A Coulomb Over-the-Barrier Monte Carlo Simulation to probe Ion-Dimer CollisionDynamics

W. Iskandar,1, ∗ X. Flechard,2, † J. Matsumoto,3 A. Leredde,2, 4 S. Guillous,5 D.

Hennecart,5 J. Rangama,5 A. Mery,5 B. Gervais,5 H. Shiromaru,3 and A. Cassimi5

1Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA-94720, USA2Normandie Univ, ENSICAEN, UNICAEN, CNRS/IN2P3, LPC Caen, 14000 Caen, France

3Department of Chemistry, Tokyo Metropolitan University,1-1 Minamiosawa, Hachiouji-shi, Tokyo 192-0397, Japan

4LPSC, Universite Grenoble Alpes, CNRS/IN2P3, 38026 Grenoble, France5CIMAP, CEA-CNRS-ENSICAEN-Universite de Caen, BP 5133, F-14070 Caen cedex 5, France

(Dated: November 9, 2018)

We present a combined theoretical and experimental study of primary and post-collision mech-anisms involved when colliding low energy multiply charged ions with van der Waals dimers. Thecollision dynamics is investigated using a classical calculation based on the Coulombic Over-the-Barrier Model adapted to rare-gas dimer targets. Despite its simplicity, the model predictions arefound in very good agreement with experimental results obtained using COLd Target Recoil IonMomentum Spectroscopy (COLTRIMS), both for the relative yields of the different relaxation pro-cesses and for the associated transverse momentum exchange distributions between the projectileand the target. This agreement shows to which extent van der Waals dimers can be assimilated toindependent atoms.

PACS numbers: 34.70.+e, 36.40.Mr, 34.10.+x, 32.80.Hd, 36.40.-c

I. INTRODUCTION

The complete understanding of primary and post-collision mechanisms involving ions, atoms and moleculesis of primordial importance in many domains of sci-ence. These processes generate interest in fundamentalphysics as well as in interdisciplinary research such asastrophysics [1], accelerator technologies, fusion plasmaphysics [2], biological and medical treatments [3, 4]. Fordecades, many theoretical and experimental investiga-tions have thus been devoted to the understanding ofcollision mechanisms particularly for ionic projectiles col-liding with atomic targets. The relative cross sectionsassociated to the different elementary processes (ioniza-tion, excitation and electron capture from the atomic tar-get) depend strongly on the collision regime, given bythe comparison of the projectile velocity (vp) to the or-bital velocity of active electrons (ve) and by the collisionasymmetry, i.e. the ratio between the projectile and tar-get atomic numbers [5–8]. It is today well establishedthat, as ionization and excitation of the target domi-nates in the high velocity regime, charge exchange (orelectron capture) is by far the most probable process atlow energy. For the latter, concomitant advances in ex-perimental and theoretical techniques have led to a quitecomplete knowledge of the collision dynamics, which isreviewed in textbooks such as [9].

Low energy collisions of ionic projectiles with molecu-lar targets such as diatomic molecules are more complex

∗ wiskandar@lbl.gov† flechard@lpccaen.in2p3.fr

and their investigation is still a theoretical and exper-imental challenge. The complexity is due to additionaldegrees of freedom such as the orientation of the moleculeand to the multiple interactions between all the compo-nents of the molecule (electrons and nuclei). It opensup new primary mechanisms, such as the screening ef-fect of both centers active electrons during the collision,and new post-collision mechanisms such as energy and/orelectron exchange between the two sites of the molecule,leading to new relaxation processes.

Primary and post-collision mechanisms can alsostrongly depend on the bond type of the molecular edificesuch as in covalently bound molecules or van der Waals(vdW) bound dimers. When an ionic projectile ionizesor captures many electrons from a covalent molecule,the latter tends to dissociate into equally charged frag-ments due to the delocalization of the valence electronsthroughout the molecule [10, 11]. In contrast, for vdWbound dimers, the long separation distance between thetwo atoms of the dimer may lead to weaker charge rear-rangement. Non-equally charged fragmentation becomesthus more efficient and can even dominate the equallycharged channel [12]. Furthermore, in the low energycollision regime, the long separation distance and theweak charge mobility between the two sites of dimerslead to projectiles preferentially scattered in the directionof the most charged fragment of the dimer [13], as op-posed to what was previously observed with N2 molecu-lar targets [11]. Another specificity of vdW bound dimerswhen compared to covalent molecules is the appearanceof new post-collision mechanisms. Many results haveshown that producing non-equally charged atoms of thedimer leads to new relaxation processes such as RadiativeCharge Transfer (RCT) and Interatomic/Intermolecular

arX

iv:1

804.

0454

4v1

[ph

ysic

s.at

m-c

lus]

12

Apr

201

8

2

Coulombic Decay (ICD) involving electron and/or energytransfer between one neutral site and one charged site ofthe dimer [12, 14–17].

For a better understanding of these primary andpost-collision mechanisms related to vdW dimer targets,several theoretical approaches can be considered. Afull quantum-mechanical treatment of collisions betweenmultiply charged ion (MCI) projectiles and moleculartargets requires solving the time dependent Schrodingerequation for a given state by taking into account the ki-netic and potential energies of all partners of the colli-sion. Such an ab initio approach is impossible in practiceand approximation methods have to be used. Classicalmodels are known to provide general insight into the in-teraction mechanisms and dependencies on the collisionparameters. For atomic targets, a simple classical modelvalid for one-electron capture was introduced by Bohrand Lindhard [20] and later elaborated by Knudsen etal. [21]. Here, an electron can only be transferred if theCoulomb force exerted by the projectile exceeds the ini-tial binding force and if the electron kinetic energy in theprojectile frame is smaller than its potential energy. Themodel predicts an energy independent cross section atlow impact energies. At high energies, the cross sectiondecreases as v−7p , in good agreement with experimentaldata [21]. Later on, interest in multiple electron processesled to the birth of the Classical Over-the-Barrier Model(COBM). The COBM, mainly developed by Ryufuku etal [25], Mann et al [26], Barany et al [27] and Niehaus[28], is based on the idea that electrons can transit fromthe target to the projectile at given internuclear distancesfor which the height of the potential barrier between thetwo nuclei is lower than the Stark shifted binding energyof the electrons. This model has been successfully usedfor MCI projectiles colliding atomic targets in the low en-ergy regime (vp ranging from 0.01 to 1 a.u., typically) topredict cross sections associated to the primary collisionprocesses [28, 29] as well as to post-collision processes[30, 31].

For molecular targets, a three-center COBM based onthe model described by Barany et al [27] has recentlybeen developed by Ichimura and Ohyama-Yamaguchi[32]. This model was used to investigate multiple electroncapture from diatomic molecules in collisions with slowMCIs. The results were found to be consistent with ex-perimental results for N2 molecules [11], showing a domi-nant dissociation into equally charged fragments for evencapture multiplicity or with only one electron differencebetween the two charged fragments for odd numbers ofelectron capture [32]. Moreover, the calculation repro-duced the higher probability to end up with the mostcharged fragment located on the molecule farther sitefrom the projectile, due to the strong polarization of theN2 molecule in the presence of the MCIs [32, 33]. Besidethe treatment of covalent molecular targets, the authorshave applied the same analytical methodology to rare gasdimer targets by adding an adjustable electron screeningparameter [34]. The calculations have shown an increase

of charge asymmetry in the ion pair distribution whenincreasing the screening parameter.

