Chemical Physics 411 (2013) 26–34

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Chemical Physics

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Dynamics of two-stage direct three-body recombination of ions

0301-0104/$ - see front matter � 2012 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.chemphys.2012.11.016

⇑ Corresponding author. Tel.: +7 499 1374104; fax: +7 499 1378258.E-mail address: rusin@chph.ras.ru (L.Yu. Rusin).

1 The question of the prevailing mechanism of three-body interactions has a ratherlong history. While considering the mechanisms of termolecular reactions involvingintermediate compounds, Kassel [1] pointed out correctly that the formation ofintermediate complexes AB or AR in the processes (2) and (3) could require a three-body collision (like the process (1) requires). Moreover, Kassel’s discussion ofTolman’s calculations [2] showed that the model of direct termolecular collisions (i.e.,the model (1)) could in principle explain the observed rates of third order reactions.

Vladimir M. Azriel, Lev Yu. Rusin ⇑, Mikhail B. SevryukInstitute of Energy Problems of Chemical Physics, The Russia Academy of Sciences, Leninskii Prospect 38, Bldg. 2, Moscow 119334, Russia

a r t i c l e i n f o a b s t r a c t

Article history:Received 24 September 2012In final form 25 November 2012Available online 8 December 2012

Keywords:Direct three-body recombinationTwo-stage elementary processesIon approach stageEnergy removal stageTime lag between reaction stagesKinematic parameters

Single-stage three-body recombination of ions and recombination via intermediate complex formationcan be considered as two limiting cases of a general mechanism involving two stages separated by sometime interval T, namely, energy removal by the third body and closest approach of the ions. Any of thesecan occur first. Particular dynamics of nascent molecule stabilization is determined by various kinematicparameters including two collision energies and the lag T. In this paper, recombination dynamics of theCs+ and Br� ions in the presence of the Xe atom is studied by quasiclassical trajectories. If T vanishes (one-stage process), stabilization proceeds chiefly via energy transfer in encounters of Xe with both ionsapproaching each other. If the energy removal stage is late, dynamics changes drastically, stabilizationoccurring via interaction of Xe mainly with only one ion. This three-body recombination mechanismmay be of considerable importance at low pressures in gas medium.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

The reactions of three-body recombination of atoms, ions, orsmall radicals determine, to a large extent, kinetics of various gasphase processes such as combustion, oxidation, or explosions.These reactions are also responsible for the properties of naturaland artificial non-equilibrium media such as, for instance, atomic,ionic, and molecular interactions in the upper layers of the atmo-sphere or in low temperature plasmas where the recombinationprocesses control the concentrations of active particles. The studiesof these reactions in chemical kinetics have been confined by aphenomenological description of a three-body interaction as of adirect single-stage reaction

Aþ Bþ R! ABþ R ð1Þ

or of reactions involving intermediate complexes1:

Aþ B! AB–; AB– þ R! ABþ R ð2Þ

or

Aþ R! AR; AR þ B! ABþ R: ð3Þ

As is noted in Ref. [3], the literature on chemical kinetics givespreference to the recombination mechanisms that include variousintermediate complexes. One of the reasons for this preference is acomparison of the frequencies of double and triple collisions in agas within the framework of the hard sphere model. Such a com-parison implies a considerable predominance of double collisionsover triple ones. In fact, in the kinetics literature, there are de-scribed several mechanisms of three-body processes. The analysisof these mechanisms in the paper [3] shows that in most cases,at least for atoms and small radicals, recombination proceedingvia a collision of all the three particles makes an appreciable orsometimes even prevailing contribution to the total recombinationrate. At the same time, the dynamics of triple interactions has beenexplored quite poorly compared with that of pair collisions.Dynamics of the chemical processes accompanying three-body col-lisions is of great interest. In particular, dynamics of the ion recom-bination reactions and especially of reactions of direct three-bodyrecombination of ions is intriguing.

It is hard to find out the actual mechanism of an elementaryrecombination process while being based just on a formal exami-nation of the termolecular interaction. This mechanism is deter-mined by the specific interaction potentials of the reagents, i.e.,by the structure of the potential energy surface (PES). It is in gen-eral more difficult to construct the PESs for termolecular reactionsthan those for bimolecular ones. Moreover, three-body interactionsare characterized by a considerably more complicated collection ofthe kinematic parameters than bimolecular processes (see Sec-tion 2 below), and each kinematic parameter has its own distribu-tion function. The absence of suitable experimental dynamical datais also an important reason for an insufficiently complete under-standing of the mechanisms of three-body reactions.

V.M. Azriel et al. / Chemical Physics 411 (2013) 26–34 27

On the other hand, the main dynamical features of three-bodyrecombination can be investigated, to a large extent, on the basisof the principle of microscopic reversibility. This principle enablesone to connect two directions of an elementary reaction, for in-stance, collision induced dissociation (CID) of a molecule andrecombination of the dissociation products in the presence of thethird body. All the three phenomenological recombination mecha-nisms Eq. (1)–(3), at least in the case of recombination of atoms orsingle-atom ions, can be governed by the same PES and proceedindependently of each other but with different collections of thekinematic parameters.

