exclusively due to the REC process. Moreover, for the experimental parameters of the present work the REC process cross section decreases roughly with the square of the principal number n2 ,

Fig. 2. The three graphs show angular differential RR cross sections for different parameters. Solid lines are interpolated data sets yielded by the RECAL calculator described in the present work, while symbols denote rigorously calculated values [13]. On the left, data is presented for the K-RR of U92þ for several collision energies. The center plot shows K-RR cross sections for bare projectile ions with different atomic numbers and a kinetic energy of 547 MeV=u. On the right, cross sections for the recombination into different orbitals are plotted for U92þ with 547 MeV=u. Overall, very good agreement is found.

Please cite this article in press as: M.O. Herdrich et al., Fast calculator for X-ray emission due to Radiative Recombination and Radiative Electron Capture in relativistic heavy-ion atom collisions, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.04.022

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Fig. 3. (a) An X-ray photon spectrum that was recorded during the heavy-ion atom collision experiment described in [19]. The experiment was performed using a U91þ beam with a kinetic energy of 400 MeV=u and H2 as a target. A Ge(i) X-ray detector was placed under an observation angle of 120 with respect to the beam axis. The displayed signal contains only events that occurred in coincidence with hits on a particle detector for down-charged projectiles. (b) The term diagram shows energy levels of He-like uranium with one electron in the K-shell and the other in an excited state. Transitions to the ground state and their contribution to the characteristic K a1 and K a2 lines are indicated.

resulting in a negligible contribution of n > 3 orbitals to the relaxation cascades. This means that a theoretical estimation of the K a cross section will be mainly determined by the L- and M-REC whose peaks together with the one of K-REC are also clearly visible as separated entities in the spectrum. As a first step, a simulation was performed using Geant4 [21] containing the most important components of the experimental setup to verify our understanding of incident REC radiation and to test our model of the Ge(i) X-ray detectors. The REC radiation for the Monte Carlo simulation was computed by RECAL as follows: To take the electron of the initially H-like projectile ion into account, an effective nuclear charge was applied, that was determined using detailed theoretical descriptions of the binding energies taken from [22]. The electron–electron interaction in the final state of the He-like ion was neglected, assuming that for example the sum of the REC cross sections for capture into the 21 P 1 and 23 P 2 states is equal to REC into 2p3 of a U92þ ion having a nuclear 2

charge equal to the effective nuclear charge of U91þ . Note that the energy splitting due to the electron–electron interaction is not resolved by our X-ray detectors. Finally, for capture into the K-shell of U91þ the RECAL cross section was divided by two to account for the already filled vacancy. The geometry of the setup was modeled after the experiment and consisted of a germanium crystal, a Pb/Cu slit collimator (to reduce the Doppler broadening by limiting the observation angles covered by the detectors) and vacuum windows made of beryllium or stainless steel (see Fig. 4a). Note that in order to reproduce the energy-dependent detector efficiency inferred from the relative intensities of the measured REC peaks the thickness of the detector crystals had to be adjusted by a factor of 0:85. This seems to be a systematic deviation for all simulated detectors that has not been fully understood yet. However, since the factor was the same for all three detectors, even though the position of the REC peaks at the different observation angles differs significantly due to the Doppler shift (90 : K-REC at 256 keV; 150 : K-REC at 151 keV), we assume that this effect is not due to a wrong estimation of the REC cross sections yielded by RECAL. Additionally, the simulated spectra were convoluted with a Gaussian distribution to incorporate the intrinsic detector resolution of 700 eV (90 ), 800 eV (120 ) and 500 eV (150 ) at 60 keV photon energy. Fig. 4b exemplarily shows the comparison between the experimental data recorded by the detector placed under 120 and the spectrum that was simulated using the respec-