For further investigation of the collision dynamics withatomic dimer targets, we have used a similar approachbut including additional ingredients. To perform a morecomplete treatment of the collision, our model comprisestwo distinct stages, the way-in and the way-out, as pro-posed by Niehaus for atomic targets [28]. Secondly, weused Monte Carlo (MC) simulations to facilitate boththe theoretical treatment and the comparison with theexperimental data. These developments allow the pre-diction of the final ion pair production, give access tocapture multiplicity on each site as a function of the im-

pact parameter ~b in the molecular frame, and providethe transverse momentum exchange between the projec-tile and each center of the dimer all along the interactionpath. In our previous papers [13, 14], the predictions ofthis model have already been successfully compared toexperimental results, leading to a better understandingof primary and post-collision mechanisms involved withdimer targets. Only few details were provided on the cal-culations and the model itself. In the following sections,a complete description of the different steps of the MC-COBM calculations is given, followed by a comparisonwith experimental data obtained with projectile veloci-ties ranging from vp ∼ 0.3 to 0.4 a.u..

II. CLASSICAL OVER-THE-BARRIER MODEL

For the sake of simplification, we will start in Sec.II Aby presenting the MC-COBM for atomic targets. Themethod is very similar to the one initially proposed byNiehaus [28] as charge exchange probabilities are calcu-lated using the same ingredients. However, it includesnew features such as randomly generated trajectories ofthe projectiles instead of an integrated cross section cal-culation, and an estimation of the transverse momen-tum exchange. We also introduce the notion of effec-tive charge of the projectile and of the target to providea simplified formulation of the equations found in [28].The adaptation of this model to an atomic dimer targetis then presented in Sec.II B.

A. Atomic Target

1. Principle of the COBM

Within the COBM model, the collision of an ionic pro-jectile, noted Aq+, with an atomic target, noted B, isprocessed in two distinct parts that will be referred toas the way-in and the way-out (Fig.1). The way-in andthe way-out correspond respectively to the stage with theprojectile approaching the target and to the stage withthe projectile moving away from the target. In the model,the projectile trajectory is approximated by a straightline.

3

Way-In Way-Out

Molecular State

Way-In:

Determination of the number of electrons

shared between the projectile and the target.

Way-Out:

Determination of the number of electrons captured

by the projectile and recaptured by the target.

-15 -10 -5 0 5 10 15

-15

-10

-5

0

Projectile Aq+

Target B

(b)

Electrons of the Target

-15 -10 -5 0 5 10 15

-15

-10

-5

01 electron shared(c)

-15 -10 -5 0 5 10 15

-15

-10

-5

01 electron still shared + 1 electron captured(e)

-15 -10 -5 0 5 10 15

-15

-10

-5

01 electron captured + 1 electron recaptured(f)

Target B+

Projectile A(q-1)+

-15 -10 -5 0 5 10 15

-15

-10

-5

0(d)

Po

ten

tial E

nerg

y [

a.u

.]

Internuclear Distance [a.u.]

2 electrons shared

b: Impact

Parameter

Potential Barrier

B

Aq+ A(q-q’)+

+ ++q (q- ) q'q'B A +BA +

Way-In Way-Out

2outR

1outR

2inR

1inR

(a)

b

FIG. 1. (a) Schematic view of the interaction between a projectile Aq+ and a target B with NB shared electrons involved in theway-in (NB=2 in this example), as the projectile is approaching the target, and where 0 to NB electrons may be captured bythe projectile in the way-out, as the projectile is leaving the target. (b to f) Schematic representation of the potential barrierevolution in the way-in and in the way-out of the collision. In the way-in, the approach of the projectile towards the targetlowers the potential barrier height between them (b). At a critical internuclear distance called sharing radius Rin

i for eachelectron i, the potential barrier may become lower than the electron binding energy on the target (c), causing this electron tooccupy a molecular state induced by the proximity of the projectile and the target. The maximum number of electrons shared

will be reached at the impact parameter distance ||~b|| where the way-in stage ends and the way-out stage starts (d). In theway-out, the projectile will leave the target causing an increase of the potential barrier height. For each shared electron i, at agiven distance called capture radius Rout

i , the model will determine the probability for this active electron to be captured bythe projectile or to be recaptured by the target (e, f).

On the way-in, as the projectile gets closer to the tar-get, the Coulomb potential barrier between the targetand the projectile decreases with the internuclear sepa-ration of the two partners of the collision. An electron isthus transferred to a molecular state when the potentialbarrier becomes lower than its initial binding energy tothe target. This transfer occurs at critical internucleardistances between the projectile and the target calledsharing radii. Each electron, numbered i (i increasingwith the binding energy of the electron), has its ownsharing radius Rin

i depending on its initial binding en-ergy to the target IBi and on the projectile and target

effective charges qAini and qBin

i :

Rini =

i+ 2√qAini × qBin

i

IBi(1)

In the COBM model, shared electrons are transferredto molecular states where they do not contribute to thecharge screening between the projectile and the targetnuclei. This means that in the way-in, the projectileeffective charge keeps its initial value qAin

i = q while the

target effective charge qBini = i increases by one unit

each time one electron is shared. Taking into account

the Coulomb interaction with the projectile, the bindingenergy of each electron i in its molecular state is alsoestimated using the relation:

Emoli = IBi +

qAini

Rini

(2)

When the projectile reaches the minimum internuclear

distance corresponding to the impact parameter ||~b||, theway-in stage is ended and the way-out stage starts. Atthis transition point, the maximum number of sharedelectrons, noted NB , has been reached. On the way-out, as the projectile goes away from the target, theCoulomb barrier height increases with the internuclearseparation between the projectile and the target. Whenthe Coulomb barrier reaches the binding energy Emol

i ofa shared electron i, this electron is then either capturedby the projectile or recaptured by the target. The corre-sponding critical distance Rout

i is called capture radius.We consider that each electron captured by the projec-tile in the way-out modifies both the projectile and thetarget effective charges. When dealing with an electroni in the way-out, the effective charges of the projectileand of the target are then respectively qAout

i = q− ci and

4

qBouti = i + ci, where ci is the number of electrons that

were previously captured by the projectile. The value ofRout

i given by Eq.3 can thus be different from the one ofRin

i :

Routi = Rin

i

(√qAouti +

√qBouti )2

(√qAini +

√qBini )2

. (3)

For each crossing of the projectile with a sphere of radiusRout

i (Fig.1(a)), the capture probability of the electron iby the projectile is then estimated using Eq.4a. Thisprobability is simply based on the multiplicities of quan-tum states associated with the effective principal quan-tum numbers, ni and mi, that can be populated by theelectron i in the case of a capture by the projectile or ofa recapture by the target, respectively. These effectivequantum numbers ni and mi are reals and are simply es-timated in the framework of the Bohr model using Eq.4band Eq.4c, where m0 is the principal quantum number ofthe outer shell electron on the target. They depend onthe final binding energy of each electron once capturedby the projectile EA

i (Eq.4d) or recaptured by the targetEB

i (Eq.4e).

