Experimental electronic stopping cross section of tungsten for light ions in a large energy interval

Electronic stopping cross section of tungsten for light ions was experimentally measured in a wide energy interval (20 to 6000 keV for protons and 50 to 9000 keV for helium) in backscattering and transmission geometries. The measurements were carried out in three laboratories (Austria, Germany and Sweden) using five different set-ups, the stopping data deduced from different data sets showed excellent agreement amongst each other, with total uncertainty varying within 1.5 - 3.8\% for protons and 2.2 - 5.5\% for helium, averaged over the respective energy range of each data set. The final data is compared to available data and to widely adopted semi-empirical and theoretical approaches, and found to be in good agreement with most adopted models at energies around and above the stopping maximum. Most importantly, our results extend the energy regime towards lower energies, and are thus of high technological relevance, e.g., in fusion research. At these low energies, our findings also revealed that tungsten - featured with fully and partially occupied f- and d-subshells, respectively, can be modeled as an electron gas for the energy loss process.


Introduction
When an energetic ion penetrates matter, a retarding force acts on the ion as it starts losing energy due to collisions with target electrons and nuclei. The average energy loss of the ion per unit path length is called stopping power, S. This quantity has been under scientific investigation for over a century [1] and is nowadays of utmost importance in practically any technological or scientific application involving charged particles, like ion beam material modification and analysis [2,3], semiconductor industry [4,5], hadron-therapy [6,7] and fusion research [8,9]. Ion-solid interaction has also provided notable insights into solid state physics [10,11] and, using sufficiently slow ions, into nonequilibrium physics [12][13][14].
For low energy ions, i.e., when the ion velocity v is comparable or smaller than the Fermi velocity vF of the target (v ≤ vF), the electronic energy loss is dominated by excitation of valence electrons. Thus, a simple but powerful model of electronic energy loss describes the target electrons as Free Electron Gas (FEG) [15]. This approach predicts S to be proportional to the ion velocity, so that S = Q(Z1,rs).v.
The friction coefficient Q may be calculated using, e.g., density function theory [16] and depends on the projectile atomic number Z1 and on the one-electron Wigner-Seitz radius rs = (3 4 ⁄ ) 1/3 (ne is the FEG density corresponding to the number of valence electrons Nval per atom of the material) [17].
At high ion energies (i.e., v >> vF), the ion can be considered only a weak perturbation of the target electronic system, and the process is commonly modeled by first-order theories [18][19][20]. To avoid a trivial dependence on the material mass density, it is convenient to normalize S by the atomic density n of the material. The resulting quantity is commonly referred to as stopping cross section, ε = S/n.
We report experimental electronic stopping cross sections of W for light ions in a wide energy range (20 keV to 6 MeV for protons and 40 keV to 9 MeV for helium). We have deduced data from energy spectra recorded in both backscattering (BS) and transmission (TR) geometries. The choice of W as the material of interest is motivated by several facts. W is expected to form a major plasma facing component (PFC) in the divertor of next generation fusion devices. To model the expected material modification knowledge on energy deposition by plasma species is urgently required, but no data at low energies is available. Also, ion beam analysis as a commonly employed tool in characterization of PFC's [21] requires accurate reference data. Finally, W features a complex electronic structure with a shallow but complete f-subshell and half-filled d-states which yields information complementary to recent studies indicating the inapplicability of a FEG-model to describe electronic stopping in early transition and rare-earth metals [22]. We also compare our obtained results to semi-empirical (ICRU [23] and SRIM [24]) and ab-initio approaches (Montanari et al. [25], CasP [26] and DPASS [27]).

