Universal deceleration of highly polar molecules

To understand how to extract the anion beam properties from the neutral beam properties, we will now study in more detail the anion formation process of equation (1).

The anion formed is usually a so-called dipole-bound anion. Its electron is bound with an electron affinity (binding energy) in the 1–100 meV range, which is much lower than the electron affinity of a valence-bound anion in the 0.1–1 eV range. Indeed, a dipole-bound anion exists simply because the neutral molecule dipole moment is large enough to bind an electron through the electron-dipole potential. Roughly speaking, when the dipole is sufficiently large, an excess electron can be bound into a very diffuse molecular orbital close to the positive pole. By studying the capture of negatively charged mesons in matter, Fermi and Teller were the first to predict that the long-range dipolar potential can bind an electron, provided that the dipole moment is larger than 1.625 D [19]. Various molecular effects, such as the length of the dipole, the presence of a short-range repulsive potential in the molecular core of the anion, and the rotational degrees of freedom of the anion turned out to increase this cutoff value. It is now commonly accepted that any molecule with a dipole moment above 2.5 D can form a dipole-bound anion [20]. In addition, there is conclusive evidence for long-range binding to quadrupolar molecules [21]. Thus the number of molecules that can be formed in this way ranges from simple molecules (LiH, LiF, SrF, BaF, HCN, SiO, BaO, AgCl, BrK ...) to molecules of chemical interest (HCN,CH₃CHO, H₂SiO,) to big polymers or large biological molecules and even DNA bases. Indeed, for big molecules individual dipoles add up, and the final one can easily be bigger that 2.5 D. Obviously, some small but interesting molecules (NH, CH, OH, CN, CO, SO, NO, ...) have lower dipole moments, but we can combine them to form molecules (such as CH₃CN or C₂H₂N₂O) that have dipole moments larger than 2.5 D, which, once decelerated, can be dissociated at threshold to form cold fragments [22]. However, this last method is far from being universal and we will not consider it here.

Weakly bound anions are best produced by the Rydberg electron transfer technique described by equation (1) [23]. The formation of dipole-bound anions via Rydberg electron transfer typically occurs over a narrow range of effective principal quantum numbers, n, of the Rydberg atom, and the excess electron-binding energy, E_b (electron affinity (EA)), of a dipole-bound molecular anion can be estimated using the empirical relationship

$$\begin{eqnarray*}&&{{E}_{{\rm b}}}/eV\approx \frac{23}{n_{{\rm max} }^{2.8}}\end{eqnarray*}$$

where ${{n}_{{\rm max} }}=n-{{\delta }_{n}}$ is the effective principal quantum number of the Rydberg that produces the maximum yield of dipole-bound anions (${{\delta }_{n}}$ being the quantum defect) [24].

The most surprising fact in this relationship is that the charge-exchange process is strongly nonresonant since the cross section reaches its maximum value when the binding energy of the Rydberg atom is much larger than that of the created negative ion (${{E}_{b}}\ll \frac{1}{2n_{{\rm max} }^{2}}$ in atomic units).

The physical explanation of this phenomenon was given by [24] and has been recently studied in detail in [25–29]. We now recall the main results. The process, sketched in figures 2 and 3, describes the ionic interaction between ${{M}^{-}}$ and ${{A}^{+}}$ and the covalent interaction between M and the Rydberg atom, A*. In the center of mass frame, at relative velocity v, the neutral molecule M crosses the electron orbits of A $^{*}(n)$ at the crossing distance ${{R}_{c}}\approx 2{{n}^{2}}$, such that ${{E}_{{\rm ionic}}}({{R}_{c}})={{E}_{{\rm covalent}}}({{R}_{c}})$. A (small) Landau–Zener nonadiabatic transfer probability, ${{p}_{n}}({{R}_{c}})$, exists because there is an avoided crossing between the ionic and covalent curves caused by the exchange interaction ($M,{{A}^{+}},{{e}^{-}}$ play the same role as $H,{{H}^{+}},{{e}^{-}}$ in the H₂ molecule).