Pi =n2i

m2i + n2i

(4a)

ni =qAouti√2EA

i

(4b)

mi =qBouti√2EB

i

− qBouti√2IBi+ci

+m0. (4c)

EAi = Emol

i − qBouti

Routi

(4d)

EBi = Emol

i − qAouti

Routi

. (4e)

To summarize, the electrons i = 1 to i = NB will firstpopulate molecular states in the way-in, as the projectilecrosses the Rin

i radii. On the way-out, i going from NB

to 1, these electrons will be redistributed on the projec-tile or on the target, according to their capture probabil-ity. This redistribution occurs at a distance Rout

i , witha probability Pi to be captured by the projectile and aprobability (1-Pi) to be recaptured by the target.

2. Monte-Carlo Simulation

The model proposed by Niehaus [28] provides capturecross sections by combining the electron capture proba-bilities and the ranges of impact parameter leading to allpossible capture multiplicities. This approach is not any-more adapted to a more complex target comprising twosites, as in the case of a dimer target. We therefore choseto combine the COBM method with Monte Carlo (MC)simulations. The MC simulation allows to set randomly

Aq+

B

1inR

2inR

A(q-q’)+

q+ (q-q')+ q'+A +B A +B

Way-in Way-out

(1;2)( )P in

z

x

y 3inR

3outR

2outR

1outR

Transition phase

b

(2;3)( )P in

(3;3)

( )P in out

(3;2)( )P out

(2;1)

( )P out

FIG. 2. Schematic view of the interaction between a projectileAq+ and a target B with NB electrons involved in the way-in (NB=3 in this example) and where 0 to NB electrons canbe captured by the projectile in the way-out. The transversemomentum exchange, due to a Coulomb repulsion along theinteraction between the projectile and the target, is calculatedfor all successive steps in each of the three (way-in, transitionstep and way-out) stages of the collision.

the projectile coordinates noted A(xA, yA) in the planeperpendicular to the projectile propagation axis z. In thecase of an atomic target, the target position is simplydefined as B(xB=0, yB=0, zB=0). For each simulatedevent, the successive crossing points between the projec-tile trajectory and spheres of radius Rin

i centered on thetarget are determined, up to i = NB (Fig.1(a)). Thesame approach is used in the way-out, as the projectilecrosses the different spheres of radius Rout

i from i = NB

to i = 1. For each capture crossing point of the way-out, a random number (uniform distribution between 0and 1) is compared to the capture probability given byEq.4a to determine whether the electron i is captured bythe projectile or recaptured by the target. Integratingall the events leading to a given capture multiplicity pro-vides then directly the associated cross section. As it willbe shown in the following, the use of the MC simulationgreatly simplifies the treatment of molecular targets andprovides better insight of specific processes through thestudy of the corresponding 2D probability map in impactparameter [13]. In addition, this method gives easy ac-cess to differential cross sections in transverse momentumexchange.

3. Transverse Momentum Exchange Calculation

To get further information about the collision dy-namics, the transverse momentum exchange due to theCoulomb repulsion between the projectile and the targetalong the collision is calculated. This information is di-rectly connected to the projectile scattering angle whichprovides additional points of comparison with experimen-tal data.

Our model of momentum transfer is based on the par-titioning of the collision into a series of charge exchange.The projectile trajectory is thus segmented into steps,defined by the successive sharing and/or capture radii,with their corresponding projectile and target effectivecharges, qAi and qBi . The transverse momentum ex-

5

changes calculated for each step are then cumulated toobtain the total momentum exchange of the collision. Westill assume projectile straight line propagation (in the zdirection), which is an acceptable approximation in therange of velocities considered here. The projectile po-sitions at crossing points with the different sharing andcapture radii, noted zini in the way-in and zouti in theway-out, are calculated using the corresponding sharingradii Rin

i or capture radii Routi and the distance between

the two collision partners in the transverse plane:

zini = zB −√

(Rini )2 − (xA − xB)2 − (yA − yB)2

zouti = zB +√

(Routi )2 − (xA − xB)2 − (yA − yB)2 (5)

As the projectile and the target start to share elec-trons, the two partners of the collision exert on each othera repulsive force due to their effective charge. The esti-mation of the resulting transverse momentum exchangevector all along the collision is divided in three stages:the way-in, from zin1 to zinNB

, the way-out, from zoutNBto

z →∞, and a transition stage (Fig.2) between the way-inand the way-out introduced for convenience.

In the way-in, for each shared electron i the trans-verse momentum exchange is integrated along the cor-responding step using Eq.6a, from tini , the time of themolecularization of the electron i, to tini+1, the time ofthe molecularization of the next electron (i + 1). Thiscalculation takes into account the effective charge of theprojectile qAin

i and of the target qBini as well as the dis-

tance between the two centers in the transverse plane(corresponding to the modulus of the impact parameter~b(xA − xB ; yA − yB) for ion-atom collisions). The timestini and tini+1 are determined using the initial velocity ofthe projectile vp (propagating in the z direction) and itsz coordinates, zini and zini+1, given by Eq.5.

∆~P(in)⊥(i;i+1) =

∫ tini+1

tini

dtqAini qBin

i

(b2 + (vp ∗ t)2)32

~b

=qAini qBin

i

vpb2(F (tini+1)− F (tini )

)~b (6a)

with

F (t) =t√

b2

v2p

+ t2

tini =zinivp

and tini+1 =zini+1

vp

Similarly, the momentum exchange occurring duringthe transition stage is given by Eq.6b. This stage com-prises only one step, the time period between tinNB

and

toutNB, corresponding respectively to the molecularization

and capture treatments of the last shared electron num-

bered NB .

∆~P(in→out)⊥(NB ;NB) =

qAini qBin

i

vpb2(F (toutNB

)− F (tinNB))~b (6b)

with tinNB=zinNB

vpand toutNB

=zoutNB

vp

On the way-out, the transverse momentum exchangeis calculated using Eq.6c for the NB steps related to eachcapture or recapture of the electron i. The last step ex-tends from tout1 to tout0 → ∞, as both the projectile andtarget may be charged after the collision.

