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Classical and quantal methods in atomic and molecular collisions
Nuclear Instruments and Methods in Physics Research B 241 (2005) 48–53 www.elsevier.com/locate/nimb
Classical and quantal methods in atomic and molecular collisions A. Dubois a,*, J. Caillat a, J.P. Hansen b, I. Sundvor b, F. Fre´mont c, P. Sobocinski c, J.-Y. Chesnel c, R. Gayet d, J. Fu e, M.J. Fitzpatrick e, W.F. Smith e, J.F. Reading e a
d
Laboratoire de Chimie Physique-Matie`re et Rayonnement (LCP-MR), Universite´ Pierre et Marie Curie, Unite´ Mixte de Recherche du CNRS, UMR 7614 Paris, France b Institute of Physics, University of Bergen, N-5007 Bergen, Norway c Centre Interdisciplinaire de Recherche Ions Lasers (CIRIL), Unite´ Mixte CEA-CNRS-ISMRA-Universite´ de Caen Basse-Normandie, Caen, France Centre Lasers Intenses et Applications (CELIA), Universite´ de Bordeaux I, Unite´ Mixte de Recherche du CNRS, UMR 5107 Talence, France e Texas A&M University, College Station, TX 77843, USA Available online 19 August 2005
Abstract We present theoretical calculations of electron capture cross-sections for ion–atom and ion–molecule collisions in the intermediate impact energy range. The results stem from intensive computations based on quantal, semiclassical and classical approaches. We focus our attention on (i) electron transfer cross-sections for quasi-one-electron ion– atom/molecule systems and (ii) energy distributions of the fragments produced in multiply charged ion–H2 collisions. The results from the different approaches are compared systematically, also with experimental data, when available. 2005 Elsevier B.V. All rights reserved. PACS: 34.10.+x; 34.50.Gb; 34.70.+e Keywords: Collision; Electron transfer; Fragmentation; Theory
1. Introduction *
Corresponding author. Tel.: +33 1 44276631; fax: +33 1 44276226. E-mail address: [email protected] (A. Dubois).
The level of sophistication reached by atomic and molecular scattering experiments, e.g. [1], requires continuous efforts from the theoretical side,
0168-583X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.07.005
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
both in the development of novel models and in the improvement of the reliability and convergence of the computations based on well-established approaches [2]. For example cross-section prediction for scattering processes involving initial excited atomic or molecular species [3] represents a severe challenge, while of great importance in various applications, e.g. the modelisation of the plasmawall interactions in tokamak devices. This is especially true for strongly coupled collision systems which require non-perturbative treatments. Within the semiclassical close-coupling approach, appropriate for intermediate impact energies, fantastic improvements have been performed, partly due to the continuous increase of computer power, in finite Hilbert basis set (FHBS) and grid schemes [2,4–6]. Nevertheless, ultimate convergence of these methods for the calculations of specific inelastic channel cross-sections in ion–atom collisions may be awkward to reach, due to technical reasons or/and inherent problems of present close-coupling approaches [7–9]. When the semiclassical or quantal approaches fail to converge satisfactorily or cannot actually be applied, classical models are important tools to estimate crosssections at a reasonable cost of development and calculations. They also provide an interesting picture for the interpretation of the experimental results [10]. Classical approaches, as the wellknown Classical Trajectory Monte Carlo method, are especially appropriate for the description of asymmetric collision systems, in the intermediate to high-energy regime [11]. However in its original form [12], this method can only describe (quasi-) one-electron collision systems, since three-body atomic or molecular species are generally unstable classically. In this contribution, we present electron transfer cross-sections for the benchmark system, H+– H, the neutral target being initially in the ground state or in excited states. Comparisons of crosssections obtained from converged finite Hilbert basis set (FHBS) treatment, continuum distorted waves approximation and classical simulations are discussed. We extend these comparisons to ion-molecule collisions to study (i) electron transfer in Z2þ –Hþ 2 collisions and (ii) the fragmentation of H2 induced by multiply charged ion impact.
