Independent-atom-model coupled-channel calculations strengthen the case for interatomic Coulomb decay as a subdominant reaction channel in slow O 3+ -Ne 2 collisions
IIndependent-atom-model coupled-channel calculations strengthenthe case for interatomic Coulomb decay as a subdominant reactionchannel in slow O -Ne collisions Dyuman Bhattacharya and Tom Kirchner ∗ Department of Physics and Astronomy,York University, Toronto, Ontario M3J 1P3, Canada (Dated: August 25, 2020)
Abstract
We report on electron removal calculations for 2.81 keV/amu Li and O ion collisions with neondimers. The target is described as two independent neon atoms fixed at the dimer’s equilibrium bondlength, whose electrons are subjected to the time-dependent bare and screened Coulomb potentialsof the classically moving Li and O projectile ions, respectively. Three mutually perpendicularorientations of the dimer with respect to the rectilinear projectile trajectories are considered andcollision events for the two ion-atom subsystems are combined in an impact parameter by impactparameter fashion and are orientation-averaged to calculate probabilities and cross sections for theion-dimer system. The coupled-channel two-center basis generator method is used to compute theion-atom collision problems. We concentrate on one-electron and two-electron removal processesresulting in the Ne + (2 s − ) + Ne , Ne + (2 p − ) + Ne + (2 p − ) , and Ne + (2 p − ) + Ne channels rightafter the collision, which can be associated with interatomic Coulomb decay, Coulomb explosion,and radiative charge transfer, respectively. We find that the calculated relative yields are in fairagreement with recent experimental data for O -Ne collisions if we represent the projectile bya screened Coulomb potential, but disagree markedly for a bare Coulomb potential, i.e., for Li impact. In particular, our calculations suggest that interatomic Coulomb decay is a significantreaction channel in the former case only, since capture of a Ne( s ) electron to form hydrogenlikeLi is unlikely. ∗ [email protected] a r X i v : . [ phy s i c s . a t m - c l u s ] A ug . INTRODUCTION Rare-gas dimers are much studied objects of the microworld with fascinating structuraland dynamical properties. Their (van der Waals) bonds are weak and their internucleardistances large so that the two atoms appear to be (quasi-) independent and the electronsoccupy (very weakly distorted) atomic states. However, it has been demonstrated that chargeand energy transfer between the two sites are possible and do happen after excitation byphoton or charged-particle impact. Perhaps the most celebrated example of such a processis interatomic Coulomb decay (ICD), which is initiated by the removal of an inner-valenceelectron from one atom by the impinging particle or radiation. ICD then involves the transferof the excitation energy to the other atom, its release in the form of (low-energy) outer-shellelectron emission, and the fragmentation of the system of two singly-charged ground-stateions produced in this way.ICD was predicted in 1997 based on ab-initio calculations [1]. The first experimental evi-dence was reported in a study of photoexcited neon clusters in 2003 [2] and was unequivocallyconfirmed for neon dimers one year later [3]. Since then, a large number of theoretical andexperimental studies have provided further data and insight (see, e.g., Ref. [4] and referencestherein). ICD is now considered to be a ubiquitous process in a variety of systems, and theassociated low-energy electron emission is deemed to play an important role in the radiationdamage of biological tissue (see, e.g., Ref. [5] and references therein).ICD in neon dimers subjected to slow multiply-charged ion impact was reported in Ref. [6].More specifically, kinetic energy release (KER) spectra for the Ne + (2 p − ) + Ne + (2 p − ) fragmentation channel were recorded in coincidence with the final projectile charge state, andpeaks in those spectra were associated with three different processes based on an analysisinvolving some of the potential energy curves of the dimer system. ICD resulting from theprimary removal of one Ne( s ) electron was one of these processes. The other two wereradiative charge transfer (RCT) and Coulomb explosion (CE). The latter corresponds to thedirect production of Ne + (2 p − ) + Ne + (2 p − ) in the collision by electron capture of one p electron from each atom, while the former is the result of a two-electron capture processfrom one atom, producing a transient state which relaxes radiatively to the same Ne + (2 p − ) + Ne + (2 p − ) channel as CE and ICD, but involves higher KER values. Relative yields