1 Introduction

Superfluid helium nanodroplets (HNDs) possess the unique ability to pick up virtually any gas-phase atoms or molecules (called dopants) they collide with [1,2,3] and allow them to coagulate or react [4,5,6,7,8,9,10,11] at their characteristic temperature of 0.37 K [12]. This has been widely exploited, e.g., for spectroscopy [13,14,15,16,17,18], studying chemical reactions [19,20,21,22,23,24] or synthesizing nanomaterials [25,26,27,28,29,30,31,32]. Whereas for most dopant atoms and molecules, a location in the interior of the HND is energetically favorable [33,34,35,36,37,38], alkali metal atoms are a prominent exception from this rule [39,40,41,42]. They are found to reside in surface dimples due to a short-range Pauli repulsion between their s valence electrons and surrounding helium atoms counteracting the van der Waals attractive force(s) [43, 44]. While this holds true for small alkali clusters, Stark and Kresin used a classical model to predict that with increasing cluster size, the attractive forces grow faster than the repulsive ones, enabling sufficiently large alkali clusters to become submerged in the HND [45]. Stark and Kresin calculated the critical sizes \(n_{c}\) as 23, 21, 78, 131 and 625 for submersion of \(\hbox {Ak}_{n}\) where \(\hbox {Ak}=\hbox {Li}\), Na, K, Rb and Cs, respectively. The last two values are interpreted as submersion of \(\hbox {Rb}_{n}\) with \(n>100\) and a lack of submersion into the HND for cesium. Lithium does not form large enough neutral clusters on the surface of HNDs that could eventually submerge into the droplets. Their binding energy to the HNDs is so low that the conversion from small high-spin state clusters to their ground state configuration will boil them off. The largest \(\hbox {Li}_{n}^{+}\) cluster that we have observed so far is the trimer. Following a proposition of Stark and Kresin to experimentally verify their predictions [45], An der Lan et al. conducted two studies utilizing HNDs doped with sodium [46] and potassium [47] by exploiting the electron ionization (EI) and electron excitation (EA) characteristics of HNDs. Whereas incident electrons with energies between 19.8 and 24.6 eV cannot directly ionize He atoms, they may produce metastable excited He* or \(\hbox {He}_{2}\)*, which can ionize almost any dopant atom or molecule by transfer of its excitation energy upon collision. Unlike \(\hbox {He}^{+}\) or \(\hbox {He}_{2}^{+}\) and most dopants, which migrate toward the interior of the HND, He* or \(\hbox {He}_{2}\)* will—like alkali atoms and small alkali clusters—locate at or near the droplet surface for the same reasons [48,49,50]. By monitoring the ion yield of alkali clusters as a function of cluster size and electron energy, the authors were able to identify a transition in the ionization characteristics consistent with a submersion into the HND. The experimentally determined submersion sizes \({n_{c,{\mathrm {exp}}}} = 21\) and \(\sim 80\) for \(\hbox {Na}_{n}\) and \(\hbox {K}_{n}\), respectively, are in remarkably good agreement with the theoretical predictions. However, whereas the surface \(\rightarrow \) interior transition of \(\hbox {Na}_{n}\) is sharp, the submersion of \(\hbox {K}_{n}\) appears to be more gradual. For the next larger alkali metal, rubidium, the transition is expected to be even more diffuse [47]. Here, we apply the concept used by An der Lan and co-workers to rubidium clusters in an effort to experimentally determine their critical submersion size.

2 Experimental

The experimental apparatus is described in detail elsewhere [46, 51]. Briefly, HNDs are produced in a supersonic expansion of ultra-pure helium (He 6.0, 99.9999% purity) at a temperature of 9.4 K and stagnation pressure of 22 bar, yielding an average size of \(\sim 5\times 10^{\mathrm {5}}\) He atoms per droplet [51]. The HND beam passes a skimmer, followed by two differentially pumped pickup chambers in one of which Rb vapor (Sigma-Aldrich, 99.6% purity) is added from a resistively heated oven kept at \(80{}^{\circ }\hbox {C}\). While we cannot measure the doping pressure right above the heated oven, we can estimate it to lie between the Rb partial pressure in the doping chamber measured at \(2.5\times 10^{\mathrm {-6}}\) mbar and, probably closer to, the vapor pressure of Rb at \(80^{\circ }\hbox {C}\), which is \(\sim 7.5\times 10^{\mathrm {-5}}\) mbar [52]. The doped droplets then enter a differentially pumped chamber where they are subject to an incident electron beam (energy spread \(> 1~\hbox {eV}\)). Ions ejected from the HNDs upon EI/\(\hbox {EA}\) [1] are analyzed in a high-resolution time-of-flight mass spectrometer (ToF-MS) which typically reaches a resolving power of up to \(m/\Delta m \approx 5000\) [53], but is limited to \(m/\Delta m \approx 2000\) in this study due to the large mass range extending to over 17000 amu or 200 Rb atoms. Several mass spectra with acquisition times between 30 min and 20 h were recorded at electron energies between 21.6 and 160 eV, along with an electron energy scan between 8 and 38 eV. Resulting cluster distributions shown in this paper were extracted from the recorded mass spectra with the IsotopeFit analysis software where possible [54].

