Bimodal velocity and size distributions of pulsed superfluid helium droplet beams

Abstract

We report detailed measurements of velocities and sizes of superfluid helium droplets produced from an Even–Lavie pulse valve at stagnation pressures of 20–60 atm and temperatures between 5.7 and 18.0 K. By doping neutral droplets with Rhodamine 6G cations produced from an electrospray ionization source and detecting the positively charged droplets at two different locations along the beam path, we determine the velocities of the different groups of droplets. By subjecting the doped droplet beam to a retardation field, size distributions can then be analyzed. We discover that at stagnation temperatures above 8.0 K, a single group of droplets is observed at both locations, but at 8.0 K and below, two different groups of droplets with different velocities are detectable. The slower group, considered from fragmentation of liquid helium, cannot be deterred by the retardation voltage at 9 kV, implying an exceedingly large size. The faster group, considered from condensation of gaseous helium, has a bimodal distribution when the stagnation temperatures are below 12.3 K at 20 and 40 atm, or 16.1 K at 60 atm. We also report similar size measurements using low energy electrons for impact ionization, and this latter method can be used for facile in situ characterization of pulsed droplet beams. The mechanism of the bimodal size distribution of the condensation group and the reason for the coexistence of both the condensation and fragmentation groups remain elusive.

I. INTRODUCTION

Superfluid helium droplets formed from either a continuous source or a pulsed source are typically polydispersed, and for different applications, different size ranges are desired.^1–11 The size distributions of superfluid helium droplets have been determined using a variety of methods, including diffraction,¹² deflection,¹³ depletion,^1,14–18 and, for charged droplets, electrostatic manipulation.^19–25 Using diffraction gratings and relying on the wave nature of particles,¹² Schoellkopf and Toennies measured the sizes of neutral helium clusters containing less than 100 atoms. Based on the degree of deflection of a droplet beam upon collisions with a well characterized atomic beam, Harms et al. also characterized the size distribution of clusters containing thousands of atoms.¹³ The depletion method relies on evaporation of helium atoms from droplets upon collisions with a variety of species, including neutral molecules with spectroscopic signatures, rare gas atoms, or electrons.^1,14–18 Although, ultimately, the detected particles in all three methods of diffraction, deflection, and depletion are ionized helium ions, the derived information is directly related to the size of neutral droplets.

Charging of neutral droplets offers an alternative method of size determination.^19–25 Electrostatic manipulation through deflection or retardation can then be used to determine the sizes of the charged droplets directly, and with sufficient information on the charging mechanism, information on the neutral droplets can also be derived. Both positive and negative charges have been deposited into superfluid helium droplets through direct electron capture or ionization,^26–28 or doping with cations or anions,^22,29,30 or with ionization after doping of neutral molecules.^25,31–33 The ionization mechanisms can be resonant or non-resonant multiphoton or single photon processes,^{27,28,31–35} or electron impact (EI).^27,35,36 Photoionization with standard lasers is particularly suitable for droplets doped with neutral dopants that have lower ionization thresholds than helium, but for neat droplets, an extreme UV light source or EI is needed.^{11,27,28,34,35} All charging methods are known to decrease the size of the droplets; hence, the resulting sizes are considered as lower limits of neutral droplets, unless the size of the cluster is sufficiently large and the reduction in size amounts to just a negligible fraction of the total size, or when the loss of helium atoms is properly taken into consideration.

In this report, we use cations from an electrospray ionization source (ESI) to charge up neutral droplets for size and velocity characterization. By detecting the doped droplets at two different locations along the beam path, we can measure the velocity of the droplet beam. This information is then used to resolve the size distribution from a retardation experiment.^{24,30,37–39} Similar to our previous report on doping of green fluorescing proteins (GFPs),³⁹ we once again observe two different groups of droplets with different velocities at stagnation temperatures below 10.2 K, although the current experiment uses a new Even–Lavie pulse valve,⁴⁰ while the previous valve was a General Valve, series 99 from Parker Hannifin.^14,18 When the two peaks are separated in arrival time at a detector located farther downstream from the source, we can resolve the size distribution of the faster group—the condensation group. Surprisingly, we observe a bimodal distribution when the stagnation temperature is below 12.3 K at 20 and 40 atm or 16.1 K at 60 atm. The slower group—the fragmentation group—is proven too large to be measurable using a retardation potential of 9 kV. The average size of the condensation group, whether bimodal or single-mode, is in agreement with reports obtained using the depletion method at stagnation temperatures above 6 K.^16,18

