Merging two molecular beams of ND₃ up to the Liouville limit

The study of molecular collisions in crossed molecular beam experiments at energies well below 1 K has emerged as a new and exciting research frontier. Yet, designing experiments to reach sufficiently low collision energies (and sufficiently high energy resolutions) has proven a challenging task. In recent years, several novel experimental approaches to study low-energy collisions have been developed and successfully implemented, and a variety of new methods are being pursued that hold great promise to accomplish this feat in the future. Low interaction energies have been achieved by using a small beam crossing angle^1,2 or by controlling the reagent’s velocity using a decelerator.³ The lowest collision energies are reached when the relative velocity between both collision partners approaches zero. This can be achieved using intrabeam collisions,^4–8 i.e., by inducing collisions between atoms and molecules that are copropagating in a beam originating from a single source, or by merging two individual molecular beams by tangentially overlapping one beam on the other beam’s path.^9–12 Typically, merging is achieved using beam manipulation tools that use the force induced by inhomogeneous magnetic or electric fields employing the Zeeman or Stark effect, respectively.

Beam merging is a well-known technique in collision studies involving ion beams,^13,14 but its successful application in collision experiments with neutrals has only been reported recently. The merged beam approach is particularly appealing as the two beams containing the collision partners are produced independently, allowing for collision studies between a large number of atomic and molecular species within a large range of collision energies. However, this technique hinges on a key property of the particles in the two beams: they must have a different interaction with the force that merges the beams. Ideally, particles in one of the beams should be completely inert to the merging force, such that particles in the other beam can be manipulated and overlapped without affecting the first beam’s path.

In successful merged beam implementations thus far, this problem was circumvented using an appropriate choice of system. Narevicius and co-workers merged beams of metastable He or Ne atoms with beams of Ar or H₂ using a magnetic quadrupole guide.^9,15–17 Whereas the unpaired spins of the metastable atoms offered a significant magnetic dipole moment for efficient manipulation, the singlet electronic states of Ar and H₂ guaranteed complete insensitivity to magnetic fields. Osterwalder and co-workers developed a dual-guide merged beam experiment in which beams of molecules with an electric dipole moment were overlapped with beams of atoms with a magnetic dipole moment using curved electric and magnetic focusers, respectively.¹⁸ Recently, Tang et al. reported on a merged beam experiment using beams of NO and ND₃ that were merged by bending the ND₃ molecules into the NO beam’s path using an electrostatic curved hexapole.¹² The much smaller dipole moment of NO compared to ND₃ ensured that the deflection of NO by the curved hexapole was negligible. Beam merging using miniaturized electrode designs has also been achieved using Rydberg states that offer extremely large electric dipole moments.^11,19

The implementations mentioned earlier unfortunately exclude an important and interesting class of systems: the scattering of two molecules that each possess a significant (electric) dipole moment. These systems are of particular interest, as the long-range and anisotropic dipole–dipole interaction offers an exquisite “handle” to control and steer collisions.²⁰ At sufficiently low temperatures, small external electromagnetic fields can vastly change the interaction, yielding distinctive opportunities to engineer interaction Hamiltonians and to steer collision pathways. In addition, the scattering of two identical molecules offers a wealth of opportunities to study resonant energy transfer using (near)-matching energy levels. Recently, a peculiar and highly interesting scattering mechanism was predicted to occur in low-energy collisions between dipolar molecules that possess near-degenerate rotational energy levels with opposite parity.¹² A local maximum in the cross section was predicted at collision energy that connects to the Langevin region at high energies and the Wigner regime at low energies. Collisions involving molecules such as OH, ND₃, H₂CO, NH (a¹Δ), CH, SH, and CO (a³Π) all possess a significant dipole moment and the required energy level structure and are, therefore, prime candidates for low-energy scattering experiments.

Merging two dipolar molecules with near-matching velocities appears fundamentally impossible, however, as the beam manipulation tools required to steer one beam will necessarily also affect the other. This problem is more formally expressed as Liouville’s theorem, which states that the phase space density of a Hamiltonian system remains constant along every trajectory.^21,22 For two beams of identical particles, the “system” encapsulates both beams. Since the phase space density has to remain constant, one cannot overlap (i.e., merge) the two separate areas of the phase space, as this would increase the density. Merging (overlap in the spatial coordinate) is then only possible by allowing the packets to heat up (non-overlap in the velocity coordinate), setting a lower limit for the collision energy that can be obtained. This situation is also found in trapping experiments: simply adding a new packet of molecules to an existing sample of trapped molecules is not possible without heating or losing the already trapped molecules.

Certainly, for specific systems, one could contemplate strategies to circumvent this “Liouville limit,” making smart use of the energy levels of a molecule as its dipole moment can effectively vanish for certain electronic or rotational energy levels. For instance, NH (a¹Δ) radicals or CO (a³Π) molecules are easily manipulated using electric fields due to the pronounced Stark effect by virtue of close lying Λ-doublets, whereas NH (X³Σ⁻) and CO (X¹Σ⁺) are immune to electric fields.^23,24 Laser-induced population transfer from one electronic state to the other can, therefore, in principle be used to effectively switch off the dipole moment just before merging and to switch it on again thereafter. Similar ideas may be considered exploiting the Stark splitting of rotational energy levels in an external electric field according to their quantum number M. Molecules that possess states with M = 0 experience (at least in first order) no Stark effect, and population transfer to a state with M = 0 may allow for perfect merging. For instance, ND₃ can be prepared in the M = 0 component of the $JKp=11+$ lower inversion-doublet component of the rotational ground state via a microwave induced $11−,|M|=1→11+,M=0$ transition in the presence of an electric field.²⁵ (Throughout this paper, we follow the common convention that J refers to the rotational angular momentum of the molecule, K refers to the projection of J on the symmetry axis of the molecule, and M refers to the projection on the laboratory frame Z axis (parallel to the applied electric field). The label p, either + or −, indicates the parity of the tunneling split wave function at no electric fields applied.) Alternatively, the $21+,M=0←11−,|M|=1$ transition can be induced using THz radiation at a frequency around 600 GHz.²⁶ Finally, some molecules that have both an electric and magnetic dipole moment have the interesting property to lose their magnetic dipole moment when prepared in a given electronic state. The OH radical is an interesting example within this category. In its X²Π_3/2 electronic ground state, OH has a significant electric and magnetic dipole moment,²⁷ whereas in the X²Π_1/2 spin–orbit excited state only the electric dipole moment remains. This may allow for the merging of OH (X²Π_1/2) and OH (X²Π_3/2) radicals using a magnetic guide.

