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Physics Procedia 66 (2015) 16 - 21

C 23rd Conference on Application of Accelerators in Research and Industry, CAARI 2014

Electron Removal Processes in Proton-Methane Collisions

Arash Salehzadeh, Tom Kirchner*

Department of Physics and Astronomy, York University ,4700 Keele Street, Toronto Ontario M3J1P3, Canada

Abstract

We investigate electron removal processes in the proton-methane collision system in the 20 keV to a few MeV energy regime. The two-centre basis generator method is used within the independent electron model to study this problem. Results for net capture and ionization are compared with available experimental and theoretical studies as well as with results obtained from Bragg's additivity rule. Good agreement is observed at high energies. At lower energies the situation is less clear. However, for ionization at intermediate energies we achieve good agreement with experimental results when we take target excitation into account.

© 2015The Authors.Publishedby ElsevierB.V. Thisis an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Selection and peer-review under responsibility of the Organizing Committee of CAARI 2014

Keywords: ion-molecule collisions, target ionization, electron capture, basis generator method, (screening corrected) additivity rule

1. Introduction

Collision systems with hydrocarbon molecule targets have been the subject of interest due to a high number of applications in various fields. While there is a wide range of experimental data available for different collision systems with hydrocarbon molecules, theoretical studies are scarce and challenging due to the presence of many electrons as well as the multi-centre nature of hydrocarbon targets.

In a previous work [1], electron removal and fragmentation processes in p-H2O collisions were investigated. The calculations were carried out by applying the two-centre basis generator method (TC-BGM) [2] within the independent electron model (IEM) to the ion-molecule system. The results obtained show consistency with

* Corresponding author. Tel.: 416-736-2100, ext. 33695; fax: 416-736-5516. E-mail address: tomk@yorku.ca

1875-3892 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Selection and peer-review under responsibility of the Organizing Committee of CAARI 2014 doi: 10.1016/j.phpro.2015.05.004

experimental data. In this work, we use similar methods to calculate net ionization and capture cross sections in the p-CH4 collision system. Experimental studies for net ionization in p-CH4 collisions have been reported some time ago [3, 4, 5, 6]. In a more recent paper [7] those cross sections as well as their uncertainties have been combined to yield "recommended data". Similarly, for net capture experimental cross sections are available at different impact energies [3, 8, 9]. On the theoretical side, the continuum distorted-wave with eikonal initial-state (CDW-EIS) approach has been applied to calculate q-fold ionization cross sections [10].

Atomic units are used throughout this work, unless stated otherwise.

2. Theory

We are interested in the p-CH4 collision system at impact energies above 20 keV. These collisions are fast enough compared to the molecular time scale that we can safely neglect the molecular rotations and vibrations. The projectile is assumed to follow a classical straight-line trajectory, which is characterized by the impact parameter b and a constant velocity. We address this collision system within the IEM in which the full many-body time-dependent Schrödinger equation (TDSE) is approximated by a set of single-electron orbital equations

\wißr(t) )=[#;»+Vp(oiiv^co ) (i)

for the initially occupied molecular orbitals (MOs)

\¥Yaß7(t1) ) = | . (2)

The molecular orientation is specified by the Euler angles ( (X , ß , J )■ The original coordinate system

(a = ß = / = 0 ) is taken to be that of [11]. HTaßy is the target Hamiltonian which includes an effective potential

on the Hartree-Fock level. Furthermore, VP (t) is the potential that describes the interaction between an electron and the projectile.

