Scaling for state-selective charge exchange due to collisions of multicharged ions with hydrogen

In this article we evaluate state-resolved charge exchange cross sections for Be 4 + , ?> B 5 + , ?> C 6 + , ?> N 7 + , ?> and O 8 + ?> projectiles colliding with atomic hydrogen employing two different methods: the classical trajectory Monte Carlo and the eikonal impulse approximations. These cross sections are used to extend previously derived scaling laws for n-, nl-, and nlm-distributions to highly excited final levels with 4 ≤ n ≤ 9 , ?> covering energies in the range 50 − 2000 keV / ?> amu. Present total and partial capture cross sections are in agreement with available experimental and theoretical data for these collision systems. Besides, the proposed scaling rules are also verified by other theories, becoming a useful instrument for plasma research.


Introduction
Charge exchange (CX) in collisions of multiply charged ions with hydrogen atoms has attracted renewed attention in the last years. Such an interest is motivated by the importance of this process, also called electron capture, in both laboratory and astrophysical plasma environments [1]. In the case of laboratory magnetic confined plasmas, the CX between multicharged ions and neutral H plays an important role to estimate the temperature of the plasma and the density and charge of ionic impurities [2,3], while in astrophysical plasmas it was found to be responsible for x-ray emission from comets due to the scattering of solar wind ions with cometary neutral gases, having been also detected in the heliosphere and in planetary atmospheres [4][5][6].
When multiply charged ions collide with H( s 1 ) at intermediate and moderated high impact energies, electrons are mainly transferred to excited states of the ion, which then decay via photon emission giving rise to specific emission lines. These emission spectra, extensively employed as a diagnostic tool in plasmas [7], strongly depend on the nldistribution of transferred electrons [8,9], making the stateselective electron capture cross section a key magnitude for the analysis of the experimental data. However, partial CX cross sections into different nl-sublevels are less studied than their corresponding total and partial n values and consequently, several plasma studies assume approximated l-distributions for the different levels [10][11][12]. In this article we focus precisely on these state-selective CX cross sections, extending previously derived scaling laws for n-, nl-, and nlmdistributions [13] to deal with final capture levels up to the n = 9 shell.
Two different theoretical methods are employed: the classical trajectory Monte Carlo (CTMC) method and the eikonal impulse (EI) approximation. Each of these methods can be applied in a different region of impact energies. The CTMC method is a classical theory that results appropriate for impact velocities v in the range v 1 3 a.u. [14], while the EI approach is a high-energy method, which can be extended to describe CX in the intermediate velocity range as well, i.e. for velocities v 2  a.u. [13]. We use both approaches to evaluate state-selective CX cross sections for different ion-H collisions with the aim of providing reliable scaling rules that cover a wide range of incidence energies, from intermediate to high impact velocities. Notice that for highly charged ions colliding with H( s 1 ), many scaling rules for total capture, and even for partial capture to a given n shell, have been proposed [15][16][17][18]. But for nl-distributions the studies on scaling laws are scarce [13,19,20]. Therefore, present state-selective scaling rules are useful tools to estimate partial contributions coming from specific final capture states, avoiding tedious calculations, specially for capture to highly excited levels.
In this article, the CTMC and EI methods are applied to evaluate CX cross sections for different bare ions-Be , 4+ B , 5+ C , 6+ N , 7+ and O 8+ -impinging on neutral hydrogen in its ground state. The range of considered projectiles is of interest in fusion and astrophysical research. Most of these ions are found as impurities in laboratory fusion plasmas, like beryllium and carbon that are expected to be present in ITER [21,22], or nitrogen that is routinely introduced in the ASDEX tokamak to protect the plasma-facing components [23,24]. Also in astrophysics, ions of carbon and oxygen are considered the predominant sources of soft x-ray emission in collisions involving the solar wind [4,25,26].
This article is organized as follows. The theoretical methods used in this work to investigate scaling rules for partial CX distributions are presented in section 2. In section 2.1 we summarize the parametric dependence of the CX cross section to a specific final state, obtained within the EI approximation, which is here employed to derive scaling laws for highly excited final states. While in section 2.2 we show how total capture cross sections derived within the CTMC and EI approximations merge in the intermediate velocity regime, allowing us to combine both theories to produce a join CTMC-EI approach that provides reliable results for a wide range of impact velocities, from 1 a.u to 6 a.u. Scaling rules for the corresponding partial cross sections are investigated in section 3, where scaled n-, nl-and nlm-distributions are presented and discussed in sections 3.1, 3.2 and 3.3, respectively. Finally, in section 4 we outline our conclusions. Atomic units (a.u.) are used unless otherwise stated.