49
2. Theoretical approaches The different theoretical approaches that we have used to calculate the cross-sections presented in the next section have been widely discussed in the literature, cf. for review [2] and for example [13–18]. Only few general considerations will be outlined in the following. Our two different semiclassical FHBS approaches to deal with ion–atom [14] and ion–molecule [15] collisions are based on the same ground: (i) the impact parameter approximation and (ii) the solution of the time-dependent Schro¨dinger equation is expressed as an expansion onto asymptotic orbitals, i.e. centered on the isolated target (T) and projectile (P), augmented by plane-wave electronic translation factors [2], X T rÞeiei t Wð~ r; tÞ ¼ cTi ðtÞuTi ð~ X P 2 þ cPj ðtÞuPj ð~ r ~ RTP Þeiej t ei~v~riv t=2 . The functions uX, centered on X, are either atomic orbitals (AO) for atoms/ions or molecular orbitals (MO) for multi-center collision partners; they correspond to negative energies eX for bound states and to positive ones for pseudo-states, spanning the (discretized) electron continuum. For ion–molecule collisions, the energy range considered in the present study allows for an important simplification: the internal motion of the molecule are decoupled from the collision stage and only MO obtained for a frozen molecule (e.g. at the equilibrium internuclear distance) are used in the expansion [15]. In spite of this approximation, the close-coupling approach is complex in its computer implementation and requires very intensive computations. On the other hand, for ion–atom collisions, various versions of this representation have been successfully developed [4] and one may expect to the reach convergence. However, possible sources of difficulty and instability in the use of very large basis set have been identified [8,9]: (i) possible linear dependence of the states spanned by the expansion and (ii) presence of electron capture channels in which fairly narrow energy regions of the continuum of the electron are important. Our particular procedure, described in details in [14], allows to avoid these difficulties
50
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
and can handle very large basis sets (e.g. 400 states on each center in the present calculations). An important part of the results presented in the next section are obtained from two models based on classical mechanics to describe the dynamics of all particles involved in the collision: the Classical Trajectory Monte Carlo (CTMC) method [12] and a quasi-classical, often called Fermi Molecular Dynamics, (FMD) method [17,18]. In the CTMC method, the HamiltonÕs equations of motion are solved numerically, the
atom (ion) to which the electron is initially bound being described by a microcanonical ensemble of orbits. When the collision is over, the electron can be found ejected or bound to the target or the projectile core in a continuum of negative energies: procedures to define the equivalence between these distributions and the discrete energy spectrum of the atom have been proposed [2,19,20]. This method has been developed for one-electron or quasi-one-electron ion–atom collisions, and requires special approximations to deal with the
Table 1 Intra K- and L-shell electron transfer cross-sections in p–H collisions
1s ! 1s
1s ! n = 2
2s ! 1s
2s ! n = 2
2p0 ! 1s
2p0 ! n = 2
2p±1 ! 1s
2p±1 ! n = 2
50 keV
50 keV
75 keV
100 keV
145 keV
CDW FHBS-h FHBS-g CTMC
0.99 0.99
6.95 0.99 1.00 0.83
1.82 1.16 1.14 1.00
6.39-1 1.22 1.22 1.17
1.48-1 1.26 1.26 1.42
CDW FHBS-h FHBS-g CTMC
0.89 0.88
2.01 0.88 0.88 1.34
5.03-1 1.07 1.05 1.04
1.66-1 1.14 1.12 1.67
3.45-2 1.18 1.17 1.87
CDW FHBS-h FHBS-g CTMC
0.98 0.97
1.44 0.98 0.97 0.77
3.68-1 1.20 1.18 1.21
1.24-1 1.27 1.26 1.54
2.69-2 1.30 1.29 1.47
CDW FHBS-h FHBS-g CTMC
1.18 1.20
7.79-1 1.21 1.22 1.21
1.50-1 1.28 1.27 1.69
4.21-2 1.23 1.23 1.97
7.37-3 1.24 1.25 2.52
CDW FHBS-h FHBS-g CTMC
0.53 0.53
3.95-1 0.51 0.52 0.74
9.44-2 0.60 0.59 0.63
2.91-2 0.65 0.63 0.50
5.31-3 0.71 0.69 0.28
CDW FHBS-h FHBS-g CTMC
0.73 0.74
5.56-1 0.75 0.77 1.18
7.43-2 0.69 0.67 1.24
1.62-2 0.67 0.67 1.19
2.06-3 0.58 0.77 0.88
CDW FHBS-h FHBS-g CTMC
0.92 0.94
8.72-2 0.93 0.93 0.98
2.02-2 0.96 0.95 0.53
6.24-3 0.93 0.93 0.40
1.17-3 0.89 0.89 0.31
CDW FHBS-h FHBS-g CTMC
1.62 1.63
1.28-1 1.68 1.67 1.88
1.44-2 1.63 1.63 1.20
3.36-3 1.50 1.43 0.74
4.72-4 1.21 1.09 0.63
The CDW cross-sections are presented in 1017 cm2 (i.e. 6.39-1 means 6.391018 cm2); FHBS-h, FHBS-g and CTMC give the ratio between the cross-sections from these theories and the CDW cross-sections. The state-to-state cross-sections can be obtained by request.
