Molecular electronic stopping cross section for H and He beams colliding with water: projectile charge state contribution

When an ion beam impinges on a target material, electron stripping and capture processes involve different charge fraction states in the beam, such that each projectile charge state produces a different energy-loss. In this work, the energy deposition of swift hydrogen and helium ion beams colliding with a water target in the gas phase is studied. The electronic structure of the molecular target is represented in terms of core, bond, and lone pair orbital decomposition within a Harmonic Oscillator representation. In this way, the stopping cross section becomes only a function of the orbital mean excitation energy, I 0i . The ion beam charge fraction compositions colliding on water is determined from the work of Wedlund et al (2019) Astronomy & Astrophysics, 630, A36) by accounting for the electron transfer cross sections. We find that the larger the projectile charge state, the larger the electronic stopping cross section and that the beam charge fraction determines the position of the maximum of the electronic stopping curve. Also, in agreement with the experiment, evidence is given on the dominant role of the largest projectile charge state in defining the stopping cross section for high energy collisions, while for low collision energies it is the lowest charge state together with all possible charge states contributing at the maximum of the electronic stopping cross section curve. Our results are reported and compared to available experimental data showing an excellent agreement to the available literature.


Introduction
When energetic particles pass through a target material, they lose energy with an eventual stop of the projectile that induces damage to the material.This energy loss allows to determine the penetration depth of the projectile, which is of importance in semiconductor doping using ions beams [1], cancer treatment by radiotherapy and dosimetry [2], electronic devises [3], and the study of irradiation effects in various materials [4].As the beam slows down, several processes take place, such as target excitation, ionization, and charge exchange.
The problem and history of the stopping power is directly linked to the birth of quantum mechanics.In the same year as Niels Bohr published his model of the hydrogen atom [5], 1913, he proposed a classical model for the slowing down of charged particles [6][7][8][9].With the advent of quantum mechanics, Bethe provided a full quantum treatment of energy deposition [10,11] as one of the first applications of perturbation theory.Later on, Fermi dealt with the problem at low collision energies [12] by studying the energy deposition of muons.Another approach to study energy deposition was given by Lindhard [13] by accounting for the response of the material through a dielectric-function formalism.
Energy deposition studies for screened projectiles goes back to Dalgarno and Griffing [14], followed by the works of Ferrell and Ritchie [15], Kim and Cheng [16], Arnau and Echenique [17], Moneta and Czerbniak [18], Sigmund [19], Azevedo et al [20], and lately Artacho et al [21] and Reeves et al [22].For a review of energy deposition, see [23].Recently, Cabrera-Trujillo [24] proposed a treatment for the study of the electronic stopping cross section for a hydrogen beam colliding with atomic targets with excellent results when compared to the experimental data.However, the approach combines two different theories, one at low collision energies (Electron-Nuclear Dynamics) that accounts for the electron transfer process and the other at high collision energies (the standard Bethe theory) that considers the excitation and ionization processes.Nevertheless, while penetrating matter, the projectile ion may carry, at least temporarily, some bound N 1 Z 1 electrons, since electron capture and loss by the projectile are processes with a high probability [9,[25][26][27].Thus, for a given projectile velocity, the ion beam may consist of projectiles with varying number of electrons [28,29].
The quantum mechanical treatment for partially stripped projectiles carrying bound electrons was proposed by Cabrera-Trujillo et al [30].The approach has successfully been applied to several different projectile-target systems by several authors [20,[31][32][33][34][35][36][37][38][39][40][41][42][43][44].In this work, we extend that approach to H and He ion beams when colliding with water in the gas phase.First, we determine the projectile and target atomic form factors used to incorporate the projectile electronic structure into the electronic stopping cross section arising from individual electrons.Second, we use core, bond, and lone-pair (LP) orbital decomposition to describe the target molecular electronic structure within a Harmonic Oscillator basis set to obtain a simple expression for the stopping cross section.The approach provides a straightforward numerical expression for the electronic stopping cross section for structured projectiles in an ionic ground state, which is easily implemented and provides good comparison to experiments.
The layout of our work is the following.In section 2, we provide a summary of the theory of stopping power for an ion with electronic structure impinging on a target incorporating the projectile ground state and the beam's charge fraction.In section 3, we discuss and compare our results with other theories and experiments.Finally, in section 4, we present our summary and perspectives.