Yet, despite the fundamental restrictions imposed by the Liouville theorem and potential work-arounds, one may still consider experimental designs that spatially overlap particles originating from two separate molecular beams and accept the consequence that the lowest collision energy achievable is compromised. This can be accomplished by merging two electrostatic guides carefully, as has been demonstrated for miniaturized structures on a chip.^28–30 Alternatively, novel structures can be designed where a single electrode is shared by two independent beam manipulation elements, as demonstrated in a honeycomb hexapole structure by Shimizu et al.³¹ Using a pioneering idea that was first introduced by Osterwalder and co-workers,³² an electrostatic hexapole (6 electrodes) can gradually be converted into two quadrupole focusers of 4 electrodes each using two Y-shaped electrodes. The resulting “beam splitter” was successfully used to split a beam of ND₃ molecules into two components. Later, the same group demonstrated the merging of two ND₃ beams by operating the beam splitter in reverse.³³ To date, however, no collision experiments have been reported using this device.

Here, we present a novel implementation of a merged guide, inspired by the guide presented by Osterwalder et al., that is specifically optimized for ND₃–ND₃ and ND₃–NH₃ collisions. We define quantitative criteria for the electrode design and implemented numerical optimization procedures to come as close as possible to the “Liouville limit.” We describe technical aspects of our design that ensure optimal high voltage compatibility. We experimentally tested the performance of the device using conventional and Stark-decelerated beams of ND₃ and NH₃, which compares favorably with the performance predicted by numerical trajectory simulations. We present preliminary collision data, including near-matching velocities of both collision partners, demonstrating the feasibility of using this device for low-energy ammonia–ammonia collisions.

This article is organized as follows: In Sec. II, we summarize the original design by Osterwalder and co-workers. In Sec. III, we describe the criteria and underlying design concepts for our merged guide and describe differences compared to the design used by Osterwalder et al. In Sec. IV, numerical optimization procedures are presented to tailor the design to reach optimal overlap in phase space for the two merging packets, i.e., strategies to achieve merging up to the Liouville limit. The mechanical implementation of our merged guide is described in Sec. V, with particular emphasis on electrode production, mounting procedures, HV compatibility, and interfacing within our existing crossed beam setup involving a Stark decelerator. Experiments to test the performance of the merged guide using beams of ND₃ and NH₃ are presented in Sec. VI, and comparisons with predictions from numerical trajectory simulations are made. Finally, we present a preliminary collision experiment in Sec. VII that demonstrates the feasibility of low-energy merged beam collision experiments using our guide.

Let us first describe the design by Osterwalder et al., as it served as inspiration for the merged guide we developed. Osterwalder and co-workers started with two independent quadrupoles that are gradually brought together.³² If the quadrupole polarities are chosen such that they are mirrored, the two neighboring sets of electrodes from each quadrupole can ultimately merge using Y-shaped electrodes [see Fig. 1(a)]. In this figure, the first half of the Y-shaped electrodes are omitted, since the two ideal quadrupoles along arbitrary trajectories can be maintained during this phase. The beam merging effectively occurs when the two half-overlapped quadrupoles start to morph into a single hexapole, as shown in Fig. 1(a).

The resulting 3 × 2 array of rods still forms two ideal quadrupole wells, with a single set of electrodes shared by both quadrupoles. Next, the rods that just merged are moved outwards to gradually morph the geometry and potential into that of a single hexapole, as illustrated in Fig. 1(a). In principle, this idea can be used to merge any n-pole with an m-pole to make an (n + m − 2)-pole guide.

To construct this intricate shape, Osterwalder and co-workers fabricated the merged guide using 3D printing techniques. It was found that direct 3D printing of metal leads to porous samples with a rough surface unsuitable for high voltage (HV) applications; an experience we gained ourselves as well. Instead, the guide was printed from plastic, which was electroplated afterward to make the surface electrically conductive. This production method is extremely quick, versatile, and unlimited in the complexity of the electrode shapes. However, the bulk plastic unfortunately has inferior HV and heat resistance properties compared to metal. Although the successful application of an impressive voltage difference of 22 kV between neighboring rods was reported, there is a severe risk of high voltage arcing or surface currents along the plastic support structure, damaging the setup.

Luckily, although the electrodes have a complex shape, it is still possible to manufacture them through conventional, subtractive methods like CNC milling. With this approach, it is unfortunately not possible to produce the entire guide in one go. Instead, each electrode has to be manufactured separately in addition to a mounting system to electrically isolate, support, and align the electrodes, adding complexity to the setup. Yet, this will be our approach, as optimal HV performance is essential to our experiments. In addition, we made substantial changes to the merged guide design to optimally suit low-energy collision experiments in combination with a Stark decelerator.

In determining the precise shape of the electrodes, two main questions need to be addressed: (1) How can the geometry of the merged guide be defined, and how is it parametrized? (2) What parameters can be modified to optimize its performance? These questions will be addressed in the following sections.

Let us start by defining a right-handed Cartesian coordinate system in which the z axis points vertically upward while the x axis points along the main beam propagation direction. The origin is placed at the center of the end of the merged guide, where the electrode geometry is that of a hexapole.

To fully define the geometry of the merged guide, it is sufficient to define the shape of every consecutive (y, z) cross section or “slice” along the main propagation direction, as depicted in Fig. 1(b). This can be achieved most conveniently in the following way. First, the mirror symmetry between the top and bottom of the design should be recognized.³⁴ Next, one can impose that the distance between the geometric center of adjacent rods equals three times the rod radius, r_rod, ensuring that the electrode surfaces are always one rod radius apart.³⁵ With this restriction, we have to define the displacement of one rod only, as the other rods follow accordingly.

