Classical-trajectory model for ionizing proton-ammonia molecule collisions: the role of multiple ionization

We use an independent electron model with semi-classical approximation to electron dynamics to investigate differential cross sections for electron emission in fast collisions of protons with ammonia molecules. An effective potential model for the electronic orbitals is introduced, and utilized in the context of the classical-trajectory Monte Carlo (CTMC) approach for single-electron dynamics. Cross sections differential in electron emission angle and energy are compared with experimental data. Compared to previous scattering-theory based quantum-mechanical results the time-dependent semi-classical CTMC approach provides results of similar quality for intermediate and high ionized electron energies. We find some discrepancies in the total cross sections for q-fold ionization between the present model and independent-atom-model calculations. The double ionization cross sections are considerably larger than recent experimental data which are derived from coincidence counting of charged fragments. The calculated triple ionization cross sections exceed the experimental coincidence data for q = 3 by several orders of magnitude at intermediate energies.


I. INTRODUCTION
Collisions of protons and multi-charged ions with molecules has become an active field of research.A number of approaches has been used for the water vapour target given its significance for radiotherapy.Among the quantum-mechanical methods from stationary scattering theory are the Born approximation [1,2], as well as versions of the continuum distorted wave (CDW) theory [3][4][5][6][7][8][9].Within the semi-classical approximation to the nuclear motion there were attempts to solve the time-dependent Schrödinger equation (TDSE) within a mean-field approximation, or density functional theory [10][11][12].A numerical solution of the TDSE with a three-center model potential was given in Ref. [13].Another quantummechanical approach is the independent-atom model (IAM) where accurate cross sections obtained in an independent-electron model (IEM) for collisions with constituent atoms are combined to form cross sections for the given molecular target [14][15][16][17].The development of methods based on the TDSE to produce ionization spectra, i.e., cross sections differential in electron energy and/or angle, has been slow, focusing so far mostly on collisions with simple atomic targets or molecular hydrogen.Therefore, many studies resort to the classical treatment of electron motion, which can be viewed as replacing the quantum Liouville equation by its ℏ = 0 limit, i.e., reducing the electronic motion problem to classical statistical mechanics.
In the context of ion-molecule collisions the classical trajectory Monte Carlo (CTMC) methods have been developed for molecular targets by a few groups [18][19][20].They can be realized at the level of an IEM, or pushed to an N -electron approach, although the inclusion of the electron-electron interactions remains a big problem.Therefore, the N -electron calculations often remain effectively close to single-electron models [21], except that they allow to obtain multiple-electron event analysis directly without the statistical approach required by the IEM analysis [22][23][24].
The CTMC-IEM approach for ion-molecule collisions has recently been extended to the level of time-dependent mean-field theory in order to deal with highly charged projectiles, and in order to be able to calculate capture processes correctly at lower energies [25].Recently differential electron emission in 250 keV proton-water molecule collisions were reported together with new experimental data, as well as CDW theory (with eikonal initial state) [26].
Our motivation for looking in detail at proton-ammonia collisions using the CTMC-IEM approach is as follows.The molecular targets H 2 O, CH 4 , NH 3 represent 10-electron systems with a central atom surrounded by hydrogen atoms, and going from a planar towards different spatial geometries.Fast proton collisions with these targets have been investigated experimentally, where the focus is on the charged fragments produced in these collisions.Recently emphasis was placed on multiple ionization events, and some serious discrepancies were found in the case of the ammonia target [27].In contrast to the case of water vapor [28,29], where multiple ionization is found to be a significant contributor, and the case of methane [30] where double ionization was clearly identified, the experimental ammonia results did not support the case of a direct two-electron ionization process.
A recent analysis of the total multiple ionization cross sections for the three collision systems within an IAM approach [17] highlights the controversy from one theoretical point of view.Here we would like to address the role of multiple ionization both in electron emission (where the distinction between net (total) and single ionization is sometimes blurred in the literature), and also from a total cross section point of view.We construct a potential model for the NH 3 molecule in analogy to what was done for the water molecule [13,18,20] and then apply the CTMC approach within the IEM.
The layout of the paper is as follows.We begin in Sec.II with a short summary of the CTMC-IEM.For the p − NH 3 system at 250 keV collision energy and higher we found the time-dependent mean-field model [25] to give practically the same results for differential electron emission as the static CTMC model, and thus we focus on the latter in the present work.We provide in Sec.II A also the expressions for doubly differential cross sections (DDCS) in ejected electron energy and direction for both the typical case of net ionization, and for the specific case of single ionization.In Sec II B we discuss the construction of the potential model.Results are presented and compared with experimental data and selected previous calculations in Sec.III, first for differential electron emission (in Sec.III A) and then for total cross sections (in Sec.III B).The paper ends with conclusions in Sec.IV.Atomic units, characterized by ℏ = m e = e = 4πϵ 0 = 1, are used unless otherwise stated.