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
classical instability of N > two-body centers ðHe; Hþ 2 ; H2 ; . . .Þ. Alternative methods have been developed to avoid these approximations and keep all repulsive terms in the Hamiltonian of the system. To describe the ion–molecule collisions considered in the following, we have adopted the FMD method in which extra two-body constraint potentials are added to the Hamiltonian to mimic the shell structure of many-electron atoms/molecules (Pauli constraint) and to avoid the electrons to collapse onto the nucleus (Heisenberg constraint) [17,18]. Finally note that in the classical approaches the electronic initial and final states are not treated likewise with respect to the energy: the initial state has a well defined energy while the energy of the final one is defined within a given interval. This asymmetry in the definition of the states does not allow the models to satisfy the detailed balance principle [19]. The FHBS and CDW approaches do satisfy detailed balancing, which in fact provides an important and rigorous check of the numerical procedures.
Capture cross section (10-16 cm2)
In Table 1, we present intra K- and L-shell electron transfer cross-sections stemming from CDW, FHBS and CTMC calculations for H+–H collisions, over a range of impact energies for which the three methods can be considered as reliable.
2+
+
Ar
He - H 2
2+
+
- H2
12
8
8
4
4
0 0
0.4
0.8 Velocity (a.u.)
1.2
1.6
0
0.4
0.8
1.2
1.6
0
Capture cross section (10-16 cm2)
The CDW results are used as references, not because we think that they are superior in reliability, but rather since they are produced by well tested codes so that they are the least likely to be in dispute. All CTMC data result from statistically converged calculations: the standard deviation is better than 5%, except for the smallest crosssections. The FHBS-g and FHBS-h results stem from intensive (up to about 800 states) calculations including angular momenta up to l = 4 and 5 respectively, each l, m being expanded over 18–22 basis states. To test further the convergence, cross-sections obtained from a basis restricted to 15 states per l, m are listed for 50 keV (1st column). In Table 1, we attempt to give the reader some feel about the convergence of the different FHBS calculations: overall the agreement is excellent, considering the number of basis states and angular momenta included in the basis sets. However, it is also shown that convergence gets more awkward to reach for collisions involving initial excited states and as the impact energy is raised. This is in agreement with the work of Toshima [9]. Considering the comparison between the results from the three methods, Table 1 shows that they reproduce the expected rapid decrease of the cross-sections as the energy is increased. However, the ratio of any particular FHBS cross-sections to the CDW results is not unity. But the factor by which it differs from unity, is quite slowly varying as the projectile energy is increased: there are no
3. Results
12
51
Velocity (a.u.)
2þ þ Fig. 1. Capture cross-sections versus impact velocity (in atomic units) in He2þ –Hþ 2 (left) and Ar –H2 (right) collisions. The coupled channel results are shown with solid line (—), the classical results by dotted line (- - -). The experimental data are from [21].
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
105 keV
intensity (arb. unit)
300
200
100
0 450
intensity (arb. unit)
severe oscillations in the FHBS results indicating lack of convergence. Perhaps plotted on a logarithmic scale the results might seem to be in good agreement but the ratio method of this presentation does reveal significant disagreement. The only experimental results available concern collisions involving ground state hydrogen: they are however not accurate enough to differentiate between our results. The present work then poses a severe challenge to each method and is an invitation to other authors to check or refute our results so that eventually the true cross-sections will emerge. Let us turn to ion–molecule collisions. In a first example we consider two similar one active electron systems, respectively He2þ –Hþ 2 , and Ar2þ –Hþ . We present in Fig. 1 capture cross-sec2 tions stemming from semiclassical coupled-channel and quasi-classical calculations [15], together with experimental results from [21]. The close-coupling results are converged within 10% or less and follow indeed the general trend of the measurements. The same conclusion can be drawn for the classical results and the agreement between both sets of calculation is quite good, especially for Ar2+ projectile. However the increase of the cross-sections from decreasing velocities observed for He2+ can be considered as a failure of the classical model: classically the electron transfer is mostly resonant and there exists no such state for He+ while Ar+(3p) is resonant. This limit of the classical model explains the overestimation of the cross-sections for low energy He2þ –Hþ 2 collisions and the agreement for the other system. Finally, we believe that the differences with the experimental data are not related to a failure of the methods but have technical reasons: from the procedure described in [21], one may expect the vibrational excitation of Hþ 2 prior the collision stage. This effect is not taken into account in the present results [15]. We now consider the study of the molecular fragmentation induced by electron transfer in the O5+–H2 collision system, which has been recently studied [22]. The theoretical results stem from the quasi-classical (FMD) approach described above but extended to two-electron molecular targets [18]. Fig. 2 shows, for three typical impact energies, the experimental [22] and simulated energy
2.5 keV
300
150 x 10
0 300
0.5 keV intensity (arb. unit)
52
200
100
0 1
10
100
fragment energy (eV) Fig. 2. Energy distributions of the H+ fragments produced in O5+–H2 collisions, for 105, 2.5 and 0.5 keV energies. The angle of detection of the protons is 30 with respect to the incident projectile beam direction. The results from the classical model are shown with dashed line (- - -), the measurements from [22] with solid line (—). In the middle panel, the ·10 indicates that the experimental data are multiplied by 10 for clarity.