Recombination of atomic ions,

Aþ þ B� þ R ! ABþ R; ð4Þ

is a simple reaction that can be explored by this approach. This isthe process reverse to CID of some molecules with an ionic bond[4–6]. However, due to various circumstances, the reaction Eq. (4)has been examined insufficiently.

One of the CID reactions whose dynamics has been studiedexperimentally in crossed molecular beams best of all is collisioninduced dissociation of some alkali halide molecules (see e.g. thepapers [4–12]) proceeding as a diabatic process with the formationof ionic products:

MXþ R !Mþ þ X� þ R; ð5Þ

! RMþ þ X�; ð6Þ

2 In the framework of statistical dynamics, one considers the dependences of theprocess probability on some chosen kinematic parameters under averaging over theremaining parameters. On the other hand, if averaging over kinematic parameters isabsent or minimal in the calculation procedure at hand, one gets the opportunity toexplore detailed dynamics of the process [20,21,23].

where e.g. MX = CsF, CsCl, CsBr, CsJ or other alkali halides that breakup mainly into ions. The cross section of the channel Eq. (6) is muchsmaller than that of the channel Eq. (5). Besides CID reactions inthree-atom systems [4–12], there are also known four-atom sys-tems exhibiting dynamics typical for the diabatic CID processes[13–15].

In Refs. [16–21], we constructed quasiclassical trajectory mod-els for direct three-body reactions corresponding to so-called cen-tral [16–18,20,21] and non-central [19] encounters of therecombining ions (see Section 2 below). These models were basedon the well determined interaction potentials in the prototypicalsystems involving the Cs+ and Br� ions and the rare gas atomR = Kr, Xe or the mercury atom R = Hg. We have revealed [16–21]that three-body recombination exhibits a great manifold ofdynamical mechanisms whose occurrence depends on the particu-lar collection of the kinematic parameters of the three-body colli-sion. For instance, the ion recombination probability was shown todepend strongly on both the collision energy of the recombiningparticles and the energy of the third body (see Section 2 below).Statistically, on the qualitative level, these dependences manifestthemselves the same way for central and non-central encountersof the recombining pair. However, from the quantitative viewpoint,the recombination probability in the non-central encounters isabout 5–6 times smaller on the average [19]. Both the energies af-fect considerably the dynamical mechanism of recombination. Thiscan be seen in the opacity functions describing the dependence ofthe recombination probability on the impact parameter bR of thethird body with respect to the center-of-mass of the recombiningpair. For central encounters of the ions at small values of the im-pact parameter (bR < 6 au), stabilization of the product moleculeis achieved by energy transfer to the third body from both therecombination partners [18]. At large impact parameters (bR > 6au), the nascent molecule stabilizes mainly in the interaction ofthe third body with one of the recombining ions [18].

These results have been obtained for the recombination reac-tion model that assumes the simultaneous presence of all the threeparticles in a small neighborhood of the center-of-mass of the sys-tem where the interaction of the particles is sufficiently strong.

This concept corresponds to single-stage direct three-body recombi-nation, and the hypothesis of a simultaneous collision of the threeparticles is in fact the only restriction imposed in the papers [16–21] on the analysis of a three-body collision yielding recombina-tion. However, such an approach to dynamics of three-bodyrecombination—where all the three particles are considered to‘‘arrive’’ at the strong interaction zone simultaneously—narrowssignificantly the domain of the conditions under which recombina-tion can happen. This model implies the conclusion (perhapsnot always justified) that an effective course of the three-bodyreaction Eq. (4) requires a sufficiently high pressure in a gasmedium [22].

In the present paper, we propose a more general, in our opinion,dynamical model of direct three-body recombination by exampleof the recombination reaction of the Cs+ and Br� ions in the pres-ence of the Xe atom. In the framework of this model, one considersthree-body collisions with an arbitrary ‘‘arrival’’ time T of the thirdbody with respect to the computed duration s of the approach ofthe recombining particles to the center-of-mass of the ion pair.This time is expressed below as the percent amount of s and canbe both positive (in the case where the energy removal by the thirdbody is delayed) and negative (in the opposite case where the en-ergy removal occurs in advance).

The article is organized as follows. After this introduction, inSection 2, we briefly describe the PES employed (this PES allowsone to reproduce quantitatively correctly the CID dynamics inthe CsBr + Xe system), the process model we use (due to the pres-ence of the time lag T, this model differs considerably from that inRefs. [16–21]), and the calculation procedure. In Section 3, we pres-ent the results of trajectory calculations of dynamics of ionencounters followed by the stage of recombination product stabil-ization by the Xe atom: this paper studies mainly the conse-quences of a delay of the third body rather than of its comingfirst (which does not affect statistical dynamics2 of the reaction al-most of all). The limiting cases of the model with a delay are the phe-nomenological Schemes (1) and (2). Section 4 is devoted to adiscussion of the results obtained, and concluding remarks are givenin Section 5.

2. Potential energy surface, model of the process, and trajectorycalculation procedure

We explore the dynamics of the three-body recombination

Csþ þ Br� þ Xe! CsBrþ Xe ð7Þ

on the PES that governs the two-channel CID process Eqs. (5) and(6)

CsBrþ Xe! Csþ þ Br� þ Xe;! XeCsþ þ Br�:

This PES describes quantitatively correctly the dynamical experi-mental data obtained in crossed molecular beams and characteriz-ing the excitation functions of the CID reaction products as well asthe angular and energy distributions of these products (the differ-ential and double differential cross sections) [10–12]. The samePES was used for our previous studies of the reaction Eq. (7)[16,18–21].