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However, once the crossing is overcome, the anion must survive all the other crossings, and mainly the first ones that occur with higher potential covalent curves linked to $A(n^{\prime} \gt n)$ (see figure 2). The product of all these probabilities can be performed using the integral approximation with a probability, ${{W}_{n}}={{e}^{-\int _{{{R}_{c}}}^{+\infty }\frac{{{n}^{3}}}{\sqrt{2{{E}_{{\rm b}}}}{{R}^{2}}}{{p}_{n}}(R){\rm d}R}}$ [25]. The total cross section is given by $\sigma =2\pi \int _{b=0}^{b={{R}_{c}}}b{{p}_{n}}(b){{W}_{n}}(b)\ {\rm d}b$, where b is the impact parameter. Typically, the value of the total cross section is about $\sigma \sim {{10}^{-12}}$ cm². More precisely, the experimental formation rate, $K=\int _{0}^{+\infty }v\sigma (v)\ {\rm d}v$, ranges from 10⁻⁷ cm³ s⁻¹ for the Butanal (C₄H₈O with a dipole μ of 2.7 D, ${{E}_{{\rm b}}}\approx 1.3$ meV and ${{n}_{{\rm max} }}=33$) to 10⁻⁸ cm³ s⁻¹ for the Acetonitrile (CH₃CN with a dipole of 3.9 D, ${{E}_{{\rm b}}}\approx 18.6$ meV and ${{n}_{{\rm max} }}=13$) [24, 26, 27]. The total cross section slightly depends on the relative velocity of the reactants, which is shown in figure 4 using the formula from [25].

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Another important feature of this reaction is the attraction between the products ${{M}^{-}}$ and ${{A}^{+}}$ under the effect of the Coulomb potential, $V(r)\propto 1/r$. This leads to a conical trajectory in the center of mass, and we can extract anion beam properties from the scattering angle

$$\begin{eqnarray}&&\Theta =2\;b\int _{{{r}_{-}}}^{+\infty }\frac{1/{{r}^{2}}}{\sqrt{1-2V(r)/\mu {{v}^{2}}-{{b}^{2}}/{{r}^{2}}}}\ {\rm d}r,\end{eqnarray} \tag{ 2 }$$

where ${{r}_{-}}={{R}_{c}}$ is the distance of smallest approach. From this relation we can extract the differential cross section given by $\sigma (\Theta )=-{{p}_{n}}(b){{W}_{n}}(b)\frac{b}{{\rm sin} (\Theta )}\left\|\frac{{\rm d}b}{{\rm d}\Theta }\right\|$. Figure 5 illustrates the behavior of the differential cross section calculated for different molecular velocities.

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The formula for the cross section used here, provided by [25], is based on an s-orbital anion wave function, dipole approximation, and a zero-range pseudopotential model for the interaction between the excess electron and the anion core. It is sufficient for our purposes and enough to provide a qualitative description of the ion-pair formation. We simply point out that it significantly underestimates, by almost a factor of 10, the absolute values of the cross sections [26, 27]. The model does not account for the rovibrational state of the molecule. However, it is expected that the dipole-bound negative ion formation is a relatively nonperturbative process, and the produced anions must have exactly the same nuclear configurations as their neutral parents [30]. Only in very specific cases, such as for some floppy molecules like the water dimer, the molecule configuration can be strongly modified [31].

The coupling between the electron motion and the molecular rovibration can limit the lifetime of the anion through, for instance, an autodetachment process: ${{M}^{-}}(N^{\prime} ,J^{\prime} )\to M(N,J)+{{e}^{-}}$. However, because of the conserved nuclear configurations in the formation process, which can be written as $M(N^{\prime\prime} ,J^{\prime\prime} )+{{A}^{+}}(nl)\to {{M}^{-}}(N^{\prime\prime} ,J^{\prime\prime} )$, the autodetachment has to be energetically allowed to occur—that is, the rovibrational internal energy has to be larger than the binding energy of the anion. Thus for molecules with a ground vibrational state and low rotational quantum numbers, long-lived anions must be formed, as confirmed by [32, 33]. For instance, in the case of CH₃CN, the data indicate that the creation of long-lived ions requires that target molecules be in states with rotational quantum numbers $J\lt 20$. Further measurements demonstrate that the longest anion lifetime ($\gt 50\;\mu$s) is limited by black body radiation-induced photodetachment.

In conclusion, the high formation rates indicate that a supersonic or cryogenic beam of highly polar molecules lying in their low rovibrational states can be efficiently transformed into anions, thanks to a charge exchange process.

