J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)442–445
000–000

Classical Trajectory Monte Carlo Simulations
for Charge Transfer Processes
YAMADA Ichihiro
National Institute for Fusion Science, Toki, 509-5292, Japan(Received: 10 December 2003 / Accepted: 11 June 2004)

Abstract
I have calculated electron transfer cross sections for slow, highly charged ion-atomic hydrogen collisions byusing the classical trajectory Monte Carlo method. Similar calculations for alkaline metal targets are also carried out
using a simple one-electron model. I compare the results to predictions by the extended classical over barrier modeland available experimental data. The calculated cross sections are in good agreements with data. A simple scaling
formula for the cross sections of electron transfer collisions between slow ions and hydrogenic targets is proposed.
Keywords:
charge exchange, electron transfer, electron capture, cross section, classical trajectory Monte Carlo simulation,
CTMC

1. Introduction

2. Classical trajectory Monte Carlomethod

Charge transfer cross sections in slow, highly charged
ion-atomic hydrogen collisions are important for studying thephysics of edge and diverter plasmas, and charge exchange
diagnostics. However, it is difficult to measure the absolute
cross sections with high accuracy due to the experimental
difficulties in preparing ground state pure atomic hydrogentargets and determining the absolute target density.
Furthermore, it has been known that the well-establishedscaling law for charge transfer cross sections [1], which isbased on the extended classical over barrier model (ECBM)[2,3], cannot be adopted for atomic hydrogen and alkaline
metal targets that have only one valence electron. Therefore,
I have constructed a simulation code based on the classicaltrajectory Monte Carlo (CTMC) method to calculate chargetransfer cross sections from atomic hydrogen and alkaline
metal atoms. It is noted that both of the two classicaltreatments of the ECBM and CTMC are not adequate for low
charge state (q ≤ 5, where q is the initial charge state ofincident ion) ion-atom collisions because quantum featuresappear strongly originated from the discreteness of the energy
levels of both of the collision partners in such collisionsystems [2,4]. I also note that since inner electrons in alkaline
metal targets are deeply bounded to core nuclear, thecontribution of inner electrons to electron transfer processes
is negligibly small, and one electron transfer process isdominant [1-3].

The basic concept of the present CTMC code is similar
to that developed by Olson and coworkers [5,6]. The CTMC
method is pure classical, but no approximation is required. It
has been successfully applied to investigate ion-atomcollisions. In the present code, the coupled equations ofmotion are numerically solved for three-body, threedimensional system by using the traditional fourth-order
Runge-Kutta method with variable step size [7]. The threecharged particles involved in collisions, incident ion, target
core and electron are treated as point charges. The target core
is initially fixed at the origin. The incident ion starts from
xi0 = –10q au and approaches the target along the x-axis. The
impact parameter is given by the Monte Carlo way. The initialconditions of the target electron are also determined with theMonte Carlo technique. I use two classical models torepresent one-electron hydrogenic systems. One is thesimplest Bohr’s atom model and the other is somesophisticated microcanonical ensemble model (MEM) argued
by Abrines and Percival [8,9] and Olson and Salop [5]. The
latter describes the spherically symmetric hydrogen atom bya microcanonical ensemble of elliptic Kepler’s orbits. In theBohr’s model, the orbital radius and momentum of target
electron are consistent with the expectation values ofprobability distributions of quantum position and momentum
respectively. On the other hand, the MEM well reproduces
the momentum distribution obtained from quantum theory,
whereas there are some differences between the spatial©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: yamadai@nifs.ac.jp1442

Yamada I., Classical Trajectory Monte Carlo Simulations for Charge Transfer Processes

distributions predicted by the classical MEM and quantum
theory. I compare the results obtained with the two models in
the next chapter. Once the trajectories are calculated, complete
collision information can be extracted from the results. In
order to obtain cross sections with the statistical uncertainties
less than ±2%, about one to two million trajectory calculations
are needed, and it takes typically 1–5 hours with a DEC alphaworkstation.