Parametric dependence of the EI capture cross section to a given final state
The prior version of the EI approximation is a distorted wave method that makes use of the exact impulse wave function to describe the final collision channel, while the initial channel is represented by means of the eikonal wave function [27]. It has been successfully applied to describe CX in asymmetric collisions, where the nuclear charge of one of the particles is larger than the other, at high and intermediate impact velocities. In particular, the prior EI approach was found to provide proper electron capture predictions for both, multicharged ions colliding with light atoms (H and He) [13] and proton-atom collisions [28].
Following previous articles [13,19], we use the prior EI approach to derive a scaling rule for the asymmetric reaction: where the charge of the fully stripped projectile, Z P , is larger than the one of the target nucleus, Z T (with Z e For this reaction the EI transition matrix reads [28]: where v  is the impact velocity and W K K momentum of the projectile and P m its final reduced mass.
The functions L ,   where z Z n P P = represents the mean value of the final electron velocity and is the projectile-electron Coulomb factor coming from the asymptotic conditions, with Γ the gamma function. In equation (3) the function nlm s  depends on the scaled parameters ṽ and W , denoted with tilde, which are defined as v v z P= and In this article we will use this approximation, together with equation (3), to derive state-selective scaling laws for capture to highly excited levels, with main quantum numbers in the range n 4 9.   As stated in [13], the approximated relation of equation (6) starts to fail for impact velocities lower than 2 a.u. when the condition z 1 P > is not verified.
The present CTMC calculations were computed using a superposition of ten standard microcanonical distributions, fitting the radial probability density of the atomic hydrogen [29]. Nuclear straight-line trajectories and a statistic of 2 10 5 electrons have been considered. The trajectories have been integrated up to t v 500 . max 1 = -Notice that capture to low n levels (n 3  ) is well characterized by the usual microcanonical distribution, but for higher n levels, like those considered in this article, the hydrogenic density-fitted one provides a better description of the electron capture process [30,31].
Concerning the EI approach, each electron capture cross section to a given n l m , , state was calculated separately, involving a three-dimensional integral on the momentum space, as given by equation (2), and a further integration on the final angular distribution of the scattered projectiles. For highly excited final levels, such numerical calculations were almost prohibitive twenty years ago, forcing us to use extrapolation rules [13], but now they can be done in a reasonable CPU time (for example, for the state n l m , , 9, 8, 0 ( ) ( ) = of carbon, it takes less than 3' in a PC I7 computer). In this article we evaluated EI total capture results by adding n l m , , ( )-cross sections up the shell n = 9, while the contribution of final levels with n 10  was estimated by using an extrapolation rule based on the n-dependence of the Brinkman-Kramer cross section, as given by equation (12) of [13].
In figure 1(a)  , and in the case of beryllium, the accord also extend to lower energies. In order to quantify the level of concordance of both theories, we can mention that for C 6+ projectiles the differences between CTMC and EI total capture cross sections at 100 keV amu −1 are around 15%, increasing up to 27% at 200 keV amu −1 . This fact allows us to combine both theoretical methods by merging them around 100 keV amu −1 to build a join CTMC-EI approach, which provides realistic cross sections in a wide velocity region, ranging from intermediate to high incidence velocities.
Total cross sections derived within the join CTMC-EI method are shown in figure 1(b) and compared with results obtained from other theories and with available experimental data. For all the projectiles the curve corresponding to the join CTMC-EI approximation is in good accord with values derived from different theories, like semiclassical close-coupling approaches with molecular (MOCC) [33] or atomic (AOCC) [32,35] wave functions, or the first born approximation (FBA) [34. Join CTMC-EI results are also in agreement with available experimental data for carbon and oxygen projectiles in the intermediate energy regime [18,36]. Such a concordance is an indication of the adequate description of the CX process obtained by means of the merge of the CTMC and the EI methods.
The same criterion is followed in the next section to combine the CTMC and EI theories also at the level of stateselective cross sections, where the differences between CTMC and EI n-resolved cross sections at 100 keV amu −1 are 20% at most for Be 4+ projectiles, decreasing up to a relative error lower than 10% as Z P augments. At low velocities, the join CTMC-EI results verified that the most populated n-level, n , max qualitatively agrees with the Z P 3 4 scaling law [37], but the value of n max slowly decreases as the velocity increases, tending in all the cases to the ground level at very high velocities.