II. THEORY
We provide a brief summary of the CTMC-IEM with static target potential, as developed for the water molecule in Ref. [20].Within the semiclassical impact parameter approximation the proton trajectories follow straight lines, and provide an explicitly time-dependent potential for the electronic orbitals bound in a multi-center potential which is described in the next subsection.The total Hamiltonian of the system is a sum of single-particle Hamiltonians for the electronic orbitals, i.e., H = j h j (t), with orbital-independent h j (t) ≡ h(t).For reference we use the single-center Hartree-Fock orbital energies of Moccia [31]: ϵ 1a 1 ≈ −15.52, ϵ 2a 1 ≈ −1.12, ϵ 3a 1 ≈ −0.415, and ϵ 1ex = ϵ 1ey ≈ −0.596 (given in Hartree units), and spin degeneracy leads to double occupation.

A. CTMC-IEM approach
The single-particle Hamiltonians for the evolution of orbitals can be written as Here Z p = 1 for protons and R(t) is the (straight-line) proton trajectory.The model potential v mod (r) is evaluated for a random orientation of the molecule with ensemble averaging implied.
For the fast collisions considered in this work neither vibrational nor rotational motion of the target molecule is taken into account.At this point one has two options: (i) quantum mechanical evolution of the time-dependent orbitals (cf.Ref. [13]); (ii) CTMC calculations in the spirit of the ℏ = 0 approximation.We follow the latter approach in this work.
The electronic orbitals are simulated by ensembles of classical trajectories which satisfy Hamilton's equations.In contrast to the time-dependent mean-field approach of Ref. [25], which was also applied for the present work, but which did not yield significantly different results, the methodology is then straightforward CTMC in the impact parameter approximation.Ensembles of electron trajectories representing MOs are analyzed both for net and for multiple differential electron emission.
Initial conditions for the trajectories are based on the microcanonical distribution.In quantum simulations one has to make sure that the orbitals are placed in eigenstates of the target Hamiltonian.In the classical Liouville approach, however, one is not forced to apply such a condition [32].The ℏ = 0 approximation to the quantum Liouville equation requires for stable evolution only that the initial distribution be a pure function of energy [33].Within the microcanonical approach one constructs stable initial orbital distributions for a given external potential by combining a chosen initial value for the orbital energy with a distribution over angular momenta consistent with the microcanonical ensemble [34].Examples of such classical microcanonical distributions were described, e.g, for the water molecule in Ref. [18] and for the uracil molecule in Ref. [19].
The analysis whether a trajectory contributes to ionization or capture after proton and molecule are separated by a large distance is as follows: at the final proton-molecule distance of about 500 atomic units the energy of the test electron representing an MO is calculated with respect to projectile and target.If both energies are positive the event is counted as ionization, and the information about ionized electron energy and direction is recorded.
The number of trajectories is rather large due to the sampling of different impact parameters and random molecular orientations.The sampling is required since the experimental data are insensitive to this information, i.e., projectile deflections or target recoil motion are not recorded.Most works in the literature report net differential ionization cross sections by summing over the contributions from all MOs.The single probability for ionization from orbital j is defined in terms of the number of ionizing trajectories n ion j as A binning procedure allows to obtain such a single probability differential in electron energy and scattering angles.The net differential cross section can be written as The interpretation of this expression is that an electron has been recorded with a given energy and direction (obviously with finite resolution) independent of how many electrons were ionized overall.The factor of two accounts for spin degeneracy in each MO.This analysis corresponds to the typical experimental situation.
Expressions for differential cross sections in the IEM specific for q-fold ionization were derived in Ref. [20].For q = 1 we quote This expression describes events where an electron has been recorded at given energy and direction, while no other electrons were found in the continuum.We report in Section III A results for both cases, i.e., Eqs. ( 3) and ( 4).