The calculated electron transfer cross sections are listed
in Table I and plotted in Fig. 2. The figure shows clearly thatthe electron transfer cross section σH can be scaled as,

σH = 4.6qP –2(1)

where q is the initial charge of projectile and P is theionization potential of target electron, and both σH and q areexpressed in the atomic units. I show the comparison between
the scaling formula and absolute experimental cross sections
measured by the NICE group for 1.5q keV I6-30+-Na, Rb andCs collisions in Fig. 3. The experimental uncertainties havebeen estimated to be about 25–35%. Experimental data forhighly charged Aq+-H collisions (q ≥ 5) reported by someanother groups are also plotted in the figure [13-19]. The
experimental cross sections are found to be well reproducedby the scaling relation deduced from the present CTMC
calculations. Of course, the scaling relation can be also
determined from the available data and it is expressed as σHExp= 4.7qP –2 that agrees well with eq. (1).
Similar qP –2 scaling has been seen for multielectron
targets such as rear gas atoms, but the coefficient is about 2.7
times larger [1],3. Results and discussion
I have calculated total electron transfer cross sections inhighly charged ion-atomic hydrogen collisions at the collisionvelocity of 0.1 au. For incident ions, highly charged Iq+ (q =5–53), and bare Os76+ and U92+ ions are considered to study
the initial charge dependence of cross sections. In addition toatomic hydrogen, I have carried out the calculations foralkaline metal targets (Na, K, Rb and Cs), which can beassumed to be quasihydrogenic, at the collision energy of 1.5q
keV to study ionization energy dependence of cross sections.
Since the NICE group has measured electron transfer crosssections in 1.5q keV Iq+-alkaline metal targets [10,11], thendirect comparisons are possible. In such slow ion-atomcollisions, it has been well known both experimentally andtheoretically that the energy dependence of charge transfer
cross section is very weak and that the value is almost
constant [4,12]. I have calculated electron transfer crosssections at different collision velocities for Rb target to verify
it. The results are listed in Table 1. The differences between
them are very small, less than 2%, as expected. In addition tothe energy dependence, I have studied model dependence withtwo classical model atoms, the Bohr’s atom model and MEM.
Figure 1 shows the comparison of the cross sections obtainedwith the two model atoms. It is found that the difference is
not significant whereas the cross sections calculated with theMEM are somewhat smaller than those by Bohr’s model. Ibelieve that the MEM is more realistic and suitable because
it reproduces well the momentum distribution of target
electron. Therefore, I use the MEM throughout thecalculations below.

σm = 4πqP –2.

(2)

The scaling law for multielectron targets is based on theECBM by Niehaus [2] and proposed by Kimura et al. [1],and has been confirmed to reproduce data well. Incident
highly charged ion is frequently called ‘absorbing sphere’. In
the absorbing sphere picture, it is considered that once target
enters inside the sphere with a critical radius aroundprojectile, target electrons are absorbed into the projectile withthe probability of almost unity due to the strong Coulomb
field generated by the incident ion. If the closet approaching
distance between projectile and target core is longer than thecritical radius, no electron is transferred. Following the ECBM
and Kimura’s paper, the critical radius rm is approximated to⎯
be rm = 2√ q P –1 for highly charged ion-atom collisions, andthen the cross section is deduced geometrically as σm = πr m2 .
I examine the validity of the absorbing sphere picture. The

Table 1 Electron transfer cross sections in au calculated with the CTMC method for I5+–53+, Os76+, U92+-quasihydrogenic targets.

qH1)

Na2)

K2)

Rb1)

Rb2)

Cs2)

5102030
40
53
76
92

7.34 × 101
1.66 × 102
3.35 × 102
5.11 × 102
6.70 × 102
8.47 × 102
1.17 × 103
1.36 × 103

5.30 × 102
1.18 × 103
2.48 × 103
3.86 × 103
5.12 × 103
6.68 × 103
9.59 × 103
1.16 × 104