II. EXPERIMENTAL SETUP

Figure 1 shows the experimental setup, which is largely similar to that reported in our previous publications,^30,39 with the exception of the new pulse valve (Digital Technology Trading & Marketing, Ltd., E-L-5-8-C-Unmounted Cryogenic Copper Even–Lavie valve). The effective nozzle diameter of the valve is 50 μm, and its temperature is maintained by a cryostat (Sumitomo, SRDK-415D2). Downstream from the pulse valve is a skimmer of 5 cm long and 2 mm in diameter (Beam Dynamics). The source chamber is pumped by one diffusion pump (Varian VHS-6) and is usually at 9.7 × 10⁻⁷ Torr when the pulsed valve is off.

FIG. 1.

Experimental setup showing the electrospray ionization source, the filament and grid for electron impact ionization, and the two different locations of the Daly detector. The retardation electrode is a mesh that can be biased to 9 kV.

Further downstream is a biased grid, above which is a bare filament that supplies hot electrons for ionization. The bias of the grid is referred to as the grid voltage in this work and is the nominal kinetic energy of the ionizing electrons. By surrounding the droplet beam with the grid, the number of passes of the electrons through the droplet beam and hence the ionization yield are significantly increased, but the kinetic energy of the thus produced ions is also raised by the grid voltage.³⁰ To minimize the potential of double ionization by one single electron, ideally, the kinetic energies of the ionizing electrons should be limited to below 48 eV.^19,41 We set the grid at 50 V while the filament was biased at 7–8 V, and the resulting ionization yield and therefore the signal-to-noise ratio (S/N) is sufficient for the experiment.⁴²

An alternative method of charging the neutral droplets is through doping with ions from an electrospray ionization source while keeping the electron source grounded. In Fig. 1, the ion source (Bruker Daltonics, BioTOF) includes an electrospray ionization section, a quadrupole mass spectrometer, and a quadrupole ion trap. During the experiment, the mass spectrometer is set as an ion guide, and all ions from the spray capillary are sent to the ion trap. A pulsed exit gate releases the ions from the ion trap, and an einzel lens focuses the released ion beam into the ion bender to join the droplet beam. Without any bias on the ion bender, the collector/stopper electrode can be connected to an operational amplifier and a resistor to count the total number of ions coming from the ESI source.^24,30,39 The region containing the ionization grid and the ion bender is pumped by a turbomolecular pump (Edwards 255H) and is typically at 5 × 10⁻⁷ Torr when the pulsed valve is off.

The ions for doping are from the laser dye Rhodamine 6G (R6G⁺, Exciton, molecular formula: C₂₈H₃₁N₂O₃Cl, cation from ESI: C₂₈H₃₁N₂O₃⁺) with a molar mass of 443 g/mol. Only singly charged cations are known to form from ESI;⁴³ hence, no mass selection is needed in the quadrupole mass spectrometer. The sample was dissolved in methanol with a final concentration of 10⁻³M. The flow rate was set at 600 μl/h, and the total number of ions released from the ion trap arriving at the collector electrode was ∼10⁷ ions/pulse.

Downstream from the ion bender, a retardation electrode is used to determine the size distribution of the charged droplets.^19,39,44,45 The electrode is a metal plate with a mesh covered hole (2.5 cm in diameter), and it can be biased at a positive voltage up to 9 kV. For a group of charged droplets moving at a known velocity, their kinetic energies are directly proportional to their sizes, and only droplets with sufficiently large sizes can overcome the positive voltage and go through the mesh for detection.

