Reactivity in ion-neutral high density media

. The recent development of hybrid traps technology has revolutionized the ﬁeld of cold chemistry. Recently, new trap conﬁgurations have been built in order to explore more involve chemical processes, among them is the three-body recombination involving ions and neutrals. Di ﬀ erent ion-neutral-neutral reactions at low collision energies: 0.1 mK - 10 mK, are studied by means of a very recent method for classical trajectory calculations in hyperspherical coordinates for treating three-body collisions [J. Pérez-Ríos, S. Ragole, J. Wang and C. H. Greene, J. Chem. Phys. 140 , 044307 (2014)]. On the other hand a classically derived threshold law is obtained and numerically conﬁrmed. The derived threshold law predicts molecular ions should dominate over molecular neutrals as the most products formed.


Introduction
The development of hybrid traps technology has brought the opportunity to study different kinds of chemical reactions involving neutrals and charged particles at cold temperatures ∼ 1 mK. Hybrid trap technology relies on the combination of neutral atom traps and ion traps, in order to make a dual trap region where ions and neutrals overlap in space. On the other hand, by switching among different kinds of neutral traps or ion traps configurations, a broad spectra of ion-neutral chemical reactions can be explored. In particular, one interesting hybrid trap configuration is done by an optical dipole trap for neutrals, whereas a four-pole Paul trap is employed for trapping ions [1]. The scheme of this trap is shown in Fig.1, where also it is shown the specific Paul trap configuration employed by Härter et al [1], as well as the typical potential that an ion feels in the Paul trap. This hybrid trap configuration allows to study the same ion-neutral chemical reactions in a quite broad range of densities, and hence exploring the reactivity of the reactants as a function of the density.
In cold ion-neutral chemistry, most of the theoretical and experimental efforts have been devoted to the study of charge transfer reactions between ions and atoms, and more recently in molecular ion-atom systems [2]. Here, however we will focus on the study of three-body recombination (TBR) processes. TBR is a non-radiative few-body process in which three atoms collide, and eventually leading to the formation of a molecule as a product state, i.e., A + A + A → A 2 + A. Nevertheless, in ion-neutral-neutral TBR the reaction has two different channels: first, B + + A + A → A 2 + B + , and second, B + + A + A → AB + + A. These two different reaction channels are studied for 87 Rb + -87 Rb -87 Rb and 138 Ba + -87 Rb -87 Rb, by means of a recent method based on classical trajectory calculations a e-mail: jperezri@purdue.edu b e-mail: chgreene@purdue.edu (CTC) in hyperspherical coordinates [3,4]. On the other hand, it is reported a classical threshold law for ion-neutral-neutral TBR, which is numerically tested, as well as it has been experimentally corroborated [5]. The green conductors provide extra DC fields to compensate the micromotion of the ion in the Paul trap, which leads to a very high control on the kinetic energy of the ion. Panel (c): typical potential that an ion sees in a Paul trap at a given time.

Theoretical Approach
A recently developed method for the calculation of TBR cross sections based on classical trajectories [3] has been adapted for the study of ion-atom-atom recombination. The method relies on mapping the three-body problem into a 6-dimensional space (after separating out the trivial center of mass motion), where the cross section emerges as a generalization of the well-known two-body cross section, since the impact parameter can be defined as a 5-dimensional vector imbedded in the 6-dimensional space [3]. This approach is totally general and in principle it is suitable for solving the N-body problem. The position and momentum vectors in the 6-dimensional space is represented in hyperspherical coordinates [6,7], which leads to a very efficient sampling of the phase-space. The validity of a classical approach for ion-neutral-neutral collisions at cold temperatures may be questioned. However, the long-range ion-neutral interaction is dominated by the charge induced dipole moment interaction −C 4 /r 4 , where C 4 = α/2, and α is the neutral atom polarizability. This long-range interaction is more attractive than its analog for neutral-neutral interactions, i.e., the van der Waals interaction. Thus, the typical length scale of ion-neutral systems will be larger than its equivalent for neutral-neutral interactions. As a consequence, the energy in which the dynamics will be totally quantal is lower in ion-neutral than in neutral-neutral interactions. A more detailed analysis of this point can be found in [4]. In order to have a more quantitative argument, the angular momentum distributions of the TBR process for 138 Ba + -87 Rb -87 Rb have been calculated at different collision energies, and the results are shown in figure 2. In this figure, it is noticed that even at collision energies ∼ 100 μ K, still many partial waves will contribute to the collision processes. Therefore, classical trajectory calculations might be an adequate approach for describing ion-neutral-neutral three-body recombination. Figure 2. Total angular momentum distribution associated with the TBR for 138 Ba + -87 Rb -87 Rb at different collision energies. The total angular momentum l is given in a.u., whereas the collision energy is given in K.