∆~P(out)⊥(i;i−1) =

qAouti−1 q

Bouti−1

vpb2(F (touti−1)− F (tini )

)~b (6c)

with touti =zouti

vp, touti−1 =

zouti−1vp

and tout0 →∞

The total transverse momentum exchange vector ∆~P⊥is obtained by summing the (NB-1) steps of the way-in,the step of the transition stage and the NB steps of theway-out:

∆~P⊥ =

NB−1∑i=1

∆~P(in)⊥(i;i+1) + ∆~P

(in→out)⊥(NB ;NB) +

1∑i=NB

∆~P(out)⊥(i;i−1)

(7)

The effective charges of the projectile qAin(out)

i and of

the target qBin(out)

i used in Eq.6a, 6b and 6c depend onthe number of molecular electrons in the way-in and inthe transition phase. In the way-out, they depend onboth, the number of molecular electrons i and the num-ber of captured electrons ci. As in the model of Niehaus,the charge of an electron captured by the projectile or re-captured by the target in the way-out will be accountedfor in the effective charge of its new center (projectile ortarget). In a classical approach, this choice can be easilyjustified by the fact that the classical orbital radius ofthe captured or recaptured electron is then smaller thanthe internuclear distance between projectile and target.It is more difficult to deal with the electrons that areshared by both centers after their molecularization in theway-in. In the COBM cross sections calculation as pro-posed by Niehaus, shared electrons are pure spectatorsuntil capture by the projectile or recapture by the tar-get (Sec.II A 1). But for the calculation of the transversemomentum exchange, the partial screening effect of the ishared electrons on the target effective charge might playan important role. To take into account the effective dis-tribution of the electrons around the molecule in eachsegment of the trajectory, the screening could be intro-duced as a position-dependent function in integrals 6a,6b and 6c, as it is a common practice in nuclear stoppingtheories. For the sake of simplicity and to limit as muchas possible the number of free parameters, we introducein the calculation of the transverse momentum transfer

6

a single screening parameter, noted S, whose effect onlydepends on the number of shared electron i in each pieceof the trajectory. The value of S can be adjusted be-tween 0 and 1 according to different screening scenarios(see Sec. IV B). The effective charges of the projectileand of the target are then respectively given by:

qAini = q and qBin

i = i× (1− S) (8a)

in the way-in (and in the transition phase with i = NB)and by

qAouti = q − ci and qBout

i = i× (1− S) + ci (8b)

in the way-out. Note that the screening parameter Sis only introduced into the model for the calculation oftransverse momentum exchange, while capture cross sec-tion calculations simply use the effective charges given inSec.II A 1.

B. Collisions with Atomic Dimer Targets

The MC-COBM model can be applied to dimer targetswhen making the approximation that the dimer can betreated as two independent atoms fixed in space. Thisapproximation is justified by the low electron mobilitybetween the two atoms of the dimer and by its largeinternuclear distance. The same approach as for a sim-ple atomic target can then be used, by considering thecollision of the projectile with the two centers B and Cof the dimer as two separate collisions with two atoms.The only indirect effect of one site on the other is thechange of the projectile effective charge during the colli-sion, as this change is then caused by captured electronsfrom both centers of the dimer. This indirect influenceof a second atomic center combined with the increase ofthe degrees of freedom due to the geometry of diatomicmolecular target are the major motivations for using MCsimulations. All the possible trajectories of the projec-tile with respect to the positions of the two centers ofthe dimer (Fig.3) are taken into account by generating,in a random way, the position (xA, yA) of the projectilein the transverse plane and the orientation of the dimeraround its center of gravity arbitrarily fixed at (0, 0, 0)coordinates. The orientation of the dimer is obtainedby sampling randomly the two parameters cos(θ) and φ,where θ is the polar angle of the dimer axis with respectto the z axis and φ is the angle of the dimer axis in thetransverse plane (x, y) with respect to the x axis. Forthe internuclear distance between centers B and C, weused the values 5.86 a.u. and 7.18 a.u. for Ne2 and Ar2,respectively. As the collision duration is very short com-pared to the vibrational period of the dimer target, thesedistances are considered as constant.

x

z

y

B

C

xB

yB

zB

Aq+

B

B

x

y

B

C

Aq+

B b

Plane perpendicular to the projectile beam axis

(a) (b)

xA

yA

FIG. 3. Representation of the collision between a projectileAq+ and a dimer target B-C in a 3D Cartesian coordinatesystem (x, y, z) (a) and in the transverse plan (x, y) (b). (xB ,yB , zB) are the coordinates of the center B of the molecule, θBis the angle of the molecular axis with respect to the projectilebeam axis (z axis), φB is the projected angle of the molecularaxis in the transverse plane with respect to x axis and (xA, yA)are the coordinates of the projectile in the transverse plane.

1. Capture cross sections calculations

The COBM model is then applied as described inSec.II A for atomic targets, but considering two atomiccenters whose electrons can be shared and captured sep-arately. In a first step, the number of electrons poten-tially occupying a molecular state in the way-in, notedNB for the center B and NC for the center C of thedimer, are determined. These shared electrons are nownumbered iB and iC , depending on their center of origin.The associated sharing radii and the binding energy ofelectrons iB and iC are calculated using Eq. 1 and 2.As the projectile enters the way-out of one of the twocenters, the capture by the projectile or the recaptureby the target of the molecular electrons may occur ac-cordingly to Eq.4a at the corresponding capture radii.Note that within this model, the electrons are shared be-tween the projectile and their center of origin, but notbetween the two centers of the target. Shared electronscan thus only be recaptured on their initial atomic site orcaptured by the projectile. As mentioned previously, thepossible change in charge of the projectile is taken intoaccount to determine the next crossing radii with the twocenters, for capture or molecularization. When treatingthe electrons of center B using Eq.1 to Eq.4e, i is thusreplaced by iB and the target effective charge is simplyqBiniB

= iB on the way-in. When dealing with an electroniB , we now have to account for the number of electronspreviously captured by the projectile from targets B andC, still noted respectively ciB and ciC . The target B ef-

fective charge thus becomes qBoutiB

= iB + ciB on the way

out, and the projectile effective charge qAiniB

= q − ciCon the way-in and qAout

iB= q − ciB − ciC on the way-out.

The same formulas apply to electrons iC from the tar-get C by swapping the subscripts or superscripts B andC. Because of the different possible orientations of thedimer, molecularization and capture of electrons from B

7

and C can occur in different order. This implies thatthe evaluation of the crossing radii and the determina-tion of capture or recapture of the electrons have to bedone sequentially to know at anytime iB , iC , ciB and ciC ,as the projectile progresses along the collision axis. Atthe end of the collision, we keep track of the number ofelectrons captured from each center of the dimer. Thecorresponding charge distribution on the ionized dimeris associated to a capture channel noted (qB , qC)c whereqB and qC are the charge states of the center B and Cafter the collision. In addition, the number iB and iC ofthe individual captured electrons are recorded, giving ac-cess to their initial atomic shell [14], as well as the initialcoordinates of the projectile and of the dimer sites B andC for the study of impact parameter dependence [13].

2. Transverse momentum calculations

The transverse momentum exchange along the collisionis also calculated separately for the center B and for thecenter C, using the methodology described in Sec.II A 3.Compared to the atomic case, we now have to account fora projectile effective charge including the contributionsfrom shared and captured electrons from both centers ofthe dimer. This implies a segmentation of the projec-tile trajectory in as many steps as defined by the totalnumber of crossing radii involved when considering bothcenters. When dealing with an electron iB from the cen-ter B, the projectile and target effective charges are thusdefined as

qAiniB

= q − ciC and qBiniB

= iB × (1− S) (9a)

in the way-in (and in the transition phase with iB = NB)and by

qAoutiB

= q − (ciB + ciC )

and qBoutiB

= iB × (1− S) + ciB (9b)

in the way-out, where S is the partial screening parame-ter introduced in Sec. II A 3. Again, the same formulasapply to electrons from the target C by swapping thesubscripts or superscripts B and C. The sum of the con-tributions from both sites provides the total transversemomentum exchange induced by the collision betweenthe projectile and the dimer.