The PES in question is the sum of the three pairwise diatomicinteraction potentials and the cross term:

28 V.M. Azriel et al. / Chemical Physics 411 (2013) 26–34

UðR1;R2;R3Þ ¼ UCsþ�XeðR1Þ þ UCsþ�Br� ðR2Þ þ UBr��XeðR3Þþ UcrossðR1;R2;R3Þ;

where R1, R2, and R3 are, respectively, the internuclear distances inthe pairs Cs+–Xe, Cs+–Br�, and Br�–Xe. For the interaction of the Cs+

and Br� ions, we use the so-called truncated Rittner (T-Rittner)potential model most frequently employed for an analytic represen-tation of the interionic interactions in the alkali halide molecules[24–26]:

UCsþ�Br� ðR2Þ ¼ A2 expð�R2=q2Þ �1=R2 � ðaCsþ þaBr� Þ=ð2R42Þ � C2=R6

2:

The pairwise interactions of each of the ions with the xenon atomare described by expressions of identical forms [27]:

UCsþ�XeðR1Þ ¼ A1 expð�R1=q1Þ � aXe=ð2R41Þ � C1=R6

1;

UBr��XeðR3Þ ¼ A3 expð�R3=q3Þ � aXe=ð2R43Þ � C3=R6

3:

Here Ai and qi are the parameters of the Born–Mayer repulsivewalls, aCsþ , aBr� , and aXe are the polarizabilities of the Cs+, Br� ionsand the Xe atom, respectively, and Ci are the van der Waals disper-sion constants. The cross term (corresponding to the polarizationinteraction in the system of the Xe atom and the Cs+–Br� dipole) is

UcrossðR1;R2;R3Þ ¼ aXe1R4

1

þ 1R4

3

!aCsþ þ aBr�

R32

1� aCsþ þ aBr�

2R32

!

!

TablThe(atom

Th

CCB

a Rtrosc

b Rsom

c Rd A

þaXeR2

1 þ R23 � R2

2

2R31R3

3

1� aCsþ þ aBr�

R32

2

:

All the expressions for the pairwise diatomic potentials UCsþ�Br� ,UCsþ�Xe, UBr��Xe and the cross term Ucross above are presented inatomic units. The values of the parameters Ai, qi, and Ci determin-ing the PES topology are given in Table 1. These values yield thediatomic properties (such as the dissociation energies and vibra-tional frequencies) consistent with the experimental data [27–30]. The polarizabilities of the particles are aCsþ ¼ 16:48,aBr� ¼ 32:46, and aXe = 27.2 au3 [25,31]. The PES thus constructedenters the Hamiltonian of the system in the quasiclassical proce-dure of trajectory calculations (see e.g. Refs. [16,32]).

As was already pointed out in the introduction, the three-bodyrecombination model proposed in the present paper assumes thatthe product stabilization does not require a simultaneous or almostsimultaneous collision of all the three particles. Instead, stabiliza-tion can also occur in the case where the third body ‘‘arrives’’ anarbitrary time interval later or earlier than the instant of the closestapproach of the recombination partners. This means that directthree-body recombination may be considered as a two-stage pro-cess, one of the stages being an encounter of the recombining par-ticles and the other one, a collision with the third body which takesaway energy from the recombining pair or from any of the partners.

e 1potential parameters for the Cs+–Br�, Cs+–Xe, and Br�–Xe pairwise interactions

ic units) in the Cs+ + Br� + Xe system.

e pair Ai qi Ci

s+–Br� 127.5a 0.7073a 87.36b

s+–Xe 318.5c 0.6494c 490.0d

r�–Xe 62.84c 0.877c 297.3d

ef. [28] (quantum mechanical calculations which agree well with the spec-opic data).efs. [29,30] (the value for the isoelectronic pair Xe–Kr of inert gas atoms, with

e corrections).ef. [27] (combining rules for the Born–Mayer parameters).calculation according to the Slater–Kirkwood formula.

Each of these stages can be either the first one or the final one. Inprinciple, under appropriate conditions, inelastic encounters ofparticles may yield the formation of a complex of various nature[22] with a lifetime sufficient for a collision with the third bodyremoving an excess of energy. From this viewpoint, the formationof a genuine complex (especially a complex with a lifetime long en-ough) is the limiting case corresponding to the Schemes (2) or (3).However, the emergence of such an intermediate complex is notnecessary for recombination, and the formation of some boundstate of the two particles (a state that lives only during the colli-sion) without its stabilization into a real complex by the third par-ticle can be sufficient. Within this scenario, energy may be takenaway from the recombination product in the interaction of thethird body with any of the partners of the process. Such a situationtakes place e.g. in three-body recombination of atomic ions whoselong-ranged interaction is governed by the Coulomb potential �1/R(R being the interionic distance) excluding a centrifugal barrier.