Since the final aim is the production of a cold sample of neutral molecules, it is crucial to follow the velocity properties of the molecular anions' M⁻ beam. Indeed, the charge transfer is expected to modify the initial trajectory of the neutral parent and lead to heating and deviation of the anion beam. Here, we calculate these mechanical effects in order to optimize the experimental configuration. First, it is worth noting that for a too-small relative velocity, ${\boldsymbol{v}} ={{{\boldsymbol{v}} }_{A}}-{{{\boldsymbol{v}} }_{M}}$, between the Rydberg atoms, A, and the molecule M, M⁻ and A⁺ do not have enough energy to be separated and will form a bound ionic pair, M⁻ - A⁺. On the other hand, M⁻ and A⁺ are dissociated if the relative velocity is equal to or greater than the critical velocity, ${{v}_{{\rm crit}}}=\sqrt{2({{E}_{{\rm reactant},\;\operatorname{int}}}-{{E}_{{\rm product},\;\operatorname{int}}})/\mu }$ where ${{E}_{{\rm product},\;\operatorname{int}}}$ is the internal (electronic + rovibrational) energy of the products (with similar notation for the reactants). Such a threshold is visible in figure 4. As the reaction (1) is endothermic, there is no reaction at zero relative velocity, which has strong experimental implications. In particulars, this means that A cannot flow within the supersonic beam in which M is seeded—by exciting the carrier gas in Rydberg state, for instance—because of the small relative velocity between the reactants.

Therefore, the Rydberg atoms must come from a different source, such as another beam or a vapor cell. The choice of the Rydberg system will strongly affect the efficiency of the process and the final beam properties. For instance, the anion formation rate, $K\sim v\sigma (v)$, decreases with decreasing Rydberg atom velocity [34] (see also figure 4). This effect is also enhanced by the post-attachment electrostatic interactions between the produced ions.

To get more insight about the process, we calculated the trajectories of an ideal monocinetic Rydberg beam orthogonal to a monocinetic molecular beam, a situation that is summarized in figure 6. Our results are given for BaF, a molecule that has a dipole moment of 3.17 D, seeded in a supersonic beam whose mean velocity differs by the use of the carrier gas. The calculation itself, performed in the center of mass, gives the scattering angle and the relative velocity of the products for a specific choice of the reactant velocities and impact parameter b. For a same relative velocity, and provided that the electron exchange is possible because the condition $b\leqslant {{R}_{c}}$ is fulfilled, note that the scattering angle increases with the impact parameter. It is found that there is a maximal scattering angle when $b\to {{R}_{c}}$ and no deviation when $b\to 0$. As this system has an axial symmetry, all the velocity vectors corresponding to a single value of b describe a cone, as illustrated in figure 6. Therefore, in the frame of the laboratory, the reaction leads to a redistribution of velocities (see figure 7; i.e., a global deviation and heating of the anion beam with respect to the molecular beam). Quantitatively the deviation is given by the change of direction of ${{{\boldsymbol{v}} }_{{{{\boldsymbol{M}} }^{-}}}}$ corresponding to $b\to 0$. Figure 8 shows that the deviation of BaF⁻ is always small for a fast supersonic beam. The heating process is evaluated by the maximal extents of transverse velocity ($\Delta {{v}_{M\bot }}$) and longitudinal velocity ($\Delta {{v}_{M\parallel }}$) of the molecular beam with respect to the deviated axis of the anion beam. As shown in figure 9, there is a global trend of little heating for a fast supersonic beam. It is noteworthy that the anion velocity tends to be less altered when the relative velocity is large because it is not much affected by the Coulomb effect. Anyway, our calculation for BaF unambiguously indicates that the use of a fast molecular beam, as can be provided by He as a carrier gas, should lead to the smallest increase in temperature.

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In conclusion, a fast, neutral molecular beam—such as that produced using He seeded gas in a supersonic expansion—will remain almost identical especially with a beam temperature that is almost unchanged when transformed into its anionic counterpart by Rydberg electron transfer.