Optimizing z_inner(x) is crucial to maximizing the transmittance throughout the merged guide and overlapping the two beams with minimal collision energy, i.e., with minimal slushing of both packets when they evolve from the ideal quadrupole trapping field into the hexapole geometry. As a starting point, let us assume a quadratic function, $zinnerrrod(x)=a−bx2$⁠, ensuring that the merging starts relatively fast but ends smoothly. For a merging segment of length L, the curve should pass through the points (−L, 1.5) and (0, 3), yielding a = 3, $b=1.5L2$⁠. This curve was used to draw the merged guide shown in Figs. 1(b) and 1(c).

After drawing the merged guide in Autodesk Inventor, the 3D electrode geometry was imported into SIMION³⁶ to calculate the position dependent electric field throughout the guide. The spatial resolution of SIMION was set to 0.1 mm. The field strength along several cross sections of the merged guide is shown in Fig. 2. Panels (a)–(c) show cross sections at three different positions along the x axis to illustrate how the two quadrupole wells gradually deform and merge into one hexapole field. In panel (d), the field strength for the (x, y) cross section at z = 0 is shown, illustrating how the trapping well evolves in the longitudinal direction. It is seen that the two quadrupole wells merge into a single hexapole well smoothly.

Before optimizing the exact shape of the merged guide, there are two changes to the merged guide concept we would like to make.

The first relates to the evolution of the wells along the longitudinal direction. Figure 2 shows that both field minima follow a curved path. For our application in scattering experiments, it is actually preferred that one of the beams propagates through the guide in a straight line, for a number of reasons. First, in our experiments, one beam of molecules is prepared by a Stark decelerator. It is beneficial to keep this exceptionally well defined beam aligned with our detector as a reference, even when the merged guide is turned off. This will aid calibration of the new merged guide and any subsequent collision experiments. In addition, it will ease troubleshooting by compartmentalizing the complexity of the total experiment. Second, a straight beam path ensures transmittance at a wide range of velocities, enabling access to a large range of collision energies. Third, with a straight beam path, molecules like NO that have a smaller dipole over mass ratio than ammonia can still be used as a scattering partner.

Therefore, to deflect the Stark decelerated beam as little as possible, one of the arms of the merged guide should follow the central axis of the Stark decelerator. This means that one well along every (y, z) cross section should be centered around the z axis, which can be achieved by warping our previous design in the y-direction. From Fig. 2(d), the positions of the field minima can be found directly. Fitting a third order polynomial bound by the position of the quadrupole well, y_Qwell = 1.5r_rod, on one end and the position of the hexapole well, y_Hwell = 0, on the other yields a smooth function that describes the required displacement for every (y, z) slice of the merged guide. Applying this warp to the previous design leads to a new merged guide, whose electric field is depicted in Fig. 3. Note that one of the beams can now pass in a straight line, as desired.

The second change we propose requires a bigger deviation from the original design. Up until now, to move the beams in each well closer together, the center two rods had to shift outwards. This does not only bring the beams closer, but at the same time lowers the barrier between them. This can result in some particles already spilling over the barrier instead of their trajectories being curved in the right direction. If it were possible to maintain a high barrier while moving the minima closer together, the two beams would be separated better, and they could be manipulated with a smaller radius of curvature. To achieve this, the center electrodes can be made thinner.

Decreasing the size of the center electrodes only works up to a certain width, as the electrodes will be prone to electrical discharge if their edges become too sharp. We assume a minimal radius of curvature of 0.5 mm to be safe with respect to arcing, yielding a minimal width of 1 mm for the center electrodes. The merging now takes place in two phases, as illustrated in Fig. 4. First (panels a to b), the outer electrodes move inwards while the center electrodes decrease in width from 4 mm diameter rods to 1 mm wide “fins.” During this phase, the distance between the two beams is reduced by almost a factor of two, while the barrier between them stays intact. Second (panels b to c), the fins start moving outward, lowering the barrier and bringing the two beams into overlap. It should be noted that after the first phase, the outer rods are already closer to the center than in a standard hexapole. To correct for this and still have confining potential, the fins do not retract completely. If the fins are retracted further than 3 mm from the center plane, we again form two potential wells, this time in the vertical z-direction.

With these design changes in mind, Sec. III C will specify the exact dimensions of what we refer to as the finned merged guide. In Sec. IV, the remaining free parameters will be optimized numerically to maximize the performance of the merged guide for our anticipated low-energy merged beam collision experiments.

Like before, the design of the finned merged guide can be fully specified by defining each (y, z) cross section along the x-direction. This time, however, we choose to parameterize the (y, z) cross section as a function of a new parameter t. The first and second phases of the merging process are described by the interval t ∈ (−1, 0) and t ∈ (0, 1), respectively. The geometry can later be optimized by tuning the position of each slice along the x axis, which is captured by the function x_slice(t).

Equations (2) and (4) define the main features of the proposed merged guide geometry. To draw the complete design, one has to find similar equations that link points between the three panels of Fig. 4, such that smooth 3D shapes can be drawn along the parametrized curves. Care should be taken to ensure that each electrode surface is smooth everywhere, especially in those regions where the fields are strongest. We will not discuss this process here, as these details hardly affect the fields along each beam, and hence the performance of the merged guide.

Now, after warping the design in the y-direction by Eq. (5), one of the guiding arms again runs along the x axis, while the other arm doubles in distance with respect to the x axis.

Together, Eqs. (2), (4), and (5) define a family of merged guide designs characterized by their finned center electrodes, with the only remaining freedom to choose x_slice(t). To ensure that the merged guide performs optimally in a certain experiment, x_slice(t) can be numerically optimized specifically for the parameters pertaining to the experiment. This will be described in detail in the next section, focusing on the merging of two packets of ND₃ molecules.

We specifically optimize the finned merged guide for a planned experiment on low-energy ND₃–ND₃ collisions. This experiment can be regarded as the successor of a recent merged beam experiment using the NO-ND₃ system reported by Tang et al. In that experiment, a packet of NO (X²Π_1/2, J = 1/2f) radicals was prepared using a Stark decelerator and merged with a packet of ND₃ $(X1A1′,JKp=11−)$ molecules that was prepared using a curved hexapole. The curved hexapole had a bending radius of 0.641 m, a bending angle of 40°, and ended a distance of 65 mm before the laser detection region.¹² Packets of ND₃ with a fixed velocity of 370 m/s were guided by this curved hexapole by applying voltages of ±12 kV to the hexapole rods. The collision energy was scanned by producing different velocities of the NO packet using the Stark decelerator. The relatively small dipole moment of 0.16 D for NO compared to the 1.5 D moment for ND₃ ensured that both beams could be merged without significantly affecting the NO packet for all velocities used.

