Understanding the $L$-shell ionization mechanism through osmium atoms bombarded by 4-6 MeV/u fluorine ions

The measurement of emitted x-rays from targets has resulted in major advances in radiation physics [1], plasma physics [2], atomic and nuclear physics [3], and the particle-induced x-ray emission (PIXE) technique [4, 5]. Thus far, the PIXE method has used light ions such as protons or alphas [6, 7, 8, 9, 10, 11, 12, 13]; however, there is an increasing interest to employ heavy ions since their cross sections are larger and have, thereby, better sensitivity [14]. Nevertheless, this potentiality is discouraged by discrepancies observed between the theories and experiments. Although these inconsistencies are often attributed to multiple ionization phenomena [15, 16, 17], they do not account for all the discrepancies observed, for example, in experiments with an 8-36 MeV Si-ion beam on targets of Au, Bi, Th, and U with thicknesses between 12 and 40 $\mu g/cm^{2}$ [18]. On such occasions, theoretical approaches have been modified to include the $L$-subshell coupling effect as well as the saturation of the binding effect at the united atom limit in addition to the multiple ionization [18, 19]. Even though closeness between the experiments and theory is achieved, differences remain, suggesting that other physical processes are involved.

It is well-known that for asymmetric collisions, $Z_{P}/Z_{T}<1$, the direct ionization (DI) is dominant, whereas for symmetric collisions, $Z_{P}/Z_{T}\approx 1$, the electron capture (EC) process becomes increasingly important. Although the present collisional system –F ions ($Z_{P}=9$) impinging on Os ($Z_{T}=76$)– is asymmetric ($Z_{P}/Z_{T}$= 0.1184), the measurements cannot be accounted for without considering the EC contribution to the $L$-shell vacancy production [20]. This phenomenon can be understood by considering the ratio of the projectile velocity $V_{P}$ to the orbital velocity of the $L$-shell electrons $V_{L}$. In our full-relativistic calculations, the mean velocities (in a.u.) of the osmium $L_{i}$ sub-shells are $v_{i}$= 26.6, 37.7, and 32.8 for $i=1,2,3$, respectively. Therefore, 0.33 $\leq V_{P}/V_{L}\leq 0.58$, and the collision is asymmetric but in the slow velocity regime. Hence, we included the EC contribution along with the DI, the multiple ionization, and the vacancy sharing. However, contrary to our expectations, the EC does not account for all the discrepancies found between our measurements and the theoretical descriptions. In the final stage of this work, we found that the presently available atomic parameters are not properly described. We modified these values iteratively until a good agreement between the theory and experiment was achieved.

To reveal the $L$-shell ionization mechanism of the collisions of 4-6 MeV/u fluorine ions on an ultra thin osmium target, we provide experimental details in Section II. The theoretical methods employed to describe the ionization are discussed in Section III. Section IV addresses the effects of the single- and multiple-hole atomic parameters required for the derivation of the subshell-ionization cross sections from the measured x-ray production cross sections. Section V describes the inclusion of ionization by the LK capture processes, and Section  VI summarizes the major findings.