Ideally we would like to transform about 100% of the molecules into anions. Assuming a molecular beam with a velocity in the lab frame ${{v}_{M}}\approx 1000$ m s⁻¹, a realistic Rydberg cloud size of $D\approx 1$ cm, and a typical formation rate of $K={{10}^{-7}}$ cm³ s⁻¹, it would require a Rydberg density $\rho ={{v}_{A}}/KD\approx {{10}^{12}}$ cm⁻³, which must be produced during the necessary collisional time of $\tau =D/{{v}_{M}}\approx 10$ μs. This is a strong limitation because the density, or similarly the Rydberg separation ${{R}_{0}}\sim {{\rho }^{1/3}}$, that can be sustained during this time is given by the equation $\tau \sim 20$ μs $\times \frac{\sqrt{M(amu)R_{0}^{5}(\mu {\rm m})}}{{{n}^{2}}}$ [35, 36]. It is thus advisable to use heavy atoms like Cs. As an example, 5 μs Penning ionization time is obtained for Cs(43p) at a density of $2\times {{10}^{9}}$ cm³ [37]. We could reach longer Penning ionization times, and thereby a higher density, by using the Rydberg state without resonant dipole-dipole interaction, as in the van der Waals $(ns)$ case, or by using a repulsive force between atoms. Atoms in the s state are also advantageous because the efficiency of the ion-pair (anion) production depends not only on the principal quantum number, n, but also on the orbital angular momentum, l, of the Rydberg electron; an s electron should produce the highest production rate [26].

However, Penning ionization always occurs [38] due to black body radiation or irregularity in the arrangement of the atoms [36]. Clearly, for molecules having a dipole moment close to the critical 2.5 D, as is required for high Rydberg states to form their anionic counterpart, it is trickier to get high efficiency in the formation process. However, for Rydberg states such that $n\lt 20$—in other words, for molecules with a dipole moment bigger than ∼3 D—such a high Rydberg density ($\rho \approx {{10}^{12}}$ cm⁻³) can be produced [39, 40].

Collisions of molecules with other molecules, ions, Rydberg or ground-state atoms might degrade the beam [41, 42]. Estimation of these effects is difficult. If some study of collisional effects with atomic negative ions exists [43], the collisional properties of dipole-bound negative ions have only been studied partially by Dunning's group [44, 45].

Interaction between ${{M}^{-}}$ and ${{A}^{+}}$ can lead to collisional destruction of the anion or simply heating of the beam. Let us assume that a 0.1 nA BaF beam (this will be limited by the space charge effect, as discussed later in the paper) with a 1-mm radius moving at 1760 m s⁻¹ corresponds to a density of roughly 4 × 10¹¹ m⁻³ and a mean interparticule separation of $\sim 100\;\mu$m. It is thus expected that going through the 1-cm cloud, the distance of minimal approach between ${{M}^{-}}$ and ${{A}^{+}}$ atoms is greater than 1 μm, a distance for which the anions see a weaker electric field than the field required to detach the electron [23]. We can conclude that, except for very low-binding-energy anions where the detachement field is especially low, no electron detachment would occur through a collision with ${{A}^{+}}$. However, deflection of the ${{M}^{-}}$ through such Coulomb interaction has to be evaluated. A simple estimation, which will be confirmed by the study of the space charge effect, is that the ${{M}^{-}}$ velocity change during the 1 ns needed to travel 1 μm at the closest distance from ${{A}^{+}}$ is on the order of 1 m s⁻¹.

Regarding collisions between M and ${{M}^{-}}$, destruction of the dipole-bound anion has been seen in collisions with polar targets as a result of rotational energy transfer. The measured reaction rates are as large as the typical anion formation rate, $\sim {{10}^{-7}}$ cm³ s⁻¹ (cross section 10⁻¹² cm²) [44, 45]. Fortunately, the density of the neutral molecules is much smaller than the density of the Rydberg atoms, and the anion formation rate will dominate. The collisional cross section between the formed anions, M⁻, and other nonpolar neutral species such as the atomic gas can be evaluated to be at most given by the Langevin rate, leading to the typical rate of $\sim {{10}^{-9}}$ cm³ s⁻¹ [46], which is a factor hundreds lower than the typical anion formation rate. We mention that electron transfer in collisions between dipole-bound negative ions and the attaching targets, such as SF₆, have been studied and the reaction rate is also small and limited by the geometrical size [47]. If we take the Langevin rate as an upper limit, the ratio of Rydberg atoms versus non-Rydberg atoms has to be as large as possible, and if possible greater than 1%. This value is easily achievable using a standard laser system for the Rydberg excitation [36, 48].