The dynamics of an elementary process is determined by thecollection of kinematic parameters. These parameters affect theprobabilities of trajectories of any given type corresponding tothe process in question. Three-body recombination is describedby a more complicated collection of kinematic parameters com-pared with bimolecular reactions AB + C. The first difference is thata collision of three particles is characterized by two interactionenergies, namely, the collision energy Ei of the recombination part-ners (the ions in our case) and the kinetic energy ER of the thirdbody with respect to the center-of-mass of the ion pair. Anotherpeculiarity is the presence of two impact parameters, the impactparameter bR of the third body with respect to the center-of-massof the recombining pair and the impact parameter bi of the encoun-ter of the ions. The value of bi determines whether the collision ofthe ions is central (head-on) or non-central, the first possibilitycorresponding to bi = 0 (or to bi close to zero) and the secondone, to bi > 0. The recombination probability for central encountersof the ions and that for non-central ones differ drastically [19]. Thelist of the essential kinematic parameters includes also the mutualorientation angles of the momenta Pi of the three particles. If oneconsiders two-stage recombination, one more kinematic parame-ter is to be introduced. This additional kinematic parameter isthe time shift T of the two stages (with respect to the single-stageprocess). The time shift T can be equal to zero for a simultaneouscollision of all the three particles according to Eq. (1) and be posi-tive or negative in the actual presence of two stages.

The coordinate frame used in the calculations is shown in Fig. 1.For more visuality, the origin O of the frame is set to coincide withthe center-of-mass of the ion pair rather than with that of thewhole system. The mass point representing the third body initiallylies in the OXY coordinate plane. The relative momentum Prel of ap-proach of the xenon atom and the center-of-mass of the recombin-ing pair is parallel to the OX coordinate axis. The initial direction ofthe relative momentum Pini of approach of the two ions is deter-mined by the polar angle H measured from the OX axis and the azi-muthal angle U in the spherical coordinates. These two angles areMonte Carlo selected in such a way as to ensure the unbiased prob-ability distribution of all the spatial configurations of the collisionof the three particles. For non-central ion encounters, one has alsoto select an additional independent angle c (not shown in Fig. 1)characterizing the position of the oriented plane passing throughthe Cs+OBr� line and parallel to Pini. In the settings where the im-pact parameters bi or bR are to be varied, they are selected in theunbiased way as well (i.e., with a uniform distribution of b2) withbi,max = 40 au and bR,max = 100 au.

The mathematical procedure of integrating the trajectories inthe present paper is similar to the procedure in Refs. [16–19].The initial internuclear distance dini between the Cs+ and Br� ionswas always set equal to 250 au. In the case of a one-stage process,

O

Prel

bR

bi

Cs+

Br-

R

Z

Y

XΘ

Φ

Pini

O is the Cs+ – Br–

center-of-mass.

R = Xe in the paper.

Fig. 1. The coordinate frame OXYZ for representing direct three-body recombina-tion. Here bR is the impact parameter of the R atom with respect to the center-of-mass O of the ion pair whereas bi is the impact parameter of the ions themselves.The initial momentum Prel of the third body with respect to the center-of-mass ofthe recombining pair is parallel to the OX coordinate axis. The direction of the initialmomentum Pini of approach of the ions is determined by the polar angle H and theazimuthal angle U.

2 4 6 8 10

2468

10

24

68

10

0.0000

0.0008

0.0016

0.0024

0.0032

0.0040

Rec

ombi

natio

n pr

obab

ility

ER (Xe - CsBr), eV

E i(Cs

+ - Br- ), eV

2 4 6 8 10

20.0000

0.0005

0.0010

0.0015

0.0020

0.0025

Rec

ombi

natio

n pr

obab

ility

a

b

V.M. Azriel et al. / Chemical Physics 411 (2013) 26–34 29

given the distance dini, the relative kinetic energy Ei of the ions, andthe third body energy ER, we determined the initial distance drel be-tween the atom and the center-of-mass of the ion pair according tothe following rule: assuming that bi = bR = 0, the ions constitute anisolated Coulomb subsystem, and the Xe atom moves under noforces, we required all the three particles to achieve the center-of-mass of the system simultaneously. In other words, if the ionsachieve the center-of-mass O of the Cs+–Br� pair in time s = s(Ei,dini), then drel = (2ER/lR)1/2 s where lR is the reduced mass of theatom and the ion pair. An explicit formula for s(Ei, dini) is presentedin the paper [16]. The additional kinematic parameter T (the shiftof the recombination stages) expressing the delay of the third bodyor its moving ‘‘ahead of schedule’’ was taken into account by a suit-able increase or decrease, respectively, in the initial distance be-tween the Xe atom and the center-of-mass of the ions. Forexample, a 10% increase in drel makes the time it takes the atomto cover the distance to the center-of-mass of the ion pair 10% lar-ger. The quantity T is measured in percents of the time s. As wasalready pointed out in the introduction, T is regarded as positiveif the third body is delayed and as negative if the third body ‘‘ar-rives’’ before the closest approach of the ions.