To probe the doped droplets, a Daly dynode-type detector consisting of a dynode that is biased at −18 kV and an opposing channeltron is used.^30,39 The assembly is shielded by a half-cylinder with a 2.5 cm dia. opening to prevent the strong field of the dynode from penetrating into the rest of the chamber. The dynode can be placed at two different locations labeled P₁ and P₂ in Fig. 1, separated by a distance of 25 cm. From the arrival times of the droplet beam on the dynode at the two different locations, the velocity of the droplet beam can be determined, which offers the necessary information for size measurements using the retardation method. The dynode region is pumped by another turbomolecular pump (Varian TV301) and is typically at 5 × 10⁻⁸ Torr with or without the droplet beam.

III. RESULTS

A. Time profiles of doped droplets

Figure 2 shows the time profiles of R6G⁺ doped superfluid helium droplets detected at the P₁ (a) and P₂ (b) positions at a stagnation pressure of 40 atm. The same detector was used, so the experiments at the two positions were performed separately. The horizontal axis refers to the flight time relative to the opening time of the pulse valve. Results from two different source temperatures (5.7 and 14.2 K) and two different settings on the pulse duration (10 and 20 µs) are shown. The dashed lines in the insets are fitting functions (see Sec. III C for details) of the profiles. When the pulse duration is changed from 10 to 20 µs, the overall shape of the time profiles does not change much, but the relative intensities of the doped droplets increase. We also notice that the signal strength exhibits more variations on a day-to-day basis at 10 µs than at 20 µs, prompting us to suspect that 10 µs is probably the threshold opening time for stable operation of the droplet source. For these reasons, most experiments reported in Secs. III B–III E of this work were performed with a duration of 20 µs, unless otherwise specified.

FIG. 2.

Time profiles of R6G⁺ doped droplets detected at the two different positions P₁ (a) and P₂ (b). Two different settings of the pulse duration are shown from two different stagnation temperatures. The stagnation pressure was 40 atm. The horizontal axis refers to the time between the trigger to the pulse valve and the arrival time at the detector. The values from the vertical axis are representative of the relative signal intensities at the two positions. The condensation (cd) and fragmentation (fr) groups are labeled, and the corresponding fittings of modified Gaussian functions are also shown by the dashed lines.

The time profiles change gradually from 5.7 to 14.2 K, and for clarity, only the traces at the two temperatures are shown. The stagnation pressure has a limited effect on the profiles when changed from 20 to 60 atm. Overall, the time profiles obtained at lower source temperatures (5.7 K) arrive at the detector later, reflecting the lower velocities of the droplet beam. Below 10.2 K, two separated groups are observable at the P₂ position, while at and above 10.2 K, only one major group is discernable at both P₁ and P₂ positions.

These time profiles are in general agreement with our previous work on doping of GFPs into superfluid helium droplets,³⁹ although a different pulse valve with a different orifice and pulse duration was used. The presence of two groups at pulse durations of 10 and 20 µs further confirms that these groups are not due to prolonged opening times of the pulse valve.

Depending on the expansion isentrope, the formation of superfluid helium droplets has been categorized into three regimes: the supercritical, the subcritical, and the intermediate regimes.⁴⁴ At 20 atm, the temperature range of supercritical expansion is below 9 K, while the subcritical range is above 12 K. It is, therefore, no surprise that above 8.0 K, only one clearly visible group formed via condensation of gaseous helium is present in the recorded beam profiles at both P₁ and P₂ positions. However, at lower stagnation temperatures, it seems that the expansion contains two components: in addition to the slower group from fragmentation of liquid helium, the faster group from condensation persists even in the supercritical regime.

The presence of two different co-existing formation mechanisms in droplet sources has been suspected in continuous sources prior to our work.^19,42 In 1990, Buchenau et al. used a chopper to measure the speed of a continuous beam produced at 8–20 atm and 5–20 K.⁴² The authors reported the presence of condensation and fragmentation groups among the multiple groups of ions produced via electron impact ionization. Later on, Jiang and Northby used both electron impact ionization and electron attachment to charge up neutral droplets¹⁹ and used a similar method of retardation as ours to analyze the droplet sizes formed at 10–80 atm and 6–18 K. The authors reported at least two main groups of droplets with two different kinetic energies.

