Assessing electronic energy loss of heavy ions detected in reflection geometry

We study energy loss spectra of bromine ions scattered from silver thin films in the energy range of 4 to 36 MeV in forward reflection geometry. The different contributions to the energy loss are analyzed by complementary Monte Carlo simulations. We assess the dependence of the scattering yield and nuclear and electronic losses on the penetration depth of detected ions. For scattering from larger depth, we observe decreased elastic losses and increased trajectory length in comparison with predictions from single scattering, with increasing effects for lower energies. To investigate the entanglement of energy loss and depth information, electronic stopping cross sections are deduced from the experimental spectra by two different approaches: (a) assuming a single scattering model and (b) making use of Monte Carlo simulations. The data obtained from the two approaches are compared, and we assess the relative contributions from nuclear stopping and from the effect of multiple scattering on trajectory length. Results of both methods are discussed in the context of composition depth profiling and compared with data from literature and with Stopping and Range of Ions in Matter (SRIM) predictions resulting in good agreement.

imperative for a wide range of research fields and manifold technologic applications including astrophysics, 1 medical physics, 2 and materials modification. 3 Associated with this energy deposition, an energetic ion experiences a decelerating force when it penetrates matter. This force is commonly denoted as stopping power S and is defined as the average loss of kinetic energy per unit distance in the material (dE/dx). This specific energy loss can be separated in two contributions: the electronic and nuclear stopping power S e and S n , resulting from interaction with the electronic system or the atomic nuclei of the target, respectively. 4 A quantity commonly used to avoid a dependence of S on the atomic density N of the material is the stopping cross section ε = 1 N S. Both S and ε are characteristics for a given ion-material combination and show a strong dependence on ion energy and charge. At high energies, the dependence of electronic stopping power on the ion energy has been accurately described already in the early days of atomic physics. 5,6 At lower energies, the description is complicated by the fact that ions may not be fully stripped, as well as the fact that the quantized nature of electronic states in the solid is limiting excitations to less tightly bound electronic states. In consequence, even for light ions, theoretical predictions by static theories can be inaccurate, 7 and computationally expensive calculations taking the dynamics of the interaction into account are necessary for an accurate description of experimental observations. 8,9 Such predictions of the stopping power of a material for, in particular, heavy ions are essential for predicting particle range in ion implantation and energy deposition in irradiation. In Ion Beam Analysis (IBA), 10 where the energy loss of detected ions is the common observable quantity, the specific energy loss permits to obtain accurate depth perception. Beams of heavy ions are successfully employed in Heavy Ion Elastic Recoil Detection Analysis (HIERDA) to enhance the detection sensitivity for light elements in heavy matrices, while simultaneously providing the total composition of a sample. 11 The accuracy of sample composition depth profiles in HIERDA is, however, hampered by several different factors: commonly, the ions employed in HIERDA have energies, where both predictions from theory and tabulated experimental reference data are scarce. 12 This situation renders predictions based on extrapolation from existing data by Stopping and Range of Ions in Matter (SRIM), 13 the most commonly employed option, despite the associated drawbacks. 14 The specific relevance of the scarcely available data can be additionally affected by the following complication: stopping power data for heavy ions are commonly obtained in dedicated experiments performed in transmission geometry. [15][16][17][18] In this approach, a (sufficiently thin) self-supporting foil is irradiated by a monoenergetic ion beam, and the energy of the transmitted ions is detected. This method permits a rather straightforward extraction of the electronic stopping, in the absence of large-angle scattering, and thus, nuclear losses. The fact that HIERDA is, however, performed in reflection geometries has several important consequences for the applicability of such energy loss data obtained in transmission: at first, in HIERDA, at least one close collision significantly deflecting the incoming ion has to occur.