tive parameters. Overall, a good agreement for the K-REC and LREC is achieved. The M-REC-peak was slightly underestimated by the simulation for all setups. As mentioned above, the data sets for REC into the 3d-orbitals were not available for the simulation, however the expected error should be below 1% and therefore cannot explain the discrepancy. After optimizing the simulation parameters, in a second step, the absolute detection efficiency of the setup was estimated by simulating a continuous energy distribution and normalizing the detected photon count per energy with the number of emitted photons. The experimental X-ray spectra measured in coincidence with the ion-counter signals were divided by the simulated efficiency to yield the efficiency-corrected X-ray spectra. To obtain the absolute emission cross sections of the K a1 and K a2 lines the normalization was then performed by calculating the area under the X-ray-peaks and dividing them with the contents of either of the REC-peaks. That way for every REC-peak at each observation angle a value for the characteristic transition cross section could be determined. Using of the Lorentz-transformation the cross sections were translated into the projectile system. The resulting emission cross sections of the K a1 and K a2 peaks are plotted in Fig. 5. The standard deviation of the estimated K a1 cross sections for all angles and REC normalizations is 3% and 4% for the K a2 transition, respectively. Within the range of this uncertainty our expectation to find a nearly isotropic distribution of characteristic transitions K a1 (21 P 1 ; 23 P2 ! 11 S1 ) and perfectly isotropic K a2 (23 P1 ; 23 S1 ! 11 S1 ) of the He-like uranium ions [23] is sufficiently satisfied. As mentioned above, a rough estimation for the K a cross sections can be made by assuming, that most of the observed K a radiation arises from electrons captured into the Lshell and M-shell of the ions. The 2p3 -REC populates the 21 P 1 and 23 P2 states of the resulting 2

He-system. As a consequence of the statistic weights of these states the population occurs with a branching ratio of 3 : 5 [24]. While the 1 P1 state exclusively decays to the ground state via a magnetic transition, according to Surzhykov et al. [25] only 70% of the 3 P2 electrons contribute to K a1 . The other 30% transition to the 3 S1 state and therefore have to be attributed to K a2 instead (see Fig. 3b). From these considerations follows that 81:3% of the 2p3 2

REC lead to K a1 emission, which according to RECAL results in a total cross section of 0:128 barn=sr. A rigorous multi-

Please cite this article in press as: M.O. Herdrich et al., Fast calculator for X-ray emission due to Radiative Recombination and Radiative Electron Capture in relativistic heavy-ion atom collisions, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.04.022

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Fig. 4. (a) The figure shows a simulated particle trace created in Geant4 for all REC photons detected by a cylindrical Ge-crystal. It visualizes the geometry of the setup used for the simulation of the X-ray detector placed under an observation angle of 120 . (b) The bottom figure shows the comparison of the resulting simulated spectrum with the experiment data. The simulated curve is normalized to the measured L-REC intensity.

configuration Dirac–Fock calculation of branching ratios for Mshell decays into the L-shell and ground state yields an additional 0:045 barn=sr contribution from the M-REC. The averaged experimental cross section is 0:185 barn, which implies a deviation of 6:87%. The value lies within the statistical error margins and a slight underestimation is to be expected since populations through cascades from higher levels (n > 3), as well as capture into the 3dorbial have been neglected. Approximately 25% of electrons captured into 2s1 form the singlet state 21 S0 [24], that will only decay 2

into the ground state via two-photon transitions [26] and therefore does not contribute to K a2 . Likewise, with a branching ration of 1 : 3 electrons captured into the 2p1 -orbital form a triplet state

the branching ratios used in the approximation have a uncertainty of around 5% themselves and transition cascades of up to three decays have been stacked for the calculation this deviation is still within an acceptable range. Note that inner shell transitions with small energy splitting, e.g. 23 P0 ! 23 S1 , have not been taken into account, because their probability is much smaller compared to the other ones. Overall, the deviation of the combined K a transition cross section from the theoretical approximation is within 10% and therefore, it can be stated that a normalization can be performed using the RR/REC data from RECAL to successfully determine the cross section of other spectral features.