B. Model potential
The target potential is modelled as a sum of central potentials for each atom of the molecule: Here r N and r H i are the distances from the active electron to the nitrogen and the i = 1, 2, 3 hydrogen nuclei of the NH 3 molecule.We follow the nuclear geometry obtained in singlecenter self-consistent field (SCF) calculations [31]: the N-H bond lengths are 1.928 a.u., the polar angles of the H atoms are 108.9• and the azimuthal angles are given as 90 • , 210 • , and 330 • , respectively.The central potentials in Eq. ( 5) are assumed to take the forms where N N = 6.2775, α N = 1.525,N H = 0.9075, and α H = 0.6170.These choices were made in the following way: The parameters N H , α H for the hydrogen atoms are the same as in previous work for H 2 O in which an analogous model potential was used [13,18,20,35].The asymptotic large-r behavior of the model potential should be −1/r.This fixes the screening charge parameter of the nitrogen atom to N N = 9 − 3N H = 6.2775.The remaining parameter α N was determined such that the energy eigenvalues of the Hamiltonian p 2 2 + V mod are in reasonable agreement with the SCF results from [31].

A. Doubly-differential cross sections
In Fig. 1 we show DDCS results for an impact energy of E p = 250 keV.The ℏ = 0 approach to electron dynamics should work well at not-too-low ionized electron energies, and we find that this is the case for all experimentally observed cases.The net ionization results (solid lines, calculated with Eq. ( 3)) agree well with the experimental data with some factor-of-two deviations at E el = 200 eV electron energies and forward angles.
One of the major objectives of this work is to illustrate the role of multi-electron processes within the CTMC-IEM.The dashed lines show the results for single ionization, i.e., an  4).The data points are the experimental results of Ref. [36], as reported in Refs.[1,7].
electron is detected at given energy and angle, and it is the only electron in the continuum.
We observe that the results are of very similar shape and typically reduced by a factor of about two.This implies that at 250 keV impact energy double (or higher multiple) ionization makes roughly an equal contribution to the net ionization cross sections.This is perhaps overlooked at times, based on the thought that protons should predominantly lead to single ionization only.The ammonia molecule is an extended object and has a number of electrons that are bound on the scale of an atomic ground-state hydrogen electron.
The experimental data were explained previously using quantum-mechanical stationary scattering theory methods, which work well for low-energy electron emission.Senger [1] used a plane-wave Born approach adopted from proton-atom scattering to represent MOs as linearly combined atomic orbitals.Results were reasonable for ionized electron energies above 20 eV with some noticeable shortfall for backward scattering, particularly for E el = 200 eV.
Mondal et al. [7] extended a three-Coulomb-wave model and used the single-center SCF orbitals of Moccia [31] to describe the initial MOs.
Tachino et al. [4] extended the theory to the post and prior versions of the CDW-EIS model (EIS=eikonal initial state).They used the Moccia orbitals, as well as a linearly-combined atomic orbital SCF representation, both giving very close results at the intermediate impact energy of E p = 250 keV.The post and prior versions of the CDW-EIS model did show, however, remarkable differences.There is some discussion in the literature as to which form ought to be favoured, and the situation may actually depend on the collision energy.The prior version appears to have been favoured (e.g., commented upon in Ref. [20]).
In Fig. 2 we compare the results for net ionization from Fig. 1 with these CDW-EIS results and with the three-Coulomb-wave model cross sections of Ref. [7].The post and prior forms provide very close results at low electron energies, and display some undulatory behaviour at E el > 100 eV and forward angles where the two models make somewhat different predictions.
For E el = 20 eV the CTMC results agree with the experimental data which are also described reasonably well by three-Coulomb-wave model of Ref. [7] (which overestimates them a bit) and the CDW-EIS results of Ref. [4] which fall a bit short at the largest angles.
At higher electron energies the correspondence principle allows the CTMC model to demonstrate its strength, since it employs a more realistic multi-center interaction.For emission angles θ el ≥ 30 degrees it follows the data very well, while the CDW-EIS results show some weakness in the backward direction.The present results for E el = 50 eV agree in shape with the results of Ref. [7].