distributions of the H+ fragments produced after double capture processes and transfer-excitation. An excellent agreement is observed between the experimental data and the simulations: the shape and the location of the peak(s) are well reproduced by the calculations. For the highest energy a well
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
defined peak around 9.5 eV is observed and corresponds to a pure Coulomb explosion mechanism [23]. Note that, as expected, this proton emission is found nearly isotropic both in the experimental data and in the theoretical predictions. For the medium energy the spectrum presents two well separate peaks: a broad low energy one and a small contribution of energetic protons. These two peaks are well described by the simulations and interpreted by the analysis of the projectile trajectories (not shown): the low energy protons originate from soft (distant) collisions while the fast protons are emitted after hard (head-on) collisions. For decreasing energies the contribution of fast protons becomes dominant in the forward direction, the dissociating protons being repelled by the slowly incoming projectiles. In conclusion the classical approach provides an excellent description of the complex dynamics of the three heavy particles system, coupled with the dynamics of the two electrons. Note however that the quasiclassical calculations predict double capture crosssections larger than the single capture ones, in contradiction with the results obtained for similar systems [22].
References [1] R. Do¨rner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, H. Schmidt-Bo¨cking, Phys. Rep. 330 (2000) 95.
53
[2] B.H. Bransden, M.R.C. McDowell, Charge Exchange and the Theory of Heavy Ion Collisions, Clarendon Press, Oxford, 1992. [3] D. Dowek, J.C. Houver, I. Reiser, J. Salgado, A. Svensson, J.W. Thomsen, N. Andersen, S.E. Nielsen, A. Dubois, Phys. Rev. A 54 (1996) 970. [4] W. Fritsch, C.D. Lin, Phys. Rep. 202 (1991) 1. [5] C.N. Cabello, L.F. Errea, L. Ferna´ndez, L. Me´ndez, A. Macı´as, I. Rabada´n, A. Riera, J. Phys. B 36 (2003) 307. [6] T. Minami, C.O. Reinhold, D.R. Schultz, M.S. Pindzola, J. Phys. B 37 (2004) 4025. [7] J.B. Wang, J.P. Hansen, A. Dubois, J. Phys. B 33 (2000) 241. [8] J. Kuang, C.D. Lin, J. Phys. B 29 (1996) 1207. [9] N. Toshima, Phys. Rev. A 59 (1999) 1981. [10] H. Kollmus, R. Moshammer, R.E. Olson, S. Hagmann, M. Schulz, J. Ullrich, Phys. Rev. Lett. 88 (2002) 103202. [11] R.E. Olson, J. Wang, J. Ullrich, in: Proc. 18th Int. Conf. on Phys. of Elect. At. Coll. (AIP Conference Proceedings), 1993, p. 520. [12] R. Abrines, I.C. Percival, Proc. Phys. Soc. London 88 (1966) 861. [13] Dz. Belkic, R. Gayet, A. Salin, Phys. Rep. 56 (1979) 279. [14] J. Fu, M.J. Fitzpatrick, J.F. Reading, R. Gayet, J. Phys. B 34 (2001) 15. [15] J. Caillat, A. Dubois, I. Sundvor, J.-P. Hansen, Phys. Rev. A 70 (2004) 032715. [16] R.E. Olson, A. Salop, Phys. Rev. A 16 (1977) 531. [17] C.L. Kirschbaum, L. Wilets, Phys. Rev. A 21 (1980) 834. [18] J.S. Cohen, Phys. Rev. A 56 (1997) 3583. [19] R.L. Becker, A.D. MacKellar, J. Phys. B 17 (1984) 3923. [20] I. Fourre´, C. Courbin, Z. Phys. D 38 (1996) 103. [21] H. Bra¨uning, I. Reiser, A. Diehl, A. Theiß, E. Sidky, C.L. Cocke, E. Salzborn, J. Phys. B 34 (2001) L321. [22] P. Sobocinski, J. Rangama, G. Laurent, L. Adoui, A. Cassimi, J.-Y. Chesnel, A. Dubois, D. Hennecart, X. Husson, F. Fre´mont, J. Phys. B 35 (2002) 1353. [23] C.J. Wood, R.E. Olson, Phys. Rev. A 59 (1999) 1317.