3 Results/discussion

3.1 Energy scan

Fig. 1
figure 1

Normalized yield of selected cluster ions as a function of electron energy. The \(\hbox {He}^{+}\) curve is offset for better visibility. This curve is expected to increase in a linear fashion above 25 eV, and its flattening above 27 eV is caused by detector saturation. In the shaded region between 19.8 and 24.6 eV, \(\hbox {He}^{+}\) as well as the two smaller Rb cluster ions display a prominent hump caused by a Penning process. While this feature is gradually disappearing toward larger Rb clusters, an increasingly steep gradient in relative ionization efficiency toward higher electron energies is noticeable with increasing cluster size

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We will start our analysis with the results of the recorded electron energy scan. The relative yield of selected ions as a function of electron energy between 8 and 38 eV is shown in Fig. 1. The notable features can be explained by considering that the energy range can be divided into three different regimes:

  1. 1.

    At electron energies below 19.8 eV, no He excitation is possible. Negligible ion yields of smaller Rb ions can be observed, most likely due to direct EI of surface-bound Rb clusters or the finite energy spread of the incident electron beam.

  2. 2.

    Between 19.8 and 24.6 eV (shaded area in Fig. 1), several resonances for electronic excitation of He exist. Metastable excited He* or \(\hbox {He}_{2}\)* are heliophobic and will migrate toward the droplet surface. They may ionize dopants located there or near it via Penning ionization, explaining the considerable yield of smaller \(\hbox {Rb}_{n}^{+}\) in this region. Subthreshold ionization of He is also possible via a Penning process where two electronically excited He atoms are required [55].

  3. 3.

    Electrons with an energy in excess of 24.6 eV may directly ionize He, and while excitations are still possible, they become increasingly inefficient relative to the direct ionization of He [56, 57]. A nascent \(\hbox {He}^{+}\) will migrate toward the droplet center by resonant charge hopping between neighboring He atoms [58,59,60]. A dopant (cluster) located in the HND interior may be ionized directly upon encountering \(\hbox {He}^{+}\) or \(\hbox {He}_{2}^{+}\), the latter being typically formed after some \(\sim 10\) hops [59]. In this regime, a strong increase in the relative yield of larger \(\hbox {Rb}_{n}^{+}\) is observed.