Scaling law for partial n-resolved cross sections with n4
In this section we focus on the partial CX cross section into a given final n level, , n s which is defined as . where U n is expected to be an universal function of W , depending on the main quantum number n. In previous articles [13,19], this scaling rule was successfully tested for partial capture into the fundamental and first excited levels, i.e., n 1 4.
= -In this work the rule is extended to consider final levels in the range n 4 9.
= -In addition, we found that when the scaled cross sections S n for the different levels are plotted together, all the CTMC-EI results can be gathered in a relatively narrow band, indicating a weak dependence on n of the universal function, i.e. U W U W n (˜) (˜)  for n 4 9,   as shown in figure 2(a). In this figure we have split the W range in two panels (upper and lower) in order to display clearly the different results.
Furthermore, we provide a simple expression to represent the approximate function U W , (˜) which is valid for final levels with principal quantum numbers in the range n 4 9.
= -The proposed function (for n s expressed in cm 2 ) reads:   = -Lines, scaled nl-populations for different n shells, derived within the join CTMC-EI method considering charges Z 4 8. P = -given n-shell through equation (7) considering velocities v 2  a.u., as well as lower energies (i.e. E 70 keV ⪆ amu) if the inequality z 1 P > is satisfied. Notice that all the CTMC-EI results displayed in figure 2(a) (more than 500 S n -values that cover an extended cross section range of almost 9 orders of magnitude) can be approximated by equation (8) with a maximum relative error of 40% at most, decreasing as the impact velocity augments.
Other theoretical methods also verify the universal behavior given by equation ( (8), as shown in figure 2 (b), where scaled partial cross sections S n , defined by equation (7), are plotted as a function of W for different sets of theoretical data: those of Igenbergs et al [35] and Toshima and Tawara [41] obtained with the AOCC; those of Schultz et al [38] and Olson et al [40] obtained with the CTMC; those of Mandal et al [39] and Saha et al [43] obtained with the continuum distorted wave (CDW); and those of Belkić et al [34] obtained with the FBA. We observe that the different theoretical and experimental data lie again inside a narrow band around the universal curve of equation (8), assuring the validity of the proposed scaling. We have also included in figure 2 (b) results for C 6+ projectiles and final capture levels n 4 8 =extracted from the ADAS website [42], considering the energy range E 70 keV  amu. Differences between the recommended ADAS data for the intermediate and high energy regions and the ones derived from the universal curve were found lower than 15% and 10%, respectively. Notice that the scaling rule of equation (8) becomes specially useful for high W values and highly excited n levels, where theoretical results are particularly scarce.
In order to illustrate the utility of the n-resolved scaling law, we compare in figure 3 partial cross sections for Ne 10+ projectiles, available in the literature, with results derived from equations (7) and ( (8)), considering final n levels close to the most populated one. Results of figure 3 confirm that the proposed scaling provides reasonable predictions for projectiles with Z 8 P > .