Due to the difficulties that arise from the multi-centre nature of the target our approach is based on two ideas: First, we use the spectral representation of the target Hamiltonian

HlPr=Y^a \ Kßr >< Kßr (3)

where SA are the energy eigenvalues corresponding to the MOs h.aßy ■ Second, we expand the initially populated MOs in a single-centred, orthonormal basis:

1 Kßy ) = Tß^J7 \Aim ). (4)

Our collisional calculations are restricted to those molecular orientations in which the system is invariant with respect to a reflection about the y-axis (assuming that the collision plane is the x-z plane). This symmetry is realized when two of the hydrogen atoms are in the collision plane and the other two hydrogen atoms are mirror images with respect to this plane. Thus, within this framework we can only consider four orientations. Given that we have found rather similar results for them, this should not be a serious limitation of our approach. The TDSE (1) is solved by expanding the solutions in a TC-BGM basis set:

\^<(t) ) = Tjalaßy \ ^ (t) ). (5)

The TC-BGM basis includes all KLM shell atomic orbitals (AOs) on the carbon atom (to represent the molecular target according to Eq. (4)), all hydrogen KLMN shell states on the projectile as well as a set of pseudostates to represent the continuum.

To find the net probabilities, we first obtain state-to-state transition amplitudes. The transition amplitudes

to the projectile states | kp ) are given by

4 (tf H kJ^A ) > (6)

and correspondingly, net capture is calculated as

Pp = II2 (7)

with sums that run over all ( N ) electrons and all projectile states included in the basis. For net ionization we use the unitarity of the problem:

Pon = N - Pcap "ZZlAfl2. (8)

The target amplitudes A^ are obtained in two ways. First, we use

4(tf) = ( Kfr 1 V*Py(tf) > (9)

where only the MOs that form the molecular ground state are considered. Thus, at the final time, the electrons are either in the target ground state, bound to the projectile or in the continuum. Consequently, target excitation is completely neglected. Alternatively, we use

4 (tf ) = ( kTlWapy(tf ) ) (10)

where | kT ) denote the target AOs included in the basis. Since these states overlap with excited MOs target excitation is approximated.

Net cross sections can be found from the net probabilities in the following way:

°aPr = Pr(b )db. (H)

Similar to the probabilities, these cross sections are orientation-dependent. Experimentally, however, one cannot control the molecular orientation. Hence, we need to find orientation-averaged cross sections. To this end, we average the orientation-dependent net probabilities for the four considered orientations

pv (b)=4(pwi+...+p^w. (12)

Then, by assuming that the average probability is orientation-independent the orientation-averaged cross section is obtained in the following way:

^vg = \Pavg (b)db = 2n[bPavg (b)db. (13)

The validity of the above approximation was verified for both capture and ionization.

In addition to using the molecular TC-BGM we studied the collision system by applying Bragg's additivity rule, which states that the molecular net cross sections are the sums of atomic ones. Thus, for CH 4 the ion-molecule problem is reduced to the sum of five ion-atom collisions

°CH4= + 4°H . (14)

The results are shown and discussed in the next section.

3. Results and Discussion

Our results for the net ionization and capture cross sections as functions of impact energy are shown in Figures (1) and (2) respectively. For net ionization, both the results obtained from the molecular TC-BGM and from Bragg's additivity rule are consistent with the experimental data at high energies. However, at low and intermediate energies the calculated cross sections are well above the experimental ones. For our molecular method when

equation (10) is used instead of equation (9) to calculate the target transition amplitudes (i.e., when excitation is taken into account), the cross sections are reduced. A remarkable improvement is obtained when excitation is considered with a larger basis set i.e., when the N shell target states are added to the TC-BGM basis, but are not used to represent the initially populated MOs. At the final time, electrons can be bound to those target states which means that more room for excitation is allowed. As is evident from Figure (1), for E > 100 keV we obtain almost perfect agreement with the experimental results with this method to analyze the TC-BGM calculations.

„ 12

"k 8 b

6 4 2 0

2 5 10 2 & 10 2 5

E (keV)

Figure 1: Net ionization cross section as a function of impact energy for p-CH4 collisions. The solid line shows the results obtained from analysis (9). The dashed line shows the results obtained from using equation (10). For the cross sections shown by the dash-dotted line equation (10) has been used and target states of the N shell are also included in the TC-BGM basis. The dotted line shows the cross section obtained by Bragg's additivity rule. The long dashed lines are the CDW-EIS cross sections from [10]. The experimental data ( • ) are the recommended cross sections from [7] where the experimental data from [3, 4, 5, 6] with their corresponding uncertainties have been combined.