The following criterion was used to establish that the Cs+ andBr� ions had recombined and the trajectory had been over: theinternuclear distance between the ions is less than 30 au, the totalinternal energy of the ion pair is negative, and the internuclear dis-tance between each of the ions and the third body is greater than250 au. The recombination probabilities P(I), where I denotes theparticular collection of kinematic parameters, are defined to bethe ratios Ns/N = Ns(I)/N(I), where Ns is the number of ‘‘successful’’trajectories (i.e., those yielding recombination) and N is the totalnumber of trajectories integrated.

2468

10

46

810

ER (Xe - CsBr), eV

E i(Cs

+ - Br- ), eV

Fig. 2. The histograms of the dependences of the recombination probability for theCs+ and Br� ions in the presence of the Xe atom on the ion encounter energy Ei andthe third body energy ER (for non-central ion approaches) for T = 0 (histogram (a))and T = 10% (histogram (b)). As is pointed out in Section 2, T is measured in percentsof s, the computed duration of the ion approach (under the assumption that theions constitute an isolated Coulomb subsystem with zero impact parameter).

3. Calculation results

The revealed dependences of the recombination probabilitieson the interaction energies, impact parameters, and spatial orien-tation angles of the initial velocities of the particles are almost‘‘symmetric’’ for T > 0 and T < 0. Therefore, as was already notedin the introduction, we present below the results of the statisticalcalculations of recombination dynamics mainly for the case T > 0,i.e., for the situation where the third body is delayed with respect

to the ion approach instant. The recombination dynamics data atT > 0 are compared with the results obtained for a simultaneouscollision of all the three particles, i.e., at T = 0. Although a two-stageprocess of recombination of the ions is possible for any sign of thequantity T, the differences in detailed dynamics for all the threecases T < 0, T = 0, and T > 0 can be significant (see Section 4 below)and require further studies.

Fig. 2 shows two histograms of the recombination probabilitiesfor both the interaction energies in one-stage and two-stage three-body collisions. These histograms are calculated for non-centralion encounters with the time gap between the stages equal toT = 0 (histogram (a)) and T = 10% (histogram (b)). The histogramsare drawn on the basis of integrating 2.5 � 105 trajectories and2 � 105 trajectories, respectively, for each ðER; EiÞ pair. One seesin this figure that the two histograms differ mainly in the valuesof the total probability which is about 1.6 times less for T = 10%than the probability for one-stage recombination. The structureof both the histograms is almost the same.

30 V.M. Azriel et al. / Chemical Physics 411 (2013) 26–34

The sufficiency of a sample of 2.5 � 105 or 2 � 105 trajectorieswas confirmed in test calculations where 105 trajectories wereintegrated for each ðER; EiÞ pair and almost the same recombina-tion probabilities were obtained. Recall that the impact parametersbi, bR and the angles H, U, and c (see Section 2) for each pair of theinteraction energies range in the intervals 0 6 bi 6 bi;max ¼ 40 au,0 6 bR 6 bR;max ¼ 100 au, 0 6 H 6 p, 0 6 U < 2p, 0 6 c < 2p andwere generated as bi ¼ f1=2

1 bi;max, bR ¼ f1=22 bR;max, H = arccos

(2f3 � 1), U = 2pf4, c = 2pf5, where f1, f2, f3, f4, f5 are independentrandom variables uniformly distributed between 0 and 1.

A more detailed information concerning the effect of the recom-bination stage shift on the recombination probability for four col-lections of Ei and ER values is presented in Fig. 3 for the sampledrange of delays T from 0 to 50%. Curves 1–3 exhibit a strongdependence of the recombination probability on the value of T, thisdependence weakening considerably as the third body energy in-creases. The dependence of the recombination probability on thetime gap between the process stages becomes less pronounced(as the Xe atom energy grows) synchronously with the decreasein the recombination probability itself. Note also that for the samevalue ER = 1 eV, the dependence of the recombination probabilityon T gets very weak as the ion encounter energy Ei increases from1 to 5 eV (curves 2 and 4). Such trends in the dependences of therecombination probability on the ‘‘arrival’’ time of the third bodyare caused by the peculiarities (considered below) of the dynami-cal behavior of the system. To a great extent, these peculiaritiesmanifest themselves in the opacity functions P(bR) and P(bi) thatexpress the dependences of the reaction probability on the impactparameters.

Fig. 4 presents typical opacity functions for the impact parame-ters bR (Fig. 4(a)) and bi (Fig. 4(b)) at the energies Ei = ER = 1 eV andat T = 10%. The shapes of such dependences for most of the sam-pled combinations of kinematic parameters with ER lying between0.2 and 1 eV are almost the same. A near identity of the shapes ofP(bR) for T = 10% and for T = 0 [18] is accompanied with a strong (bya factor of 2–3) increase (as T grows) in the value of bR where thisopacity function attains its maximum. However, as ER rises up to5 eV, the bR opacity function changes its shape drastically and be-comes bimodal as is shown in Fig. 5(a). A change in the shape of thefunction P(bR) gets noticeable already for ER = 2 eV and increases asthe energy ER grows. The ‘‘splitting’’ of the opacity function

0 10 20 30 40 500.000

0.001

0.002

0.003

0.004

0.005

0.006

4

3

2

1

Rec

ombi

natio

n pr

obab

ility

Time delay T of the Xe atom arrival, %

Fig. 3. The dependences of the probability of recombination of the Cs+ and Br� ionson the delay T in the collision of the ion pair with the Xe atom for four collections ofthe ðEi; ERÞ values. Curve 1: Ei = 1 eV, ER = 0.2 eV. Curve 2: Ei = 1 eV, ER = 1 eV. Curve3: Ei = 1 eV, ER = 5 eV. Curve 4: Ei = 5 eV, ER = 1 eV. The quantity T is measured inpercents of s.

intensifies considerably as the ER value increases to 10 eV and asthe time T of the recombination stage shift grows. Fig. 5(b) presentsthe opacity function P(bR) for the energies Ei = 1 eV and ER = 10 eVat T = 50%. Note that the shape of the bi opacity function keepsalmost the same as the parameters in question are varied.