forthese processes were determined for three different projectile species: O , Ar , and Xe impact, contributing 20% to the total yield.These findings were supported by classical over-the-barrier model (COBM) calculationspublished along with the data. The calculations were based on an independent-atom-model(IAM) description of the ion-dimer collision problem using bare Coulomb potentials for theprojectiles. For the O -Ne system they resulted in at most qualitative agreement with themeasurements; in particular the ICD channel appeared to be too weak (contributing just8.2% to the total yield), while the CE yield was found to be significantly stronger than in theexperiment. Given that the O -Ne system was the only one that showed evidence for ICD,an independent calculation based on a higher-level theory is desirable. This is the motivationfor the present work.Our calculations are also based on the IAM, but the ion-atom collisions are computedin a quantum-mechanical coupled-channel framework using the two-center basis generatormethod (TC-BGM) for orbital propagation [7]. We combine electron removal probabilities inan impact parameter by impact parameter fashion for three perpendicular orientations ofthe dimer with respect to the rectilinear projectile trajectories and then orientation-averagethe results to calculate absolute yields, i.e., cross sections, for the processes of interest. As itturns out, it is crucial to describe the O ion by a screened Coulomb potential and takeinto account that its s subshell is occupied.Our model is explained in Sec. II. In Sec. III we present and discuss our results incomparison with the experimental data and the previous COBM results. The paper ends witha few concluding remarks in Sec. IV. Atomic units, characterized by (cid:126) = m e = e = 4 π(cid:15) = 1 ,are used unless otherwise stated. II. MODEL
The basic assumptions of our theoretical model are that (i) the projectile ion travels ona straight-line classical trajectory with constant speed v , and (ii) the target system canbe described as two independent atoms, fixed in space during the collision at a distancethat corresponds to the equilibrium bond length R e of the neon dimer. We use the value R e = 5 . a.u. [6, 8]. Following the work of, e.g., Lühr and Saenz for collisions involving H +2 [9] and H [10] we consider three perpendicular orientations of the target with respect to the3rojectile path: In orientation I, the dimer is aligned parallel to the projectile beam axis.In orientation II it is perpendicular to the projectile beam in the scattering plane, while inorientation III it is perpendicular to the scattering plane (see Fig. 1 of Ref. [9] for a sketch ofthe geometry). We calculate electronic transition probabilities for the processes of interest asfunctions of the (scalar) impact parameter b , measured with respect to the center-of-mass ofthe dimer, for these three orientations and construct an orientation-average for each process j according to P ave j ( b ) = 13 (cid:0) P I j ( b ) + P II j ( b ) + P III j ( b ) (cid:1) . (1)This orientation-averaged probability is then integrated over the impact parameter to calculatethe cross section σ ave j = (cid:90) P ave j ( b ) d b = 2 π (cid:90) ∞ bP ave j ( b ) db. (2)In the following subsection we describe how the ion-atom problem is computed. Thesubsequent Sec. II B deals with the combination of the ion-atom results to obtain probabilitiesand cross sections for the ion-dimer system. A. Ion-atom collision calculations
The ion-atom collision calculations are carried out at the level of the independent electronmodel (IEM) using the well-tested TC-BGM [7, 11]. The single-particle Hamiltonian isassumed to be of the form ˆ h ( t ) = − ∇ + v T ( r ) + v P ( r , t ) (3)with a spherically-symmetric effective target potential v T , which includes the nuclear Coulombpotential (with charge number Z T = 10 for Ne) and ground-state Hartree screening and ex-change potentials obtained from the optimized potential method (OPM) of density functionaltheory [12]. The projectile potential v P is a bare Coulomb potential with charge number Z P = 3 for Li projectiles and a screened Coulomb potential of Green-Sellin-Zachor [13]form for O : v P ( r , t ) = v P ( r P ) = − r P (cid:20)
51 + H ( e r P /d −