The ion yield curves shown in Fig. 1 already contain strong evidence for an exterior \(\rightarrow \) interior transition of Rb clusters. Small Rb molecules and clusters show a distinct local maximum in ion yield around 23 eV, indicating efficient Penning ionization via He* and \(\hbox {He}_{2}\)* and thus an exterior location. This feature gradually weakens for larger clusters, whereas the increase in relative ion yield beyond \(\sim 25\hbox { eV}\) gets progressively stronger for larger \(\hbox {Rb}_{n}^{+}\). The shapes of the ion yield curves of larger \(\hbox {Rb}_{n}^{+}\) (e.g., \(\hbox {Rb}_{9}^{+}\) in Fig. 1) compare favorably to those of heliophilic species recorded in separate experiments at the same apparatus (Fig. S1) and are thus consistent with an interior location of larger Rb clusters. We carefully checked for alternative explanations that could arise due to the fact that the enhanced pickup cross section of larger droplets favors the formation of larger Rb clusters. Since HNDs have a size distribution of their own, it is conceivable that the observed change of ionization characteristics for larger \(\hbox {Rb}_{n}^{+}\) is indicative of the HND size rather than the dopant location. We were, however, unable to find any evidence for a strong influence of the droplet size on our measurements (cf. Fig. S1). The apparent transition from an exterior to an interior location appears to proceed gradually with no abrupt changes immediately noticeable from the ion yield curves, so it is difficult to identify a critical size at which a Rb cluster is fully submerged in the HND. An der Lan and co-workers noticed that the transition in \({\hbox {K}}\) clusters [47] was much more diffuse than for \({\hbox {Na}}\) [46], suggesting it would be even more diffuse for \({\hbox {Rb}}\) [47], in good agreement with our observation. In their study of potassium clusters, An der Lan et al. fitted the ion efficiency curves of \(\hbox {K}_{n}^{+}\) clusters via a superposition of the ion efficiency curves of \(\hbox {K}_{2}^{+}\) and \(\hbox {K}_{90}^{+}\). The contribution from \(\hbox {K}_{2}^{+}\) exhibits a resonance at about 22 eV and is remarkably similar to the cross section of He* formation and thus indicating Penning ionization of small neutral potassium clusters contribution to the ion yield of \(\hbox {K}_{2}^{+}\). In contrast, the contribution from \(\hbox {K}_{90}^{+}\) is sharply increasing at electron energies higher than 25 eV, which suggests that large neutral potassium clusters ionized via charge transfer from \(\hbox {He}^{+}\) or \(\hbox {He}_{2}^{+}\) inside the HND are predominantly forming \(\hbox {K}_{90}^{+}\). The relative contributions of \(\hbox {K}_{2}^{+}\) and \(\hbox {K}_{90}^{+}\) that reproduced the ion efficiency curves of a given potassium cluster ion were then used as a measure of the degree of cluster submersion. Unfortunately, this approach is not feasible for our data since the ion yields are fairly low, especially for large clusters, making a fitting of the ion yield curves difficult and prone to large statistical errors. We will therefore attempt a different, but perhaps superior method for the determination of the critical submersion size by using mass spectra recorded at a fixed electron energy, which feature a much better signal-to-noise ratio.

Fig. 2
figure 2

Lower-mass portion of a cluster size distribution extracted from a mass spectrum of Rb clusters grown in large HNDs (\({N}_{\mathrm{He}}\sim 5\times 10^5\)), recorded at an electron energy of 21.6 eV. Prominent local anomalies corresponding to cluster sizes of higher stabilities and/or electronic shell closures are marked in the mass spectrum. The notable features in this mass range are equally observed in mass spectra recorded at higher electron energies of 40, 70 and 160 eV

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3.2 Mass spectra

Before we attempt to identify the critical size for submersion of Rb clusters in HNDs, we shall give a brief overview over some of the recorded mass spectra. In Fig. 2, we show an excerpt of a mass spectrum of Rb-doped HNDs recorded at an electron energy of 21.6 eV extending to about 4000 amu or 45 Rb atoms. The notable features in this region were qualitatively very similar in all mass spectra. We find several local anomalies in the cluster size distribution, such as strong odd–even oscillations, most notably up to \(n=16\), as well as abrupt drops in the abundance of \(\hbox {Rb}_{n}^{+}\) after so-called magic number clusters [61,62,63,64,65], e.g., after \(n=5\), 9, 21 and 41. These local anomalies can be understood by considering that both excitation and charge transfer from He to the dopant are highly exoergic processes. The emanating cluster ion is generally formed in a highly excited, “hot” state, leading to fragmentation as a possible way of dissipating excess energy. This process leads to a depletion of less stable cluster ions and a corresponding enrichment of particularly stable sizes in the mass spectra. Whereas the overall abundance distribution is mostly governed by the pickup statistics (i.e., the size distribution of neutral dopant clusters), variations in this envelope directly reflect the relative stability of the corresponding cluster ions. The observed odd–even oscillations and magic numbers are equally found for all other \(\hbox {Ak}_{n}^{+}\) (\(\hbox {Ak} = \hbox {Li}{-}\hbox {Cs}\)) [62, 66,67,68,69,70] and can be explained in terms of an electronic shell model where the cluster ion is described as an (\(n - 1\)) electron system. Odd-numbered \(\hbox {Ak}_{n}^{+}\) and especially magic number clusters are considered particularly stable due to an even number of electrons (i.e., only paired electrons), whereas the observed magic numbers additionally correspond to the complete filling of electronic shells (similar to rare gas atoms) [71].