III. POST-COLLISION TREATMENT OF THEMC-COBM DATA

For a meaningful comparison of the MC-COBM resultswith those obtained experimentally, several additionalsteps are required. First, one has to determine the frag-mentation channels expected for each capture channelprovided by the model. Then, the trajectories of the twoionic fragments from the dimer and the response functionof the experimental setup have to be simulated.

A. Estimation of Relaxation Channels

The MC-COBM allows us to determine the numberof electrons captured by the projectile from each cen-ter of the dimer and thus, to deduce the charge distri-bution on the dimer (qB ;qC)c for each simulated eventright after the collision. A first step before any compari-son with the experimental results consists in identifyingthe relaxation channels leading to the fragmentation ofthe molecular ion. The resulting fragmentation channelsare noted (q1;q2)f , where the subscript f stands here forfragmentation, unlike the subscript c, indicating a cap-ture channel. This step is essential as the charge statesq1 and q2 may not be the same as qB and qC , and becausethe experiment is sensitive to charged fragments only.

Several scenarios can be considered for the relaxationmechanisms of a given capture channel (qB ;qC)c towardsfragmentation channels (q1;q2)f . One of these scenarioscorresponds to the capture of electrons from both centersof the dimer, leaving the dimer in a dissociative statewhich directly relax in two charged fragments. Here, thedissociation occurs directly via Coulomb Explosion (CE)and the charge distribution on the dimer remains thesame ((qB ;qC)c ≡ (q1;q2)f ) and other scenarios have tobe considered when the projectile captures electrons fromonly one center of the dimer, leaving the dimer in anasymmetric charge distribution with one charged centerand one neutral center. Here, the collision can eitherpopulate a dissociative state or a non-dissociative state ofa specific channel. When a dissociative state of the dimerwith one neutral site is populated, the simulated eventis excluded from the comparison with the experimentaldata as neutral fragments are not detected. As it willbe discussed in the following sections, the determinationof this probability to populate unbound states with oneneutral center is non trivial.

For the non-dissociative states, the two centers of thedimer remain bound until relaxation towards a dissocia-tive state via intermediate mechanisms such as ChargeTransfer (CT), Radiative Charge Transfer (RCT) or In-teratomic Coulombic Decay (ICD) [12, 14]. These pro-cesses lead, respectively by charge exchange and energytransfer between the two centers of the dimer, to theproduction of two charged fragments to which the ex-perimental detection system is sensitive. The relaxationprocess of a given capture channel is determined afterthe study of the molecular ion potential energy curves(PECs), as previously discussed in [12, 14]. In most cases,direct crossings between PECs of molecular states asso-ciated with a channel (qB ;0)c and molecular states asso-ciated with (qB-1;1)∗c lead to the (qB-1;1)f fragmentationchannel through CT. In the present collision systems, thisis the case for the (4;0)c and (3;0)c capture channels. Asshown in [12, 14, 37], such direct crossings do not existfor the one-site double capture channel (2;0)c and thelatter relaxes towards the (1;1)f fragmentation channelthrough RCT.

For single capture (1;0)c, the non-dissociative states

8

are stable and do not dissociate, except for the case of in-ner valence shell single electron capture from a Ne dimer.In that specific case, the excitation energy is sufficientfor the ICD process to occur, resulting in the populationof the (1;1)f fragmentation channel with the emissionof a low energy electron, as previously observed in [14].With Ar9+ and Xe20+ projectiles, the experimental re-sults have shown no sign of ICD and the MC-COBM doesnot predict any significant inner-shell single electron cap-ture from Ne dimers. The ICD process will thus not bediscussed further in the present article.

After simulating many events (typically 108 per colli-sion system), the MC-COBM data are sorted accordingto the relaxation process and to the final fragmentationchannel inferred using the method described above. Thisclassification is kept in the following steps of the sim-ulation, providing the relative production of the differ-ent relaxation mechanisms and fragmentation channels(Sec.IV A) and the associated transverse momentum ex-change distributions (Sec.IV B).

B. Simulation of the Spectrometer and Analysis ofthe Data

This section describes briefly the principle of the ex-perimental setup and how the trajectories of the chargedfragments inside the spectrometer are simulated. The fulldetails of the COLTRIMS experimental apparatus aredescribed in Refs. [12–14, 35]. The projectile ion beamcrosses the dimer target provided by a supersonic gasjet in the center of the recoil ion momentum spectrome-ter. After the collision, recoil ions and dimer fragmentsresulting from charge transfer are collected using the uni-form electric field of the spectrometer and detected by amicrochannel plates delay-lines detector giving both thedetection time and position of detected ions. The co-incidence TOF (time of flight) map of both fragmentsfrom a dimer dissociation is used to identify and selectthe fragmentation channels. The fragments momenta arecalculated from the positions and TOF data by imposingmomentum conservation restriction for optimal resolu-tion and false coincidence events suppression [35]. TheKinetic Energy Release (KER), deduced from the frag-ments momenta in the molecular frame, provides thenan identification of the relaxation mechanisms leading toa specific fragmentation channel. By separating all theprocesses, the relative intensities of all relaxation mecha-nisms is finally determined. In addition, as shown in [13],the momenta of the two fragments give access to the ini-tial orientation of the dimer target and to the transversemomentum exchange induced by the collision.

The geometry of the spectrometer, the size of the col-lision region, and the electric field applied to extract thefragments all the way to the detector have been imple-mented in last step of the MC-COBM simulation. Thisfinal treatment of the MC-COBM events ensures an un-ambiguous and fair comparison between experiment and

theory. The spatial extension of the collision region issimulated by generating randomly the initial position ofthe dimer using a 3D Gaussian distribution of FWHM 0.6mm corresponding to the overlap between the projectilebeam and the gas jet. Each process leading to a givenfragmentation channel releases a specific kinetic energyfor the two charged fragments [12, 14]. For dimers, thisKER depends, to first order, on the charge state of thetwo fragments and on the internuclear distance at whichthe dissociation takes place. In the simulation, we haveused the distributions in KER obtained experimentallyfor each process leading to a specific fragmentation chan-nel: for each simulated event, the KER is randomly se-lected according to the proper experimental distribution.The momentum vector of the fragments in the frame ofthe dimer center of mass is deduced from the KER andfrom the orientation of the dimer. The momentum ex-change between the projectile and the target providedby the MC-COBM calculation is then accounted for todetermine the momentum vector of each fragment in thelaboratory frame. For each event, the trajectory of thetwo charged fragments inside the extraction field of thespectrometer is calculated using these initial parameters:position, momentum, mass and charge of the fragments.The TOF and the position on the detector of the two ionsare finally convoluted with the detector response function(position and TOF resolution of respectively 0.5 mm and0.5 ns FWHM) and recorded.

