Helium Nanodroplets: Formation, Physical Properties and Superfluidity

Abstract

In this introductory chapter, we begin by informing the reader about the fascinating history of superfluidity in bulk liquid helium. This is followed by relating attempts in using liquid helium as a low temperature matrix for spectroscopy. After a brief review of the thermodynamic properties of helium in Sect. 1.2, the different types of free jet expansions used in experiments to produce clusters and nanodroplets of different sizes are described in Sect. 1.3. First it is shown how they depend on the nature and location in the phase diagram of the isentropes which determine the course of the expansion. Depending on the four regimes of isentropes, different number sizes and distributions are expected. Next in Sect. 1.4, the results of theoretical and, where available, experimental results on the total energies, excited states, radial density distributions, and temperatures of clusters and droplets are discussed. Finally, in Sect. 1.5 the theoretical and experimental evidence for the superfluidity of nanodroplets is briefly reviewed. For more information on the production and characteristics of nanodroplets, the reader is referred to the chapters in this book and to the reviews in Appendix.

1.1 History

1.1.1 History of Superfluidity in Helium

The first evidence for the existence of helium was a new spectral feature in the Fraunhofer spectrum of the sun discovered in 1868 by the French astronomer Jules Jansen and quickly confirmed in the same year by English astronomers. It was thought then to be another alkali atom and was named after the sun Helios with an ending of “um” as with all the then known alkali atoms. Helium is the second most abundant element in the universe after hydrogen. It is, of course, prevalent in the sun where it is produced by the nuclear fusion of hydrogen. On the heavy planets Saturn and Jupiter it makes up 60–70% of their atmospheres. Because of the earth’s light mass, helium easily escapes and is virtually not present in our atmosphere.

Helium was first liquified by Kamerlingh Onnes in 1908 with a six-stage cooling cascade in Leiden, Holland. Three years later, the ultralow temperatures enabled Kamerlingh to discover superconductivity in mercury at 4.17 K. Thirty years later, in 1938, the superfluidity of liquid helium was discovered at temperatures below 2.141 K by Pyotr Kapitza in Moscow [1] and simultaneously by Allen and Misener in Cambridge [2]. Both results were published back to back in Nature. The fascinating stories behind the race to discover superfluidity have been recently reviewed [3,4,5,6]. In the same year, László Tisza published a Nature article where he postulated that superfluid helium consists of two interpenetrating liquids called the superfluid and normal fractions (two fluid model) [7]. Also in 1938, Fritz London proposed that Bose–Einstein Condensation (BEC) [8] could explain the superfluidity of helium [9].

Superfluidity manifests itself through many different apparently unrelated phenomena, such as a vanishingly small viscosity, the fountain effect, the ability to creep out of a container defying gravity, and the extraordinary ability to conduct heat much more efficiently than even the purest metals. These many strange properties are all macroscopic. These properties inspired Fritz London in 1954 to proclaim that “superfluid helium, also called liquid helium II, is the only representative of a particular fourth state of aggregation” [10].

In 1941 when Nazi Germany invaded Russia, Lev Landau, then in Tbilisi, was the first to formulate a microscopic theory of superfluidity by assuming that the elementary excitations in a superfluid are dominated by highly coherent phonons at low energies [11]. Consequently, he postulated that the phonon dispersion curves in the superfluid are different than in an ordinary liquid. For one they are sharply defined, as in a solid, and further exhibit a maximum which he called a “maxon” and a minimum called a “roton” (Fig. 1.1). He also pointed out that the sharpness of the dispersion curve and the roton would allow the frictionless motion of a particle at velocities below 58 m/s. The sharp dispersion curves coupled with the two-fluid model of Tisza [7] and the suggestion by London relating Bose–Einstein Condensation with superfluidity [9] provided a unifying framework within which many of the above macroscopic observations could be described in the following years.