Classical and quantal methods in atomic and molecular collisions A. Dubois a,*, J. Caillat a, J.P. Hansen b, I. Sundvor b, F. Fre´mont c, P. Sobocinski c, J.-Y. Chesnel c, R. Gayet d, J. Fu e, M.J. Fitzpatrick e, W.F. Smith e, J.F. Reading e a
d
Laboratoire de Chimie Physique-Matie`re et Rayonnement (LCP-MR), Universite´ Pierre et Marie Curie, Unite´ Mixte de Recherche du CNRS, UMR 7614 Paris, France b Institute of Physics, University of Bergen, N-5007 Bergen, Norway c Centre Interdisciplinaire de Recherche Ions Lasers (CIRIL), Unite´ Mixte CEA-CNRS-ISMRA-Universite´ de Caen Basse-Normandie, Caen, France Centre Lasers Intenses et Applications (CELIA), Universite´ de Bordeaux I, Unite´ Mixte de Recherche du CNRS, UMR 5107 Talence, France e Texas A&M University, College Station, TX 77843, USA Available online 19 August 2005
Abstract We present theoretical calculations of electron capture cross-sections for ion–atom and ion–molecule collisions in the intermediate impact energy range. The results stem from intensive computations based on quantal, semiclassical and classical approaches. We focus our attention on (i) electron transfer cross-sections for quasi-one-electron ion– atom/molecule systems and (ii) energy distributions of the fragments produced in multiply charged ion–H2 collisions. The results from the different approaches are compared systematically, also with experimental data, when available. 2005 Elsevier B.V. All rights reserved. PACS: 34.10.+x; 34.50.Gb; 34.70.+e Keywords: Collision; Electron transfer; Fragmentation; Theory
1. Introduction *
Corresponding author. Tel.: +33 1 44276631; fax: +33 1 44276226. E-mail address: [email protected] (A. Dubois).
The level of sophistication reached by atomic and molecular scattering experiments, e.g. [1], requires continuous efforts from the theoretical side,
0168-583X/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2005.07.005
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
both in the development of novel models and in the improvement of the reliability and convergence of the computations based on well-established approaches [2]. For example cross-section prediction for scattering processes involving initial excited atomic or molecular species [3] represents a severe challenge, while of great importance in various applications, e.g. the modelisation of the plasmawall interactions in tokamak devices. This is especially true for strongly coupled collision systems which require non-perturbative treatments. Within the semiclassical close-coupling approach, appropriate for intermediate impact energies, fantastic improvements have been performed, partly due to the continuous increase of computer power, in finite Hilbert basis set (FHBS) and grid schemes [2,4–6]. Nevertheless, ultimate convergence of these methods for the calculations of specific inelastic channel cross-sections in ion–atom collisions may be awkward to reach, due to technical reasons or/and inherent problems of present close-coupling approaches [7–9]. When the semiclassical or quantal approaches fail to converge satisfactorily or cannot actually be applied, classical models are important tools to estimate crosssections at a reasonable cost of development and calculations. They also provide an interesting picture for the interpretation of the experimental results [10]. Classical approaches, as the wellknown Classical Trajectory Monte Carlo method, are especially appropriate for the description of asymmetric collision systems, in the intermediate to high-energy regime [11]. However in its original form [12], this method can only describe (quasi-) one-electron collision systems, since three-body atomic or molecular species are generally unstable classically. In this contribution, we present electron transfer cross-sections for the benchmark system, H+– H, the neutral target being initially in the ground state or in excited states. Comparisons of crosssections obtained from converged finite Hilbert basis set (FHBS) treatment, continuum distorted waves approximation and classical simulations are discussed. We extend these comparisons to ion-molecule collisions to study (i) electron transfer in Z2þ –Hþ 2 collisions and (ii) the fragmentation of H2 induced by multiply charged ion impact.