The $L$-shell x-ray production cross sections in the Os elements fusing the ¹⁹F ions (charge states $q=6+$, $7+$, $8+$) in the 76–114 MeV energy range have been measured in the atomic physics beamline at the Inter-University Accelerator Centre, New Delhi. The heavy ions of fluorine –F⁶⁺ (76 and 84 MeV), F⁷⁺ (90 MeV) and F⁸⁺ (98 and 114 MeV)– were obtained from the 15 UD Pelletron accelerator. The chamber has provision for two silicon surface barrier (SSB) detectors at $\pm$ 7.5^o and two x-ray detectors at 55^o and 125^o to the beam direction, respectively. The target was mounted on a steel ladder forming a 90^o angle to the beam direction. The vacuum inside the chamber was $\sim 10^{-6}$ Torr. The spot of the ion beam at the target had a diameter of approximately 2 mm. The spectra were taken at different positions of each target. Details of the experimental setup and detection system are given by Kumar et al. [21]. The ultra-thin target of ₇₆Os was prepared on the polypropylene backing using an ultra-high vacuum deposition setup at IUAC, New Delhi. The thickness of the target was measured using the Rutherford Back-scattering (RBS) method and its spectrum is given in Fig. 1. The target turned out to be very thin, only 1.09 $\mu$g/cm². The beam current was kept below 1 nA to avoid pile up effects and damage to the target. The spectra were collected for a sufficiently long time to get good statistical accuracy. The $L$ x-ray spectra of natural Os bombarded by F^q+ at different projectile energies (76-114 MeV) is shown in Figs. 2 and 3. The spectra were analyzed with a fitting method considering a Gaussian line shape for the x-ray peaks and a suitable background function. From the figures, it is clear that all major $L$ x-ray components are well resolved by the Si(Li) detector. The details about the data acquisitions and the terms related to the projectile velocity are given by Oswal et al. [22].

The measured $L$ x-ray production cross sections for the major peaks namely $L_{l}$, $L_{\alpha}$, $L_{\beta}$, and $L\gamma$ were obtained using the following relation

where $Y_{x}^{i}$ is the intensity of the $i$th x-ray peak, $A$ is the atomic weight of the target, $\theta$ is the angle between the incident ion beam and the target foil surface, $N_{A}$ is the Avogadro number, $n_{p}$ is the number of incident projectiles, $\epsilon$ is the effective efficiency of the x-ray detector, $t$ is the target thickness 1.09 $\mu$g/cm² and $\beta$ is a correction factor for the absorption of the emitted x-rays inside the target.

The absorption correction factor for the absorption of the emitted $L$ x-rays in the target is written as

where $\mu$ is the attenuation coefficient inside the target and its unit is $cm^{2}/g$ [23]. The value of $\beta$ is $\geq$ 0.99 for the target thickness used in the present measurements. The energy loss calculation using the SRIM code [24] for the incident beam within the target suggests negligibly small energy loss for the target thickness and the beam energies used in the present work. The ion beam changes its charge state during its passage through the target.

The role of projectile charge state in this collision regime for 4-6 MeV/u is found to be negligible [22]. Integrated charge in a Faraday cup measured by a current integrator has been used to count $N_{p}$ (see discussion in Section VI). The energy loss calculation using the SRIM code shows that 76 and 114 MeV fluorine ions lose 2.30 and 1.91 keV in the Os target, respectively. The peak areas $Y_{x}^{i}$ are evaluated using the computer program CANDLE [25]. This software is an improved version of the Levenburg-Marquardt [26] non-linear minimization algorithms for the peak fitting. The energy calibration of the detector is performed before and after the in-beam measurements. A semi-empirical fitted relative efficiency curve for the present measurement is available in Oswal et al. [22].

The percentage error in the measured x-ray production cross sections is about $10-15\%$. This error is attributed to the uncertainties in different parameters used in the analysis, namely the photo peak area evaluation ($\sim\!5\%$ for the $L_{\alpha}$ x-ray peak and $3\%$ for the other peaks), ion beam current ($\sim\!7\%$), target thickness ($\sim\!3\%$). In the energy region of interest, the error of the absolute efficiency values $\epsilon$ ranges between 5 and 8%.

To calculate the direct Coulomb ionization cross section, we have employed (i) the coupled-states relativistic semi-classical approximation (RSCA-CSM), (ii) the shellwise local plasma approximation (SLPA) with fully relativistic electronic structure calculations for Os, and (iii) the ECUSAR theory, which accounts for the energy (E) loss and the Coulomb (C) deflection of the projectile, the perturbed-stationary state effect through a united and separated atom (USA) treatment of the ionization process, and by considering the relativistic (R) nature of the $L$-subshell electronic states. All these models are briefly described below.