In [1] for the p-H2 O problem, target excitation was neglected. Unlike for the p-CH4 problem, the target excitation channel seemed unimportant as the model predicted the experimental data fairly well. In the present case, despite the good agreement at intermediate and high energies, the situation is different at low energies. The results obtained from the analysis including target AOs up to the N shell show an unphysical behaviour as no maximum is obtained down to E = 20 keV. Overall, all of the molecular TC-BGM models behave poorly at low impact energies. One reason could be a lack of convergence of the TC-BGM basis set. Another reason may be the approximation involved in the spectral representation of the molecular Hamiltonian. Only five MOs (that represent the molecular ground state) have been included in the sum which may be insufficient, but is difficult to improve upon given that the MOs are represented in a single-centre basis (equation (4)). Some numerical issues may also contribute to this problem at low energies particularly for the larger basis set. However, the results provide a strong indication of the significance of target excitation processes.

Also included in Figure 1 are results obtained from the CDW-EIS method [10]. The model predicts the

experimental data well for E > 100 keV, although the cross section curve lies slightly above the experimental data points. Overall, all of the models have limitations at low impact energies and appear to be restricted to intermediate and high energies.

Figure 2: Net capture cross section as a function of impact energy for p-CH4 collisions. The solid line shows the cross sections obtained from the molecular TC-BGM. The dotted curve shows the cross sections obtained from Bragg's additivity rule. The dashed curve represents the SCAR result. The experimental data shown by (• ], (□] and (A] are from [3], [9] and [8], respectively.

For capture (Figure 2), we consider the impact energy range of 20-200 keV. The molecular TC-BGM predicts the overall behaviour better than Bragg's additivity rule, although at energies below 100 keV the results are above the experimental data. Bragg's additivity rule predicts the experimental results well for E > 60 keV while at lower energies the model is not applicable. The reason is that in Bragg's rule we have to consider four p-H collisions for which electron capture becomes resonant toward low impact energies, whereas there is no such resonance in p-CH 4 . Therefore, in order to use Bragg's additivity rule in this energy regime one has to make corrections. One suggested model, originally applied to electron-molecule scattering, is the screening corrected additivity rule (SCAR) [12]. It has been argued in [12] that at low energies individual atoms cannot be considered as independent scatterers and multiple scatterings within a molecule take place. To that end, screening coefficients have been introduced and multiplied to each atomic cross section to account for the overlaps between the atoms:

^=Ew (15)

with 0 < si < 1. Even though it is not obvious that the same model with the same coefficients should be applicable

to ion-impact collisions, we use it here for the electron capture channel. It it interesting to see (Figure (2)) that the results are significantly improved compared to the standard Bragg results.

In conclusion, our molecular method seems to be restricted to intermediate and high impact energies for the p-CH 4 collision system. The results suggest that target excitation processes are important. Similarly, Bragg's additivity rule is consistent with the experimental results at high energies while there are discrepancies at low and intermediate energies. For capture at low energies the application of the SCAR model leads to significantly improved results. This extension of Bragg's additivity rule will be the subject of further studies.

Acknowledgments

This work has been supported by the Natural Sciences and Engineering Research Council of Canada. We thank Britta Vincon for contributions at an early stage.

References

[1] M. Murakami,T. Kirchner, M. Horbatsch, H.J. Ludde, Single and multiple electron removal processes in proton

water-molecule collisions, Phys. Rev.A85(2012)052704.

doi:10.1103/PhysRevA.85.052704.