Stopping power
2.1.The Bethe theory for structured projectiles within the First born approximation Experimentally, it is known that a beam contains all the charge states as a function of the beam energy [28].As proposed by Allison et al [28,29,45], the electronic stopping cross section for all charge fractions in the beam is given by that is, a statistical average over the beam charge fractions with f N 1 being the fraction of the ion beam for the projectile carrying N 1 electrons (N 1 = 0, 1, KZ 1 ) and Z 1 being the projectile nuclear charge.Here ) is the electronic stopping cross section for a projectile with i number of electrons.The stopping cross section is defined as the energy loss per unit length, dE/dx, per target density number n t , i.e. S v n Both, the electron stripping and capture cross sections for a projectile colliding with a given target are defined by the beam charge fraction f i .Notice that we have neglected the contribution of negative ions within the material.The theory to determine S v e i ( ) for a projectile with a number of fixed electrons, i = N 1 was developed by Cabrera-Trujillo et al [30].Consequently, here we only summarize the principal equations that lead to the present implementation.Consider a projectile ion moving with a velocity v, mass M 1 , nuclear charge Z 1 e, and carrying N 1 bound electrons described by the electronic eigenstate |n 0 〉, colliding with a stationary target with mass M 2 and N 2 bound electrons in an initial state denoted by |m 0 〉.By assuming that the projectile, as it penetrates the target, remains in its ground state carrying N 1 electrons obeying an stripping process known as Bohr's adiabatic criterion [8,46], then the stopping cross section contribution arises from the target excitation, i.e.

S v e
Here are the Generalized Oscillator Strengths (GOS) for the target as defined by Bethe [10], E mm 0 D are the excitation energies of the target, and • are the atomic form factors [47,48] for the electronic transition between the initial and final state of the target (or projectile).Here, q is the momentum transferred during the collision from the projectile to the target with its minimum and maximum values given by momentum and energy conservation implies that the momentum transferred to the target contains a screened contribution from the projectile electronic structure through the ionic ground state atomic form factor.
Our next assumption is that the target's N 2 electrons contribute to the electronic stopping cross section through a Bragg-like rule.This assumption is justified by the independent particle approach [49] such that and F m m i 2 0 is the GOS of the i-th electron in the target.Equation (6) gives the stopping cross section for a projectile with N 1 Z 1 electrons that collides with a target with N 2 electrons.

H and He atomic form factor
We shall consider the case of hydrogen and helium beams in their ground state.For the case of a bare projectile, there are no atomic form factors, i.e.M 0 n n 1 0 0 = , as there are no electrons.For a hydrogenic projectile (H or He + ), the wave function is given by where Z * = 1 for hydrogen and Z * = 2 for He + .For the case of a helium atom (two electrons), we use its variational hydrogenic-like solution Thus, the projectile atomic form factor, M n n and Z 1 = 2 while for the case of neutral helium, Z Z 16

Molecular target description
For the case of the molecular target, we use a core, bond, and lone-pair quantum chemistry description of a molecule through a Floating Spherical Gaussian Orbital (FSGO) basis set [50,51].In this representation each electron is described by a Gaussian orbital for either a core (1s 2 ), bond, or lone pair orbital in the molecule, Here ρ i is the orbital radius and R i is the position of the Gaussian center.The set of molecular parameters {ρ i , R i } is obtained by variationally minimizing the energy.The FSGO approach predicts reasonably well the electronic structure and geometry of a molecular system [51].
The connection of the FSGO model with the treatment of energy deposition resides in equation (11), which corresponds also to the ground state wavefunction of the Harmonic Oscillator (HO).Consequently, it is natural to assume a HO basis set to study the target electronic structure [49].We have found that in the HO approach, the GOS has the analytical expression where  with ω 0i the oscillator frequency [49], which connects directly with the FSGO approach as m 2 .
1 3 In this description, the mean excitation energy for the target becomes I 0i = ÿω 0i , Therefore, the stopping cross section can be determined from equations (7), (10), and (12) by defining x q 2 i ´-


where = [ ], S 0 = 0.76199564 × 10 −15 eV cm 2 , and Here, ò = E p /I 0i is the reduced projectile collision energy and where its integer part corresponds to the maximum of excitations induced in the target by the projectile.This expression is straightforward to evaluate numerically and it gives the stopping cross section for a projectile (with nuclear charge Z 1 and a fixed number of bound electrons N 1 ) that collides with a target electron bound to a molecule consisting of N 2 electrons, each with a mean orbital excitation energy I 0i given by equation (13).The integration of equation ( 14) is carried out straightforwardly by implementing a 96 points Gauss-Legendre quadrature [52] in a FORTRAN 95 code.The orbital mean excitation energies used in this work for H 2 O are those reported in [53] and are given in table 1 for completeness.Thus, equation (14) gives the contribution to the stopping cross section for an ion with N 2 electrons at a given velocity v.