Following the spectrometer simulation, the last stepconsists in analyzing the simulated data using the sameprocedure as for the experimental data. This method en-sures that all the relevant apparatus effects are includedin the simulations prior confrontation with the experi-mental results.

IV. MC-COBM VS EXPERIMENTAL RESULTS

Part of our experimental results on collisions betweenMCIs projectiles and rare gas dimers have already beenpresented in previous publications [12–14]. The results ofMatsumoto et al. [12] were obtained using Ar9+ projec-tiles colliding on Ar2 targets. They have shown evidencefor a large charge asymmetry in the dissociation of Ardimers following multiple-electron capture, as opposed towhat was previously seen using covalent molecules suchas N2 [10, 11]. Moreover, for the fragmentation channel(1;1)f , an unambiguous separation between two-site cap-ture, related to CE process, and one-site capture, relatedto RCT process, was observed using the KER distribu-tion. Later on, a comparison between the MC-COBMmodel and these experimental results was performed [13].A very good agreement has been obtained in terms of rel-ative production of the different dissociation mechanismsand associated fragmentation channels.

For the double electron capture channel, the relativecontributions of CE, RCT, and ICD given by the cal-culations for three different collision systems involving

9

O3+, Ar9+ and Xe20+ projectiles and Ne2 targets werealso compared to the experimental data [14]. The calcu-lations reproduced reasonably well the CE versus RCTcontributions for both Ar9+ and Xe20+ projectiles, witha similar dependence of these contributions on the pro-jectile charge state. Moreover, the appearance of the ICDprocess observed experimentally in neon dimers was alsopredicted by the model for the low charge O3+ projec-tiles. However, for this collisional system O3+ + Ne2,the estimation by the model of the relative production ofICD, CE and RCT processes did not reproduce quanti-tatively the experimental results. This indicates some ofthe limits of this model for low projectile charge stateswith a hydrogenic approximation which becomes rudi-mentary for a precise estimate of the capture probabili-ties in the way-out of the collision (Eq.4a).

Measurements of the angular correlation between thescattered projectile and the recoiling fragments combinedwith model calculations have also been investigated forthe Ar9+ + Ar2 collision system. This study providedaccess to atomic site sensitivity, showing that electroncapture from near-site atoms is strongly favored [13], asopposed to what was previously observed with N2 cova-lent molecules [11].

To provide a more complete and stringent test of theMC-COBM approach, we present in the following an ex-haustive comparison of experimental data with calcula-tions for four collision systems involving Ar9+ and Xe20+

projectiles at 15 qkeV colliding with Ar2 and Ne2 targets.

A. Capture and Fragmentation Channels

For want of anything better, we simply considered inprevious calculations [12–14] that 50 % of the one-siteelectron capture yield resulted directly in the populationof unbound states with the emission of one neutral frag-ment. There are two physical mechanisms which supportthis assumption.

The first mechanism is the intersystem crossing, whichcouples some vibrational bound states with the contin-uum states of another spin multiplicity by means of spin-orbit interaction. For the X2+

2 molecules with X=Ar,Ne,the singlet states correlating to the asymptotic chan-nels X2+(1D)-X and X2+(1S)-X are energetically abovethe triplet dissociation limit X2+(3P)-X. The intersys-tem crossing will thus takes place, in competition withthe RCT relaxation mechanism. As a radiative transi-tion, the latter is quite slow. The RCT lifetime for Ar2+2is typically 5.0 ns [36] and there is sufficient time forthe intersystem crossing to take place efficiently. Thisis confirmed indirectly by the observation of radiativeemission, for which no RCT transition is observed at theenergy corresponding to the singlet states for Ar2+2 [37].

Dissociation of Ar2+-Ar and Ne2+-Ne states occurs di-rectly if the vibrational energy exceeds the potential welldepth of the state populated by the collision. By con-sidering that vibrational energy can be provided to the

target through momentum exchange with the projectileand that transverse momentum exchange dominates overlongitudinal momentum exchange, the maximum vibra-tional energy brought to the target can be estimated. Forcollisions leading to one-site multiple capture such as the(2;0)c capture channel, the measurement of the trans-verse momentum exchange gives thus new insight on theprobability to produce unbound states with neutral emis-sion. As it is shown in the next section, for (2;0)c capturechannels, the maximum transverse momentum exchangeobserved experimentally is ∼20 a.u.. It corresponds toa maximum energy exchange of ∼75 meV for Ar2 tar-gets and of ∼150 meV for Ne2 targets. On the otherhand, the binding energy in the Franck Condon region(at internuclear distances of R=7.18 a.u. and R=5.86a.u. for Ar2 and Ne2 targets, respectively) can be ap-proximated by accounting for the polarizability term in−αq2/2R4, as shown in [37]. We find binding energies of236 meV for the Ar2+-Ar states and of 123 meV for theNe2+-Ne states. This is in both cases too high to lead toan efficient dissociation. As the binding energy evolveswith the square of the charge, a similar situation is ex-pected for the (3;0)c and (4;0)c capture channels. A moreelaborate calculation should be performed to provide abetter estimate, but this simple and classical compari-son indicates a possible small contribution for Ne2 and anegligible amount in the case of Ar2. This small amountof dissociation adds to the singlet dissociation discussedabove, yielding a total dissociation fraction that shouldnot exceed 50%. .

In the MC-COBM calculation, we thus now considertwo different scenarios for the population of dissocia-tive states for capture channels involving one-site elec-tron capture. In the first scenario, there is no directproduction of dissociative states comprising one neutralfragment, meaning that 100% of one-site multiple cap-ture events populate transient non-dissociative molecu-lar states, all relaxing eventually by the emission of twocharged fragments. In the second one, 50 % of the one-site electron capture yield results in direct fragmentation(q;0)f with one neutral center, to which the experimen-tal setup is insensitive. For all four collision systems,the relative yields of the different fragmentation chan-nels obtained experimentally are compared in Fig.4 tothe MC-COBM results for the two scenarios discussedabove.

As discussed in Sec. III.A, each fragmentation channel(q1;q2)f can be fed by different relaxation mechanisms(qB ;qC)c corresponding to a given number of electronsremoved from each site of the dimer during the collision,prior to any possible charge redistribution. The (1;1)ffragmentation channel can result from two-site DoubleCapture (DC) (1;1)c leading directly to Coulomb explo-sion (CE), but also from one-site DC (2;0)c populatingnon-dissociative molecular states relaxing through radia-tive charge transfer (RCT). These two processes can bedistinguished experimentally by their different KER dis-tribution [12, 14] and their contributions are shown sepa-

10

FIG. 4. Relative yield of the different fragmentation channels obtained experimentally (white) and using the MC-COBMcalculations. MC-COBM results are given for two different scenarios: when considering no direct production of dissociativestates with one neutral fragment (light gray) and when considering equiprobable populations of dissociative states and transientnon-dissociative states (gray). The contributions of the initial capture channels are indicated when relevant.

rately. Transient non-dissociative molecular states popu-lated by one-site triple capture (TC) and quadruple cap-ture (QC), denoted (3;0)c and (4;0)c, lead to the (2;1)fand (3;1)f fragmentation channels, respectively, throughcharge transfer occurring at direct crossings with excitedstates (CT) or with radiative emission (RCT). These pro-cesses are expected to result in KER distributions higheror close to the ones obtained for the direct two-site cap-ture channels (2;1)c and (3;1)c. As shown in[35], bothCT and RCT are here weak channels and one-site multi-ple capture could thus not be clearly isolated experimen-tally from two-site capture for TC and QC fragmentationchannels.