Sample preparation
Two different types of tungsten samples, i.e., bulk and self-supporting foil targets were selected for the present work. Backscattering (BS) experiments were performed on a commercially available (MaTecK GmbH, 99.9 % nominal purity) polycrystalline tungsten foil (0.5 mm thick and cut in 10x10 mm 2 ). Prior the analysis, the sample was cleaned with acetone followed by isopropanol (99 %) in ultrasonic bath. As information on the possible presence of impurities on and in the sample is essential for the reliability of the final deduced stopping data [28], the sample quality was repeatedly assessed by coincidence time-of-flight-energy heavy elastic recoil detection analysis (ToF-E ERDA), using 36 MeV I 8+ as probing beam at Uppsala University (information on the technique and data analysis are found elsewhere [29] [30,31] -this latter technique is used for a better depth-resolved oxygen quantification. A more detailed description of the manufacturing process and sample quality is given elsewhere [32].

Experimental set-ups
Experimental backscattering and transmission measurements were carried out using in total five setups. Thus, we cover a broad energy range and are able to check the consistency of the final results in overlapping energy ranges, decreasing thus possible systematic uncertainties arising from e.g., the adopted experimental technique and energy calibration of the accelerator. For this reason, not only different set-ups and energy ranges were selected but also different geometries were adopted. Low energy backscattering measurements were carried out at the AN-700 van de Graaff accelerator of Johannes Kepler University (JKU) [33] in Austria for D + and H + ions, and at the new multi-purpose setup for low-energy ion scattering assembled at the third beam line (BL3) of the 350-KV Danfysik implanter at Uppsala University (UU) [34] in Sweden, for H + and He + ions. Additionally, measurements in both geometries, BS and TR, were performed using the ToF-MEIS system [35] at the implanter (UU) for helium as projectile. Additional measurements in BS geometry for protons and helium at energies around 1 MeV were carried out at the µ-beam line of 3 MV Tandem Accelerator at the Helmholtz-Zentrum Dresden Rossendorf (HZDR) in Germany. Finally, high energy stopping data were deduced from backscattering measurements conducted at the 5 MV NEC 15SDH 2 Tandem accelerator of Uppsala University using H + and He +,2+ as probing species [21]. A summary of employed laboratories, set-ups (main scattering chamber), samples (bulk or self-supporting foil), methods (backscattering or transmission), projectile primary energy ranges, geometries and detector resolutions is given in Table   1. Table 1. Summary of experimental conditions (i.e., set-ups, samples, experimental method, energy range, geometries, and detector resolution) used in this present work to deduce the electronic stooping data of W.
* BS and TR are, respectively, backscattering and transmission geometry (see text and references therein). ** α and θ are incident and scattering angles with respect to sample normal and beam direction, respectively. ***Detector resolution at FWHM in keV and averaged over the respective energy chain.

Evaluation procedures
In the backscattering approach, apart from the experiments employing self-supporting foils, we rely on the fact that the height H of an energy spectrum contains information on the stopping cross section factor [ε], which in turn, includes the electronic stopping cross section ε of the projectile on the way in and on the way out of the target [36,37]. Direct evaluation of the electronic stopping from a single BS energy spectrum would require precise and accurate knowledge on the relevant experimental  parameters [38,39]. Therefore, we performed relative measurements between the sample of interest,

