486 Nuclear Instruments and Methods in Physics Research B27 (1987) 486-490

North-Holland, Amsterdam

CLASSICAL AND QUANTAL MODELS FOR ION-ATOM COLLISIONS

B.H. BRANSDEN

The University of Durham, Durham, England

As part of a project for the evaluation of the classical trajectory Monte Carlo model for charge exchange and ionisation, a

comparative study of classical and quanta1 cross sections for reactions between He ‘+ ions and neutral lithium ions is described.

1. Introduction

As long ago as 1966, Abrines and Percival [l] devel- oped a purely classical model of charge transfer and ionisation which can be applied to systems of three particles interacting through Coulomb potentials. In this classical trajectory Monte Carlo (CTMC) model, the exact classical solutions of the three-body problem at a given energy are averaged over an ensemble of target bound state orbits. This ensemble is chosen to provide the same distribution in momentum as that determined by the nonrelativistic wave function of the initial state of the target atom. The initial applications of the model to proton-atomic hydrogen scattering were encouraging and prompted extensive later work by Olsen [2] and Olsen and Salop [3] who calculated cross sections for charge exchange and ionisation in collisions between fully and partially stripped positive ions and hydrogen atoms. Comparison with experiment suggests that cross sections of useful accuracy can be completed from the CTMC model for impact velocities in an external IJ~ 5 u < 100, where v0 is the Bohr velocity of the bound electron in the initial state of the target atom. It is thought that the lower velocity limit is imposed by the quasimolecular nature of ion-atom collisions at low energies. At the higher velocities for which the cross sections are small, it becomes difficult to obtain results which are statistically significant, and in addition, above some energy the classical model must fail because, for example, it cannot reproduce the correct high energy behaviour of the ionisation cross section, given by the Born approximation.

Some allowance can be made for the complex struc- ture of a partially stripped ion by treating it as being equivalent to a bare nucleus with an effective charge, and in this approximation the CTMC model for Coulomb interactions can be employed [2,3]. In order to employ more realistic effective potentials between the active electron and a positive ion, McDowell and his co-workers [4] have recently generalised the CTMC model, again considering three-body systems, but taking

0168-583X/87/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

the interparticle interaction to be non-Coulombic. This new model has been applied to calculate exchange and ionisation cross sections for the systems He+ + H, HC + He, He2+ + He and He+ + He+ [5] by introducing an effective potential between the active electron and the He+ core which is designed to reproduce the spectro- scopic data for the neutral He atom. Both these and other applications of the model [6] provide results which are sufficiently encouraging to warrant further investi- gation. The same conclusion can be reached from inde- pendent, but similar, work of Reinhold and Falcon [7] who have studied the Hf + He and Ht + Li+ systems. At this stage it seems important to attempt to assess the range of validity of the generalised CTMC model by making a close comparison of CTMC cross sections for a number of systems with those calculated from the standard quanta1 expansion model as well as with the experimental data. In this paper, I shall report on the first stage of such a set of comparative studies by Ermolaev and co-workers [8,9]. The example chosen is the He2+ + Lie system and a comparison has been made between ionisation and charge transfer cross sec- tions calculated from the CTMC model and those calculated from the two-centre quanta1 expansion model in the energy range 50-400 keV (charge transfer) and 50-2000 keV (ionisation). The energies quoted are for 4He ions in the laboratory system. Details of the models are given in the next section and the results are sum- marised in section 3.

2. The He’+ + Lie system

Below impact energies of about 200 keV (lab.) the charge transfer cross section is dominated by capture of the 2s valence electron of Lie

He2+ + Li’(2s) - He+( nr) + Li+. (I) To treat this reaction and also the corresponding excita- tion and ionisation reactions, the valence electron of lithium was taken to move in an effective potential

B. H. Bransden / Models for ion -atom collisions 487

V,(r) due to the Li+(ls*) core. Above 200 keV charge transfer from the K shell is much more important than capture from the L shell. Similarly below 1000 keV only ionisation of the valence electron is important but above this energy K shell ionisation is the dominant process. The cross sections for two-electron capture and ionisa- tion are small compared with single electron cross sec- tions, over the energy region of interest, and have been neglected. Capture and ionisation from the K shell are again treated as single electron processes, the active electron moving in a potential V,(r). Total cross sec- tions are obtained by adding the K and L shell contri- bution calculated independently.

2.1. Effective potentials

The general form of effective potential employed [lo] for the valence electron is:

v(r) = -+ - +(a,+a,r+rr,r’) eeX’+ V,(r),

(2)

where V,(r) is a core polarisation term decreasing at large r like rm4. In the quantum calculations VP was omitted and slightly different parameters were used than in the corresponding CTMC calculations. How- ever, both potentials are practically identical for the purpose of capture and ionisation calculations, never differing in the important regions of r by more than 3%. For K shell capture a potential of the same form was used in the quanta1 calculations (with, of course, different parameters) and in the CTMC calculations a Coulomb potential was employed with an effective charge chosen to give the correct K-shell energy.