To study low-energy ND₃–ND₃ collisions, the finned merged guide will replace the curved hexapole. We impose to the design that the distance between the end of the finned merged guide and the laser detection region remains 65 mm, as this is the distance needed to incorporate a high-resolution velocity map imaging (VMI) detector.³⁷ Traversing this distance, both packets should optimally be fully merged, i.e., we impose to the design that both beams in this region interact with the lowest possible collision energy and best spatial overlap. In addition, we also impose on the design that there is a minimal contribution of collisions that occur within the merged guide where both packets only partially overlap. This will ensure that we study collisions under well-defined conditions, as these in-guide collisions occur in the presence of large inhomogeneous electric fields, substantially complicating the interpretation of the scattering signals.¹² We furthermore assume a fixed velocity of 370 m/s for the curved arm of the merged guide and that we can apply a maximum of ±12 kV to the electrodes.

To assess the fitness of any x_slice(t), the trajectories of 500 particles are simulated through the corresponding geometry. These particles have a mean velocity of 370 m/s and are initialized at the entrance of the curved arm of the merged guide with incident angle α. Both the speed and direction of the particles are normally distributed, with a standard deviation of 10 m/s and 0.5°, respectively. Spatially, they are uniformly distributed along 1 mm. In order to reduce the computational cost of these trajectory simulations, the z-dimension is ignored, i.e., only particles that move in the (x, y) plane are considered.

In order to numerically optimize x_slice(t), two things are required: (1) A parametrization of x_slice(t) with a reasonable initial guess. (2) An objective function, which evaluates the chosen x_slice(t) with a single number or fitness. Optimization routines can then be used to vary the initial guess for x_slice(t) and to find the local minima of the objective function. This procedure is discussed below.

It should be noted that not all of these parameters are independent. To make this function continuous at t = 0, b₁ = b₂ should hold. Furthermore, to place the end of the merged guide at the origin, x(1) = 0, one finds b₂ = −(a₄ + a₅ + a₆). This leaves 6 independent parameters that can be represented as a single vector, $a⃗=(a1,a2,a3,a4,a5,a6)$⁠. The starting angle of the particles, α, is nonzero. It is either fixed at a value close to 6° or added to $a⃗$ as a seventh parameter.

Using $a⃗0$ as initial parameters, the resulting electric field strength along the (x, y) cross section of the finned merged guide is shown in Fig. 5. As intended, one of the arms is aligned with the x axis, while the other follows a circle with R = 0.641 m.

To evaluate the performance of the design, it is instructive to look at the transverse (y, v_y) phase space distribution of both packets at various longitudinal positions along the beamline. These are displayed in Fig. 6(a) as simulated from $a⃗0$⁠. Particles guided through the curved arm are displayed as blue dots, while the particles guided through the straight arm are displayed as orange dots. The dark colors represent the particles when they exit the merged guide, while the light colors in panel (b) represent the same particles at the plane of detection, 65 mm behind the merged guide. As can be seen, this geometry actually performs poorly, as the particles guided through the curved arm have a large transverse velocity at the exit of the merged guide, even though they are roughly centered around y = 0. This causes them to drift to positive y values during the 65 mm free flight further downstream, missing the point of detection and limiting the overlap with the collision partner. Clearly, the particles “fly off track” upon exiting the merged guide, and the curvature of the guide should be modified to prevent this.

Taking the poor performance of the initial guess as an example, it is clear that designing a structure that gradually converts two wells into a single well requires careful consideration. For our intended application, a proper merged guide should result in a transverse phase space distribution from the curved arm that is located as close as possible to the phase space from the straight arm, which is centered around (y, v_y) = (0, 0). (Note that perfectly overlapping both phase space distributions is forbidden by the Liouville theorem unless their combined phase space volume is maintained.) To achieve a design through numerical optimization that meets this requirement, a single score for the performance of a geometry is required. An algorithm can in turn minimize this score by varying $a⃗$⁠. The function that calculates this score is called the objective function, f_obj. Our requirements for f_obj can be summarized as follows:

1. f_obj is small ⇔ many particles reach the detection laser, which we modeled with a diameter of 1 mm.

2. f_obj is small ⇔ the detected particles have a low transverse velocity.

3. f_obj should be continuous, even when all particles miss the detection laser.

4. f_obj should not get skewed by a few “bad” particles that are not detected.

Requirements (1) and (2) are competing in the Liouville limit since the particles that travel straight already occupy the phase space region centered around (y, v_y) = (0, 0). Consequently, it is hard to find an unambiguous objective function. In general, different choices for the objective function are possible, each of them leading to a slightly different solution.

We numerically optimized $a⃗$ starting with $a⃗0$ and imposing a fixed starting angle of α = 6°. The optimizer converged to a solution represented by vector $a⃗1$⁠, thereby decreasing the value of the objective function by over five orders of magnitude. The exact parameters can be found in Table I. Again, we evaluate the performance of this solution by looking at the (y, v_y) phase space distribution of the resulting packets that are shown in Fig. 7. It is seen that with solution $a⃗1$⁠, the two packets are dramatically closer at the exit of the guide, and the velocity distribution v_y of the packet from the curved arm is nicely centered around v_y = 0. Upon exiting the guide, the two packets propagate together in free flight toward the plane of detection, yielding the phase space distributions shown in Fig. 7(e). During this flight, both packets overlap spatially and are distributed near-symmetrically around the origin, although a strict boundary remains between them as imposed by Liouville’s theorem. Instead of overlapping in both the spatial and velocity coordinates, the two packets swirl around one another.

Solution $a⃗1$ represents the optimal merging strategy within the geometrical constraints of the finned merged guide. On closer inspection, however, solution $a⃗1$ may still be further improved. The phase space distribution of the two packets is still separated by a little distance throughout, whereas the Liouville theorem in principle allows for a situation in which both packets touch each other. In addition, it is seen that the secondary beam (blue dots) has already crossed the straight beam axis (y = 0) and is now being focused back by the electrodes along the outer bend, i.e., it has already made a half-oscillation in phase space. The primary beam has formed a tail that has started to move upward to relatively large values of v_y.