The first work on pulsed superfluid helium droplets using the Even–Lavie pulse valve also reported two groups with different velocities and very different sizes.⁴⁰ Pentlehner et al. compared the time profiles of resonantly excited laser induced fluorescence (LIF) and non-resonantly excited Rayleigh scattering signals from phthalocyanine doped droplets, and based on the earlier timing of the LIF signal, the authors concluded on the existence of the second group of droplets. This group disappeared at higher stagnation temperatures and was the predominant component in Rayleigh scattering, while the smaller faster group was only observable when resonantly excited in the LIF experiment.

These reports from both continuous and pulsed sources,^19,40,42 in combination with our previous^24,30,39 and current observations, seem to allude to the universality of both condensation and fragmentation processes in the formation of superfluid helium droplet beams at low source temperatures.

A few observations on the effect of pulse duration are puzzling to us. At 14.2 K, a longer opening time always results in a slightly faster arrival time, but the intensity change is much more dramatic: at the P₁ position, the intensity at 10 µs drops by 1/4 of that at 20 µs, but at the P₂ position, the intensity drops by more than 7/8. This could be related to the more extensive loss of smaller droplets formed at shorter pulses due to collisions with ambient gases, although size measurements did not reveal any substantial difference under both durations. Another puzzling observation is the farther separation between the two groups at the P₂ position, 5.7 K, and 10 µs duration because the speed of the condensation group at 10 µs seems faster than that at 20 µs, opposite to the behavior of the fragmentation group and opposite to the observation at 14.2 K. In fact, the condensation group requires two modified Gaussian functions (see Sec. III C) to fit properly, as shown by the dashed lines, and variations of the relative intensities of the two components can create a shift in timing for the group. However, even with this decomposition, the first peak of the group at 10 µs is still faster than the first peak at 20 µs. In the later retardation experiment, the two peaks seem to have different dependence on the retardation voltage, but since they are not quite resolvable, we treat the two peaks as one group in the following discussion.

B. Velocities

From the arrival times of the doped droplets to the two different detector positions, the velocity of the doped droplets can be derived. Figure 3(a) shows the resulting velocities for both the condensation group (open symbols) and the fragmentation group (filled symbols). The intensity of the fragmentation group at 10.2 K and above is too low for a confident measurement of the arrival time; hence, only velocities at 5.7 and 8.0 K are shown. The uncertainty is primarily determined by the uncertainty of the arrival time when fitting the profile of each group (see Sec. III C). The continuous line represents the theoretical value v calculated using the enthalpy at the stagnation temperature assuming ideal gas behaviors,

$$v\approx \sqrt{\frac{5k_{B}T}{m}},$$ (1)

while the dashed lines are results using the Bernoulli equation⁴⁵ based on the mass density from Ref. 46,

$$v\approx \sqrt{\frac{2P}{\rho }}.$$ (2)

In Eq. (1), k_B, m, and T are the Boltzmann constant, the atomic mass of helium atoms, and the stagnation temperature, and in Eq. (2), ρ is the mass density of liquid helium at the corresponding pressure P [labeled in Fig. 3(a)] and temperature T. Also shown in the same figure is the velocity obtained from EI of neutral droplets at a stagnation pressure of 20 atm. The voltage on the grid, at 50 V, accelerates the resulting ionized droplets, and this effect is generally more obvious for smaller droplets produced at higher stagnation temperatures.

FIG. 3.

(a) Velocity distributions derived from the arrival times of the different groups. Velocities of the condensation groups (cd) are represented by open symbols and those of the fragmentation groups (fr) by filled symbols. The velocities derived from electron impact ionization were obtained at a stagnation pressure of 20 atm. The calculation assuming an ideal gas expansion is shown as a continuous black line. The calculations from the Bernoulli equation, labeled with the stagnation pressures, are shown as dashed lines. (b) Comparisons of velocities reported from a few droplet sources: Gomez,¹⁶ Verma,¹⁸ Bierau,²² Alghamdi,³⁹ and Schilling.⁴⁸

The dependence of beam velocity of the condensation group on the source temperature only qualitatively follows the ideal gas behavior of Eq. (1), while the actual velocities are slower than predicted. If we believe that the reported temperature measured from a silicon diode attached to the nozzle may not reflect the true temperature of the expanding gas,¹⁸ a reduction in temperature, not inflation as expected due to incomplete thermal equilibrium between the gas in the nozzle and the cryostat, by 10% (60 atm) to 40% (20 atm) is necessary to reproduce the experimental values.