Fig. 3
figure 3

a Higher-mass portion of cluster size distributions extracted from mass spectra of Rb clusters grown in large HNDs (\({\sim }\) \(5\times 10^5\)), recorded at electron energies of 21.6 and 40 eV, respectively. The gradual submersion of Rb clusters into the HND is evident from the decrease in ion yield between 60 and 100 atoms in the 21.6 eV mass spectrum. The 40 eV mass spectrum is governed by Poissonian pickup statistics, possibly limited by a decrease in detection efficiency toward larger masses. b Ratio of ion yields recorded at 21.6 and 40 eV for Rb. A gradual decline of ion yield ratio corresponding to a submersion into the HND is noticeable starting around 50 Rb atoms. Rubidium clusters larger than \(\sim 100\) atoms are assumed to be fully submerged. Also shown is a re-evaluation of K data by An der Lan et al. [47] at electron energies of 21.5 and 27.5 eV, suggesting a similar submersion transition around 80 K atoms

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Figure 3a shows a comparison of two cluster distributions extracted from the higher-mass portion (\(\sim \) 3300–17000 amu or 40–200 Rb atoms) of mass spectra recorded at electron energies of 21.6 eV and 40 eV, respectively. Apart from the electron energy, all other experimental settings (i.e., HND source settings, Rb oven temperature, ion optics and ToF-MS settings) were identical. The cluster size distributions are fairly similar until \(\sim 60~\hbox {Rb}\) atoms (see discussion about Fig. 2), but markedly different toward higher numbers of Rb atoms. The cluster ion yield at 21.6 eV first increases toward 60 Rb atoms but drops off sharply afterward and levels out at negligible values around \(\sim 95~\hbox {Rb}\) atoms. It should be noted that a distinct drop after \(n = 93\), which is also apparent in the 40 eV distribution, coincides with an electronic shell closure, indicating a magic number cluster. In contrast, the ion yield recorded at 40 eV continues to rise and displays a broad maximum around \(\sim 95~\hbox {Rb}\) atoms before gradually declining with the last discernible cluster ion signals at \(\sim 200~\hbox {Rb}\) atoms. The 40 eV mass spectrum more closely resembles a log-normal shape, which is expected for the size distribution of neutral clusters due to Poissonian pickup statistics [72] into differently sized HNDs that follow a log-normal size distribution as well [73]. It is possible that the cluster size distribution actually extends further but is discriminated due to a declining detector efficiency and/or ion transmission in the high-mass region.

3.3 Critical submersion size

As previously discussed, at an electron energy of 21.6 eV, dopant ionization occurs primarily at or near the HND surface via Penning ionization by He* or \(\hbox {He}_{2}\)* while at 40 eV, dopant ionization primarily occurs in the droplet interior via charge transfer from \(\hbox {He}^{+}\) and \(\hbox {He}_{2}^{+}\). While cluster ions larger than \(\sim 100~\hbox {Rb}\) atoms are notably absent in the 21.6 eV mass spectrum, they are abundant in the 40 eV spectrum. Since all experimental parameters except the electron ionization energy are equal, neutral clusters of at least the size indicated by the 40 eV spectrum must also be present in the 21.6 eV experiment but fail to show up as ions. This fact is taken as an indication for the full submersion of Rb clusters around \(\sim 95~\hbox {Rb}\) atoms. Similar to the energy scan curves, it appears difficult to make out a precise critical submersion size \({n_{c,{\mathrm {exp}}}}\) from the presented mass spectra. As previously discussed, this is not surprising, as a fairly diffuse transition is expected for rubidium [47]. In an attempt to more precisely determine \({n_{c,{\mathrm {exp}}}}\), we calculated the ratio of ion yields of the previously discussed mass spectra at 21.6 and 40 eV for every cluster ion. The results, shown in Fig. 3b, are similar if mass spectra at different low and high energies are used for analysis. Higher ion yield ratios indicate a relatively more efficient Penning versus charge transfer ionization, pointing toward a surface location of the corresponding neutral cluster, whereas lower values suggest the corresponding neutral cluster is submerged in the HND to a higher degree. Similar to the mass spectra, we see a sharp decline starting around 60 Rb atoms, which becomes less steep around 80 Rb atoms and finally levels out around 100 Rb atoms.