For a direct recombination reaction proceeding in two stageswith a delay of the energy removal stage, the dependences of therecombination probability on the orientation angles of the veloci-ties of the particles involved in the reaction exhibit new features.These dependences are very different from those for central ionencounters as well as for non-central ones at T = 0. Fig. 6 presentsthe dependences of the recombination probability on the spatialorientation angles H and U of the velocities of the three particlesat the collision energies Ei = ER = 1 eV for T = 0 (panels (a) and (b))and for T = 10% (panels (c) and (d)). These figures imply that a10% time gap between the recombination stages affects consider-ably the behavior of the dependence of the recombination proba-bility on each of the two angles. The U angle dependence (panels(b) and (d)) changes especially drastically: the local maximum atU = 180� becomes global and much more pronounced, the twominima at U = 90� and U = 270� getting much more pronouncedas well. The changes in the shapes of the dependences in questionbecome even more noticeable as the energy Ei and the quantity Tincrease.

The distributions of the product molecules in the internal ener-gies for various values of T > 0 differ but slightly from the distribu-tions for central and non-central ion interactions at T = 0 [18,19]. Inother words, one finds a strongly non-equilibrium distributionin the vibrational energy and a nearly equilibrium distribution inthe rotational energy of the CsBr molecule.

4. Discussion

As one can see in Fig. 2, the recombination stage shift deter-mined by the parameter T does not yield essential qualitative mod-ifications in the dependence of the total recombination probabilityP(Ei, ER) on both the collision energies. However, a 10% time gapbetween the two stages of the process reduces the probabilitiesby about a factor of 1.6. Note that the maximal values of the prob-abilities P(Ei,ER) are attained in central ion encounters at T = 0. Asone takes the impact parameter bi of ion collisions into accountfor the same value T = 0, the total recombination probability de-creases by about a factor of 5–6 [19]. As one then proceeds to treat-ing recombination as a two-stage process with T = 10%, theprobability gets still smaller by about a factor of 1.6. A further in-crease in T leads to a further decrease in the total recombinationprobability. Fig. 3 indicates clearly a strong dependence of therecombination probability not only on the value of the time shiftT but also on the energy ER of collision of the recombining pair withthe third body. These figures imply that the actual recombinationmechanism is mainly determined by the collision dynamics char-acterized by the kinematic parameters responsible for the interac-tion features of each pair of the particles.

The effect of the collision dynamics on the current mechanismof the elementary process in question manifests itself most clearlyin the opacity functions which express the dependences of therecombination probability on the impact parameters of the ionencounter (bi) and of the ion pair with respect to the third body(bR). As was noted above in Section 3, the shapes of the functionP(bi) for ER < 2 eV over a wide range of other parameters (includinga rise in T up to 50%) qualitatively coincide with that of the bi opac-ity function for a one-stage process, i.e., at T = 0. An increase in T upto 50% leads just to a decrease in the probability itself of the recom-bination process, while the shapes of the dependences P(bR) andP(bi) remain almost the same. However, Fig. 5 shows that an

0 10 20 30 40 500.00000

0.00005

0.00010

0.00015

0.00020

T = 10%

Ei = 1.0 eV ER = 1.0 eV

Cs+ + Br- + Xe → CsBr + Xe

Rec

ombi

natio

n pr

obab

ility

Impact parameter bR, au 0 8 16 24 32 40

0.00000

0.00005

0.00010

0.00015

0.00020

T = 10%Ei = 1.0 eV ER = 1.0 eV

Cs+ + Br- + Xe → CsBr + Xe

Rec

ombi

natio

n pr

obab

ility

Impact parameter bi, au

a b

Fig. 4. The dependences of the ion recombination probability on the impact parameter bR of the third body (a) and on the impact parameter bi of the ion encounter (b) for atime gap T between the recombination stages equal to 10% (i.e., s/10).

0 10 20 30 40 500.00000

0.00002

0.00004

0.00006

0.00008

0.00010

T = 10%

Ei = 1.0 eV ER = 5.0 eV

Cs+ + Br- + Xe → CsBr + Xe

Rec

ombi

natio

n pr

obab

ility

Impact parameter bR, au 0 20 40 60 80 100

0.0000000

0.0000008

0.0000016

0.0000024

0.0000032

0.0000040

T = 50%

Ei = 1.0 eV ER = 10.0 eV

Cs+ + Br- + Xe → CsBr + Xe

Rec

ombi

natio

n pr

obab

ility

Impact parameter bR, au

a b

Fig. 5. The dependences of the ion recombination probability on the impact parameter bR of the third body for ER = 5 eV and T = 10% = s/10 (a) and for ER = 10 eV andT = 50% = s/2 (b) at Ei = 1 eV.