Such collisions come with increased local electronic losses and are known to shift the charge state of the ion significantly away from equilibrium, affecting ionization along the trajectory following the collision. 19,20 Additionally, different from straight trajectories and small acceptance angles in transmission, the influence of multiple scattering is increased in backscattering. Despite the possible need of a more explicit consideration of nuclear stopping, one can expect that also along the trajectory, on average, smaller impact parameters are probed. In consequence, in analogy to the reduced energy loss observed due to restriction to large impact parameters in channeling, 21  As indicated, the SSM is commonly employed in calculations of composition depth profiles from energy loss spectra of recoils and scattered particles using analytical tools such as Potku, 23 CONTES,24 and SIMNRA 25 in HIERDA. Although Monte Carlo (MC) tools have shown to be capable of reproducing experimental spectra with much higher accuracy, 26,27 these calculations are computationally very expensive. As an alternative, an analytical approximation for the multiple scattering contribution has been developed, 28,29 which exhibits relatively good agreement with MC data at sufficiently high energies. 30 The need of iterative fitting of simulations to extract energy loss for calculating depth profiles, even nowadays favors approaches using the SSM and analytical expressions for the scattering and recoil yield when running HIERDA as a standard analytical tool. At the same time, sufficiently straight trajectories, which would overcome the above-mentioned potential complications with insignificant multiple and plural scattering contribution, can be only achieved by ion beams in an energy range that is not accessible by most accelerator facilities. 31 In an earlier study, 32 we started investigating deviations from SSM in the case of iodine in gold, at energies as commonly employed in HIERDA. MC simulations confirmed a significant deviation from a single scattering geometry, in particular for low energies. Nevertheless, values for electronic stopping deduced from experimental energy loss data using predictions for path length and elastic losses from simulations and the single scattering approximation were found in very good agreement. Results were also found comparable with earlier published data obtained in transmission, even for ion energies of less than 100 keV/amu. These results thus indicated an applicability of the single scattering approximation for linking energy loss, stopping power, and depth scales, in HIERDA even at energies, where its basic assumptions are at least partially flawed. Data furthermore indicated that a potential influence of the geometry on electronic stopping cross sections is minor compared with other uncertainties.
In the present study, we aim to further investigate the abovementioned aspects of electronic energy loss in reflection experiments and their relevance for HIERDA. We performed complementary experiments in reflection geometry for MeV bromine ions in silver thin films, which makes our data cover a wider range of potential projectile ions in HIERDA with different relative weight of electronic and nuclear losses than previously investigated. Again, energy loss data are assessed by two evaluation models. We compare stopping power data obtained from the SSM, as employed in common HIERDA experiments, with results of a detailed data evaluation supported by MC simulations. Simulated spectra were employed to extract trajectory length and elastic losses of reflected ions. Finally, we discuss the relevance of the obtained results for accurate composition depth profiling. For the complementary MC simulations, the experimental geometry was reproduced in TRIM. 13 The collision output data file, which holds the whole particle trajectory for all projectiles, was filtered to select only ions leaving the sample within an angle of 45 ± 2.5 with respect to the incident beam direction in the scattering plane formed by incident beam axis and surface normal. Subsequently, the ion trajectories were analyzed to extract path length and electronic and nuclear energy loss experienced by each ion.