Quantitatively, the effect of pressure on the beam velocity of the condensation group depends on the stagnation temperature: at 5.7 K, the velocity increases by 29% from 20 to 40 atm and by 52% from 20 to 60 atm, while at 14.2 K, the variations are within 10% across all pressures. The velocity of ideal gases is independent of pressure, so the increasing velocity with pressure at lower temperatures cannot be modeled by Eq. (1). The stronger effect of pressure at lower source temperatures is similar to that observed by Buchenau et al.⁴² In the work of Kornilov and Toennies,¹⁵ however, non-monotonic pressure dependence of the droplet velocity has been observed below 40 atm in the temperature range of 12–20 K, and around 20–60 atm, the velocity exhibits only weak dependence on the stagnation pressure. In this aspect, our results from the pulsed sources are in general agreement with those from continuous nozzles.

Interestingly, the velocities of the condensation group at 5.7 and 8.0 K actually agree with the values from the Bernoulli equation, while the velocities of the fragmentation groups are all slower than calculation. Since the Bernoulli equation is derived from liquid expansion, this situation is unexpected. The reported velocities from Bierau et al. at 30 atm²² from 5 to 8 K seem to also agree with the Bernoulli equation, although the authors did not report the observation of two groups; hence, it is unclear if the velocities correspond to the condensation group. For expansion conditions below the critical temperature of 4.2 K from continuous nozzles, on the other hand, good agreement between the experiment and the Bernoulli equation has been reported.⁴⁵ Limited by the cooling capacity of our cryostat, we have no data for expansions below the critical temperature.

Figure 3(b) compares the velocities reported from different experiments including our own at a stagnation pressure of 20 atm (labeled “R6G⁺”). Pulsed superfluid helium droplet sources are not as reproducible as continuous sources,^2,47 and depending on the model and operational conditions of the valve, and sometimes even the assembly process, different behaviors have been reported.^37–39 We note that Gomez et al. and Schilling used a continuous source with a nozzle diameter of 5 µm at 20 atm,^16,48 Verma and Vilesov used a General Valve with an orifice of 500 µm and a stagnation pressure of 10 atm,¹⁸ Bierau et al. used a General Valve with an orifice of 800 µm and a stagnation pressure of 30 atm,²² and Alghamdi et al. used a General Valve with an orifice 500 µm and a stagnation pressure of 26 atm.³⁹ Nevertheless, the velocities are generally in agreement, and they are mostly slower than ideal gas behaviors.

C. Retardation experiment

Figure 4 shows the dependence of the R6G⁺ doped droplets on the retardation voltage recorded at the P₂ position at a stagnation pressure of 40 atm, temperatures of 5.7 and 14.2 K, and a pulse width of 20 µs. Even at a retardation voltage of 9 kV, the fragmentation group at 5.7 K shows a minimal change in intensity, implying exceedingly large sizes. The condensation group decreases in intensity with increasing retardation voltage, but at source temperatures lower than 10.2 K, a substantial amount of signal remains even at 9 kV. Furthermore, the decrease with voltage is not uniform across the asymmetric lineshape of the condensation group, with the leading edge being more dramatic. To obtain the size distribution at each stagnation condition, we analyzed the abundance of each group as a function of the retardation voltage using two different methods. The first is to fit the condensation group with one or multiple Gaussian or modified Gaussian functions⁴⁹ and use the resulting integrated area as the intensity. The uncertainties in the peak positions (the first peak when multiple modified Gaussian functions are needed) from the fitting are used in calculating the uncertainties in velocity (Sec. III B). The second approach is to visually separate the two groups in the arrival time and integrate each section separately. Regardless of the treatment method, within the experimental uncertainties, the same sizes and size ranges are obtained under the same stagnation conditions.

FIG. 4.