2.2. CTMC model

In the classical model Hamilton’s equations of mo- tion for each trajectory are solved numerically to the desired degree of precision. Initially the bound electron moves in an orbit about the target nucleus (or core) and the classical orbit (for a given energy) is specified by five parameters, which are randomly selected from given distributions. Arguments based on the impulse ap- proximation [ll] suggest that it is important to use the correct momentum distribution for the initial bound state. For a pure Coulomb interactions the microcanon- ical ensemble of target orbits provides a momentum distribution which is identical to the quantum mechani- cal one. This is not the case with non-Coulomb poten- tials, however with the sampling chosen by McDowell and collaborators’ the two distributions are very close. Naturally the spatial distributions in the classical and quantum cases differ considerably, since certain regions of space are classically inaccessible.

2.3. The quanta1 model

In the energy region under consideration the heavy particle motion can be described by a straight line trajectory at a constant impact parameter and the quantum mechanical problem reduces to solving ap- proximately the time dependent Schriidinger equation for the active electron (in atomic units)

[-~~2+~:\e(r4)+~~e(ro)-i~]~(r, t)=O, (3)

where VA, and V,, represent the interaction between ions A and B and the electron. In terms of r, the position vector of the electron with respect to the mid- point of AB, we have

rA=r-+R(t); rB=r++R(t), (4)

where the time dependence of the relative position vector R of A and B is specified by the choice of heavy particle trajectory. In the standard model [12,13] I/ is expanded in travelling orbitals centred on A and B. The expansions include terms representing atomic bound states centred at both A and B and also pseudostates which represent the atomic continua. Clearly there are practical limitations to the size of the basis employed, which should be optimised for the problem in hand. For this reason different sets were used when calculating charge exchange cross sections and when calculating excitation and ionisation.

Charge exchange cross sections for capture from the L shell have been calculated previously by Fritsch and Lin [14] and by Ermolaev and Bransden [15]. Although accurate at energies below - 40 keV, the basis sets used in the previous work were not extensive enough to provide converged results at the higher energies. After extensive investigation a basis set was found which is believed to provide converged results up to 400 keV. This included 32 terms, nine centred on the target and 23 centred on the projectile. The pseudostates generated by diagonalising the projectile or target Hamiltonians, were chosen so that the correct oscillator strengths for transitions from the low lying states were obtained. The overlap integrals between the basis and the atomic continua (in the one electron model) were also calcu- lated and used to correct the computed cross sections to allow for transitions to levels not represented explicitly in the basis. The K shell capture cross sections which dominate above 200 keV were calculated with a similar but rather larger basis of 43 terms, which was also employed in the K shell ionisation calculations.

For ionisation (and excitation) it is known that, except at low energies K 2 40 keV, target centre basis terms are much more significant than those centred about the projectile. For this reason the calculations of ionisation to the I = 0, 1 and 2 levels of the continuum were based on a 53 state expansion about the target

488 B.H. Bransden / Models for ion-atom collisions

supplemented by one basis state centred on the pro- jectile to allow to some extent for the influence of charge exchange. The efficiency of the pseudostates employed in representing the target continuum was tested [16] by a method due to Reading [17]. By adding further basis states to centre about the projectile, it was discovered that this basis is not adequate for the calcu- lation of the important I= 3 contribution to the cross

section. However the correction to the 53 state basis (which is small at the higher energies) was estimated

using a large 65 state basis at 65 keV. At this energy the ionisation and charge exchange cross sections are com- parable in size so that at higher energies for which the charge exchange cross sections are small, basis states centred at the projectile become much less important in ionisation. The contributions to ionisation leading to continuum levels with 12 4 were shown to be given by the Born approximation to sufficient accuracy, thus avoiding the time consuming close coupling calculations for these 1 values. Below 1 MeV only ionisation from the L shell of Li is important, but at higher energies the K shell contribution is significant and was calculated in a similar way.

t I I I ( IIII I 1 loo 10' 102

H%irnpact energy E keV (lab I

3. The CTMC and quantal cross sections

Fig. 1. Total cross sections for capture from the L shell of Li by He2+. Theoretical cross sections: Molecular orbital basis: MOl, Shipsey et al. [20]; M02, Sato and Kimura [18]. Atomic orbital basis: AO+, Fritsch and Lin [14]. 0 20 state and W 32 state basis, Ermolaev [S], CTMC model +, Ermolaev et al. [8]. Experimental total cross sections (containing contributions from both the L and K shells: @ Kadota et al. [21], 4 Murray et al. [22], & McCullough et al. [23], * Varghese et al. [24] $

3.1. Charge exchange cross sections DuBois and Toburen [25], I$ Shah et al. [26].

In fig. 1, the theoretical total cross sections for capture of the 2s valence electron of Li by He2+ are shown for an energy range extending to 300 keV. Up to an energy of about 200 keV, capture from the K shell is unimportant and the calculated cross sections can be compared directly with the experimental data, which does not distinguish capture from the K and L shells. The most striking feature shown in fig. 1 is the ability of the quanta1 coupled model to reproduce the experimen- tal data over the whole energy range up to 200 keV. At the lower energies the results of Ermolaev et al. [8,15] are in harmony with the earlier work of Fritsch and Lin [14] and very similar results are provided by calcula- tions using molecular orbital expansions [18]. The CTMC cross sections are also in remarkably good agreement with the quanta1 calculations and with ex- periment although there is some tendency to overesti- mate the cross section in a region close to 100 keV where ionisation cross section is peaked. The CTMC calculations have been repeated with a Coulomb poten- tial rather than the effective potential of eq. (2). These are compatible with earlier similar calculations of Olson [19] and show that the cross sections of the Coulomb approximation are within about 20% of those produced with the effective potential (2).