V.M. Azriel et al. / Chemical Physics 411 (2013) 26–34 31

increase in the energy ER of collision with the third body alters con-siderably the shape of the opacity function P(bR). Such a dramaticevolution of the opacity function is caused by a change in the col-lision dynamics of the third body with the bound ion pair. A char-acteristic feature of most of the trajectories with T > 0 is anapproach of the ions (until a repulsion between them appears)and their subsequent ricochet. In a one-stage process (T = 0), themajority of the trajectories with an effective energy removal exhi-bit an acute-angled triangular configuration with a sufficientlystrong repulsive interaction of the third body with both the ionssimultaneously or in consecutive interactions of the third bodywith the ions [19–21]. An analysis of the trajectories of two-stagerecombination has revealed that, in contrast to the recombinationreaction proceeding in a single stage, a conversion of the recombin-ing ion pair into a bound molecule in the presence of a time lag be-tween the two stages occurs mainly after the ricochet of the ions ina collision of the third body with only one of the two ions. In suchconfigurations, the energy transfer takes place at large distancesbetween the ions. This excludes an effective transfer of energyfrom two ions or via successive encounters of the atom with theions. The interionic distances at which the nascent product mole-cule is formed increase rapidly as the value of T grows which

toughens the conditions for energy transfer to the third body.Within this situation, the most effective energy exchange occursin a head-on or nearly head-on encounter of the Xe atom withone of the ions.

Studying the trajectories pertaining to various regions of theopacity function presented in Fig. 5(a) has revealed that in the do-main of the first maximum, a stable molecule is formed while en-ergy is transferred to the Xe atom in its encounter with the Cs+ ion.On the contrary, stabilization of the CsBr molecule in the domain ofthe second maximum is due to an encounter of the atom with theBr� ion. Moreover, in both the cases, nearly collinear configurationsare formed with a sufficiently strong repulsion between the atomand the corresponding ion as shown in Fig. 7. For impact parame-ters in a vicinity of the minimum of the opacity function, the timegap between the two recombination stages leads to a noticeable in-crease in the fraction of trajectories with an essentially nonlinearcollision configuration (the angle between the velocity of the thirdbody and that of the ions’ flying away from each other is less than45�).

This peculiarity of recombination with a large delay of the en-ergy removal stage manifests itself even more vividly at the energyER of collision with the third body equal to 10 eV. As in the case

0 30 60 90 120 150 1800.0000

0.0001

0.0002

0.0003

0.0004

0.0005

Ei = 1.0 eV ER = 1.0 eVCs+ + Br- + Xe → CsBr + Xe

Rec

ombi

natio

n pr

obab

ility

Angle Θ, deg

T = 0

0 60 120 180 240 300 3600.00000

0.00005

0.00010

0.00015

0.00020

0.00025

Ei = 1.0 eV ER = 1.0 eV

Cs+ + Br- + Xe → CsBr + Xe

Rec

ombi

natio

n pr

obab

ility

Angle Φ, deg

T = 0

0 30 60 90 120 150 1800.0000

0.0001

0.0002

0.0003

0.0004

0.0005

T = 10%Ei = 1.0 eV ER = 1.0 eV

Cs+ + Br- + Xe → CsBr + Xe

Rec

ombi

natio

n pr

obab

ility

Angle Θ, deg

0 60 120 180 240 300 3600.00000

0.00005

0.00010

0.00015

0.00020

0.00025

T = 10%Ei = 1.0 eV ER = 1.0 eV

Cs+ + Br- + Xe → CsBr + Xe

Rec

ombi

natio

n pr

obab

ility

Angle Φ, deg

a b

dc

Fig. 6. The dependences of the ion recombination probability on the orientation angles H and U of the initial velocities of the particles for the collision energies Ei and ER

equal to 1 eV at the recombination stage shift T equal to 0 ((a) and (b)) and to 10% ((c) and (d)).

32 V.M. Azriel et al. / Chemical Physics 411 (2013) 26–34

considered above, the two maxima of P(bR) obtained are due toencounters of Xe with different ions of the recombining pair. Herea further decrease in the recombination probability is observedwhich is caused by ‘‘difficulties’’ in the formation of a nearly collin-ear configuration that grow as the third body velocity increases. Arise in the parameter T up to 50% at an ER energy of 10 eV (seeFig. 5(b)) reduces the recombination probability by about a factorof 30 (most probably, due to the same reason).

In Refs. [20,21], a similar mechanism of product stabilization isdescribed. This mechanism is characterized by taking away energyfrom the pair of the ions which are flying past each other with ahigh repulsion energy after their closest approach. Such a mecha-nism is typical for one-stage recombination at large impact param-eters bR. However, this similarity is somewhat formal because in asingle-stage process, energy transfer occurs, as a rule, at not solarge distances between the ions moving off from each other, andthe formation of a collinear configuration is not necessary.