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with J ¼ 0 that can only decay to the ground state through the emission of two photons. The sum of all contributing total cross sections including M-REC parts with their respective fraction leads to an estimate of 0:631 barn=sr for the K a2 line. The averaged measured cross section, however, is 0:535 barn=sr. Thus, the deviation from the estimation amounts to 15:1%. Taking into account that

4. Conclusion and outlook A calculator was developed for angular differential cross section and angular polarization distribution data of Radiative Recombination and Radiative Electron Capture radiation. The fast calculator that was implemented utilizing a dense grid of precalculated dif-

Please cite this article in press as: M.O. Herdrich et al., Fast calculator for X-ray emission due to Radiative Recombination and Radiative Electron Capture in relativistic heavy-ion atom collisions, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.04.022

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Fig. 5. The two graphs show the results of the normalization for the K a1 (left) and K a2 (right) cross sections. The average for all angles (solid line) is shown as well as the estimated value from theory (dashed line) for comparison. All angles are given in the laboratory system.

ferential cross section and polarization data for the Radiative Recombination process is able to produce requested data sets by interpolation within seconds. The program yields results with less than 1% maximum point-wise deviation from rigorously calculated data sets that require up to several hours of computation time. It is therefore significantly faster than any comparable direct computation method. The available set of parameters for the K-, Land M-RR process ranges between collision energies of 1 MeV=u and 1 GeV=u and elements up to uranium. Beyond that, utilizing the database radiation characteristics for the Radiative Electron Capture process can be calculated in a fast way and with high precision as well. As such, the database is a powerful tool that can be used to provide data for simulations and calculations concerning experiments with relativistic highly charged heavy-ions. Previously, simulations have been performed to model the response of semiconductor Xray polarimeters using REC spectra generated by the fast calculator [27]. Now, within this work, it was shown that with help of the fast calculator cross sections for spectral features recorded during an experiment at the ESR could be successfully determined as well. A method was presented to normalize the measured intensities to those of the REC lines to yield cross sections for characteristic X-ray lines, that are otherwise difficult to describe theoretically. In the future this method could be applied to determine the cross sections of more interesting spectral features recorded together with the REC peaks, e.g. absolute cross sections for projectile excitation in ion-atom collisions. Overall, the availability of a system capable of producing large sets of data for the RR and RECprocess is very beneficial for the planning and understanding of future heavy-ion studies at facilities like FAIR.

Acknowledgment We acknowledge financial support of the European Union and the federal state of Thuringia via Thüringer Aufbaubank withing the ESF project (2015 FGR 0094). We would also like to thank A. Surzhykov for supplying the Mathematica implementation of the RR calculation method used in RECAL.