Within the MC-COBM model, all possible capturechannels leading to a given fragmentation channel are bynature separated (Sec.III A). For the (2;0)c, (3;0)c and(4;0)c one-site capture channels, 100 % (first scenario,light gray) or 50% (second scenario, gray) of the popu-lations given by the calculations have been attributed totransient non-dissociative molecular states. These contri-butions were added to the (1;1)f , (2;1)f and (3;1)f frag-mentation channels, fed respectively through the RCTand CT processes. For both scenarios, the MC-COBMresults are found in good agreement with the experimen-tal data of the four collision systems. The relative popu-lations of the fragmentation channels of interest are wellreproduced with maximum deviations remaining below10%. If the agreement is somewhat better when using thefirst scenario, considering that 50% of the one-site multi-ple capture results in the emission of a neutral fragment

does not affect strongly the final distributions shown inFig.4.

For QC channels, Fig.4 clearly shows the preferencefor the asymmetric fragmentation channels (3;1)f overthe symmetric one (2;2)f , specific to vdW bound atomicsystems and resulting from the low electron mobility be-tween the two atoms of the dimer. This feature is repro-duced using the two scenarios.

For the DC channels, we clearly see the evolution withthe charge states of the projectile of the relative con-tributions of the RCT and CE processes associated re-spectively to the (2;0)c and (1;1)c capture channels. Theexperimental results show that for the high charge stateof the projectile Xe20+, the CE process following a two-site double capture dominates, while for the lower chargestate projectile Ar9+, the RCT process associated to aone-site double capture takes over. This behavior canbe intuitively understood using simple geometrical con-siderations. As shown in table I, the sharing radii givenby the COBM model for double electron capture fromthe 3p shell of Ar and 2p shell of Ne using Xe20+ pro-jectile are about two times larger than the internucleardistance of the Ar2 (7.18 a.u.) and Ne2 (5.86 a.u.) targetsand should favor the removal of electrons from both sitesof the dimer. For Ar9+ projectiles, the double electronsharing radii are only slightly larger than the internucleardistance of the Ar2 and Ne2, and one-site double capturebecomes more probable than two-site double capture.

This dependence is qualitatively well reproduced bythe model, with a two-site double capture contribution

11

TABLE I. Sharing radii Rini (in a.u.) estimated by the COBM

model, using Eq.1, for double electron capture by Xe20+ andAr9+ projectiles from atomic Argon and Neon targets.

Projectile Ar target Ne targetRin

2 [3p−2] Rin2 [2p−2]

Xe20+ 14.42 9.73Ar9+ 10.32 6.96

increasing with the projectile charge when using both sce-narios. For Ar9+ projectiles, the second scenario (50% ofunbound states, dark gray) results in one-site over two-site double capture ratios in good agreement with theexperiment. But the first scenario, with no direct pop-ulation of dissociative channels (2;0)f , tends to overesti-mate one-site double capture. For Xe20+ projectiles, theagreement worsens, as with both scenarios, one-site dou-ble capture is overestimated. These observations do notallow to conclude on the validity of one of the scenariosover the other, but indicate more likely shortcomings ofthe MC-COBM model when calculating the one-site andtwo-site double capture cross sections, in particular forXe20+ projectiles.

B. Transverse Momentum Exchange

The good agreement between the experimental resultsand the MC-COBM data shown in the last section moti-vated to further test the capability of the model to repro-duce the experimental results. We focus in this sectionon the distribution of the transverse momentum exchangearising from the Coulomb repulsion between the collisionpartners.

For the experimental data, the transverse momentumexchange between the projectile and the center of mass ofthe dimer is inferred from the sum of the vector momentaof the two ionic fragments in the laboratory frame. Forthe MC-COBM ones, the momentum exchange inducedby the collision is calculated as described in Sec.II B, us-ing three different values of the charge screening param-eter S in Eq.9a and 9b. The post-collision treatmentof the data described in Sec.III B is then performed toaccount for the spatial extension of the collision regionand the response function of the apparatus. As in [13],the dimer orientation is selected with θB between 60◦

and 120◦ for both the experimental and the MC-COBMdata, so that the dimer axis is quasi-perpendicular to theprojectile beam axis.

For all the processes and fragmentation channels dis-cussed in the previous section, the transverse momentumexchange distributions obtained experimentally (thickblack line) are compared to the the MC-COBM ones(color lines) in Fig.5 for Ar9+ projectiles and in Fig.6 forXe20+ projectiles. The MC-COBM distributions shownfor the fragmentation channels (2;1)f and (3;1)f wereobtained by summing the contributions of the (3;0)c

and (2;1)c capture channels for (2;1)f , and of the (4;0)cand (3;1)c capture channels for (3;1)f , using the relativeyields given by the gray columns of Fig.4 assuming a 50%population of unbound states for one-site capture. Notethat choosing the other scenario (light gray columns ofFig.4, no unbound state population for one-site capture)lead to very similar distributions (not shown here). Thiscan be explained by the weak dependence of transversemomentum exchange on the charge repartition within theionized target, as can be seen when comparing the dif-ferential cross sections obtained for channels (1;1)c and(2;0)c in Fig.5 and Fig.6.

The first MC-COBM momentum exchange calculation,with S = 1 (blue dashed lines), assumes a full chargescreening. Within this unrealistic model, the chargeof shared electrons is allocated to the target effectivecharge. The second calculation, with S = 0 (red or darkgray), corresponds to the picture of the collision given byNiehaus, with shared electrons acting as pure spectatorswithout any screening contribution. The last one, withS = 0.5 (green or gray), is an intermediate view assumingpartial screening of the target by the shared electrons.

When assuming full screening of the target (S = 1),the momentum exchange distribution is systematicallystrongly underestimated by the model. Consideringshared electrons as pure spectators (S = 0), as for thecrossing radii and cross sections calculations, lead con-trariwise to an overestimation of the transverse momen-tum exchange. However, a target screening parameterS = 0.5 reproduces remarkably well the experimentaldistributions. The mean values and the widths of themomentum exchange distributions, as well as their in-crease with the number of captured electrons are al-most perfectly reproduced by the model for both sys-tems Ar9++Ar2 and Ar9++Ne2. An excellent agreementis also obtained with S = 0.5 for Xe20++Ar2 collisions,where we only note a small underestimation of the meanmomentum exchange for quadrupole electron capture.For Xe20++Ne2 collisions, a screening parameter value of0.5 seems too strong and lead to a small but systematicunderestimation of the momentum exchange during thecollision. Nevertheless, considering the extreme simplic-ity of the model, the overall agreement between its pre-dictions and the experimental data remains very good forall systems when considering an empirical charge screen-ing parameter S close to 0.5 for the transverse momentumtransfer calculation. It is in contradiction with the initialassumptions of the model of Niehaus used for the estima-tion of sharing and capture radii, where shared electronsare considered as pure spectators.