We empirically found that we can further improve on this design by shortening the length of the guide and limiting the amount of time both packets experience the potential well of the final hexapole. In our optimization routine, this can be achieved by omitting the final slices of the parametrization by limiting the range of t to t ∈ (−1, t_stop). For each choice of t_stop the numerical optimization was performed again, yielding new solutions. Three are presented here, with increasingly smaller t_stop: $a⃗2$⁠, $a⃗3$ and $a⃗4$ (see Table I for more details). The phase spaces at the exit of the merged guide are shown in Figs. 7(b)–7(d), while those at the plane of detection are shown in Figs. 7(f)–7(h).

The design parameterized by $a⃗2$ results in a phase space distribution that outperforms solution $a⃗1$⁠, as is also represented by the lower value for f_obj. The two packets are almost touching each other and spatially overlap near perfectly during the free flight upon exiting the guide, with a minimum value for v_y (and hence the collision energy). The total length of the design is 145 mm, only 6 mm shorter compared to the design pertaining to solution $a⃗1$⁠.

Yet, when considering the construction of a finned merged guide, additional requirements beyond the objective function may play a role. In our envisioned experiment, limiting the amount of collisions that can already take place inside the guide is essential. As mentioned before, these in-guide collisions can substantially complicate the interpretation of the scattering results, as the cross sections for the scattering between two dipolar molecules can increase by orders of magnitude in the presence of an external electric field. Ideally, the two packets should barely overlap upon exiting the guide but have a phase space distribution such that full spatial overlap is only achieved in the free flight section downstream from the guide.

Referring back to Fig. 7, solution $a⃗2$ clearly offers the best overlap of both packets in phase space, yet it scores less well with respect to our additional requirement of limiting the overlap inside the guide. A significant part of the secondary beam still crosses y = 0 before exiting the guide, and the probability for collisions to occur inside the guide is relatively high. In this respect, solutions with an even lower value for t_stop yield phase space distributions with a larger separation between the packets, together with an increased transverse velocity of the secondary beam (see solutions $a⃗3$ and $a⃗4$ in Fig. 7). Although the values for f_obj are less optimal for these solutions (but still reasonable), these solutions efficiently prevent in-guide collisions. They produce a phase space distribution for the secondary beam that lags behind the distribution of the primary packet (y < 0), but with a positive mean transverse velocity v_y > 0. During the free flight section between the guide and the detection point, the secondary beam overtakes the primary beam, and full spatial overlap is achieved at the expense of a slightly larger mean collision energy. In addition, these solutions substantially reduce the overall length of the guide, reducing the longitudinal spreading of the packets as they propagate through the setup.

All designs presented in this section are tailored to the experimental constraints incorporated into the optimization, including a 370 m/s forward velocity. Considering these designs and the rationale to gauge their performances, we chose to build the design represented by $a⃗3$⁠. As for our intended experiments, this design offers a good compromise between efficient merging close to the Liouville limit and more pragmatic problems related to the interpretation of the experimental scattering results we may obtain. The manufacturing of the finned merged guide according to this design and its implementation into our existing crossed beam setup will be described in detail in Sec. V.

The numerically optimized design of the merged guide has a separation between both entrance quadrupoles of only 6 mm, which is too small to interface the guide with a Stark decelerator or conventional beam sources. The guide was therefore extended at either entrance with ideal quadrupoles. On the straight arm, the electrodes were extended with a straight quadrupole of length 107 mm, whereas the curved arm was extended with a curved quadrupole with a 641 mm radius of curvature, ending at 15° with respect to the straight axis. The latter value followed extensive numerical optimization as described below in Sec. V C. These quadrupole extensions are fully integrated into the design of each electrode; further extensions are possible by placing additional beam manipulation elements before the merged guide entrances depending on requirements imposed by existing setups (see also Sec. V C).

Figure 8 shows an exploded view of the resulting six individual electrodes of the finned merged guide annex entrance quadrupoles. Panel (a) shows the exit side where the six electrodes of the hexapole are visible, whereas panel (b) shows the two quadrupole geometries at the entrance side. The beam axes at the entrance and exit are indicated by the red dots. For clarity, a zoom-in of the exit as well as a cut through the center plane of the construction are shown in panels (c) and (d), respectively.

The shape of the individual electrodes was first determined using the numerical optimization procedures described earlier. The 3D shape of all electrodes could be drawn in Autodesk Inventor 3D CAD software using Eq. (6) and the geometrical constraints discussed in Sec. III C. The resulting geometry files are then directly read by a 5 axis CNC milling machine at Radboud University workshops. The electrodes were milled from aluminum 7075-T6 and manually polished after milling for excellent high voltage compatibility.

In designing the mechanical support for the six electrodes, it is critical to (1) ensure proper alignment of the electrodes with respect to each other and (2) ensure excellent high voltage compatibility. For the former, we impose an alignment accuracy of ∼100 μm, whereas for the latter, we know from experience that the most critical parts causing high voltage breakdown are insulators that separate mechanical parts placed at different potentials from each other.

Both requirements are met using a mechanical electrode support concept as shown in Fig. 9. The electrodes are clamped in squared aluminum brackets with rounded edges on all sides, preventing the need for insulators across the small distance between electrodes. Like for a standard quadrupole or hexapole, our finned merged guide is operated with only two independent voltages. One set of three electrodes shares three mounting brackets, whereas the other set is mounted using only two brackets. The brackets have carefully designed and milled mounting planes, in which the electrodes lock into position, ensuring positioning of the electrodes with an accuracy defined by the high accuracy with which the mounting brackets are machined. The electrodes are tightened to the brackets using a screw from the side surfaces of the brackets. Carefully placed venting holes ensure optimal vacuum compatibility.

The two sets of three electrodes need to be aligned with respect to each other. This is ensured by locking the brackets onto an aluminum baseplate using two ceramic (aluminum oxide; Al₂O₃) rods per bracket. Again, we solely rely on machine accuracy for all parts, with no freedom to manually align the structure. The holes supporting the Al₂O₃ rods are recessed with a relatively large radius of curvature to shield the triple points (interface between ceramic, metal, and vacuum) that are known to easily initiate high voltage breakdown. A photograph of the assembly is shown in Fig. 10(a).