References [1] A. Gumberidze, T. Stöhlker, D. Banas´, K. Beckert, P. Beller, H.F. Beyer, F. Bosch, X. Cai, S. Hagmann, C. Kozhuharov, et al., Precision tests of QED in strong fields: Experiments on hydrogen- and helium-like uranium, J. Phys: Conf. Ser. 58 (1) (2007) 87, http://dx.doi.org/10.1088/1742-6596/58/1/013. [2] T. Stöhlker, A. Gumberidze, M. Trassinelli, V. Andrianov, H.F. Beyer, S. KraftBermuth, A. Bleile, P. Egelhof, T.F. Collaboration, Quantum electrodynamics in extreme fields: Precision spectroscopy of high-Z h-like systems, in: S.G. Karshenboim (Ed.), Precision Physics of Simple Atoms and Molecules, no. 745 in Lecture Notes in Physics, Springer, Berlin Heidelberg, 2008, pp. 157–163, ISBN978-3-540-75478-7 978-3-540-75479-4. [3] T. Stöhlker, D. Banaá, H. Bräuning, S. Fritzsche, S. Geyer, A. Gumberidze, S. Hagmann, S. Hess, C. Kozhuharov, A. Kumar, et al., Polarization and angular correlation studies of x-rays emitted in relativistic ion-atom collisions, Eur. Phys. J. Spec. Top. 169 (1) (2009) 5–14, http://dx.doi.org/10.1140/epjst/e200900965-0. [4] G. Weber, H. Bräuning, A. Surzhykov, C. Brandau, S. Fritzsche, S. Geyer, S. Hagmann, S. Hess, C. Kozhuharov, R. Märtin, et al., Direct determination of the magnetic quadrupole contribution to the Lyman-a1 transition in a hydrogenlike ion, Phys. Rev. Lett. 105 (24) (2010) 243002, http://dx.doi.org/ 10.1103/PhysRevLett. 105.243002. [5] T. Stöhlker, Y.A. Litvinov, A. Bräuning-Demian, M. Lestinsky, F. Herfurth, R. Maier, D. Prasuhn, R. Schuch, M. Steck, SPARC collaboration: new strategy for storage ring physics at FAIR, Hyperfine Interact. 227 (1) (2014) 45–53, http:// dx.doi.org/10.1007/s10751-014-1047-2. [6] M. Lestinsky, V. Andrianov, B. Aurand, V. Bagnoud, D. Bernhardt, H. Beyer, S. Bishop, K. Blaum, A. Bleile, A. Borovik, et al., Physics book: CRYRING@ESR, Eur. Phys. J. Spec. Top. 225 (5) (2016) 797–882, http://dx.doi.org/10.1140/epjst/ e2016-02643-6. [7] R. Maier, U. Bechstedt, F.M. Esser, HESR Technical Design Report V. 3.1.2, Technical Design Report, GSI, 2008. [8] M. Pajek, R. Schuch, Radiative recombination of bare ions with low-energy free electrons, Phys. Rev. A 45 (11) (1992) 7894–7905, http://dx.doi.org/10.1103/ PhysRevA.45.7894. [9] D. Banas´, P. Jagodzin´ski, M. Pajek, A. Gumberidze, A. Surzhykov, T. Stöhlker, Monte-Carlo simulations of the radiative recombination of ions with electrons in cold magnetized plasma, Phys. Scr. 2014 (2014) 014001, http://dx.doi.org/ 10.1088/0031-8949/2014/T161/014001. [10] N.R. Badnell, Radiative recombination data for modeling dynamic finitedensity plasmas, Astrophys. J. Suppl. Ser. 167 (2) (2006) 334, http://dx.doi.org/ 10.1086/508465. [11] T. Stöhlker, T. Ludziejewski, H. Reich, F. Bosch, R.W. Dunford, J. Eichler, B. Franzke, C. Kozhuharov, G. Menzel, P.H. Mokler, et al., Charge-exchange cross sections and beam lifetimes for stored and decelerated bare uranium ions, Phys. Rev. A 58 (3) (1998) 2043–2050, http://dx.doi.org/10.1103/ PhysRevA.58.2043. [12] J. Eichler, T. Stöhlker, Radiative electron capture in relativistic ion–atom collisions and the photoelectric effect in hydrogen-like high-Z systems, Phys. Rep. 439 (1) (2007) 1–99, http://dx.doi.org/10.1016/j.physrep.2006.11.003.