1) + 3 (cid:21) . (4)In Eq. (4) r P = | r − R ( t ) | is the distance between the active electron and the projectilenucleus, whose position vector follows the straight-line path R ( t ) = (˜ b, , vt ) where ˜ b is the4mpact parameter with respect to the target atom [to be distinguished from the impactparameter b in Eqs. (1) and (2)]. The parameters d = 0 . and H = 3 . d , taken fromTable I of Ref. [14], were determined by a modified Hartree-Fock procedure described in thatpaper. The potential (4) interpolates between − /r P for long and − /r P for short distances,as it should from the perspective of an (active) electron placed on the target atom initiallyand ionized or captured by the projectile during the course of the collision.The eight Ne L -shell electrons are propagated subject to the Hamiltonian (3) using abasis representation obtained from the TC-BGM, while the K -shell electrons are assumed tobe passive. The K -shell electrons of the O projectile ion are assumed to be passive as well,whereas the projectile L -shell electrons have to be treated with more care, as is explainedfurther below.The basis used includes the s to f target orbitals and all projectile orbitals from s ( s ) to i for Li (O ). We use atomic orbitals with real instead of the standard complexspherical harmonics as their angular parts. This has the advantage that all basis states haveeven (‘gerade’) or odd (‘ungerade’) symmetry with respect to reflections about the scatteringplane and do not mix during propagation. We denote these symmetry-adapted orbitals bythe quantum numbers nlm g and nlm u in the following. The target and projectile two-centerbasis is augmented by sets of 35 BGM pseudo states of gerade symmetry and 21 states ofungerade symmetry constructed in the usual way by operating with powers of a regularizedprojectile potential operator on the target eigenstates [7]. Calculations have been carried outfrom an initial to a final projectile–target distance of 50 a.u. on fine impact-parameter gridsto resolve the rich structure at the impact energy of E = 2 . keV/amu (corresponding to v = 0 . a.u.) which was used in the experiment [6].Figure 1 shows the single-electron removal probabilities, obtained from subtracting theasymptotic target orbital populations from unity, for the Li projectile. The probabilitiesare almost indistinguishable from the single-electron capture probabilities, i.e., direct transferto the continuum is negligible (less than 0.5%). Clearly, electron removal is stronger for theinitial Ne( p ) and Ne( p g ) electrons than for the s electrons which are more strongly bound( ε OPMNe(2 s ) = − . a.u. versus ε OPMNe(2 p ) = − . a.u.) and cannot be captured very efficientlyinto hydrogenlike Li . Qualitatively, this can be understood by comparing the bindingenergies of the relevant target and projectile orbitals and keeping in mind that capture tolower-lying states is more likely because of the Stark shifts of the target states in the projectile5 b [a.u.] P ( b )
2s removal2p removal2p g removal2p u removal FIG. 1. Single-particle probabilities for electron removal from the Ne L shell by 2.81 keV/amu Li impact plotted as functions of the impact parameter. potential. This simple argument suggests that capture of Ne( p ) electrons to projectile statesof principal quantum number n = 2 ( ε Li n =2 = − . a.u.) is the strongest channel andindeed this is what the numerical calculations show. The removal of the Ne( p u ) electronsis relatively weak, since fewer final states are available in the ungerade symmetry case. Amore detailed analysis would require to compute correlation diagrams and quasimolecularcouplings.For O impact the situation is complicated by the fact that Pauli blocking may preventsome electron capture transitions. As mentioned above, we consider both the Ne and theO K -shell electrons as passive and do not include those states in the TC-BGM basis. Thisis justified by the large binding energies of those states and their weak couplings to otherbasis states. Such an approach does not work for the occupied L shells as some state-to-state couplings are strong and simply eliminating occupied states from the coupled-channelcalculations contaminates some of the open channels. To illustrate these points, we notethat in a TC-BGM calculation with the full basis the single-particle transfer probabilityfrom Ne( s ) to O (2 s ) becomes very close to unity at some impact parameters, while testcalculations in which the (occupied) O (2 s ) state was removed from the basis resulted insizable transfer to the continuum—a process that should be ineffective at low collision energy.In order to deal with this situation we subtracted the single-particle probabilities for thetransitions Ne (2 l ) → O (2 s ) from the Ne (2 l ) electron removal probabilities and interpretedthe results as the ‘true’ removal probabilities. This seemingly naive procedure can be justified6 b [a.u.] P ( b )