The mounting concept as shown in Fig. 9 offers the additional advantage that the structure can be constructed and pre-tested with respect to mechanical and high voltage stability in a separate vacuum chamber before installing it in the experiment. During high voltage conditioning tests, we have successfully applied a voltage of ±13 kV between the two sets of electrodes for several hours, with currents below our detection limit of 100 nA. The device has been routinely used in our experiments using a voltage of ±12 kV, where additional high voltage switches were used to gate the high voltage pulses applied to the electrodes.

In our anticipated ND₃–ND₃ merged beam experiment, the straight and curved arms of the merged guide are fed by packets of ND₃ molecules produced by a Stark decelerator and conventional beam source, respectively. To accommodate sufficient space for the Stark decelerator, the separation between the two entrance quadrupoles needs to be further increased. This is accomplished by installing a straight quadrupole between the Stark decelerator and the straight arm of the merged guide and a curved hexapole in front of the curved arm. Unlike the integrated extensions mentioned in Sec. V A, these extensions are separate electrostatic guides.

The straight quadrupole has a length of 20 cm, in which the quadrupole field is generated by the rounded edges of four plate electrodes. Like in the merged guide design, these electrodes are supported by aluminum brackets mounted on a baseplate using ceramic feet. A photo of part of this quadrupole is shown in Figs. 10(b) and 10(c). Only modest electric fields are required to transport the ND₃ molecules emerging from the Stark decelerator to the merged guide, and the quadrupole was operated by applying voltages of only ±1.5 kV to the electrodes.

On the curved side, one could consider installing either a curved quadrupole or a hexapole. A quadrupole offers a linear dependence of the electric field as a function of the displacement from the beam axis, whereas for a hexapole this dependence is quadratic. Consequently, molecules with a predominantly linear Stark effect (such as ND₃) experience a linear restoring force resulting in a near-perfect oscillatory motion in a hexapole, whereas molecules with a predominantly quadratic Stark effect (such as NH₃) are best manipulated using a quadrupole. As our main objective involves ND₃–ND₃ collisions (with a possible future extension of a second Stark decelerator to load the curved arm), we opted for installing a curved hexapole. Using NH₃ in the curved arm is then still possible, but its trajectories are less nicely described.

As a quadrupole and hexapole have quite different phase space acceptances, care must be taken to design the curved hexapole such that the packet of ND₃ molecules smoothly transitions from the curved hexapole into the curved quadrupole entrance of the merged guide. From extensive numerical optimizations, we found that a curved hexapole with a 641 mm radius of curvature, spanning a 25° segment of a circle, is optimal. These parameters result in the tight focusing of a ND₃ packet traveling with our target velocity of 370 m/s into the quadrupole entrance by the hexapole, i.e., the phase space distribution has a narrow spatial but large transverse velocity spread upon entrance of the merged guide. As the quadrupole has a large velocity acceptance, this distribution is nicely matched to the acceptance region of the quadrupole, resulting in the least amount of heating. The ND₃ molecules traveled through the curved hexapole at a distance of 1.54 mm from the geometric center due to the centrifugal force. We aligned this equilibrium orbit with the geometric center of the quadrupole entrance. A photograph of the curved hexapole is seen in Fig. 10(b).

Numerical simulations were performed to find the optimal voltages to operate the curved hexapole. Figure 11 shows the normalized expected peak intensity as a function of the voltages applied to the merged guide and curved hexapole. It is seen that the maximum intensity is found near the diagonal, i.e., optimal performance is expected when the curved hexapole and merged guide share the same voltages. Deviations from this diagonal are only found for voltages exceeding ±11 kV. We therefore electrically connected the electrodes of the curved hexapole to the electrodes of the merged guide, such that both devices can share a single set of high voltage power supplies and switches.

In order to verify the performance of the merged guide in conjunction with the rest of the setup, we performed full 3D numerical trajectory simulations. Realistic time-dependent phase space distributions of ND₃ molecules were generated for each source, mimicking the molecular beam pulses on both sides of the experiment. To assess the performance in the most critical situation, both beams travel at matching velocities around 370 m/s. These distributions were then propagated through each consecutive manipulation element along each beamline, i.e., a Stark decelerator, quadrupole, and merged guide for the straight arm, and a curved hexapole in conjunction with the merged guide for the curved arm. Collisions between the two packets were simulated using the same method as Tang et al.,¹² taking the phase space overlap of both packets emerging from the merged guide into account. For this, we only considered the 65 mm free flight section from the guide exit to the laser ionization region and ignored the possibility that both packets may already partially overlap before exiting the merged guide.

The results are summarized in Table II. In this table, we list the relevant performance for different hypothetical versions of the experimental setup, such as the relative densities of the straight (I₁) and curved (I₂) beams, the collision energy (E), collision energy resolution (σ_E), and the relative number of collisions (I_col) that may be expected from the simulated overlap of both packets. For ease of comparison between the different configurations, we normalize the intensity of the Stark decelerated beam, after propagation in free flight to the detection region, to 100. For the curved beamline, the intensity that is found if the curved hexapole as used by Tang et al. is used throughout is also normalized to 100. We note that the suggestion that both these reference beams have equal intensity is not true; from experiments, consequently, the packet of molecules exiting the curved beamline has approximately a factor of ∼10 to 100 higher density than the Stark-decelerated packet.

The second line of Table II pertains to the situation where no merged guide is used, i.e., this is the experimental configuration as used by Tang et al. for the NO–ND₃ system. It is observed that if this configuration is used for ND₃–ND₃, about 85% of the ammonia molecules that exit the Stark decelerator are deflected by the curved hexapole. The particles that do pass the curved hexapole are transversely heated significantly, occupying a ring in phase space that surrounds the secondary beam. This results in a rather poor beam overlap and a relatively high minimal collision energy that is about four times larger than the minimal collision energy obtained by Tang et al. for NO–ND₃. For the curved side, the harmonic potential of the curved hexapole is ideal for molecules with a linear Stark effect like ND₃, such that the beam can be focused toward the interaction region, resulting in an intense secondary beam.