For the CDW-EIS models the shortfall in predicted backward electron emission leads to an underestimation of the doubly differential cross section by about an order of magnitude at the higher electron energies of 100 and 200 eV, while the CTMC results show very good agreement with the experimental data.The good agreement of the CTMC-IEM results at backward angles and high electron energies is gratifying and lends credibility to the experimental data.
It is likely caused by the multi-center potential used in the CTMC approach which results in a noticeable effect even after orientation-averaging of the molecule.
The three-Coulomb-wave model gives good results for 100 eV, and falls short by about a factor of two at 200 eV when compared to experiment at the largest measured scattering angles of 110 and 125 degrees.While we do not show the plane-wave Born data of Ref. [1] we note that the comparison is provided in Fig. 7 of Ref. [7].Overall, the three-Coulomb-wave model agrees with the model used in Ref. [1]: a small relative enhancement of backward electron emission at high energies appears to be a notable difference.We note that in the case of p − H 2 O collisions at E p = 250 keV improvements to the CDW-EIS model have been introduced which address the problem at backward angles [26].
Concerning the comparison of the two versions of CDW-EIS we observe that the post form shows stronger undulations in the cross sections for E el ≥ 100 eV and forward to intermediate scattering angles which are not supported by the data.The prior form, on the other hand, fares better in this respect.
The case of 1 MeV impact energy is presented in Fig. 3.A marked difference is found in the angular dependence of the DDCS.The results are of similar quality as those of Mondal et al. [7] (not shown).In some way the result for net ionization just confirms that the CTMC-IEM works at this high energy.We observe that multiple ionization plays a lesser The data points are the experimental results of Ref. [36], as reported in Ref. [1,7].
role compared to 250 keV proton impact.The single-ionization DDCS shown by the dashed lines follow the net ionization more closely in Fig. 3 than in Fig. 1.
The ratio of single to net DDCS varies with collision energy.At 250 keV impact energy this ratio varies between 0.65 at forward angles and 0.5 at backward angles for low ionized electron energies.For electron energies of 100 eV and higher it is closer to 0.5 and more uniform.For the impact energy of 1 MeV the ratio of single to net DDCS values moves towards 0.8 − 0.9.It is at the higher end for low ionized electron energies and closer to the lower bracket at energies of 50 eV and higher with small variation with respect to emission angles.
Concerning the absolute height of the net ionization DDCS we note the good agreement between the CTMC-IEM and experimental results.The 1 MeV collision energy is within the range where the deviation between the classical and quantum total ionization cross section behaviour is not yet noticeable (the difference being a logarithmic dependence missing in the classical case).
In Fig. 4 we show results for the impact energy of 2 MeV.We have not included data The data points are the experimental results of Ref. [36], as reported in Refs.[7].
for the ionized electron energy of 1,000 eV, due to statistical limitations in the present calculations.For intermediate to high electron energies (50-200 eV) the comparison with the data is very good.The contribution of single ionization towards the net DDCS is at the 80 % level with small dependence on the emission angle and energy.
For the lower electron energies of 20 eV, and particularly 11.3 eV the CTMC-IEM results clearly fall short of the experimental data.This is likely to be caused by the lack of a dipole ionization mechanism in the semi-classical ℏ = 0 approach, which is discussed in the literature, cf.Ref. [37].One can then raise the question of the total cross section at such a high collision energy: does this shortfall of low-energy electron production lead to an underestimation of the net ionization cross section?This is addressed in Section III B. We note, however, that it is difficult to estimate the total cross section from the experimental data shown in Fig. 4, since an extrapolation to low electron energies would be required.In turn, this also raises the question of absolute normalization of the experimental DDCS data.
As mentioned above, the role of double (or higher multiple) ionization is reduced at higher collision energies, i.e., there is no longer a factor-of-two discrepancy between the solid and the dashed lines as one compares the collision energies of 250 keV with 1 and 2 MeV.
Nevertheless, the main conclusion from the analysis of CTMC data is that multiple ionization contributes significantly to the net DDCS data: it is quite important at the lower energies.
Another view of the situation can be obtained by comparing integrated (total) cross sections as a function of energy to see how the net cross section is made up of single and multiple ionization contributions.