He*‘impact energy E keV(Lab)

In table 1 the partial cross sections for capture into states of He’ with principal quantum number n are

Fig. 2. Total cross sections for capture from the K shell of Li by He*+. Theoretical cross sections (Ermolaev et al. [S]): Atomic basis; ~ 2 state, @ 23 state for E < 100 keV and 43 state basis for E > 100 keV. CTMC model + Experimental total cross sections (containing contributions from both the L and K shells): & McCullough et al. [23], * DuBois and Torburen

[25], @ Shah et al. 1261.

Table 1 Comparison of quantal, aQ, and CTMC, UC, cross sections for capture into states of principal quantum number n for the reaction He2+iLi* + HeC(n)+Li+.

Lab Totaf

2 3 4 r4

50 oh 5.15 22.0 11.3 5.9 44.4 *c 6.7. 18.5 15.9 6.0 Lt5.5

65 OQ 3.89 10.2 X.16 7.25 29.5 “c 4.6 12.5 6.6 10.1 33.7

loo % 2.84 2.49 2.65 4.52 12.5 oc 2.67 6.51 3.45 4.67 17.5

shown in the quanta~ and CTMC a~pro~mations; the latter being obtained by the binning procedure of Olson [2], Capture into the 1s level is improbable and is not shown. Although differing in detail, the overall pictures presented by the CTMC and quanta1 methods are sim.i- lar and underline the usefulness of the CTMC method when cross sections of high precision are not required.

Fig. 2 illustrates the results of calculations of capture from the R shell of Li*, which dommates the charge exchange mechanism above 200 keV. For this process the only ~mport~t f4nal level of He’ is the (Is) ground state and over a two state calculation including the Hc2+ + Li(2s) and the Ne*(ls) + Lit channels provides cross sections which are close to those using au elaborate many sta.te basis. Agreement between the CThK and the quanta1 model is close and both are in harmony with experiment.

3.2. Ianisation and excifafion

The cross section for ion&don are sumrn~is~~ In fig. 3. The calculated cross sections, with the exception 5E the Born appro~a~on results, refer to ~o~sa~on of the L shell. Below - 100 k,eV the CTMC and quantaJ cross sections are in reasonable agreement, and differ sharply from the Born cross sections. At about 100 keV% the classical cross sections are considerably larger than those of the Born appro~mation and the expansion method. The statistical errors in the classical. ionisation cross sections are large at high energies and the results show considerable scatter. For this reason the apparent agreement between the experimental data and the CTMC cross sections at the higher energies may be misleading. The results of the quanta1 expansion method will nI~rnate~~ approach those of the Born approxima- tion, and the two sets of cross sections are not very different above a few hundred keV, but both are much sma&r than the measured value. The electron loss cross section shown in fig. 4 also shows that the results of the quantal model are some 30% smaIler than the data, which is represented quite well above 1.50 keV by the

CTMC cross sections. Further work is needed atthough it is difficult to understand how the quantai cr5ss

sections for ionization could be increased by the rc- quised factor, to obtain agreement with experiment.

We remark that the excitation cross section is dominated by the 2s -p 2p ionisation and that the quan- tal results are in excellent agreement with experiment, both for this and for the 2s-3d transition. The cross sections for excitation of higher states are small and are therefore difficult to calculate accurately.

To conclude, ~~hou~ the charge exchange cross sections proved by the quanta1 and classical models are in harmony over a wide energy range; this is far from

-

Fig. 3. Tatal cross section rn~~t~p~e~ by the energy for ianisa- tiun in He’+ + Li cdisians. ~e~~et~~~ cross sections (Ermolaev et al. [9]): FB(Z), l%(K) and FB(K+ t): First Born a~pr~~rnatj~~ for capture from the I, shell and (I_+ K) she& respectively. -. e- .*. Atomic orbital cross sections for ionisa- lion of L shell. + t CTMC model. ~xperirnentai data: $I Shah

et al. [I%], * DUB&S [27].

490 B. H. Bransden / Models for ion - atom collisions

160 200 240 280 320

He*‘impact energy E keV(lab 1

Fig. 4. Electron cross sections in the He’+ + Li collision. Theoretical cross sections (Ermolaev et al. [9]): n - -----a quanta1 expression

model; + + + CTMC model. Experimental data: $ Shah et al. [26].

the case for ionisation cross sections. Further investiga- tion is required both for this and for other systems to see how the quantal cross sections can be brought.more into line with the data.

I should like to acknowledge many useful discussions with Professor M.R.C. McDowell, with Dr. A.M. Ermolaev and with other members of the Durham

atomic physics group.

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