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M.O. Herdrich et al. / Nuclear Instruments and Methods in Physics Research B xxx (2017) xxx–xxx [13] A. Ichihara, J. Eichler, Angle-differential cross sections for radiative recombination and the photoelectric effect in the K, L, and M shells of oneelectron systems calculated within an exact relativistic description, At. Data Nucl. Data Tables 79 (2) (2001) 187–222, http://dx.doi.org/10.1006/ adnd.2001.0868. [14] A. Surzhykov, S. Fritzsche, T. Stöhlker, Photon polarization in the radiative recombination of high-Z, hydrogen-like ions, Phys. Lett. A 289 (4) (2001) 213– 218, http://dx.doi.org/10.1016/S0375-9601(01)00589-8. [15] A. Surzhykov, S. Fritzsche, T. Stöhlker, S. Tachenov, Polarization studies on the radiative recombination of highly charged bare ions, Phys. Rev. A 68 (2) (2003) 022710, http://dx.doi.org/10.1103/PhysRevA.68.022710. [16] G. Weber, H. Ding, M.O. Herdrich, A. Surzhykov, Towards a fast calculator for the radiation characteristics of radiative recombination and radiative electron capture, J. Phys: Conf. Ser. 599 (1) (2015) 12040–12044, http://dx.doi.org/ 10.1088/1742-6596/599/1/012040. [17] F. Biggs, L.B. Mendelsohn, J.B. Mann, Hartree-fock compton profiles for the elements, At. Data Nucl. Data Tables 16 (3) (1975) 201–309, http://dx.doi.org/ 10.1016/0092-640X(75)90030-3. [18] D. Jakubaa-Amundsen, R. Müller, A. Surzhykov, V. Yerokhin, Relativistic theory for radiative forward electron emission in heavy ion-atom encounters, Eur. Phys. J. D 68 (12), doi: http://dx.doi.org/10.1140/epjd/e2014-50574-7. [19] A. Gumberidze, D.B. Thorn, C.J. Fontes, B. Najjari, H.L. Zhang, A. Surzhykov, A. Voitkiv, S. Fritzsche, D. Banas´, H. Beyer, et al., Electron- and proton-impact excitation of hydrogenlike uranium in relativistic collisions, Phys. Rev. Lett. 110 (21), doi: http://dx.doi.org/10.1103/PhysRevLett.110.213201.

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[20] A.K. Pradhan, Recombination-cascade x-ray spectra of highly charged heliumlike ions, Astrophys. J. 288 (1985) 824–830, http://dx.doi.org/10.1086/162853. [21] Geant4: A toolkit for the simulation of the passage of particles through matter, URL: http://geant4.cern.ch/, (access: 21.11.2015). [22] W. Johnson, G. Soff, The lamb shift in hydrogen-like atoms, 1 6 z 6 110, At. Data Nucl. Data Tables 33 (3) (1985) 405–446, http://dx.doi.org/10.1016/ 0092-640X(85)90010-5. [23] A. Surzhykov, S. Fritzsche, N.M. Kabachnik, T. Stóhlker, Theoretical progress in studying the characteristic x-ray emission from heavy few–electron ions, J. Phys: Conf. Ser. 163 (1) (2009) 012008, http://dx.doi.org/10.1088/1742-6596/ 163/1/012008. [24] A. Surzhykov, U.D. Jentschura, T. Stöhlker, S. Fritzsche, Electron capture into few-electron heavy ions: independent particle model, Eur. Phys. J. D 46 (1) (2008) 27–36, http://dx.doi.org/10.1140/epjd/e2007-00269-3. [25] A. Surzhykov, U.D. Jentschura, T. Stöhlker, S. Fritzsche, K a1 radiation from heavy, heliumlike ions produced in relativistic collisions, Phys. Rev. A 74 (5), doi: http://dx.doi.org/10.1103/PhysRevA.74.052710. [26] A. Surzhykov, A. Volotka, F. Fratini, J.P. Santos, P. Indelicato, G. Plunien, T. Stöhlker, S. Fritzsche, Angular correlations in the two-photon decay of heliumlike heavy ions, Phys. Rev. A 81 (4), doi: http://dx.doi.org/10.1103/ PhysRevA.81.042510. [27] M.O. Herdrich, Photonen- und Elektronen-Emission von relativistischen Schwerionen beim Durchgang durch Materie, Friedrich Schiller Universitat Jena, 2016, Master’s thesis.

Please cite this article in press as: M.O. Herdrich et al., Fast calculator for X-ray emission due to Radiative Recombination and Radiative Electron Capture in relativistic heavy-ion atom collisions, Nucl. Instr. Meth. B (2017), http://dx.doi.org/10.1016/j.nimb.2017.04.022