Projectile beam charge fraction
Now, we turn to the study of the beam charge fraction as required by equation (1).For the case of a hydrogen beam, the charge fractions are determined from the electron capture, σ 10 , and stripping cross sections, σ 01 , as formulated by Cuevas et al [29], i.e.
: H H single capture : H H single stripping, 10 0 01 0 where σ ij is the cross section for a charge exchange process from an initial charge state i to a final charge state j.
The charge fraction for a hydrogen beam is then such that a projectile colliding with different targets may have different beam fractions, consequence of the charge exchange process.Here f 1 is the charge fraction of H + (N 1 = 0) while f 0 is the charge fraction of a neutral H (N 1 = 1).The charge-state distributions for He 2+ , He + , and He 0 have been derived by Wedlund et al [54].For the three charge states of helium, the six relevant cross sections σ ij , are where f 2 is the charge fraction for He 2+ , f 1 is the charge fraction for He + , and f 0 is the charge fraction for neutral He.These capture and stripping cross sections are determined by solving the time-dependent Schrödinger equation for all the initial projectile charge states.The calculation of σ ij is a field of research by itself [55], theoretically and experimentally.

Analysis and discussion
The theoretical charge fraction results of Wedlund et al [54] for the case of hydrogen ions colliding with water molecules as a function of the projectile reduced velocity u = v/v 0 , with v 0 being Bohr's velocity, are shown in figure 1 (A) (symbols).In the same figure, we show the analytical expression proposed by Betz [56] (dashed lines) obtained through an effective charge assumption.We find that a much better fit to Wedlund's data is obtained by the expression The solid ( f 0 ) and dotted ( f 1 ) lines are our analytical fit to Wedlund's data, equation (19), for H and H + beams, while the dashed lines are Betz's analytical expression [56], equation (18).(B) The same as panel (A), but for a helium ion beam when colliding with a water target.The solid ( f 0 ), dotted ( f 1 ), and dot-dashed ( f 2 ) lines correspond to He, He + , and He 2+ , respectively.
as shown in the same figure (thick and dotted lines).Notice that in our approach the beams have the same charge fractions at around 1.2 a.u.(35 keV/amu) while in Betz's case it occurs at around 0.6 a.u.(9 keV/amu).This is a consequence of the electron capture and stripping process occurring in the hydrogen beam produced by the interaction with the water molecule [54].
The results for the case of a helium beam colliding with water, as reported also by Wedlund et al [54] are shown in figure 1 where b = 0.0273606 = 11/402.For collision energies between 25 and 150 keV/amu (1.0-2.5 a.u. in velocity), the He + charge fraction, f 1 , is dominant, with 60% of the charge fraction while the other two charge fractions contribute with 20%, as a consequence of the large stripping cross section of neutral He and the large single electron capture cross section by He 2+ .For both H and He cases, our analytical fit is based on the assumption that f e au 0 b = -as suggested by Betz and it is adjusted through a nonlinear-squares Marquardt-Levenberg algorithm [57] for the parameters a and b with a similar procedure for the other charge states.
The electronic stopping cross section for a hydrogen beam colliding with a water target is shown in figure 2 and it is compared to available experimental data (letter symbols) [60][61][62][63] and the theoretical work of Reeves et al [22] based on the TD-DFT (double-dot-dashed line).
The red thick solid line is the average result over all the projectile charge fractions, equation (1), when using equations (6), (14), and (19).The purple solid line is the contribution from just a bare proton and the green dotted line is the contribution of the neutral hydrogen projectile.We observe a good agreement between our results and the experimental data for the region from the maximum of the stopping cross section curve towards high energies.Recall that this is only the contribution from a projectile with electronic structure in its ground state.Thus, we expect it to be lower than the experimental data in the low energy regime as the excited states of the projectile should contribute with a further energy loss through projectile excitations.Work is in progress to incorporate them in a similar approach.We notice that the TD-DFT results produce a reasonable good stopping cross section at low collision energies but then deviate at the maximum and higher collision energies when Figure 2. Electronic stopping cross section for a hydrogen beam colliding with water in gas phase.The red thick solid line is the result of equations (1), (14), and (19) when using the hydrogen beam charge fractions.The purple solid line is the result of equation ( 6) for the case of bare protons only and the green dotted line is when we have a neutral hydrogen atom, N 1 = 1.The experimental data are taken from H. Paul's compilation database [58,59].The experimental data are from: A [60], B [61], C [62], and D [63] while the blue double-dot-dashed line is the work of Reeves et al [22] (TD-DFT).
compared to the experimental data and our theoretical results.This implies that the excitation and ionization processes in TD-DFT are not properly described.Now, we turn our attention to the case of helium beams for the same target.The electronic stopping cross section for a helium beam colliding with water (H 2 O) is shown in figure 3. Again, the red thick solid line corresponds to the charge fraction averaged stopping cross section, the purple solid line to the bare helium ion contribution, the green dotted line to the He + ion projectile contribution, and the light blue dot-dashed line to the neutral helium contribution.The experimental data, represented by letter symbols, are taken from [64][65][66][67][68] and the TD-DFT work of Reeves et al [22] is shown by the blue double-dot-dashed line.We observe that our approach agrees very well with the experimental data for projectile collision energies from the maximum towards higher collision energies.Furthermore, the beam charge fraction determines the position of the maximum of the stopping cross section curve as an average over the maxima of the individual contributions of the beam charge fractions.For both, H and He beams, we find that the higher the ionic charge, the larger the electronic stopping cross section.However, the final electronic stopping cross section is the average result of these ion beam charge fractions such that at high collision energies dominates the high projectile charge while at low collision energies dominates the lowest charge state beam fraction.The TD-DFT results of Reeves et al [22] have a similar behavior than that for H beams (figure 2).They are reasonably good at low collision energies, but deviate from the experimental data and from our results from the maximum to higher collision energies.These results confirm that our approach provides the correct description for excitation, ionization, electron capture and loss involved in the energy loss process from collision energies from the maximum of the stopping curve and higher.