2s removal2p removal2p g removal2p u removal FIG. 2. Single-particle probabilities for electron removal from the Ne L shell by 2.81 keV/amuO impact plotted as functions of the impact parameter. The probabilities are corrected for thepresence of the projectile s electrons as described in the text. based on the principle of detailed balance [which asserts that the probability for a transitionfrom, say, Ne (2 s ) to O (2 s ) equals the probability for a transition from O (2 s ) to Ne (2 s ) ]and the inclusive probability formalism of Ref. [15]. The argument is presented in theAppendix.We note that we ignored Pauli blocking due to the presence of one p electron in O based on the rationale that this should be a weak effect given that five out of six states inthe p subshell are vacant.The resulting single-particle electron removal probabilities for O -Ne collisions arepresented in Fig. 2. Similarly to those of the Li -Ne system (cf. Fig. 1) they show richstructure as a function of impact parameter, but the details are quite different. Notably,all probabilities reach higher values, not far from unity for the Ne (2 p ) initial states andup to 0.6 for Ne (2 s ) , the latter to be contrasted with a maximum removal probability p rem2 s of approximately 0.3 for Li impact. Also, p rem2 s extends to significantly larger impactparameters for O than for Li projectiles, while the trend is opposite for p removal. Themain reason for the increased probabilities in the < b (cid:47) . a.u. range is the strongerbinding energy of the (vacant) O (2 p ) orbitals at -1.868 a.u. as compared to -1.125 a.u. forhydrogenlike Li (2 p ) , which makes capture (from all states) more effective. The increasedNe (2 s ) -vacancy production probability will become important for the role of ICD to bediscussed in the next section. 7 . Analysis of electronic processes resulting in ICD, CE, and RCT We now look at the neon dimer in each of the three orientations described above andcombine ion-atom probabilities in an impact parameter by impact parameter fashion tocalculate the probabilities on the right hand side of Eq. (1) for the three processes of interest.For orientation I in which the dimer is parallel to the ion beam axis the situation is simple,since the impact parameters with respect to both atoms are the same and coincide with theimpact parameter with respect to the center of mass of the dimer, i.e., ˜ b ≡ b I = b .For each value of b considered, we proceed by determining the corresponding atomicimpact parameters for orientations II and III and then carry out TC-BGM calculations atthose impact parameters to avoid interpolations when combining and orientation-averagingprobabilities for the ion-dimer system. For orientation III in which the dimer is perpendicularto the scattering plane both atomic impact parameters are the same and are given by b III = (cid:112) b + ( R e / . For orientation II the two atomic impact parameters are different. Theone with respect to the closer atom is b (1)II = | ( R e / − b | and the other one is b (2)II = b + ( R e / .As mentioned in the Introduction ICD, CE, and RCT can be associated with specificone- and two-electron removal processes [6, 16]. We calculate these processes by consideringmultinomial combinations of single-particle probabilities, i.e., by using the IEM for thecombined ion–two-atom system.Let us exemplify this procedure for the simplest case of orientation I in which both atomicimpact parameters are the same. The probability for finding one vacancy in one of the Ne( s )orbitals is given by P I2 s − ( b ) = 4 p rem2 s ( b I )(1 − p rem2 s ( b I )) (1 − p rem2 p ( b I )) (1 − p rem2 p g ( b I )) (1 − p rem2 p u ( b I )) , (5)where b = b I . This expression accounts for the requirement that all p electrons and three outof four s electrons of the two atoms remain bound. The multiplication factor of four arisesbecause each of the four initial s electrons can be the one that is removed. The s -vacancyprocess (5) can be associated with ICD.Similarly, the probability for the removal of one p electron from each atom is given by P I2 p − , p − = (1 − p rem2 s ) [2 p rem2 p (1 − p rem2 p )(1 − p rem2 p g ) (1 − p rem2 p u ) + 2 p rem2 p g (1 − p rem2 p g ) × (1 − p rem2 p ) (1 − p rem2 p u ) + 2 p rem2 p u (1 − p rem2 p u )(1 − p rem2 p ) (1 − p rem2 p g ) ] , (6)8here we have omitted the impact parameter dependence for ease of notation. The firstfactor involving p rem2 s ensures that no inner-valence vacancy is created. The three terms insquare brackets account for the removal of one electron from either the p , the p g , or the p u orbital and the whole expression is squared to ensure that one-electron removal happenson both atoms simultaneously (and independently). The probability (6) can be associatedwith