The distinctions in the dynamical mechanisms of one-stage andtwo-stage recombination lead to different dependences of therecombination probability on the orientation angles of the initialvelocities of the three particles. Fig. 6 shows that as one proceedsfrom T = 0 to T = 10% at Ei = ER = 1 eV, the local maxima and minimaof the U dependence become incomparably more pronounced,while the maximum of the H dependence is shifted towards largerH values. These rules somewhat change as one alters the Ei and ER

energies but the general trends remain the same. The shift towardslarger H angles is probably connected with the fact that for

positive values of T, stabilization of CsBr occurs almost exclusivelyat the repulsion phase of the motion of the recombining pair, andthe interionic distance can be large due to a delay of the secondstage of the process. The changes in the recombination probabilitydependences on the orientation angles are facilitated by domi-nantly collinear or almost collinear configurations of the interact-ing particles. One will be able to determine the role of themutual orientation angles H and U in a two-stage process moreaccurately from studies of detailed dynamics of the reaction (cf.Refs. [20,21]).

As was already pointed out in Section 3, an analysis of the en-ergy distributions of the recombination products shows that atwo-stage process yields equilibrium distributions in the rotationalenergy and strongly non-equilibrium distributions in the vibra-tional energy of the molecules. This result agrees completely withthe distributions that appear in one-stage recombination [18,19].

As one expects, a negative time T modifies considerably the de-tailed dynamical picture of recombination since the third body ‘‘ar-rives’’ at the approach phase of the ion motion and, as a rule,‘‘squeezes’’ between the recombining particles. In this setting, theenergy removal occurs in repulsive interaction of the third bodywith one of the ions followed by scattering of the neutral atomaway from the ion approach line. As in the case of positive valuesof T, more complicated trajectories are also possible, but stabiliza-tion of the recombining pair (i.e., elimination of an excess of en-ergy) happens as the first stage of the process followed by theion approach stage yielding the product molecule.

Fig. 7. A typical collision configuration preceding an energy transfer from the Cs+ ion to the Xe atom along with the energy characteristics of the system. Curve 1 is the totalpotential energy of the system. Curve 2 is the potential energy in the Xe–Cs+ pair. Curve 3 is the potential energy in the Xe–Br� pair. Curve 4 is the kinetic energy of the xenonatom. The dots indicate the energies corresponding to the particle configuration shown in the figure. Each trajectory step is 50 au of time.

V.M. Azriel et al. / Chemical Physics 411 (2013) 26–34 33

5. Conclusion

We have discussed the elementary process of three-bodyrecombination as a reaction consisting of two consecutive stagesseparated by some time interval. Such a mechanism enables oneto treat both the phenomenological models accepted in chemicalkinetics as limiting cases of a two-step process. These limiting casesare single-stage direct three-body recombination for T = 0 andrecombination via an intermediate complex for the stage shift Tequal to the actual lifetime of the complex. Within the model pro-posed in the present work, one of the stages is an approach of therecombining particles while the other stage consists in taking awayenergy from this pair by the third body, i.e., in stabilization of therecombination product.

For T > 0, the reaction proceeds via an approach of the activeparticles followed by stabilization of the nascent molecule by thethird body. For T < 0, the first stage is energy removal from oneof the recombination partners while the active particles areapproaching. The model considered demonstrates the possibilityof three-body recombination via two successive stages withoutthe formation of intermediate complexes at T – 0. For positive val-ues of T, recombination takes its course via a bound state of therecombining pair of the particles, the lifetime of this state being re-stricted by the minimal duration of interaction of these particles.Compared with one-stage direct three-body recombination(T = 0), this enhances greatly the phase space volume where a sta-ble product can be formed. In turn, this means that the two-stageprocess of three-body recombination without intermediate com-plexes may play a significant role not only at high pressures butalso at low pressures in a gas medium.

Indeed, it is not hard to verify that the recombination rate (thenumber of salt molecules formed per unit time in unit volume) isequal to

n2i nR

Z 1

0

Z 1

0fiðEiÞ

2Ei

li

� �1=2

fRðERÞ2ER

lR

� �1=2

AðEi; ERÞdEidER;

where

AðEi; ERÞ ¼sðEiÞ100

Z T2

T1

ðpb2i;maxÞðpb2

R;maxÞPðEi; ER; TÞdT;

ni is the (number) concentration of each of the ions, nR is the(number) concentration of the R atoms, fi(Ei) denotes the probability

density of the ion encounter energy Ei, fR(ER) denotes the probabilitydensity of the third body energy ER, li is the reduced mass of theions, lR is the reduced mass of the atom R and the ion pair, andthe time shift T ranges in the interval T1 6 T 6 T2. The coefficients/100 appears due to the fact that T is measured in percents ofthe time s in our work. As we observed (see Figs. 2 and 3) fornon-central ion encounters in the case of three-body recombinationEq. (7), the recombination probabilities P(Ei, ER, T) do not decreasevery strongly as one passes from T = 0 to T = 10% (they reduce byabout a factor of 1.6 on average). Some additional calculations showthat the same holds as one passes from T = 0 to T = �10%. Conse-quently, as one passes from the case where only very small T valuesare taken into account, say, jTj 6 0:1%, to the interval jTj 6 10%, therecombination rate increases by almost two orders of magnitude.This is especially important at low concentrations ni and nR.

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