Multiple vacancies in the target atom change the atomic parameters: the fluorescence yields and the CK yields, which in turn alter the x-ray production cross sections. In the present work, single-hole fluorescence $\omega_{i}^{0}$ and CK yields $f_{ij}^{0}$ [42], were corrected for multiple ionization using a model prescribed by Lapicki et al. [43]. Each electron in a manifold of the outer subshells is ionized with a probability $P$, which is calculated using Eq. (A3) from [43], and replacing the projectile atomic number $Z_{P}$ by its charge state $q$ [44],

with $\beta=0.9$. For the charge state $q$, we take the incident charge state of the projectile. The $\omega_{i}^{0}$ values corrected for the simultaneous multiple ionization (SMI) in the outer subshells are given by

while the $f_{ij}$ values for multiple ionization are given by

Note that the fractional rates $F_{ip}$ remain unchanged because both the partial and the total non-radiative widths are narrowed by identical factors. According to Eqs. (17) and (18), the single-hole fluorescence and CK yields depend on the energy and charge state of the projectile ion. The fluorescence and the CK yields for singly- and multiply-ionized Os is given in Table 1. It is clear from this table that in the extreme the $L_{i}$ subshell fluorescence yields are enhanced by $\sim$30%, and the CK yields are reduced up to $\sim$50% from the single-hole to the multiple-hole atom in Os. Note that the use of different sets of atomic parameters can change the x-ray production cross section by $\sim$30% or more. Recent values of $\omega_{i}^{0}$, and $f_{ij}^{0}$, compiled by Campbell [42] for the elements with $25\leq Z\leq 96$, have been used in the present work for singly-ionized atoms.

To calculate the $L$-$K$ electron capture cross sections, we used the theory of Lapicki and Losonsky [45], which is based on the Oppenheimer-Brinkman-Kramers (OBK) approximation [46] with binding and Coulomb deflection corrections at low velocities. Neglecting the change in the binding energy of the $K$ shell electron of the projectile with one versus two $K$ shell vacancies, a statistical scaling is used to calculate the electron transfer cross section for the case of one projectile $K$-shell vacancy, $\sigma_{L\rightarrow K}$, resulting in $\sigma_{L\rightarrow 2K}/2$, where $\sigma_{L\rightarrow 2K}$ is the production cross section for two projectile $K$-shell vacancies. In the present experimental condition, $v_{1}$ ranges between 12.39 and 15.17, while $v_{2L}=Z_{2L}/n_{2}=35.925$ (in a.u.), $n_{2}$ and $n_{1}$ are the principal quantum numbers of $L$ and $K$ shell electrons of the target and the projectile atom, respectively. Following Lapicki and Losonsky [45], $\sigma_{L\rightarrow 2K}$ can be obtained as

with $Z_{2L}=Z_{T}-4.15$ and

where $E_{L}$ is the binding energy of $L$-shell electron of the target (in eV), and the parameters $a_{0}$, $v_{1k}$, $Z_{p}$, and $Z_{T}$ are the Bohr radius, the $K$-shell orbital velocity of the projectile ion, atomic number of the projectile and the target atom, respectively.

Due to certain charge state distribution inside the target, the effective capture contribution will be $F(q)\times\sigma_{L\rightarrow 2k}^{\mathrm{OBK}}(\theta_{L})$, for $q=8+$ and $9+$. Here, $F(q)$ is the charge state fraction for a specific $q$. To obtain $F(q)$, we employed a two-fold procedure. First, we used a Fermi-gas-model based empirical formula for the determination of the mean charge state, $q_{m}$, inside the target [47]

where $z_{1}$ and $v_{F}$ are the projectile atomic number and the Fermi velocity of target electrons, respectively. The value of $v_{F}$ for Os is $8.19\times 10^{6}$ cm/sec. To showcase the difference between the ionization of the projectile ion inside and outside the target, we displayed the $q_{m}$ as predicted by the Fermi-gas-model [47] and by the Schiwietz model [48] in Fig. 4(A). This contrasting picture is governed by the solid surface [49, 50]. The charge state of the heavy projectiles inside the solid target is higher than for those outside. This feature has been described in detail by Chatterjee et al. [20]. In the second part of the procedure, the $q_{m}$ values inside the target are substituted by a Lorentzian charge state distribution [49] to obtain the $F(q)$ as follows