JKU -Linz
i.e., W to a reference sample with well-known stopping power, i.e., Au under otherwise identical experimental conditions [40]. Thus, the ratio of the heights of the experimental spectra, HW,expt/HAu,expt, contains information on the stopping cross section ratio εAu,expt/εW,expt. For a quantitative SCS evaluation, HW,expt/HAu,expt is compared to the ratio obtained from simulations, HW,sim/HAu,sim, using either the Monte Carlo TRBS [41] or the SIMNRA [42] codes. Such simulations account for potential differences in cross sections due to screening, multiple scattering (of higher interest towards lower energies). Agreement between height ratios of experimental and simulated spectra is obtained by variation of the W stopping power used as input in the simulation, as the only adjustable parameter [43]. Note that our BS approach also ensures acquisition of both spectra for the same primary ion charge. The energy interval of evaluation is chosen close to the kinematic onset of the spectra, reducing thus multiple scattering effects and ensuring sufficient linearity of the specific energy loss. We selected Au as reference material due to its vicinity in atomic number (hence kinematic factor), easy manufacturing process (guaranteeing high purity bulk samples), and due to its abundancy in well-established data sets [44][45][46][47] selected from the IAEA database [48]. A detailed description of the present approach (considering similar energy regimes), is given by the authors elsewhere [49]. detector [50], which can be circularly rotated around the sample, is placed with its center at 0° (and evaluation radius of ±1°) in transmission position with respect to the beam direction. The energy loss ΔE is calculated from the TOF shift between primary beam and particles transmitted through the foil.
The SCS is evaluated using the mean energy approximation [37], which is proved to be accurate enough for the present energy range [51] (texture effects in the foil are negligible). Monte Carlo simulations using TRBS are performed to account for effects of multiple scattering, and as in BS εW is adjusted so that the simulations reproduce the experimental data. In these simulations we have also considered the foil impurities, as they may influence the multiple scattering calculations (especially towards lower energies [52]). The final W stopping data is extracted from the simulation by employing Bragg's rule [53]. In Fig. 1 (b), we show a typical experimental spectrum of 100 keV helium ions transmitted through a tungsten foil of (50 ± 1) nm (black dotted line), as well as a simulation with optimized stopping power εW (red solid line). The small peak at E0 in panel (b) featured particles that directly hit the detector, due to pinholes in the foil target. By simultaneous detection of both peaks, possible systematic fluctuations in either beam energy and/or detection electronics -even if small -are canceled out in the evaluation. The result of the simulation is convoluted with a Gaussian to account for the finite energy resolution of the detector and straggling.