Results
The TBR cross section for 87 Rb + -87 Rb -87 Rb and 138 Ba + -87 Rb -87 Rb has been computed by running 10 5 trajectories per collision energy. Details about the numerical method employed to solve Hamilton's equations of motion, in conjunction with the sampling of the initial conditions, can be found elsewhere [3]. All of the calculations have assumed that neutral-neutral interactions, as well as ionneutral interactions occur along one single potential energy curve. Concretely, Rb -Rb collisions are assumed to occur through the 3 Σ potential, i.e., the spin flip transitions have been neglected [4]. On the other hand, the ion-neutral potential is described by the model potential −C 4 (1 − (r m /r) 4 /2)/r 4 , where C 4 is taken as C 4 =160 a.u., and r m represents the position of the minimum of the potential [4]. Table 1. Classical threshold law for the TBR cross section. A power law dependence of the TBR cross section as a function of the collision energy is assumed and used as a fitting function for the CTC numerical results. The error on the fitting parameters are associated with a confidence interval of 95 %.
System β (dimensionless) 87 Rb + -87 Rb -87 Rb -1.27 ± 0.13 138 Ba + -87 Rb -87 Rb -1.18 ± 0.07 Classical threshold law -1.25 From a long-range perspective, the ion-atom interaction is of course more attractive than the atomatom interaction. For any orientation, energy and impact parameter, the collision begins when the trajectories of the atoms start to deviate from uniform motion or free motion. This happens when the interaction potential where the moving atoms are currently located is comparable to the collision energy, i.e, E k ≈ C 4 /r 4 , where E k represents the collision energy. Let us assume that this value of the radius defines the maximum impact parameter associated to a TBR event, i.e. b max (E k ) = (C 4 /E k ) 1/4 . In our formalism the TBR cross section is defined as [3] σ where a unit opacity function has been assumed. Finally, substituting the value of b max (E k ) in Eq. 1, found. This prediction is tested by fitting the CTC numerical results. In particular, the fitting function employed has the form σ(E k ) = γE β k , and the results are shown in table 1. In table 1, it is observed that energy scaling law for the TBR cross section numerically obtained for both systems, 87 Rb + -87 Rb -87 Rb and 138 Ba + -87 Rb -87 Rb, are in a good agreement with the predicted energy scaling law associated with the derived threshold law. This suggests that the derived threshold law is satisfied in different systems under different dynamical conditions. On the other hand, the presented numerical results for 138 Ba + -87 Rb -87 Rb have been corroborated in hybrid trap experiments [5]. The classically derived and numerically confirm threshold law has important implications on the final product state distributions. In particular, it predicts that for ion-neutral-neutral recombination the dominant product state will be the channel associated to molecular ions, i.e., B + + A + A → AB + + A.

Conclusions
The study of TBR for 87 Rb + -87 Rb -87 Rb and 138 Ba + -87 Rb -87 Rb has been performed by using a very recent classical trajectory method in hyperspherical coordinates. On the other hand, a classically derived threshold law for ion-neutral-neutral recombination has been obtained and corroborated numerically. Finally, this threshold is consistent with experimental results of the TBR rate for 138 Ba + -87 Rb -87 Rb [5].
Chemistry is maybe one of the most important playgrounds for few-body physics, where many important reactions involve the collision of very few molecules or atoms. However, chemistry has been receiving a very little of consideration by the few-body physics community. The present work is an example of how few-body physics can elucidate some of the intriguing chemical reactions . Therefore, we hope that our work serves as a motivation to go beyond the apparent limits of few-body physics.