The effect of the target composition can also be dis-cussed by comparing the results obtained for Ne dimersand Ar dimers (Fig.6.a versus Fig.6.b and Fig.5.a versusFig.5.b). For both projectiles Xe20+ and Ar9+, the trans-verse momentum exchange produced with Ne dimers islarger than for Ar dimers. This is due to the differentouter-shell number, n = 2 for Ne atoms and n = 3 forAr atoms, of the electrons mostly involved in the charge

12

FIG. 5. Differential cross section (DCS) in transverse momentum exchange obtained experimentally (thick black line) andusing the MC-COBM for Ar9+ projectiles colliding with Ar dimers (a) and Ne dimers (b). The MC-COBM distributions,previously normalized to the experimental data, are given for three different charge screening parameters: S = 1 (blue dashedline), S = 1/2 (green line (or gray)) and S = 0 (red line (or dark gray)).

exchange process. Ne dimers lead thus to smaller sharingand capture radii than Ar dimers. For a given capturechannel, this results in smaller impact parameters andto larger Coulombic repulsion for Ne dimers, as observedexperimentally and in the calculations.

V. SUMMARY

An adaptation of the COBM model for low energy col-lisions between MCIs and rare gas dimer targets has beendeveloped. It is based on the simple representation ofthe dimer as two atoms fixed in space and on the use ofa MC simulation to integrate over the impact parame-ters and molecular orientations. An overall good agree-ment between the MC-COBM calculations and the ex-perimental data has been obtained for the four collisionsystems investigated. Despite its simplicity, the modelhas provided for each channel of interest relative crosssections whose deviations with the experimental resultsremained below 10% of the total yield. The calculation

of the transverse momentum exchange between the pro-jectile and the target has also been implemented, pro-viding a mean to study the effect of the electrons sharedduring the collision. A very good agreement has beenobtained for a charge screening parameter S = 0.5, indi-cating that shared electrons are not only spectators, asusually considered in the Niehaus model. The clear over-all success of the present model shows that rare gas vander Waals dimers can be fairly well represented as twoindependent atoms. Further investigations with differentcharge states should help us to delimit the applicabilityof the model. In a close future, the same methodologycould also be employed to investigate the collision dy-namics involving more complex targets, such as largerhomonuclear or mixed clusters.

ACKNOWLEDGMENTS

We would like to pay a posthumous tribute to Do-minique Hennecart, who played a major role in collision

13

FIG. 6. Differential cross section (DCS) in transverse momentum exchange obtained experimentally (thick black line) andusing the MC-COBM for Xe20+ projectiles colliding with Ar dimers (a) and Ne dimers (b). The MC-COBM distributions,previously normalized to the experimental data, are given for three different charge screening parameters: S = 1 (blue dashedline), S = 1/2 (green line (or gray)) and S = 0 (red line (or dark gray)).

physics in Caen and whose work triggered the develop-ment of this MC-COBM model. We thank the CIMAP

and GANIL staff for their technical support. This workwas partly supported by TMU Research Program Grant.

[1] K. Dennerl, Space Sci. Rev., 157, 57 (2010).[2] B. G. Logan, L. J. Perkins, and J. J. Barnard, Phys.

Plasmas, 15, 072701 (2008).[3] U. Amaldi and G. Kraft, Europhys. News, 36, 114 (2005).[4] A. Nikoghosyan et al., Int. J. Radiat. Oncology Biol.

Phys., 58, 89 (2004).[5] R. D. DuBois, Phys. Rev. A, 34, 2738 (1986).[6] R. Shingal, C. J. Noble, and B. H. Bransden, J. Phys. B,

20, 793 (1987).[7] S. Knoop et al., J. Phys. B, 38, 1987 (2005).[8] X. Flechard et al., Journal of Physics: Conference Series,

629, 012001 (2015).[9] B. H. Bransden and M. R. C. McDowell, Charge Ex-

change and the Theory of Ion-Atom Collisions (Claren-don Press, Oxford, 1992).

[10] I. Ben-Itzhak, S. G. Ginther, and K. D. Carnes, Phys.Rev. A, 47, 2827 (1993).

[11] M. Ehrich et al., Phys. Rev. A, 65, 030702(R) (2002).[12] J. Matsumoto et al., Phys. Rev. Lett. 105, 263202 (2010).[13] W. Iskandar et al., Phys. Rev. Lett. 113, 143201 (2014).[14] W. Iskandar et al., Phys. Rev. Lett. 114, 033201 (2015).[15] L. S. Cederbaum, J. Zobeley, and F. Tarantelli, Phys.

Rev. Lett. 79, 4778 (1997).[16] T. Jahnke et al., Phys. Rev. Lett. 93, 163401 (2004).[17] H.-K. Kim et al., Phys. Rev. A 88, 042707 (2013).[18] J. R. Oppenheimer, Phys. Rev. 31, 349 (1928).[19] H. C. Brinkman and H. A. Kramers, Proc. Acad. Sci.

Amsterdam 33, 973 (1930).[20] N. Bohr and J. Lindhard, K. Dan. Vidensk. Selsk. Mat.

Fys. Medd. 28, 1 (1954).

14

[21] H. Knudsen, H. K. Haugen, and P. Hvelplund, Phys. Rev.A 23, 597 (1981).

[22] R. E. Olson, Phys. Rev. A 27, 1871 (1983).[23] R. E. Olson, J. Ullrich and H. Schmidt-Bocking, Phys.

Rev. A 39, 5572 (1989).[24] L. F. Errea et al., Appl. Radiat. Isot. 83, 86 (2013).[25] H. Ryufuku, K. Sasaki, and T. Watanabe, Phys. Rev. A

21, 745 (1980).[26] R. Mann, F. Folkmann, and H. F. Beyer, J. Phys. B 14,

1161 (1981).[27] A. Barany et al., Nucl. Instr. Meth. B 9, 397 (1985).[28] A. Niehaus, J. Phys. B, 19, 2925 (1986).[29] A. Niehaus, Physics Reports, 186, 149 (1990).

[30] A. Niehaus, Nucl. Instrum. Meth. B, 31, 359 (1988).[31] L. Guillemot et al., J. Phys. B, 23, 4293 (1990).[32] T. Ohyama-Yamaguchi and A. Ichimura, J. Phys.: Conf.

Ser. 163, 012047 (2009).[33] T. Ohyama-Yamaguchi and A. Ichimura, Nucl. Instr. and

Meth. B, 235, 382 (2005).[34] T. Ohyama-Yamaguchi and A. Ichimura, Phys. Scr.

2013, 014043 (2013).[35] J. Matsumoto et al., Phys. Scr. T144, 014016 (2011).[36] M. Daskalopoulou and S. D. Peyerimhoff, Molecular

Physics 79, 985 (1993).[37] G. Durand and F. Spiegelmann, J. Chem. Phys. 96, 6085

(1992).