Results and discussions
In Fig. 2 (a), the experimentally deduced stopping cross section of W (blue filled symbols) for protons in the energy range between 20 keV and 6 MeV is shown: left and right triangles are results obtained with D + and H + using the 700 keV accelerator at JKU (Austria), respectively, circles are data from the implanter using BL3 at UU (Sweden), diamonds are data from the 3 MV Tandem at HZDR (Germany) and, finally, squares are results from 5 MV Tandem at UU (Sweden). See Tab. 1 for more details. Our results for protons feature a total uncertainty (i.e., random and potential systematic contributions) of ≈ 1.5 -3.8 %. For example, the low energy (protons) data set corresponding to measurements at BL3 (see Tab. 1) show an averaged uncertainty of ≈ 3.8 %, whereas high energy stopping data at T4-1 were deduced with an (averaged) uncertainty of ≈ 1.5 %. The final uncertainty was calculated following recommendations given in [54,55], and taking into account uncertainties from: i) integral beam dose (i.e., for Au and W spectra), ii) pile-up contributions and iii) reference data (for BS method), as well as iv) counting statistics (BS: spectra ratio and TR: peak position) and v) fit precision (BS and TR). For a more detailed description of how each contribution affects the deduced stopping data, we refer to Sec. 5 of [49]. To present the stopping cross section in a continuous and smooth manner, and to avoid problems with data interpolation, we fitted our εW values using the Ziegler-Biersack (ZB) function, as described in [56] (see blue solid line). The averaged fit precision was found to be ≈ 1.5 %. The agreement between different data sets is evident (blue filled symbols), ensuring consistency in the experimental and data evaluation approaches. Our results are compared also to data from literature [48] (black open symbols). In total, 65 data points from 4 different data sets [57][58][59][60]   In Fig. 2 ( [48] (black open symbols): 62 data points from 5 different data sets [60][61][62][63][64] are shown in Fig.2 (b), with stated uncertainties of 2 to 8 %. In general, good agreement between our data and literature values is achieved. Around the maximum stopping (i.e., ≈ 1 MeV), there is a systematic difference of up ≈ 6 % between the data from [61] (open upward triangles) and [64] (open circles); our data better confirm the former within ≈ 1 %.
We also compared the experimental εW data of this work and from literature with several semiempirical and ab-initio models (other lines in Fig. 2). In the former case, we employed two widely used semi-empirical approaches: the last released version of SRIM (2013) [24] and the online compiled tables for protons [65] and helium [66], which are based on the International Commission on Radiation Units and Measurements (ICRU) Report 49 [23]. Both approaches are similar in their underlying physics: the stopping power at higher energies (typically > 1 MeV/u) is generally evaluated based on a Bethe-Bloch-like formalism, whereas at lower energies (< 1 MeV/u), fitting formulas are adopted, while interpolations between these two energy regimes is usually done adopting Fano slopeplots [67,68]. Therefore, the precision of such evaluations effectively depends on the availability and reliability of the experimental data. One important advantage of such semi-empirical approaches is that their outputs can always be improved in new releases by updating their internal databases.
For the ab-initio stopping predictions, we used three different approaches. First, in the formalism by Montanari et.al. (only for protons), the electronic W output (retrieved from [25]) is calculated from two main contributions: i) from the bound electrons, responsible for the energy loss in the high energy regime using the Shell Wise Local Plasma Approximation (SLPA) [69], and ii) from the valence electrons, responsible for the energy loss at lower energies (in the FEG regime). In the second approach, the CasP code [26] is used to predict stopping values by employing the Unitary Convolution Approximation (UCA) [70] for energies in the Bethe regime and Transport Cross Section (TCS) [71,72] model for the low energy regime. Finally, the third approach consisted of adopting the recent version of DPASS tabulations (2020) [27], which evaluates the stopping power in the Binary theory [73], which is based on Niels Bohr's classical approach [74], but with improved modelling of the underlying physics (see Ref. [75] and discussions therein).
To assess in more details the capability of the each model to reproduce our experimental data in the investigated wide energy range, we show in Fig. 3 the difference between the respective models to our experimental data (  In this figure, the blue horizontal solid line is equivalent to a prediction in agreement with our ZB fit, as discussed above (other color lines represent the models -see respective legends). As one can see in  Using density function theory (DFT) calculations from [79,80] for the proton data (panel (a)), we obtain subshells, leading to an inter-shell screening effect that needs to be considered in their FEG model (see [25] and discussions therein). Concordantly, a possible influence of f-electrons on electronic stopping of H ions has also been discussed for proton stopping experiments in other transition (Pt) and rareearth (Gd) metals in the low energy regime [22], and more recently, even at energies up to the stopping maximum (i.e., Bragg peak) for several early transition metals [82]. For helium ions (panel (b)), the situation is significantly different. Even though velocity proportionally is observed (blue dashed line), for the DFT calculations [79] [83] and explained by charge exchange [84], and effects of projectile excitation and charge state on the interaction potential [85,86]

Summary and conclusions
In this work, we have experimentally evaluated the electronic stopping power of W in a wide energy range for protons (from 20 keV to 6 MeV) and helium (from 40 keV to 9 MeV) from energy spectra recorded in backscattering and transmission geometries. Measurements were carried out in five different set-ups from three laboratories (Austria, Germany and Sweden) in various energy ranges, geometries and using different samples (i.e., foils and bulks). The agreement amongst the data sets is excellent, and the final stopping data is obtained with a total averaged uncertainty (random and systematic) between  1.5 -3.8 % for protons and  2.5 -5.5 % for helium. Our results represent an improvement in the availability of experimental data, especially towards low energies (i.e., FEG regime), which are of particular interest for e.g., fusion research, where no data is available.
Our stopping results have also been systematically compared to the mostly used semi-empirical (ICRU and SRIM) and ab-initio theoretical approaches ( the agreement with SRIM is found to be worst, which is no surprise due to the lack of experimental data at low energies, this points to the necessity of future experiments at even lower energies, to permit accurate predictions of relevance for ion-materials interaction at commonly employed sputtering energies.