49
2. Theoretical approaches The different theoretical approaches that we have used to calculate the cross-sections presented in the next section have been widely discussed in the literature, cf. for review [2] and for example [13–18]. Only few general considerations will be outlined in the following. Our two different semiclassical FHBS approaches to deal with ion–atom [14] and ion–molecule [15] collisions are based on the same ground: (i) the impact parameter approximation and (ii) the solution of the time-dependent Schro¨dinger equation is expressed as an expansion onto asymptotic orbitals, i.e. centered on the isolated target (T) and projectile (P), augmented by plane-wave electronic translation factors [2], X T rÞeiei t Wð~ r; tÞ ¼ cTi ðtÞuTi ð~ X P 2 þ cPj ðtÞuPj ð~ r ~ RTP Þeiej t ei~v~riv t=2 . The functions uX, centered on X, are either atomic orbitals (AO) for atoms/ions or molecular orbitals (MO) for multi-center collision partners; they correspond to negative energies eX for bound states and to positive ones for pseudo-states, spanning the (discretized) electron continuum. For ion–molecule collisions, the energy range considered in the present study allows for an important simplification: the internal motion of the molecule are decoupled from the collision stage and only MO obtained for a frozen molecule (e.g. at the equilibrium internuclear distance) are used in the expansion [15]. In spite of this approximation, the close-coupling approach is complex in its computer implementation and requires very intensive computations. On the other hand, for ion–atom collisions, various versions of this representation have been successfully developed [4] and one may expect to the reach convergence. However, possible sources of difficulty and instability in the use of very large basis set have been identified [8,9]: (i) possible linear dependence of the states spanned by the expansion and (ii) presence of electron capture channels in which fairly narrow energy regions of the continuum of the electron are important. Our particular procedure, described in details in [14], allows to avoid these difficulties
50
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
and can handle very large basis sets (e.g. 400 states on each center in the present calculations). An important part of the results presented in the next section are obtained from two models based on classical mechanics to describe the dynamics of all particles involved in the collision: the Classical Trajectory Monte Carlo (CTMC) method [12] and a quasi-classical, often called Fermi Molecular Dynamics, (FMD) method [17,18]. In the CTMC method, the HamiltonÕs equations of motion are solved numerically, the
atom (ion) to which the electron is initially bound being described by a microcanonical ensemble of orbits. When the collision is over, the electron can be found ejected or bound to the target or the projectile core in a continuum of negative energies: procedures to define the equivalence between these distributions and the discrete energy spectrum of the atom have been proposed [2,19,20]. This method has been developed for one-electron or quasi-one-electron ion–atom collisions, and requires special approximations to deal with the
Table 1 Intra K- and L-shell electron transfer cross-sections in p–H collisions
1s ! 1s
1s ! n = 2
2s ! 1s
2s ! n = 2
2p0 ! 1s
2p0 ! n = 2
2p±1 ! 1s
2p±1 ! n = 2
50 keV
50 keV
75 keV
100 keV
145 keV
CDW FHBS-h FHBS-g CTMC
0.99 0.99
6.95 0.99 1.00 0.83
1.82 1.16 1.14 1.00
6.39-1 1.22 1.22 1.17
1.48-1 1.26 1.26 1.42
CDW FHBS-h FHBS-g CTMC
0.89 0.88
2.01 0.88 0.88 1.34
5.03-1 1.07 1.05 1.04
1.66-1 1.14 1.12 1.67
3.45-2 1.18 1.17 1.87
CDW FHBS-h FHBS-g CTMC
0.98 0.97
1.44 0.98 0.97 0.77
3.68-1 1.20 1.18 1.21
1.24-1 1.27 1.26 1.54
2.69-2 1.30 1.29 1.47
CDW FHBS-h FHBS-g CTMC
1.18 1.20
7.79-1 1.21 1.22 1.21
1.50-1 1.28 1.27 1.69
4.21-2 1.23 1.23 1.97
7.37-3 1.24 1.25 2.52
CDW FHBS-h FHBS-g CTMC
0.53 0.53
3.95-1 0.51 0.52 0.74
9.44-2 0.60 0.59 0.63
2.91-2 0.65 0.63 0.50
5.31-3 0.71 0.69 0.28
CDW FHBS-h FHBS-g CTMC
0.73 0.74
5.56-1 0.75 0.77 1.18
7.43-2 0.69 0.67 1.24
1.62-2 0.67 0.67 1.19
2.06-3 0.58 0.77 0.88
CDW FHBS-h FHBS-g CTMC
0.92 0.94
8.72-2 0.93 0.93 0.98
2.02-2 0.96 0.95 0.53
6.24-3 0.93 0.93 0.40
1.17-3 0.89 0.89 0.31
CDW FHBS-h FHBS-g CTMC
1.62 1.63
1.28-1 1.68 1.67 1.88
1.44-2 1.63 1.63 1.20
3.36-3 1.50 1.43 0.74
4.72-4 1.21 1.09 0.63
The CDW cross-sections are presented in 1017 cm2 (i.e. 6.39-1 means 6.391018 cm2); FHBS-h, FHBS-g and CTMC give the ratio between the cross-sections from these theories and the CDW cross-sections. The state-to-state cross-sections can be obtained by request.