In order to test our method, we re-evaluated the potassium data by An der Lan et al. in the critical size range, shown alongside our Rb data in Fig 3b. We find a pronounced transition around \({n_{c,{\mathrm {exp}}}} \approx 80\), in good agreement with the evaluation of An der Lan et al. [47] and the prediction of Stark and Kresin [45]. Since there is good agreement with the above-mentioned studies and the transition found with our method appears much sharper, we conclude that this method of evaluation should be reasonable, if not better than the fitting method. For Rb, however, we still observe a fairly smooth transition, making it difficult to assign a critical submersion size. Before we attempt to do so, we will discuss two effects that might skew our assignment toward smaller Rb cluster sizes. As Stark and Kresin [45] as well as An der Lan et al. [47] pointed out, there is a more delicate balance between repulsive and attractive forces in the transition region for clusters of heavier alkali species, leading to an increasingly gradual submersion. The shrinking of the HND due to evaporation of He upon pickup and coagulation of dopant Rb atoms might play an additional role and exaggerate this effect. We can estimate that for the assembly of a \(\hbox {Rb}_{100}\) cluster, \(\sim 50\hbox { meV}\) kinetic and \(\sim 0.65\hbox { eV}\) binding energy [74] per Rb atom or 70 eV in total have to be dissipated by the HND. Assuming an energy release of 0.6 meV upon evaporation of one He atom [75], this would shrink the HND by \(\sim 10^{\mathrm {5}}\) He atoms. This is considerable compared to the average size of \(\sim 5\times 10^{\mathrm {5}}\) He atoms per HND, meaning that the lower portion of the HND size distribution would get largely evaporated. Consequently, a large Rb cluster grown in a small HND would be located closer to the surface and could thus be ionized efficiently at 21.6 eV, either directly or by a Penning process, despite being large enough to be fully submerged. On the other hand, large Rb clusters are less likely to be formed in small droplets due to the smaller geometric cross section for the pickup of Rb atoms. Taking into account that such smaller droplets typically represent less than a few percent of the droplet size distribution, we conclude that the discussed shrinking of droplets should not significantly affect our assignment.

A second effect could arise due to a common problem with mass spectrometry of doped HNDs in that we can only measure the distribution of cluster ions after the ionization process. As previously discussed, He-mediated ionization is generally highly exoergic and might induce fragmentation of the emanating cluster ions, which would manifest in a mass spectrum shifted toward smaller cluster sizes. However, larger clusters are generally more robust against fragmentation since excess energy can be stored in the larger number of internal degrees of freedom, while the emanating cluster ion is cooled via evaporation of He atoms from the droplet. This is also evident from the mass spectra, which are highly structured at smaller cluster sizes, especially below \(n =15\), whereas at higher masses, the spectra show a much smoother cluster size distribution. This indicates that while smaller cluster ions are probably subject to intense fragmentation, it is diminished for larger clusters. Consequently, it is likely that the size distribution of larger cluster ions closely resembles that of neutral clusters, leading us to conclude that fragmentation should not have a big influence on our assignment.

In summary, we consider both effects discussed above to be negligible and thus experimentally assign the critical size for submersion of Rb clusters into HNDs at \(\sim \) 100 atoms. Whereas Stark and Kresin’s calculation yields 131 atoms [45], they interpret this result carefully and rather give an estimate of \(>100\) atoms as a critical submersion size, in decent agreement with our results. It appears that while the simple model works remarkably well for the smaller alkali species Na and K, its predictions become less accurate for the larger Rb, and likely Cs as well, for which a failure of submersion is predicted. Considering, however, that the model appears to overestimate the critical submersion size of Rb, it would be interesting to see whether revised theoretical work arrives at the same conclusion for Cs. This would also help to determine whether an experimental observation of Cs cluster submersion into HNDs is feasible. With our present setup, this is highly questionable, even if the critical submersion size was considerably lower than calculated by Stark and Kresin [45].

4 Conclusion

We performed an experimental study based on the studies of An der Lan et al. [46, 47] aiming to determine the critical size of rubidium cluster for submersion in HNDs. Rb clusters with over 200 atoms were observed in EI mass spectra of Rb-doped HNDs. The cluster size distribution showed well-known features of alkali clusters such as odd–even oscillations and magic numbers such as \(n = 9\), 21 and 41, which correspond to electronic shell closures. We then compared the ion yield for different cluster sizes obtained from mass spectra recorded at electron energies of 21.6 and 40 eV as a measure of submersion. We observed a rather gradual transition from an exterior to an interior location of Rb clusters as a function of cluster size. Our analysis suggests full submersion of Rb clusters larger than \(\sim 100\) atoms, in good agreement with a theoretical model by Stark and Kresin [45]. Future theoretical and experimental work is encouraged to determine whether such a transition is also feasible for cesium clusters.