Summary
The projectile electronic structure, through its ionic ground state wave-functions and the projectile beam charge fraction, play an important role in determining the electronic stopping cross section of swift ions for H and He beams colliding with a water target in gas phase.Our approach only requires the knowledge of the projectile number of electrons, the projectile wave-function screening parameters, and target orbital mean excitation energies, without adjustable parameters.Our results are in very good agreement to the experimental data from the peak of the stopping cross section curve towards high collision energies.
Our work shows that the electron capture and stripping cross sections play an important role together with the projectile atomic form factors in determining the electronic stopping cross section of structured projectiles.In particular, the projectile beam charge fraction determines the position of the maximum of the electronic Figure 3. Electronic stopping cross section for a helium beam colliding with water.The red thick line is the result of averaging over the helium beam charge fraction.The purple solid line is the contribution for a bare α particle (He 2+ ), N 1 = 0.The green dotted line is the result for He + and the light blue dot-dashed line is for neutral helium, N 1 = 2.The experimental data are from: A [64], B [65], C [66], D [67], and [68] while the blue double-dot-dashed line is the result of Reeves et al [22] (TD-DFT).See text for discussion.stopping cross section.Also, we find that the largest the ion beam charge state, the largest the contribution to the electronic stopping cross section.The inclusion of projectile excitations into our approach would provide an even better description of the energy loss process.This is work in progress and will be reported elsewhere.

Figure 1 .
Figure 1.(A)Hydrogen ion beam charge fraction when colliding with water in gas phase as reported by Wedlund et al[54] (symbols).The solid ( f 0 ) and dotted ( f 1 ) lines are our analytical fit to Wedlund's data, equation(19), for H and H + beams, while the dashed lines are Betz's analytical expression[56], equation(18).(B) The same as panel (A), but for a helium ion beam when colliding with a water target.The solid ( f 0 ), dotted ( f 1 ), and dot-dashed ( f 2 ) lines correspond to He, He + , and He 2+ , respectively.
(B) (symbols).In the same figure, we show our analytical fit

Table 1 .
(14)es of ρ i and I 0i used to evaluate equation(14)to determine the average electronic stopping cross section for water.Here ρ i is in Bohrs (a 0 ) and I 0i in eV.