7.55 × 102
1.66 × 103
3.49 × 103
5.40 × 103
7.22 × 103
9.38 × 103
1.34 × 104
1.63 × 104

8.82 × 102
1.94 × 103
4.04 × 103
6.19 × 103
8.17 × 103
1.09 × 104
1.56 × 104
1.87 × 104

8.85 × 102
1.99 × 103
4.07 × 103
6.14 × 103
8.32 × 103
1.10 × 104
1.58 × 104
1.89 × 104

9.42 × 102
2.07 × 103
4.37 × 103
6.70 × 103
8.80 × 103
1.17 × 104
1.67 × 104
2.02 × 104

1) Collision velocity of 0.1 au.
2) Collision energy of 1.5q keV.
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Yamada I., Classical Trajectory Monte Carlo Simulations for Charge Transfer Processes

Fig. 1Comparison of electron transfer cross sections
calculated with Bohr’s atom model and microcanonicalensemble model for I5-53+-H collisions.

Fig. 3Comparison between the scaling formulae obtained
from the CTMC calculations and experimental data. The
uncertainties of the data are estimated to be about±25–35%.

Fig. 2

CTMC cross sections for electron transfer processes.
The solid line shows the scaling relation determinedfrom the CTCM results, σ H = 4.6qP –2.

Fig. 4

Impact parameter dependence of the capture
probability for I5-53+-H collisions at the collision velocityof 0.1 au. From a critical radius, the capture probability
falls rapidly. The sphere of the critical radius is called‘absorbing sphere’.

account and the critical radius is estimated by the static initial
two parameters, initial charge q and ionization energy P.
However, the ionization potential of target electron changes
dynamically and drastically during the collision due to theCoulomb field of the incident ion. Especially, since atomic
hydrogen has only one electron, no cancellation mechanism
against the Coulomb field, such as dynamical screening
effects by core electrons, can be expected in the pure threebody Coulomb systems. In the case of multielectron targets,since many electrons surround the target nuclear, then thescreening effects may be able to reduce the dynamicalchanges in ionization potential of outer electrons. In order tofurther improve the ECBM, such dynamical effects need to
be incorporated into the model by some ways. Important
knowledge may be obtained from the collision dynamics

closet approaching distance can be approximated by the
impact parameter in ion-atom collisions. Fig. 4 shows theimpact parameter dependence of electron transfer probability.
As expected from the absorbing sphere model, the electrontransfer probability is almost unity within a critical impact
parameter and the probability decreases rapidly from thecritical value. However the critical radius is found to be⎯smaller and approximated by rH : 1.2√ qP –1. This is the reason
why the coefficient of the scaling law of σH is smaller thanthat for σm. The ECBM may be a good picture qualitatively
but not valid quantitatively for the electron transfer collisions
between highly charged ions and guasihydrogenic. There area few candidates for the explanation about the reason whythe ECBM fails in the description of ion-hydrogenic atomcollisions. In the ECBM, no dynamical effect is taken into3444

Yamada I., Classical Trajectory Monte Carlo Simulations for Charge Transfer Processes

analysis using the CTMC calculations.

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4. Summary
By using the CTMC code developed for threedimensional three-body Coulomb systems, I have calculated
electron transfer cross sections for the collisions of I5-53+,
Os76+ and U92+ with H, Na, K, Rb and Sc at low collisionvelocities. It has been found that the calculated cross sections
agree well experimental data, and the cross section can bescaled as σH = 4.6qP –2 for highly charged ion-hydrogen andalkaline metal collisions.

Acknowledgements
I am grateful to the NIFS NICE group for useful
discussion about electron transfer processes in ion-atomcollisions. This research was partially supported by the Japan
Society for the Promotion of Science, Grant-in-Aid for
Scientific Research, 10740212, 1998–1999.

References[1] M. Kimura et al., J. Phys. B28, L643 (1995).
[2] H. Ryufuku et al., Phys. Rev. A21, 745 (1980).
[3] A. Niehaus, J. Phys. B19, 2925 (1986).

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