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
classical instability of N > two-body centers ðHe; Hþ 2 ; H2 ; . . .Þ. Alternative methods have been developed to avoid these approximations and keep all repulsive terms in the Hamiltonian of the system. To describe the ion–molecule collisions considered in the following, we have adopted the FMD method in which extra two-body constraint potentials are added to the Hamiltonian to mimic the shell structure of many-electron atoms/molecules (Pauli constraint) and to avoid the electrons to collapse onto the nucleus (Heisenberg constraint) [17,18]. Finally note that in the classical approaches the electronic initial and final states are not treated likewise with respect to the energy: the initial state has a well defined energy while the energy of the final one is defined within a given interval. This asymmetry in the definition of the states does not allow the models to satisfy the detailed balance principle [19]. The FHBS and CDW approaches do satisfy detailed balancing, which in fact provides an important and rigorous check of the numerical procedures.
Capture cross section (10-16 cm2)
In Table 1, we present intra K- and L-shell electron transfer cross-sections stemming from CDW, FHBS and CTMC calculations for H+–H collisions, over a range of impact energies for which the three methods can be considered as reliable.
2+
+
Ar
He - H 2
2+
+
- H2
12
8
8
4
4
0 0
0.4
0.8 Velocity (a.u.)
1.2
1.6
0
0.4
0.8
1.2
1.6
0
Capture cross section (10-16 cm2)
The CDW results are used as references, not because we think that they are superior in reliability, but rather since they are produced by well tested codes so that they are the least likely to be in dispute. All CTMC data result from statistically converged calculations: the standard deviation is better than 5%, except for the smallest crosssections. The FHBS-g and FHBS-h results stem from intensive (up to about 800 states) calculations including angular momenta up to l = 4 and 5 respectively, each l, m being expanded over 18–22 basis states. To test further the convergence, cross-sections obtained from a basis restricted to 15 states per l, m are listed for 50 keV (1st column). In Table 1, we attempt to give the reader some feel about the convergence of the different FHBS calculations: overall the agreement is excellent, considering the number of basis states and angular momenta included in the basis sets. However, it is also shown that convergence gets more awkward to reach for collisions involving initial excited states and as the impact energy is raised. This is in agreement with the work of Toshima [9]. Considering the comparison between the results from the three methods, Table 1 shows that they reproduce the expected rapid decrease of the cross-sections as the energy is increased. However, the ratio of any particular FHBS cross-sections to the CDW results is not unity. But the factor by which it differs from unity, is quite slowly varying as the projectile energy is increased: there are no
3. Results
12
51
Velocity (a.u.)
2þ þ Fig. 1. Capture cross-sections versus impact velocity (in atomic units) in He2þ –Hþ 2 (left) and Ar –H2 (right) collisions. The coupled channel results are shown with solid line (—), the classical results by dotted line (- - -). The experimental data are from [21].
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
105 keV
intensity (arb. unit)
300
200
100
0 450
intensity (arb. unit)
severe oscillations in the FHBS results indicating lack of convergence. Perhaps plotted on a logarithmic scale the results might seem to be in good agreement but the ratio method of this presentation does reveal significant disagreement. The only experimental results available concern collisions involving ground state hydrogen: they are however not accurate enough to differentiate between our results. The present work then poses a severe challenge to each method and is an invitation to other authors to check or refute our results so that eventually the true cross-sections will emerge. Let us turn to ion–molecule collisions. In a first example we consider two similar one active electron systems, respectively He2þ –Hþ 2 , and Ar2þ –Hþ . We present in Fig. 1 capture cross-sec2 tions stemming from semiclassical coupled-channel and quasi-classical calculations [15], together with experimental results from [21]. The close-coupling results are converged within 10% or less and follow indeed the general trend of the measurements. The same conclusion can be drawn for the classical results and the agreement between both sets of calculation is quite good, especially for Ar2+ projectile. However the increase of the cross-sections from decreasing velocities observed for He2+ can be considered as a failure of the classical model: classically the electron transfer is mostly resonant and there exists no such state for He+ while Ar+(3p) is resonant. This limit of the classical model explains the overestimation of the cross-sections for low energy He2þ –Hþ 2 collisions and the agreement for the other system. Finally, we believe that the differences with the experimental data are not related to a failure of the methods but have technical reasons: from the procedure described in [21], one may expect the vibrational excitation of Hþ 2 prior the collision stage. This effect is not taken into account in the present results [15]. We now consider the study of the molecular fragmentation induced by electron transfer in the O5+–H2 collision system, which has been recently studied [22]. The theoretical results stem from the quasi-classical (FMD) approach described above but extended to two-electron molecular targets [18]. Fig. 2 shows, for three typical impact energies, the experimental [22] and simulated energy
2.5 keV
300
150 x 10
0 300
0.5 keV intensity (arb. unit)
52
200
100
0 1
10
100
fragment energy (eV) Fig. 2. Energy distributions of the H+ fragments produced in O5+–H2 collisions, for 105, 2.5 and 0.5 keV energies. The angle of detection of the protons is 30 with respect to the incident projectile beam direction. The results from the classical model are shown with dashed line (- - -), the measurements from [22] with solid line (—). In the middle panel, the ·10 indicates that the experimental data are multiplied by 10 for clarity.