URL http://link.aps.org/doi/10.1103/PhysRevA.85.052704

[2] M. Zapukhlyak, T. Kirchner, H.J. Ludde, S. Knoop, R. Morgenstern, R. Hoekstra, Inner-and outer-shell electron

dynamics in proton collisions with sodium atoms, Journal of Physics B: Atomic, Molecular and Optical Physics 38(14)(2005)2353. URL http://stacks.iop.org/0953-4075/38/i=14/a=003

[3] M.E. Rudd, R.D. DuBois, L.H. Toburen, C.A. Ratcliffe, T.V. Goffe, Cross sections for ionization of gases by 5-

4000-kev protons and for electron capture by 5-150-kevprotons, Phys. Rev. A28 (1983)3244-3257. doi:10.1103/PhysRevA.28.3244. URL http://link.aps.org/doi/10.1103/PhysRevA.28.3244

[4] J.G. Collins, P. Kebarle, Ionization cross sections, charge-transfer cross sections, and ionic fragmentation

patterns of some paraffins, olefins, acetylenes, chloroalkanes, and benzene with 40-100-kev protons, J. Chem. Phys. 46(1967)1082-1089.

doi: http://dx.doi.org/10.1063/L1840772.

URL http://scitation.aip.org/content/aip/journal/jcp/46/3/10.1063/1.1840772

[5] R.J. McNeal, Production of positive ions and electrons in collisions of 1-25-kev protons and hydrogen atoms

with co, co2, ch4, and nh3, The Journal of Chemical Physics 53 (11)(1970)4308-4313. doi: http://dx.doi.org/10.1063/L1673938.

URL http://scitation.aip.org/content/aip/journal/jcp/53/11/10.1063/L1673938

[6] D.J. Lynch, L.H. Toburen, W.E. Wilson, Electron emission from methane, ammonia, monomethylamine, and

dimethylamine by 0.25to2.0 mev protons, The Journal of Chemical Physics 64(6)(1976)2616-2622. doi:http ://dx.doi.org/10.1063/1.432515.

URL http://scitation.aip.org/content/aip/journal/jcp/64/6Z10.1063/L432515

[7] M.E. Rudd, Y.K. Kim, D.H. Madison, J.W. Gallagher, Electron production in proton collisions: total cross

sections, Rev. Mod. Phys.57 (1985) 965-994.

doi:10.1103/RevModPhys.57.965.

URLhttp://link.aps. org/doi/10.1103/RevModPhys .57.965

[8] J.M. Sanders, S.L. Varghese, C.H. Fleming, G.A. Soosai, Electron capture by protons and electron loss from

hydrogen atoms in collisions with hydrocarbon and hydrogen molecules in the 60-120 kev energy range, Journal of Physics B:

Atomic, Molecular and Optical Physics 36 (18)(2003) 3835.

URLhttp://stacks.iop.org/0953-4075/36/i=18/a=311

[9]C.Barnett, J.Ray, E.Ricci, M.Wilker, E.McDaniel, E.Thomas, H.Gilbody, Atomic Data for Controlled Fusion Research, Vol.1, 1977.

[10]L.Gulya's, I.To'th, L.Nagy, Cdw-eis calculation for ionization and fragmentation of methane impacted by fast protons, Journal of Physics B: Atomic, Molecular and Optical Physics 46(7)(2013)075201.

URL http://stacks.iop.org/0953-4075/46/i=7/a=075201

[11]R.M.Pitzer, Optimized molecular orbital wave functions for methane constructed from a minimum basis set, The Journal of Chemical Physics 46(12)(1967)4871-4875. doi: http://dx.doi.org/10.1063/L1840649. URL http://scitation.aip.org/content/aip/journal/jcp/46/12/10.1063/L1840649

[12]F.Blanco, G.Garc'ia, Screening corrections for calculation ofelectron scattering from polyatomic molecules, Physics Letters A317(5-6)(2003)458-462.

doi: http://dx.doi.org/10.1016/j.physleta.2003.09.016.

URL http://www.sciencedirect.com/science/article/pii/S0375960103013689.