Interaction between the negative ions and the Rydberg atoms is probably important. However, contrary to a collision with a cation, when a free or Rydberg electron approaches a negative atomic ion, it will experience a repulsive field. Therefore, in general we do not expect the charge-exchange process to occur even if for specific systems, some di-anionic resonances also play a role [49, 50]. Such cross sections will thus probably be orders of magnitude lower than the one found in the cation case, where it can reach the Rydberg geometrical cross section, $\pi a_{0}^{2}{{n}^{4}}\sim {{10}^{-11}}$ cm² for ($n\sim 20$) [51, 52]. Estimation of the cross section for all other possible processes due to interaction between ${{M}^{-}}$ and $A(nl)$ is quite difficult. However, because the relative velocity, $v\sim 1760$ m s⁻¹, between the molecule and the Rydberg is very small compared to the orbital velocity of the Rydberg electron, $\sim {{v}_{B}}/n$ (${{v}_{B}}=2.19\times {{10}^{6}}$ m s⁻¹ is the atomic unit of velocity), and the impact parameter will be smaller than ${{a}_{0}}{{n}^{3}}v/{{v}_{B}}\sim 1$ nm (for n = 30), we will be in the orbital adiabatic collision regime, where the Rydberg orbital electron adjusts itself adiabatically to the slow ion perturbation [53, 54], such that no energetic transition occurs. Therefore, we can estimate the n or l Stark mixing cross section being negligible. Finally, we mention that precise choice of the $nlm$ quantum state—for instance, to have a nonpolarizable Rydberg state—can also help further reduse these collisional rates.

Even if experiments are obviously needed to clarify these aspects, we expect, except maybe in the case of very low-binding-energy anions, that no profound effect on the formation and the temperature of the anion beam will result from collisions between anions and other particles.

Several configurations for the Rydberg atoms can be suggested, including a vapor cell, a magneto-optical trap, an effusive beam, or a supersonic beam. However, the previous simulation and the constraints in terms of density and monochromaticity indicate that a Cs (slightly) supersonic beam might be a simple choice. A Mach number, M, of 5 may be enough to limit the energy dispersion of the anion beam. We recall here that $M\sim {{(D/\lambda )}^{0.4}}$, where D is the nozzle diameter and λ is the mean-free path [55]. For example, with a nozzle diameter of D = 2 mm, a cesium temperature of 350 $^{\circ }$C should be sufficient. Using a recirculating oven [56–60] is probably a good choice for this because it has the great advantage that the effused element usage is minimized, so in principle, it need only be reloaded after thousands of hours of operation [61].

The detachment field, E, required is such that the barrier drops below the energy of the dipole (quasi-)bound state. This leads to a field of 0.1 kV cm⁻¹ for a binding energy of 0.7 meV, and 10 kV cm⁻¹ for a binding energy of 15 meV [67].

We can apply a pulsed electric field to a specific region when the particles have the smallest velocity. A very short but intense electric field would detach the electron without significantly affecting the velocity of the anion. The simplest way of doing this is probably to abruptly increase the value of the field producing the deceleration to a value higher that the detachment value just after the stopping is performed.

Another method to neutralize the anion is to photodetach it. The photodetachement process could, in principle, even produce cooling in a similar way to one-way cooling: a proper laser spatial shaping can favor detachment of slow molecules [13].

Analytic description of dipole-bound anion photodetachment has been given in [68] based on the experimental data of [69]. Because of the very low binding energy, ${{E}_{{\rm b}}}$, the photodetachment cross section can be quite large, but it decreases, as ${{\omega }^{-2}}$, where ω is the photon frequency. From [68], we can reasonably assume a photodetachment cross section of ${{\sigma }_{{\rm pd}}}\approx 100{{(\hbar \omega /{{E}_{{\rm b}}})}^{-2}}a_{0}^{2}$. It is clearly advantageous to use a far infrared laser, such as a CO₂ laser with a wavelength of 10.6 μm. Using a 100-watt laser focused on 3 mm leads to a photodetachment rate near 10⁴ s⁻¹ for a small binding energy of 1 meV, and it goes to a photodetachment rate of 10⁶ s⁻¹ for a binding energy of 10 meV.