The third line in Table II corresponds to the situation where the finned merged guide, including the quadrupole and curved hexapole to load the guide from the straight and curved arms, respectively, is installed. A large increase in density is observed in the straight arm. With the quadrupole and merged guide in place, particles are efficiently transversely focused upon exiting the Stark decelerator, yielding a large intensity increase for the primary beam. The intensity of the curved secondary beam, however, is reduced by 70%. From the simulations, we conclude that this loss is mainly caused by the fact that molecules are first guided by a hexapole field and then loaded into a quadrupole field in the first segment of the merged guide. Consequently, molecules do not experience a harmonic potential throughout, resulting in less than ideal focusing properties and a reduction in density. Yet, the overall performance of this arrangement for ND₃–ND₃ is dramatically superior to a strategy of performing a ND₃–ND₃ collision experiment using the original arrangement that was designed and ideal for NO–ND₃. The minimum collision energy of 0.17 cm⁻¹ is only about a factor of 2 larger than the energy that could be achieved for NO–ND₃, as the merged guide inevitably compromises the phase space overlap of both beams that can fundamentally be achieved. By simply comparing the product of the two beam intensities between the second and third lines, one would expect a factor of (5.55 × 0.32)/(0.15 × 1.00) ≈ 12 increase in collision signal. However, this value is even exceeded, as a factor of 23 is predicted by the simulations. This underlines the superior overlap of the two beams achieved by the merged guide, allowing for a larger probability for collisions to occur during their flight from the guide exit to the detection region.

For the last line of the table, the curved arm was loaded with NH₃ molecules rather than ND₃, which will be the preferred configuration to probe ND₃–NH₃ collisions in our experiments. It is observed that the curved arm of the merged guide performs a factor of 1.7 better with NH₃ compared to ND₃. This is perhaps surprising, as the merged guide was optimized for ND₃ in our simulations. A full analysis of this observation is beyond the scope of this work, but we hypothesize that the quadratic Stark shift of NH₃ that is present up to rather large electric fields reduces the loss of molecules during the transition from a hexapole to a quadrupole configuration. The resulting minimal achievable collision energy amounts to 0.15 cm⁻¹, whereas the gain in expected collision signal is slightly below a factor of 30.

Based on the predicted performances of the merged guide and Stark decelerator assembly as summarized in Table II, the expected beam densities and collision signal levels are sufficient to perform ND₃–ND₃ and ND₃–NH₃ scattering experiments at collision energies down to 0.15 cm⁻¹. Preliminary collision experiments for these systems using the merged guide are presented in Sec. VII.

The performance of the merged guide was experimentally tested by probing the peak intensity of the ND₃ molecules emerging from either the straight or curved arm using standard 2 + 1 REMPI at a distance of 65 mm from the exit of the guide. The straight arm was loaded with a packet of ND₃ molecules from a 2.6 m long Stark decelerator, as described in detail elsewhere.¹² The curved arm was loaded with a conventional molecular beam. Both molecular beam pulses were generated using a room-temperature NPV³⁹ by entraining ∼2% ND₃ in a carrier gas at a backing pressure of 1 bar and skimmed using a 3 mm diameter skimmer placed ∼15 cm from the nozzle orifice.

For the straight arm, the peak signal intensity was measured as a function of ND₃ velocity and voltages applied to the decelerator, quadrupole, and merged guide, both for the situation where the quadrupole and merged guide were switched on or off. Overall, good qualitative agreement with the simulations was obtained (data not shown), although the signal gain when applying voltages to the quadrupole and merged guide was over-estimated in the simulations by about a factor of 1.5. This signal gain is very sensitive to the exact laser detection volume, and we, therefore, consider this quantitative disagreement not unreasonable.

The curved arm was tested using ND₃ seeded in Xe throughout, as the design of the merged guide was optimized for a fixed velocity of 370 m/s. The peak signal intensity was measured as a function of the voltage applied to the curved hexapole and merged guide. It was observed that maximum transmission is obtained for voltages of ±11 kV, which was in quantitative agreement with the simulations. We limited our experimental tests to below ±12 kV to avoid a possible electrical breakdown.

The performance of the curved arm was further investigated by measuring full arrival time distributions of the ND₃ molecules arriving in the detection region. This distribution will generally not directly represent the velocity distribution of the original molecular beam, as the transmittance of the curved hexapole and merged guide is expected to critically depend on the velocity of the molecules. Molecules that are too fast will not follow the curvature, whereas molecules that are too slow will have sub-optimal focusing properties within the structure. In particular, efficient transfer of molecules from the curved hexapole to the merged quadrupole guide is only possible if the (transverse) phase space distribution of the molecules evolves as assumed in our simulations.

Consequently, only a narrow velocity class will be transmitted. We can take advantage of these effects and develop experimental probes to critically assess the performance of the curved arm by switching off the voltages on the guide prematurely. The idea is that with an early interruption of the guiding fields, only a subset of the molecules will arrive at the detector after propagating in free flight. These molecules originate from a narrow region in phase space (negative position but positive velocity coordinate), such that measuring the arrival time distribution for different switch-off times directly probes the phase space distribution of the molecules within the guide. We introduce the parameter t_off to define the switching off with respect to the timing of the detection laser, which is kept fixed for all measurements and defines t = 0. The arrival time distribution is then effectively measured by scanning the timing at which the valve fires.

Figure 12(a) shows the arrival time distributions as a function of t_off as a 2D plot, together with the expected distributions from simulations in panel (c). It is seen that qualitative agreement is obtained between the experimental and simulated distributions. Maximum intensity is observed for t_off ≥ −200 µs, as the molecules then experience fields throughout their entire journey through the guide. When the guide is switched off earlier, the arrival time distribution narrows, in agreement with our interpretation that only a subset of the molecular distribution can reach the detector in free flight. An interesting revival is observed for t_off ∼ −400 µs, which is also observed in the simulations. The reduced intensity observed at t_off ∼ −350 µs likely results from a part of the phase space where the molecules exit the merged guide more diverging.

Similar experiments and simulations were conducted for NH₃ molecules that pass through the curved arm [see Figs. 12(b)–12(d)]. As explained earlier, NH₃ will behave differently from ND₃ in the hexapole and merged guide, as the large inversion splitting causes a second order Stark effect up to relatively high fields. Nevertheless, beams of NH₃ can still be used in the curved arm, and we in fact observed a slightly higher signal intensity for NH₃ compared to ND₃, in agreement with Table II. The measured arrival time distributions for NH₃ are generally narrower compared to ND₃, in qualitative agreement with simulations.