distributions of the H+ fragments produced after double capture processes and transfer-excitation. An excellent agreement is observed between the experimental data and the simulations: the shape and the location of the peak(s) are well reproduced by the calculations. For the highest energy a well
A. Dubois et al. / Nucl. Instr. and Meth. in Phys. Res. B 241 (2005) 48–53
defined peak around 9.5 eV is observed and corresponds to a pure Coulomb explosion mechanism [23]. Note that, as expected, this proton emission is found nearly isotropic both in the experimental data and in the theoretical predictions. For the medium energy the spectrum presents two well separate peaks: a broad low energy one and a small contribution of energetic protons. These two peaks are well described by the simulations and interpreted by the analysis of the projectile trajectories (not shown): the low energy protons originate from soft (distant) collisions while the fast protons are emitted after hard (head-on) collisions. For decreasing energies the contribution of fast protons becomes dominant in the forward direction, the dissociating protons being repelled by the slowly incoming projectiles. In conclusion the classical approach provides an excellent description of the complex dynamics of the three heavy particles system, coupled with the dynamics of the two electrons. Note however that the quasiclassical calculations predict double capture crosssections larger than the single capture ones, in contradiction with the results obtained for similar systems [22].
References [1] R. Do¨rner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, H. Schmidt-Bo¨cking, Phys. Rep. 330 (2000) 95.
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[2] B.H. Bransden, M.R.C. McDowell, Charge Exchange and the Theory of Heavy Ion Collisions, Clarendon Press, Oxford, 1992. [3] D. Dowek, J.C. Houver, I. Reiser, J. Salgado, A. Svensson, J.W. Thomsen, N. Andersen, S.E. Nielsen, A. Dubois, Phys. Rev. A 54 (1996) 970. [4] W. Fritsch, C.D. Lin, Phys. Rep. 202 (1991) 1. [5] C.N. Cabello, L.F. Errea, L. Ferna´ndez, L. Me´ndez, A. Macı´as, I. Rabada´n, A. Riera, J. Phys. B 36 (2003) 307. [6] T. Minami, C.O. Reinhold, D.R. Schultz, M.S. Pindzola, J. Phys. B 37 (2004) 4025. [7] J.B. Wang, J.P. Hansen, A. Dubois, J. Phys. B 33 (2000) 241. [8] J. Kuang, C.D. Lin, J. Phys. B 29 (1996) 1207. [9] N. Toshima, Phys. Rev. A 59 (1999) 1981. [10] H. Kollmus, R. Moshammer, R.E. Olson, S. Hagmann, M. Schulz, J. Ullrich, Phys. Rev. Lett. 88 (2002) 103202. [11] R.E. Olson, J. Wang, J. Ullrich, in: Proc. 18th Int. Conf. on Phys. of Elect. At. Coll. (AIP Conference Proceedings), 1993, p. 520. [12] R. Abrines, I.C. Percival, Proc. Phys. Soc. London 88 (1966) 861. [13] Dz. Belkic, R. Gayet, A. Salin, Phys. Rep. 56 (1979) 279. [14] J. Fu, M.J. Fitzpatrick, J.F. Reading, R. Gayet, J. Phys. B 34 (2001) 15. [15] J. Caillat, A. Dubois, I. Sundvor, J.-P. Hansen, Phys. Rev. A 70 (2004) 032715. [16] R.E. Olson, A. Salop, Phys. Rev. A 16 (1977) 531. [17] C.L. Kirschbaum, L. Wilets, Phys. Rev. A 21 (1980) 834. [18] J.S. Cohen, Phys. Rev. A 56 (1997) 3583. [19] R.L. Becker, A.D. MacKellar, J. Phys. B 17 (1984) 3923. [20] I. Fourre´, C. Courbin, Z. Phys. D 38 (1996) 103. [21] H. Bra¨uning, I. Reiser, A. Diehl, A. Theiß, E. Sidky, C.L. Cocke, E. Salzborn, J. Phys. B 34 (2001) L321. [22] P. Sobocinski, J. Rangama, G. Laurent, L. Adoui, A. Cassimi, J.-Y. Chesnel, A. Dubois, D. Hennecart, X. Husson, F. Fre´mont, J. Phys. B 35 (2002) 1353. [23] C.J. Wood, R.E. Olson, Phys. Rev. A 59 (1999) 1317.