Dependence of R6G⁺ doped droplets on the retardation voltage. The stagnation pressure was 40 atm, and the pulse duration was 20 µs. At higher source temperatures (b), the retardation field is sufficient to block all the doped ions from reaching the detector, but at lower temperatures (a), while the condensation group can only be partially blocked, the fragmentation group shows negligible dependence on the retardation field. The condensation (cd) and fragmentation (fr) groups are labeled in the top panel, and the bottom panel only contains the condensation group.

After the retardation voltage is converted into the corresponding cutoff size based on the velocity at each temperature, the abundance of the condensation group can be plotted as shown in Fig. 5, and an average droplet size can be derived. For comparison, the variation of the fragmentation group at 5.7 K is also shown. The size distributions of the condensation group resemble exponential decays, but as expected from Fig. 4, with decreasing stagnation temperatures, an increasing amount of signal remains even at the highest retardation voltage. We used the Akaike information criterion (AIC) to determine the adequacy of a single or a bi-exponential decay fitting.⁵⁰ At stagnation pressures of 20 and 40 atm, droplets produced at 10.2 K and below are preferentially fit using double exponential functions, while at 60 atm, droplets produced at 14.2 K and below are double exponential. Limited by the range of the retardation voltage, however, for the larger size component at lower stagnation temperatures, the reduction in signal is too small to capture a significant portion of the decay, and hence, the larger size components have large errors.

FIG. 5.

Size distribution of R6G⁺ doped droplets at different stagnation temperatures. Only the sizes of the condensation groups (labeled “cd”) can be determined, while the fragmentation group (labeled “fr”) cannot be stopped by the retardation electrode. The stagnation pressure was 60 atm, and the pulse width was 20 µs. The continuous lines are exponential fittings of the size distribution: bi-exponential at 14.2 K and below, and single exponential at 16.1 K and above.

D. Size distribution

Figure 6(a) shows the resulting average sizes of the condensation group derived from Fig. 5 obtained at three different stagnation pressures: for bimodal distributions, both the larger component (lg, open symbols) and smaller component (sm, filled symbols) are shown, and the continuous line represents the average at 40 atm. The larger component is about two orders of magnitude larger than the smaller component, while the ratio of the amplitudes of the two components (inset) is in the range of 0.2–0.8 (larger/smaller). It is therefore the presence and the variation of the larger component that determines the monotonic behavior of the overall average size. The size distribution of the smaller component is marginally affected by the stagnation pressure, but the larger component is more sensitive to pressure, particularly at 60 atm.

FIG. 6.

(a) Droplet sizes derived from different stagnation conditions. Only the average sizes at a stagnation pressure of 40 atm (solid line) are shown for clarity. At temperatures below 12.3 K (20 and 40 atm) or 16.1 K (60 atm), bi-exponential functions are required to represent the size distribution, and the ratios of amplitudes of the two different components, labeled as “lg” and “sm,” are shown in the inset. (b) Comparisons of the average droplet sizes from different experiments: Verma,¹⁸ Henne,²⁰ Schilling,⁴⁸ Gomez,¹⁶ and Alghamdi.³⁹ The model of the valve, the stagnation pressure, and nozzle diameter of the different experiments are different.

Figure 6(b) shows the comparison of sizes from different literature reports,^{16,18,20,39,48} including results from EI discussed in Sec. III E. Similar to the comparison of Fig. 3, different nozzle models and exact experimental conditions affect the size distribution, but, in general, the size in the temperature range of 5.7–12.3 K varies by about two orders of magnitude, from 10⁷ to 10⁵ atoms/droplet. We note here that Gomez et al. used the depletion method,¹⁶ and the result corresponds to the size distribution of neutral droplets from a continuous source. The depletion method is also more reliable for droplets of large sizes. The pulsed source from Verma and Vilesov with a much larger nozzle (diameter: ×100) but a lower stagnation pressure (÷2) than a continuous source resulted in larger sizes above 9 K.¹⁸ The authors proposed that a longer “ripening” time in larger nozzles is the cause of larger droplets in pulsed nozzles. We note here that the sizes from our measurements are not scaled by the collision cross sections of the droplets, and depending on the model of the pickup process and size-dependent detection efficiencies,⁴¹ the actual size of the neutral droplets can vary by a factor of 4/3 to 1/3. In any case, the size range remains the same.