To probe the quantum state purity of the ND₃ molecules emerging from the merged guide, we recorded 2 + 1 REMPI spectra in the $B(ν2′=5)←X(ν2=0)$ region,⁴⁰ where ν₂ indicates the vibrational quantum number for the umbrella mode of ND₃. Figure 13 shows the REMPI spectra recorded for both the straight (upper trace) and the curved (lower trace) beamlines. To record these spectra, the same conditions for the laser were used as during scattering measurements, i.e., we optimize for high detection efficiency by choosing laser powers to just saturate the transitions. Consequently, the spectral resolution in the measured spectra is limited by power broadening. From the spectra, it can be seen that the molecules originating from both beamlines are rotationally cold and near-exclusively populate the low-field-seeking $JKp=11−$ state. In the spectrum pertaining to the straight beamline, three weak spectral features can be discerned. These features can be attributed to the $22−$ first rotationally excited state, which lies 14.5 cm⁻¹ above the $11−$ ground state and has a small initial population in the beam. This state is low-field seeking, and molecules in this level are focused by the beam manipulators. Population in this state is not observed for the curved beamline, most likely because the Stark effect of the $22−$ state is not ideal to properly guide the molecules through the curved beam elements.

To demonstrate the feasibility of performing low-energy collision experiments with our new merged guide, we performed preliminary collision experiments using the ND₃–ND₃ system. This system is of particular interest, as recently an interesting scattering mechanism was predicted to occur in collisions between two polar molecules, where each molecule possesses near-degenerate rotational levels with opposite parity.¹² 

We tested the ability to observe scattering signals for ND₃–ND₃ collisions under the most difficult conditions, i.e., when both beams travel with matching velocities resulting in the lowest collision energy. The premise is that if we can unequivocally observe scattering signals under these conditions, scanning toward higher energies should then be feasible as well, as the kinematic conditions and beam densities will become more favorable as the collision energy is increased.

We operated the Stark decelerator to produce packets of ND₃ molecules with velocities ranging between 375 and 400 m/s. The curved arm was loaded with ND₃ at a fixed velocity of 375 m/s. Isotope labeling was used in order to spectroscopically separate signals from the Stark decelerator (¹⁵ND₃) and curved beam (¹⁴ND₃). Inelastic collisions are probed by recording the collision induced population increase in the $11+$ level of the packet of ¹⁵ND₃ molecules emerging from the decelerator using 2 + 1 REMPI. The integral ion signal on a selected part of the MCP was recorded without attempting to record images using velocity map imaging.

Signals were recorded by scanning the velocities of the Stark decelerated packet using a repeated cycle addressing six different velocities. During this cycle, for each velocity, the signal is recorded for three different configurations of the experiment, averaging over 40 consecutive shots of the experiment per configuration. In the first, both beams are in full overlap, such that collisions can occur. In the second, only the straight beam is operated, generating a background signal in the $11+$ detection channel originating from this beam alone. In the third, the $11+$ background signal is recorded by operating exclusively the curved beam. This cycle is then repeated 265 times, after which the number of recorded ions per velocity and per configuration is binned.

Figure 14 presents the resulting average recorded events per shot of the experiment. The combined background of both individual beams is plotted (red data points), together with the signals recorded when both beams are brought in overlap (blue data points). Clearly, the blue data points are well above the red ones through the energy range probed. The collision signal levels are small (⁠$∼0.1$ detected collision event per shot) but are discerned with statistical relevance within a reasonable signal accumulation time. We conclude that with the merged guide presented here, low-energy collision experiments using two dipolar molecules are feasible at collision energies down to 0.17 cm⁻¹.

It is noted that the combined background signal is dominated by the Stark decelerated beam, which accounts for about 90% of the recorded background signal. This background signal mainly originates from ¹⁵ND₃ molecules that reside in the $11+,M=0$ level in the original beam. Molecules in this state experience, in first order, zero Stark shift and propagate through the apparatus without being affected by electric fields. The curved arm hardly contributed to the background level, as (i) only trace amounts of ¹⁵ND₃ are present in this beam, and (ii) molecules in the $11+,M=0$ level will not follow the curvature of the hexapole and guide. The background in the straight beam could potentially be reduced substantially by designing a merged guide that curves both beams (as in the original design by Osterwalder et al.), but this would compromise our desire to propagate the Stark decelerated beam on a straight axis throughout the entire beamline.

We have developed an electrostatic guide that allows for the optimal merging of two beams of polar molecules. Based on a previous design by Osterwalder et al., we optimized the shape of the electrodes using extensive numerical simulations to merge the two beams with an optimal overlap in phase space allowed by the Liouville theorem, i.e., yielding optimal probability for scattering at the lowest possible collision energy. Furthermore, the design allows for the merging of both beams in a field-free region upon exiting the guide, facilitating the interpretation of experimental scattering results. The merged guide was successfully used to observe scattering for the ¹⁵ND₃–¹⁴ND₃ system at collision energies down to 0.18 cm⁻¹. The merged guide presented here offers a powerful method to experimentally probe low-energy collisions between two dipolar molecules with unprecedented accuracy and in hitherto unexplored energy regimes and can help us to unravel the peculiar and highly interesting low-energy collision properties of dipolar molecules.

This work was part of the research program of the Netherlands Organization for Scientific Research (NWO). S.Y.T.v.d.M. acknowledges support from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Program (Grant Agreement No. 817947 FICOMOL). We acknowledge A. Osterwalder (EPFL Lausanne) for carefully reading the paper and for stimulating discussions. We acknowledge N. Janssen and T. Kuijs for their excellent technical support.

The authors have no conflicts to disclose.

S. E. J. Kuijpers: Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). A. J. A. van Roij: Data curation (equal); Investigation (equal); Methodology (equal); Validation (equal). E. Sweers: Investigation (equal); Methodology (equal); Resources (supporting); Validation (equal). S. Herbers: Data curation (equal); Formal analysis (supporting); Investigation (equal); Methodology (equal); Validation (equal); Visualization (supporting). Y. M. Caris: Data curation (equal); Formal analysis (supporting); Investigation (equal); Methodology (equal); Validation (equal). S. Y. T. van de Meerakker: Conceptualization (lead); Funding acquisition (lead); Project administration (lead); Resources (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author, S. Y. T. v. d. M., upon reasonable request.