B. Total cross sections
In Fig. 5 we show total cross sections for p − NH 3 collisions over a wide range of energies.
On the one hand they allow to understand to what extent the discussion of shown DDCS is representative of the entire collision problem at a given impact energy.On the other hand the comparison allows us to discuss the relationship with experimental total cross sections based on fragmentation yield measurements [27], which were analyzed recently within the IAM framework to compare them with supposedly similar target molecules, such as and CH 4 .
The data for E p = 250 keV in Fig. 5 show how the net cross section is dominated by single ionization, while σ 1 + 2σ 2 + 3σ 3 is a lower bound to the net cross section, and the CTMC result is quite close to the data point of Ref. [36].Together with the comparison of DDCS data shown in Fig. 1 we may conclude that the CTMC-IEM presents a consistent picture in the sense that about 40 % of the net ionization cross sections comes from double (and higher) direct ionization processes.Comparison with the experimental fragmentation data of Wolff et al., on the other hand, would imply that the net cross section is dominated by σ 1 alone.
At higher energies the CTMC result for 3 q=1 q σ q falls barely below the experimental data.This indicates that the energy dependence of the ℏ = 0 semiclassical approach to the electron dynamics based on the microcanonical distribution is incorrect at high energies on account of the sharp cut-off in the probability density as a function of distance.For the atomic hydrogen H(1s) target this problem was identified and different attempts were made to remedy the problem using other initial distributions than the microcanonical one [33,38,39].Another reason is the lack of a dipole ionization mechanism in distant collisions [37].
When looking at the CTMC data for σ 1 (E p ) we notice that they are similar in shape Total cross sections (in units of Å 2 ) for proton collisions with ammonia molecules as a function of collision energy.Solid blue line: present CTMC approximate net ionization result, 3 q=1 q σ q ; blue open diamonds: net ionization cross sections from Lynch et al. [36].Solid lines (black, red, green): present CTMC results for σ q with q = 1, 2, 3 respectively.Dashed lines (black, red, green): IAM results shown in Ref. [27] and Ref. [17] for σ q .Dots (black, red, green): experimental values for σ q from fragmentation yields (Ref.[27]).The results for q = 3 are associated solely with to the experimental data of Ref. [27], but lower by about one third.While at low collision energies σ 2 (E p ) can make up for this shortfall when comparing with the total (net) cross section, it fails to do so at high energies due to the faster fall-off with E p .
The comparison with the IAM results given by the dashed black line (Refs.[17,27]) shows that the IAM is making a different prediction: its values for σ 1 match the experimental data at medium to high collision energies rather well.Only at lower E p do the contributions from σ 2 start playing a more significant role for the net cross section.
The double ionization data σ 2 highlight this discrepancy between the two theories and experiment.The quantum-mechanical IAM results show a steeper fall-off for the ratio σ 2 (E p )/σ 1 (E p ) than the present CTMC results (cf.Fig. 6 in Ref. [17]).For the IAM results the ratio is inversely proportional to E p log (E p ) at the higher energies, while the CTMC result follows some power law (∼ E −0.6 p ).The experimental data have been explained in Ref. [27] in terms of decay of satellite states, i.e., not at all consistent with a direct multiple ionization picture.
The case of q = 3 may illustrate limitations of the IEM approach for singly charged projectile ions.The CTMC cross section is rather large and matches the experimental σ 2 values.This indicates that the IAM which uses atomic IEM cross sections and combines them geometrically has an advantage in this respect.The experimental data are from a single coincidence channel that corresponds to q = 3, and may be too low since two-proton coincidences together with a singly charged nitrogen atom cannot be recorded with the experimental set-up.Multiple proton coincidences were observed with a special coincidence technique in p − H 2 O collisions by Werner et al. [40].Thus it is not unreasonable to assume that the experimental data of Ref. [27] are only a lower bound for σ 3 .
Presently, we cannot offer any resolution to this controversy except to re-iterate that the experimental fragmentation data are inconsistent not only with the quantum-mechanical IAM approach, but also with the semiclassical CTMC method.Both theoretical approaches take the molecular geometry fully into account.