where the distribution width $\Gamma$ is taken from Novikov and Teplova [51] as follows

being $x=q_{m}/z,\,\,\alpha=0.23,\,\,\beta=0.32$ and $C=2.669-0.0098\,Z_{2}+0.058\,Z_{1}+0.00048\,Z_{1}\,Z_{2}$. The $F(q)$ for $q=8+$ and $9+$ are displayed with a bar chart in Fig. 4(B).

In a recent article [52], we analyzed the major sources of errors in the measurements of $L$-subshell cross sections. We concluded that these uncertainties come from inaccurate (i) determination of target thickness, (ii) counting the number of projectile particles, (iii) background subtraction during spectrum analysis, and (iv) atomic parameters used for the conversion of (a) the measured x-ray production cross sections into $L_{i}$-subshell ionization cross sections and (b) the theoretical $L_{i}$-subshell ionization cross sections into $L$-shell x-ray production cross sections. In this work, all these aspects were considered carefully. The first source of error was managed by measuring the foil thickness by the RBS method, as shown in Fig. 1. The mass thickness (target thickness in $\mu$gcm^-2) is normally measured by three techniques viz., RBS, PIXE and XRF (x-ray fluorescence). Out of these, RBS is the most accurate (see Fig. 4 of Ager et al. [53]). The second source of uncertainties was controlled by measuring the integrated charge count of the projectile ions in a Faraday cup placed behind the target for a fixed duration (100 s) and under two different conditions: (a) solely the blank target frame in place and (b) the target foil in place. The ratio between the integrated charge counts for the two different conditions is $R=\frac{nq}{nq^{\prime}}=\frac{q}{q^{\prime}}$ or $q^{\prime}=\frac{q}{R}$, where $n$ is the number of projectile ions of incident charge state $q$ in case of blank target, and of charge state $q^{\prime}$ when the target is in place. If the spectra is recorded for a long duration (say, 30 minutes), then, the total measured charge divided by $q^{\prime}$ will give the number of projectile ions passing through the target foil as required in Eq. (1). The third source of error was also taken into consideration, as can be evidenced from Figs. 2 and 3: the data points are well on the fitting profile, and the reduced $\chi$-squared values are close to 1. The fourth aspect is also addressed through a step by step evaluation, and is discussed below.

Besides the $L$-shell ionization, heavy ion collisions give rise to simultaneous ionization of the higher-shell electrons. This change of electronic environment in an atom alters the properties of the $L$ x-ray emission. As a result, the atomic parameters vary with the projectile energy as shown in Fig. 5. Although this effect was taken into account, the $L_{i}$ subshell-ionization cross sections derived from the measured $L$ x-ray production cross sections are underestimated by the ECUSAR-CSM, ECUSAR-CSM-EC, RSCA-CSM, RSCA-CSM-EC, SLPA, and SLPA-EC models, as can be seen in Fig. 6. The vacancy sharing among the $L$ subshells described by the coupled-states model (CSM) seems to have a minor role in the present collisional system. Despite the $LK$ electron capture (EC) effects are included to the various theories, the cross sections remain lower than the measured values. The EC cross sections for the $L_{i}$ subshells as a function of the impact energy are plotted in Fig. 7. According to the figure, the EC contribution is the largest for the $L_{3}$, and is almost identical for $L_{1}$ and $L_{2}$. To estimate these contributions, we used the charge-state distributions of the projectile ions inside the solid target as presented in Fig. 4.