CE.It was argued in Refs. [6, 17] that double p removal from one atom may result in thethird observed process, RCT, but not necessarily so, since the system can also dissociate asis, giving rise to one doubly-charged and one neutral fragment. The experiment was blind tothe latter channel and in the COBM calculations reported along with the measurements itwas assumed that 50% of double removal from one atom will lead to RCT while the other50% result in Ne + Ne production [6].Within the IEM, removing two p electrons from one atom while the other atom remainsintact is represented by P I2 p − = 2(1 − p rem2 s ) [( p rem2 p ) (1 − p rem2 p g ) (1 − p rem2 p u ) + ( p rem2 p g ) (1 − p rem2 p ) (1 − p rem2 p u ) + ( p rem2 p u ) (1 − p rem2 p ) (1 − p rem2 p g ) + 2 p rem2 p (1 − p rem2 p )2 p rem2 p g (1 − p rem2 p g )(1 − p rem2 p u ) + 2 p rem2 p (1 − p rem2 p )2 p rem2 p u (1 − p rem2 p u )(1 − p rem2 p g ) + 2 p rem2 p g (1 − p rem2 p g )2 p rem2 p u (1 − p rem2 p u )(1 − p rem2 p ) ] × [(1 − p rem2 p ) (1 − p rem2 p g ) (1 − p rem2 p u ) ] . (7)While this expression is lengthy, the interpretation of each term is straightforward. Thefirst square bracket accounts for the removal of two electrons from one of the atoms fromeither the same p orbital or from two different orbitals, the latter terms being multiplied bytwo factors of two to account for the fact that both electrons in a given orbital are equallylikely to be removed or not. The expression in the second square bracket takes care of therequirement that no p electron be removed from the second atom and the overall prefactorof two is there since it can be one or the other atom that gets ionized. If one rearranges theterms in Eq. (7) and compares the whole expression with Eq. (6) one obtains P I2 p − , p − − P I2 p − = 2(1 − p rem2 s ) (1 − p rem2 p ) (1 − p rem2 p g ) (1 − p rem2 p u ) [( p rem2 p ) (1 − p rem2 p g ) (1 − p rem2 p u ) + ( p rem2 p g ) (1 − p rem2 p ) (1 − p rem2 p u ) + ( p rem2 p u ) (1 − p rem2 p ) (1 − p rem2 p u ) ] ≥ , (8)i.e., the prediction that CE is stronger than RCT, even if one makes the extreme assumptionthat double removal from one atom will always result in RCT.9 b [a.u.] P ( b ) (a) -1 -2 -1 , 2p -1 b [a.u.] P ( b ) (b) -1 -2 -1 , 2p -1 FIG. 3. Probabilities for s − , p − , and ( p − , p − ) production in (a) Li and (b) O collisionswith Ne in orientation I at E = 2 . keV/amu. This can be seen in Fig. 3 in which the probabilities (5) to (7) are plotted as functions ofthe impact parameter b for both Li [panel (a)] and O [panel (b)] collisions, using thesame scales on the x and y axes to ease the comparison.As can be expected from the ion-atom single-particle probabilities shown in Figs. 1 and 2the results for the two projectiles are quite different. The p removal processes (6) and(7) are significantly stronger for Li than for O projectiles and extend to larger impactparameters. The first part of this observation may seem surprising given that the probabilitiesdisplayed in Fig. 1 (for Li ) tend to be smaller than those of Fig. 2 (for O ). However, onehas to keep in mind that both Eqs. (6) and (7) include factors of the type (1 − p rem2 p ) whichcorrespond to the fact that ten out of twelve p electrons are not removed. These factors actas effective suppression factors when the single-particle probabilities approach unity.For the s -vacancy production (5) the situation is reversed and the O projectile isoverall more effective than Li . Again, it is a consequence of the (1 − p rem2 p ) factors that theshallow maximum of the O impact s single-particle probability around ˜ b ≈ . a.u. (cf.Fig. 2) results in the main peak of P I2 s − , while the process is mostly suppressed at smallerimpact parameters.For orientation III one can summarize the situation as follows: The expressions (5)–(7)remain unchanged except that the impact parameters on the left and right hand sides arenow different, i.e., b (cid:54) = b III = (cid:112) b + ( R e / . One then sees (in Fig. 4) the probabilitydistributions which occur at impact parameters b ≥ R e / in orientation I at smaller impactparameters and stretched out over a wider interval. The structures occuring at b < R e / inorientation I are eliminated from the plot for orientation III, since the projectile never gets10 b [a.u.] P ( b ) (a) -1 -2 -1 , 2p -1 b [a.u.] P ( b ) (b) -1 -2 -1 , 2p -1 FIG. 4. Probabilities for s − , p − , and ( p − , p − ) production in (a) Li and (b) O collisionswith Ne in orientation III at E = 2 . keV/amu. close enough to the two atoms. This is why the s -vacancy production process is absent forLi projectiles [Fig. 4(a)].Orientation II in which the two atomic