IV. CONCLUSIONS
We have presented a model calculation for fast proton-ammonia molecule collisions.An independent electron model is introduced in analogy with previous works which involve the water molecule which from the point of view of MO energies has a comparable valence electron structure.This IEM was solved at the ℏ = 0 level and doubly differential cross sections were shown to be of overall competitive quality with approaches based on quantum scattering theory.These approaches follow a scheme of linearly combined atomic orbitals, and effectively do orientation averaging before the collision process is considered.The present CTMC model makes a reasonable prediction for the net ionization cross section.
The DDCS comparison with experiment and the mentioned quantum calculations is complemented by explicitly showing the contributions from singly ionizing collision events, which demonstrates that multiple ionization does play an important role within this IEM approach.Previously, a discrepancy with experiments for fragmentation yields as a function of energy was reported, particularly, with respect to the differences with water and methane targets for which direct multiple ionization appears to be the dominant contributor to double, and even triple ionization for CH 4 and H 2 O respectively [17,27].The present work shows that the discrepancy persists in an IEM approach.Since the experimental fragment collection may suffer from incomplete detection (e.g., simultaneous arrival of two protons), one may argue that the problem deserves further experimental investigation.If the sequence of similar targets H 2 O, CH 4 , NH 3 , on the other hand is confirmed to display such different behavior in terms of multiple ionization, then the agreement with experimental net DDCS data obtained by various methods would seem rather fortuitous.A further investigation in this respect using a CDW-EIS approach, e.g. that of Ref. [6], which explicitly addressed multiple ionization in proton-water collisions, would also provide more clarity and lead to some resolution.

FIG. 1 .
FIG. 1. Doubly differential cross section in units of Å 2 /(eV srad) for proton collisions with ammonia molecules at a collision energy of E p = 250 keV for ionized electron energies of E el = 10, 20, 50, 100, 200, 500 eV.Solid lines: present CTMC net ionization results obtained with Eq. (3); dashed lines are for single ionization obtained with Eq. (4).The data points are the

FIG. 2 .
FIG.2.Logarithm of the net doubly differential cross section for proton collisions with ammonia molecules at a collision energy of E p = 250 keV as compared to the CDW-EIS results of Tachino et al.[4] for ionized electron energies of E el = 10, 20, 50, 100, 200, 500 eV (from top to bottom).Solid black lines connecting binned data: present CTMC net ionization results obtained with Eq. (3); solid red curves: CDW-EIS (prior), dashed blue curves: CDW-EIS (post); green dash-dotted curves (starting at E el = 20 eV): three-Coulomb wave model of Ref.[7].The value of 2 on the y-axis corresponds to a cross section value of 10 −6 Å 2 /(eV srad).

FIG. 3 .
FIG.3.Same as in Fig.1, but for a collision energy of E p = 1 MeV.The ionized electron energies are E el = 15, 47, 207, 500 eV.The data points are the experimental results of Ref.[36], as reported

FIG. 4 .
FIG.4.Same as in Fig.1, but for a collision energy of E p = 2 MeV.The ionized electron energies are E el = 11.3, 20, 50, 100, 200 eV.The data points are the experimental results of Ref.[36], as