Scaling for nl-distributions
We investigate the nl-distributions, defined as P , nl nl n s s = with nl s the partial CX cross section into the nl subshell, which is obtained as . where U nl is assumed as an universal function. In figure 4 we plot CTMC and EI results for the P nl distributions corresponding to Be , 4+ B , 5+ C , 6+ N , 7+ and O 8+ projectiles, as a function of W , considering final n levels from n = 4 to n = 9. While the CTMC method is used to obtain the nldistributions for the lower values of W , i.e. W 1.5  a.u., the EI approach is employed to describe the high energy region, which corresponds to W 2.0  a.u. In all the cases, the P nl values derived from the CTMC approach match with those obtained with the EI theory in the intermediate region, which corresponds to scaled transferred momenta W in the range between 1.5 and 2.0 a.u. Therefore, it is possible to derive practically universal U nl functions for the different nl-subshells, which can be used to estimate partial angular momentum distributions for ionic projectiles with larger nuclear charges than the ones considered here. To build these curves we have used the join CTMC-EI method, connecting the results of both theories at an impact velocity v 2.2 » a.u. As a general tendency we found that for low W values, in the range W 0.5  a.u., the capture proceeds to the subshell with the highest orbital momentum, i.e. l n 1, =but the lvalue corresponding to the maximal contribution decreases as the scaled momentum W augments, as shown in figure 5, where the P nl distributions for the orbital quantum numbers l n 1 =and l n 2 =are displayed as a function of W . Even though all the curves for l n 1 = -(or l n 2 = -) present similar shapes, the population of the nl-subshell also depends on the main quantum number n, since as n increases, the position of the maximum slightly shifts towards lower W values and its relative contribution diminishes. This behavior is expected to hold also for higher Z P charges.
With the aim of providing a more precise representation of the P nl functions for the different nl-subshells, numerical values corresponding to the universal distributions obtained from the join CTMC-EI method are tabulated in tables 1-6 in the appendix. These tables can be easily used to compute a partial nl cross section for a given electronic transition and impact velocity. For example, 43 s for the C 6+ + H  nl C 43 5 ( ) = + process. First, we have to calculate 4 s by applying equation (7) with Z 6 P = and z P =6/4 as follows: s´As an illustrative example, in figure 6 we display partial CX cross sections, derived from the proposed scaling functions, as a function of the impact energy, considering: (a) Z 6, P = n = 4, l = 3; (b) Z 7, P = n = 5, l = 4; (c) Z 8, P = n = 8, l = 3. These particular cases were only chosen as a proof of the efficacy of the proposed scaling rule. In all the cases, the scaling-derived curve, displayed with a black line, runs close to the CTMC and EI results, as well as to the theoretical data extracted from [35,46]. Only for the O 8+ example with n = 8, the scaling slightly overestimates the data for the lowest energies, since the condition Z n 1 P > is not verified.

Scaling for nlm-distributions
Finally, we study the scaling for the nlm-distributions, defined as P , nlm nlm nl s s = which are expected to verify: where U nlm denotes again an universal function.
As an illustration, in figures 7 and 8 we plot the magnetic number distribution for the n = 6 and n = 7 shells, considering the quantization axis along the direction of the incidence velocity. In all the cases electrons are mainly captured to final states oriented parallel to the velocity, which correspond to m = 0, while the contribution of other final state orientations decreases as m increases. This is a general behavior, also observed for other n shells.

Conclusions
We have used two distinct approximations, CTMC and EI, in order to describe the CX process between multicharged ions and atomic hydrogen for a wide range of impact energies, encompassing from intermediate to high velocities. For five different bare projectiles, total and state-resolved CX cross sections derived with these methods show a good agreement with available experimental and theoretical data, being indicative of the accuracy of the considered descriptions. In addition, the fact that CTMC and EI stateselective cross sections for capture to highly excited levels, from n = 4 to n = 9, match between them in the intermediate energy region allowed us to extend previously derived scaling laws for shell and sub-shells capture distributions [13,19] along the studied energy range. In the case of the nresolved scaling, an analytical universal function in terms of the scaled transferred momentum W is suggested, while for the nl-distributions, values obtained from the proposed scaling are listed in the appendix to facilitate their interpolation. These scaling rules are important tools, not only to display partial cross sections for different multicharged projectiles in a comprehensive way, but also to predict n-, nl-, and even nlm-distributions originated by highly charged ions.

Acknowledgments
M S G and J E M acknowledge financial support from CONICET, UBA, and ANPCyT of Argentina. A J and C I financial support from the projects ENE2011-28200 and ENE2014-52432-R of the Secretaría de Estado de Investigación, Desarrollo e Innovación (Spain).

Appendix A. Appendix: P nl tables
Numerical values for the distribution corresponding to equation (9), derived from the proposed scaling rule, are listed in the following tables.