impact parameters are not the same, produceslengthier (but still straightforward) mathematical expressions and more complicated patternsfor the three processes. This orientation does allow for P p − > P p − , p − and in a quitepronounced way, in particular for O projectiles as shown in Fig. 5(b). For this projectilethe P p − , p − probability is essentially zero except at b < a.u., which can be understood byonce again inspecting Fig. 2 and noticing that all p electron removal probabilities are smallat atomic impact parameters larger than R e / . a.u. and are approaching zero rapidlytoward more distant collisions. Given that both atoms need to be ionized for this processto occur and the farther atom in this orientation is at least a distance of R e / away fromthe projectile, P p − , p − is very small. By contrast, P p − reaches substantial values, sinceboth electrons can be efficiently removed from the closer atom. In this case, the obtaineddistribution is basically symmetric with respect to b = R e / which corresponds to a head-onion-atom collision. The same is true for the s -vacancy production process, except that theshallow peak around b ≈ . a.u. is too far out to have a mirror image at small impactparameters.For Li projectiles [Fig. 5(a)] the situation is different since the p single-particle removalprobabilities extend beyond b ≈ R e / (cf. Fig. 1) and more substantial overlaps betweencontributions from the close and the far atom occur. The s -vacancy production channelcontributes in the interval (cid:47) b (cid:47) a.u., as can be expected from Fig. 1 and Fig. 3(a):Only the closer atom can provide a nonzero p rem2 s factor and it does so only when the atomic11 b [a.u.] P ( b ) (a) -1 -2 -1 , 2p -1 b [a.u.] P ( b ) (b) -1 -2 -1 , 2p -1 FIG. 5. Probabilities for s − , p − , and ( p − , p − ) production in (a) Li and (b) O collisionswith Ne in orientation II at E = 2 . keV/amu. b [a.u.] b P ( b ) [ a . u .] (a) -1 -2 -1 , 2p -1 b [a.u.] b P ( b ) [ a . u .] (b) -1 -2 -1 , 2p -1 FIG. 6. Orientation-averaged impact-parameter-weighted probabilities for s − , p − , and( p − , p − ) production in (a) Li and (b) O collisions with Ne at E = 2 . keV/amu. impact parameter is one atomic unit or smaller. The distribution is not symmetric about b = R e / because of the contributions from the (1 − p rem2 p ) factors from both atoms.Figure 6 displays the orientation-averaged probabilities weighted by the impact parameter,i.e., the integrands of the cross section formula (2) for the three processes of interest. Themost obvious differences between the plots for both projectiles are that (i) both p removalprocesses are stronger for Li [panel (a)] than for O [panel (b)] and, (ii) on a relativescale, the s -vacancy production process and the process in which one p electron is removedfrom each atom switch roles: The former is by far the weakest channel for Li , while thelatter shows less area under the curve for O , i.e., a smaller total cross section.12 ABLE I. Relative yields (in percent) for the three processes of interest. The TC-BGM resultsmarked with a star are obtained from the assumption that 100% of P p − contributes to RCT, whilein the other columns it is assumed that only 50% contributes to this channel. The COBM andexperimental data are from Ref. [6].COBM (Li ) TC-BGM (Li ) TC-BGM (O ) TC-BGM (O ) ∗ Expt. s − (ICD) 8 4 35 23 20 p − , p − (CE) 40 50 12 8 10 p − (RCT) 52 46 53 69 70 III. COMPARISON WITH EXPERIMENTAL AND COBM DATA AND DISCUS-SION
We now discuss the relative yields obtained from comparing the total cross sections forthe three processes. In order to compare the present results with the experimental data forICD, CE, and RCT and the COBM calculations of Ref. [6] we apply the same correctionas used in that work, namely we assume that only 50% of P p − contributes to RCT, while100% of P p − , p − feeds into CE and 100% of P s − into the ICD channel. The resultingrelative yields (in percent) are shown in Table I. For our O calculations we also showresults obtained from the assumption that 100% of P p − results in RCT.Clearly, the calculations for O projectiles are in better agreement with the measurementsthan those for Li impact. In particular, they give the experimentally observed orderingCE < ICD < RCT, while both the Li TC-BGM and the COBM calculation of Ref. [6]appear to overemphasize the CE channel and underestimate ICD. These two calculationsmake different predictions about which one is the strongest channel, but are nevertheless infair agreement with each other.The fact that the present results for O are in almost perfect agreement with theexperimental yields when the ‘100% assumption’ is