Two major differences between the results of our current work and those from the depletion measurements^16,18 need to be clarified. One is that our work can resolve the fragmentation and the condensation groups, and our listed values are only for the average values of the condensation group. Our detection limit, due to the practical range of the retardation voltages, makes our method incapable of measuring droplets with sizes larger than 10⁸ atoms/droplet. If both groups are included, a dramatic increase in size at lower temperatures below 10.2 K is expected, similar to those from the depletion experiments.^16,18 The second difference is the size reduction due to doping of R6G⁺ into neutral droplets; hence, our results are directly indicative of the size of cation doped droplets. However, given the small fraction of size reduction during the doping process (<10% even for the smallest droplets), we consider the sizes of the cation doped droplets similar to those of the neutral droplets. In our earlier work of Alghamdi et al. using a General Valve,³⁹ we could not fully resolve the fragmentation and the condensation group due to the limited flight distance at the P₁ position. Hence, when we noticed that with decreasing stagnation temperatures, even at a retardation voltage of 3 kV (the highest voltage achievable at the time), there were still some GFP doped ions detected, and we mistakenly assumed that the remaining large ions are due to the fragmentation group.

E. Electron impact ionization

With a nominal kinetic energy of 50 eV, most of the collisions between an electron and a droplet should produce just one cation. Although there is a chance of two electrons colliding with one droplet producing a doubly charged droplet,⁵¹ the probability does not seem to affect the resulting average size distribution, as shown in Fig. 6(b)—the resulting sizes are quite consistent with those from doping with R6G⁺. The agreement between the two methods, and the general agreement between results from our two methods and other methods in the range of stagnation temperatures above 5.7 K, demonstrate that at least for droplets containing over 10⁵ atoms/droplet, EI can be used to adequately measure the size of the droplets, as long as the kinetic energy of the electrons is kept below 50 eV.⁵¹ The acceleration effect of the ionizing grid does increase the velocity of the droplet beam, as shown in Fig. 3(a). Above 70 eV, based on our previous work, the size reduction effect of EI has to be taken into consideration when reporting the size of the neutral droplets.^37,38

IV. DISCUSSION

A. Bimodal size distribution of the condensation group

While the fragmentation group at low stagnation temperatures and the condensation group at high temperatures are expected, the presence of both, reported from our previous work on doping of GFPs in the temperature range of 5–11 K, was initially not expected.^{19,20,39,40,42,52,53} The present result reaffirms this observation using a different pulse valve under different experimental conditions. In addition, our results further show that even the condensation group can be bimodal at low stagnation temperatures. Although our measurements only directly present the size distribution of cation doped droplets, there is no reason to suspect that the bimodal distribution is due to cation doping. Rather a more reasonable extrapolation is that the neutral droplets are bimodal in size. Both stagnation temperature and pressure affect the presence of bimodal distributions. For example, at 60 atm, a bimodal distribution is observable even at temperatures of 12.3 and 14.2 K, while at lower stagnation pressures, a single exponential distribution is observed above 10.2 K. This behavior is expected qualitatively based on the isentropes of expansion. However, the reason for the presence of the bimodal distribution of the condensation group is unclear. We note here that at 12.3 and 14.2 K, no fragmentation group is present in the droplet beam at 60 atm, and yet, the condensation group still has a bimodal distribution.

The bimodal distribution of the condensation group is only clearly observable at the P₂ position, after a sufficient traveling distance to separate the fragmentation group from the condensation group. The two components of the condensation group have the same velocity, and when the contribution of the fragmentation group is not clearly separated, it is possible to attribute the larger size component to contributions of the fragmentation group, as we did in an earlier experiment.³⁹ This result shows the advantage of time resolution in using electrostatic manipulation for droplet size measurements.