The atomic parameters are used to convert the $L$ x-ray production cross section to $L_{i}$ subshell ionization cross section as discussed above. In general, such parameters are taken from various sources, where the authors have calculated them using different theoretical methods. In some cases, the theoretical predictions considerably deviate from each other. In particular, for $w_{1}$, $f_{12}$ and $f_{13}$ a spread of up to a factor of two can be observed, as shown in Table 4. In the present work, we used the most recent theoretical parameters [42]. The estimated uncertainties for $\omega_{1}$, $\omega_{2}$, $\omega_{3}$, $f_{12}$, $f_{13}$, and $f_{23}$ are 15%, 5%, 5%, 10%, 5%, and 5%, respectively. To resolve the above-mentioned discrepancy of the $L_{i}$ subshell ionization cross section between the measurements and the theoretical models, we varied these parameters iteratively until a good agreement was achieved; see Fig. 5. We present the optimized parameters in Table 1 and show their variation with the beam energy in Fig. 5. The difference between the original [42] values and the optimized atomic parameters is not significant, except for $\omega_{1}$.

There are two ways to study the $L_{i}$ subshell ionization by ion impact: by comparing the theoretical predictions (i) with the measured $L_{i}$ subshell ionization cross sections, and (ii) with the measured $L$ x-ray production cross sections. The former has been considered above, and the latter is examined below.

The measured x-ray production cross sections are compared with calculations in Fig. 8. The theoretical values in Fig. 8(A) are obtained by employing the set of atomic parameters from [42], while in Fig. 8(B) the optimized set of atomic parameters are used. The various theories underestimate the measured cross sections when the original parameters are used; however, the new set of atomic parameters allows one to obtain a better agreement.

The difference between the above mentioned methods is as follows. In the first one, an individual $L_{i}$ subshell ionization cross section is compared, while in the second one, a combination of the contributions of the various subshells is considered. Hence, the mathematical expression for the former is given by a differential operation while the later constitutes an integral. From the values in Tables II and III, we notice that the agreement between the theory and experiment is better for the $L_{i}$ subshell ionization cross sections than for the $L$ x-ray production cross sections. Thus, the differential comparison provides a better picture that the integral one.

In the present work, the $L$ x-ray production cross sections of Os were measured by 4-6 MeV/u ${}^{19}F^{q}$ ions of charge states $q=6+,\,7+$ and $8+$, and compared with theoretical $L$ x-ray production cross sections. Different ionization theories such as RSCA, ECUSAR and SLPA were used for the Coulomb direct ionization. Additionally, the contribution of the $LK$ electron capture was added to each theory. The effect of multiple ionization was also considered through the modified atomic parameters. Furthermore, $L_{i}$ ($i=1,2,3$) subshell ionization cross sections were derived from the measured $L$ x-ray production cross sections, and compared with the corresponding theoretical counterparts. Both comparisons show the best agreement with the experiment for the ECUSAR-CSM-EC model. However, certain differences are still clearly noticed. To resolve such discrepancies, the atomic parameters were optimized to obtain a good agreement between the measurements and ECUSAR-CSM-EC model. Thus, this work gives us a convincing understanding of the $L$-subshell ionization mechanism by heavy ion bombardments, while states doubts on the atomic parameters used in the conversion of the x-ray production cross sections to the ionization cross sections. Further, our work suggests the urgent need for accurate measurements and theoretical calculations of the atomic parameters used.

S.C. acknowledges the University of Kalyani for providing generous funding towards his fellowship. Financial support from the Science and Engineering Research Board (SERB), New Delhi (SB/FTP/PS-023/2014), to Sunil Kumar in terms of Young Scientist scheme, and the grant from IUAC, New Delhi (UF-UP-43302 project), are highly acknowledged. L. Sarkadi acknowledges the support from the Hungarian Scientific Research Fund (Grant No. K128621). A.M.P. Mendez, D.M. Mitnik and C.C. Montanari acknowledge the financial support from the following institutions of Argentina: Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT), and Universidad de Buenos Aires (UBA).

The authors are grateful to the Pelletron staff for the smooth conduct of the experiments and the RBS lab for the target thickness measurements.