applied to the p − channel should perhapsnot be overinterpreted given that our model has several limitations: First, reinspecting Figs. 3to 5 one observes that the orientation dependence is quite strong. This raises the questionwhether an orientation-average involving more than three orientations might yield a differentresult. While ultimately this can only be answered by additional calculations we note that in13heir work for collisions involving H +2 [9] and H [10] Lühr and Saenz also found considerableorientation dependence, but concluded that averages based on the three perpendicularorientations only were rather accurate.Second, the IAM for the dimer coupled with the IEM for the electrons of both atomsprovides of course only an approximate framework for the discussion of the collision problemat hand. In recent work for a large variety of multicenter systems, ranging from small covalentmolecules to large clusters and biomolecules, an amended IAM was explored, in which thegeometric overlap of effective atomic cross sectional areas was taken into account [11, 18, 19].However, that model has so far only been applied to net cross sections and not to the moredetailed one- and two-electron removal processes studied here. While such an extension isoutstanding, one can perhaps argue that overall geometric overlap should be small for asystem such as Ne whose internuclear distance is large, but that it would affect the threeorientations considered differently and would amount to their re-weighting in the calculationof the orientation-average. To estimate the potential effect we applied the extreme assumptionthat the orientation I probabilities are to be divided by a factor of two to account for thecomplete overlap of the atomic cross sectional areas when viewed from the position of theimpinging projectile, while the results for the other two orientations remain unchanged. Wefound that the relative yields do change, but not very dramatically. In particular, the ICDyields decrease from 35% and 23% for the two models shown in Table I to 34% and 22%.It is more difficult to estimate the error associated with using the IEM, or, in other words,the effects of electron correlations. While it is known that they do play a role in collisionalmultielectron dynamics [20–22], it is not clear how they affect the relative yields of interesthere. First-principles many-electron calculations would be required to shed light on this issue.In their absence, we can only say that the fair agreement between our O results and theexperimental data does not suggest that electron correlations are of major importance. IV. CONCLUDING REMARKS
Motivated by a recent joint experimental/theoretical work [6], we have studied specificone- and two-electron removal processes in Li and O collisions with neon dimers at E = 2 . keV/amu, representing the target system as two independent atoms and computingthe ion-atom electron dynamics at the level of the independent electron model. The coupled-14hannel basis generator method has been used for orbital propagation, taking into accountin the O case that the projectile potential is of screened Coulomb character and that the s subshell is occupied.We find that the results for both projectiles are markedly different and only the O calculations yield fair agreement with experimental data for ICD, CE, and RCT. In particular,our calculations suggest that ICD is so weak a process for bare projectiles that it might behard to measure it. This is a new piece of information given that the classical calculation ofRef. [6] predicted a somewhat higher ICD yield for Li impact.Together with the conclusion of that paper that ICD can only be observed in lowly-charged ion collisions (because s removal is overwhelmed by additional p removal for morehighly-charged projectiles) one may say that a fine balance of charge state and structure of aprojectile is required to make ICD a significant process in low-energy capture collisions. Itwould be interesting to see if one could identify an optimal projectile that maximizes the ICDyield. Future work should also be concerned with a more systematic study of the relativestrengths of ICD, CE, and RCT over a range of projectile species and energies and also fordifferent target systems, such as water clusters. A more quantitative understanding of theICD process in particular may have important implications for ion-beam cancer therapy,since the low-energy electrons it produces are effective agents for inflicting cellular damage. ACKNOWLEDGMENTS
Financial support from the Natural Sciences and Engineering Research Council of Canada(NSERC) (RGPIN-2019-06305) is gratefully acknowledged.