| DATA EVALUATION
One important question to be addressed in the present work is to what extent, and for which range of primary energies the SSM can be applied to extract reliable stopping cross sections for heavy ions in reflection geometry. Two ToF-to-energy converted spectra of scattered bromine ions with significantly different primary energies are shown in Figure 1 (black solid line). In the case of 36 MeV 79 Br presented in panel (a), one can see that the spectrum exhibits a well-defined plateau and width ΔΕ. In fact, the shape of the peak can be reproduced sufficiently well by analytical modeling using SIMNRA as shown for comparison (red solid line).
The analytical models for multiple and dual scattering are selected in SIMNRA but prove insufficient to reproduce the observed background at lower energies. The red dashed line corresponds to a SIMNRA spectrum with the employed stopping power adjusted to the width of the experimental spectrum by using a correction factor.
The shape of the acquired spectra is severely modified at lower primary energies. As illustrated in Figure 1B this procedure can be also abandoned, 35 if abstraction from any a priori scaling is desired.
Obviously, assessing the width of the spectra at lower energies comes with a significantly increased uncertainty as the background contribution due to plural scattering has to be estimated. More specifically, the determination of the exact position of the low-energy edge of the spectrum (E min ) comes with an almost constant uncertainty of at least 100-200 keV, which drastically increases the total uncertainty of the deduced data towards lower energies. As a result, the uncertainty in ΔΕ should be expected to range from 3% to 10% depending on the primary energy. Additionally, a systematic uncertainty of the thin film thickness (≈3%) and random uncertainties due to experimental statistics (≈5%) are considered. Therefore, the total uncertainty of the final deduced stopping data is estimated to be within 6-23%. In Figure 2, the path length distribution of the ions contributing to the spectrum depicted in Figure 1B  This range of nuclear losses expected by SSM is indicated by the shadowed region. However, the vast majority of the ions experience much lower nuclear losses despite the fact that they undergo many interactions with the target nuclei. A straightforward explanation to this effect is that there are more favorable ways in terms of energy and more probable trajectories for an ion to be deflected into a certain angle than trajectories containing a single large-angle scattering event.
For example, the inset in Figure 3  Kantre et al. 32 As mentioned earlier, these calculations are found to be computationally expensive and for lower scattering cross sections, that is, ions of higher energy, it is hard to obtain sufficiently good statistics, as CPU times of several days are necessary.
Therefore, we limit our calculations to low energies, where the discrepancy between experiments and the SSM is expected to be largest.
First, we focus on the ions that experience the maximum energy loss, that is, those that penetrated deep in the film. These ions are determined experimentally by the left edge of the spectrum as denoted in Figure 1 (Ε min ), and their path length and nuclear loss distributions are plotted in Figures 1 and 3, respectively (light blue histograms). The electronic stopping power can be therefore calculated by where E 0 − E min is the experimentally measured total energy loss of these ions. Average nuclear loss calculated by TRIM (Ε nucl. TRIM ) is F I G U R E 3 Comparison of nuclear energy loss distributions of ions scattered from the nearsurface and interface regions of the film as calculated by MC simulations. The range of the nuclear energy loss as expected by Single Scattering Model (SSM) is indicated by the shadowed region. Inset plot: the trajectory of an ion that reached the deepest 10% of the film, while its nuclear energy loss was only 937 keV subtracted to estimate the electronic energy loss. This value, divided by the distance traveled in the material, that is, average path length x TRIM , defines the electronic stopping power S e , which is assigned to In addition to the uncertainties from extracting experimental energy loss data, in this approach, we have to take the statistical fluctuation of data obtained from the MC simulations into account.
This contribution of this uncertainty on the deduced energy loss is estimated to range from 4% at low energies to 10% at higher energies.

| RESULTS AND DISCUSSION
The electronic stopping cross section of bromine in silver as obtained by both methods is plotted in Figure 4. At high energies, SSM appears to be approximately 8% higher than SRIM values but in good agreement with data from literature. 36 A possible systematic deviation to be expected for stopping crosssection data obtained in reflection geometry compared with data obtained in transmission would favor slightly higher data obtained in reflection for the reasons stated in the introductions. As shown by the investigation performed in the previous section, the single scattering scenario is a good approximation in this energy range, and the probability for one large scattering close to 45 is still high. The latter implies very small impact parameters that, in turn, lead to perturbation of the charge state equilibrium effectively resulting in higher electronic energy loss. Note, however, that even an excess energy loss of several tens of keV for the given film thicknesses would only induce an increase of ε on the order of 1%, that is, much lower than the uncertainty of the deduced data. Even the distorted spectra obtained at low energies (see, e.g., Figure 1B) can be employed to deduce reasonable stopping

| SUMMARY
Energy loss spectra in reflection forward scattering geometry were measured for MeV bromine ions scattered from silver thin films. In addition, simulations in the experimental geometry were performed using TRIM and the obtained MC data were used to extract information on elastic energy losses and the path length distribution of