The larger size component in the condensation group strongly skews the resulting average size, and the resulting value is actually in reasonable agreement with those from depletion measurements above 6 K.^16,18 The presence of the smaller size component, however, affects the doping statistics, particularly when dopant clusters are concerned.^25,54 In addition, droplet sizes have been considered to affect the lineshape of electronic transitions of embedded molecules.^55–60 The presence of two groups of droplets with vastly different average sizes complicates the spectroscopic analysis. Although the different velocities of the fragmentation and condensation group from pulsed sources offer an opportunity for size selection based on arrival times, unfortunately, for the two components of the condensation group with the same velocity, timing discrimination is unavailable for size discrimination. Depletion with neutral molecules or electrons can eliminate the smaller component and allow only the larger component to be detected.^54,61,62

B. Different methods of size measurements

The electrostatic method of size determination of charged droplets has the advantages of experimentally simple and with time resolution. The current report of a bimodal distribution is only possible when the condensation and fragmentation groups are separated in time. However, a major limitation of the method is imposed by the achievable high voltages. As revealed in Fig. 4 and subsequent analysis, the retardation potential fails to deter large droplets. Although a possible solution is to use electrostatic deflection instead of retardation,^22,24,30 the implementation of the deflection method requires the off-axis installation of detectors, which adds experimental complexity.

This work also demonstrates that EI under low kinetic energies offers a facile and reliable method of size characterization, particularly for pulsed sources. Among the several methods of droplet charging,^19,30,37,51 anions are considered more favorable in preserving the size of the droplets due to the lower energy cost in forming an electron bubble than forming a cation snowball.⁵¹ However, the yield of anions is less than 1/10 of that of cations reported in Fárník et al.,⁵¹ tilting the preference to cations when S/N is concerned. Moreover, when the size of the droplets is sufficiently large, similar sizes have been obtained from measurements using both cation and anions.^15,20 A potential complication of EI is the production of doubly or multiply charged droplets.^37,51 However, the yield of double charging is relatively low, particularly at low kinetic energies, and as shown in Fig. 6, the resulting average size is still reasonably reliable.

To some degree, doping with cations is analogous to doping with electrons but with a much larger cross section. In addition, the relative velocities between the cation and the helium droplets can be minimized,³⁰ further minimizing the number of evaporated helium atoms upon doping. The inherent limitation on the cation density due to the space charge effect also guarantees that only one charge can be deposited into one droplet, without the caveat of double charging. Although the interaction between the cation and helium will result in evaporation of a few thousand atoms, if the droplet sizes are more than 10⁵ atoms/droplets, this size reduction can be considered negligible. On the other hand, if the binding energy between the cation and the helium droplets is known, the size of even smaller neutral droplets can also be derived.

For large droplets containing over 10⁸ atoms/droplets, the depletion and the deflection methods remain the only practical methods of size determination,^1,13–18 although with the sacrifice of time resolution. The deflection method using another atomic beam requires rotatable detectors,¹³ not easily achievable in most standard molecular beam apparatus. The depletion method requires standard experimental hardware, but it relies on the droplets maintaining their original direction of velocity to reach the ionizer and detector;^14,16–18 hence, it is applicable to droplet sizes larger than 10⁵ atoms/droplets.

V. CONCLUSION

We present velocity and size measurements of superfluid helium droplets produced from a pulsed source at different stagnation pressures and temperatures. At lower source temperatures below 10.2 K, two different groups of droplets with different velocities can be resolved. While the faster group, considered a result of helium condensation, has velocities slower than those of ideal gases, the slower group, considered a result of liquid helium fragmentation, is slower than that derived from the Bernoulli equation. At higher source temperatures at and above 10.2 K, only the condensation group is detected. The size of the fragmentation group is beyond the detection limit of our retardation field, and the size of the condensation group is bimodal up to 10.2 K at 20 and 40 atm or 14.2 K at 60 atm. The larger component of the condensation group is about two orders of magnitude larger than the smaller group and is more sensitive to the stagnation temperature. The average size, largely determined by the larger component, agrees with most literature reports above 5.7 K. The multimode distribution in velocity and size implies a complicated formation process in pulsed nozzles, which poses challenges for modeling the doping statistics. On the other hand, our method of charging and characterization of the droplet beam, in particular, using electrons with a kinetic energy of ∼50 eV for ionization, is relatively easy to implement in most laboratories equipped with pulsed droplet sources. Instead of relying on reports from the literature, in situ measurements of droplet sizes and size distributions can be and should be performed for every experiment.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.