Appendix
Let us consider a simplified problem with just two spin-parallel electrons occupying thetarget and projectile s states | s T (cid:105) and | s P (cid:105) before the collision. We denote the solutionsof the single-particle equations for both electrons at a final time t f after the collision by | ψ s T (cid:105) and | ψ s P (cid:105) .According to Ref. [15], the inclusive probability for finding one electron in | s T (cid:105) afterthe collision while the other one is not observed is given by the one-particle density matrix15lement (cid:104) s T | ˆ γ | s T (cid:105) = |(cid:104) s T | ψ s T (cid:105)| + |(cid:104) s T | ψ s P (cid:105)| . (A.1)Not observing one electron implies that it can be anywhere but in the s target state, whichis blocked by the other electron. Hence, we can interpret P T vac ≡ − (cid:104) s T | ˆ γ | s T (cid:105) (A.2)as the probability for finding the s target state vacant after the collision. The principleof detailed balance demands that | (cid:104) s T | ψ s P (cid:105)| = |(cid:104) s P | ψ s T (cid:105)| , provided both electrons arepropagated in the same single-particle Hamiltonian. We have checked that our TC-BGMsolutions are consistent with this result. Accordingly, we can write P T vac = 1 − |(cid:104) s T | ψ s T (cid:105)| − |(cid:104) s P | ψ s T (cid:105)| , (A.3)i.e., the s target vacancy probability is obtained by subtracting the s T → s P transitionprobability from the probability that the initial target electron is not found in its initial s state. Given that target excitation is a weak process in the collision system studied in thiswork, we can interpret the latter as the target electron removal probability.The same argument applies to the initial p target electrons and can readily be extendedto several target electrons and both spin directions (given that spin flips are impossiblefor a spin-independent Hamiltonian). This justifies our procedure to determine the ‘true’single-particle removal probabilities by subtracting the probabilities for Ne (2 l ) → O (2 s ) from the original Ne (2 l ) electron removal probabilities. [1] L. S. Cederbaum, J. Zobeley, and F. Tarantelli, Phys. Rev. Lett. , 4778 (1997).[2] S. Marburger, O. Kugeler, U. Hergenhahn, and T. Möller, Phys. Rev. Lett. , 203401 (2003).[3] T. Jahnke, A. Czasch, M. S. Schöffler, S. Schössler, A. Knapp, M. Käsz, J. Titze, C. Wimmer,K. Kreidi, R. E. Grisenti, A. Staudte, O. Jagutzki, U. Hergenhahn, H. Schmidt-Böcking, andR. Dörner, Phys. Rev. Lett. , 163401 (2004).[4] T. Jahnke, J. Phys. B: At. Mol. Opt. Phys. , 082001 (2015).[5] X. Ren, E. Wang, A. D. Skitnevskaya, A. B. Trofimov, K. Gokhberg, and A. Dorn, Nat. Phys. , 1062 (2018).
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