Fig. 1.1

The dispersion curve of superfluid helium proposed by Landau. Particles moving in the superfluid at velocities less than the critical velocity of 58 m/s experience no resistance

In contrast, on a microscopic level, the physical mechanisms behind many of the unusual properties of both the quantum liquids ³He and ⁴He are yet not fully understood. Even today, the statement made in 1984 that the “connections between Bose condensation and superfluidity (in ⁴He) remain a deep and complex problem” [12] is still valid despite considerable progress in explaining many related phenomena in Bose–Einstein condensed gases [13, 14]. On the other hand, superfluid helium droplets and trapped Bose–Einstein condensates are both quantum many-body systems and have much in common [15].

Superfluidity and superconductivity are closely related [16]. Both are ubiquitous phenomena which occur in solids, in the nuclear matter in stars, as well as in nuclei [17]. and also in elementary particle physics, e.g. Higgs boson [18] The two volumes edited by Bennemann and Ketterson entitled “Novel Superfluids” list and discuss the many manifestations of superfluidity and superconductivity [19].

1.1.2 History of Helium as a Cryomatrix for Spectroscopy

The secret behind the advantages of helium as a cryomatrix for spectroscopy lies ultimately in the unique properties of the helium atom. The properties relevant to the use of droplets as a matrix are briefly summarized in the following four paragraphs:

• 1. The helium atom is the most inert of all atoms. No chemistry is known. In comparison with the simpler H atom, the He atom, with two electrons in a closed shell, is even smaller in size and has a three times smaller polarizability. The energy of the first excited state of 19.82 eV makes liquid helium transparent at wavelengths larger than 62.56 nm.

• 2. The inertness of the helium atom is responsible for the exceedingly weak van der Waals potential of the helium-4 dimer, which is just barely bound. As discussed in Chap. 2 by M. Kunitski in this book, the dimer has a mean internuclear distance of 45 $\pm$ 2 Å. This is about 3 times greater than its classically allowed maximum distance and is due to quantum tunneling. Consequently, it is the largest of all ground state diatomic molecules. Because of its low mass in comparison with other atoms and molecules, it has a large zero-point energy and therefore forms the weakest of van der Waals bonds. With alkali and alkaline earth atoms, the helium atom even has no bound state.

• 3. The extremely weak He–He bond also explains that it is the only element that, as discussed in the next section, at atmospheric pressure does not freeze at 0 K. Thus, droplets are definitely liquid. Due to its light mass, weak bonding, and bosonic characteristics, liquid ⁴He undergoes a second-order phase transition below 2.17 K, a type of Bose–Einstein Condensation called superfluidity.

• 4. The extremely weak He–He bond also means that it has a very small heat of evaporation. In vacuum it evaporates very rapidly until the temperature is so low that further evaporation cannot occur. As discussed in Sect. 1.4.4, this property explains the low temperature of 0.38 K for ⁴He droplets and 0.15 K for ³He. Droplets are thus an ideal cryostat with a well-regulated constant temperature.

Most of the remarkable features listed above have been known for nearly a century through the extensive experiments carried out at the Physics Institute of the University of Leiden by Kamerlingh Onnes, his successor Willem Hendrik Keesom, and colleagues in the early half of the last century [20]. Only the superfluid nature of ⁴He is more recent. Thus, one may ask why hasn’t helium been used as an ideal matrix earlier?

One of the first attempts was undertaken in 1964 by Jortner et al. who deposited N₂ and O₂ on the surface of liquid helium to produce colloidal aggregates. These were excited by a ${}^{210}P_{0}\, \alpha$-source and the electronic emissions ${N}_{2}\left({A}^{3}{\sum }_{n}^{+}\to {X}^{2}{\sum }_{g}^{+}\right)$ and ${O}_{2}\left({C}^{3}{\sum }_{u}^{-}\to {X}^{3}{\sum }_{g}^{+}\right)$ were observed [21]. The first attempt to inject atoms into the liquid made use of ion and electron emission from a hot filament. Later discharges were used as sources of ions and the ions were neutralized with electrons in situ. In 1973, Eugene Gordon et al. carried out a series of experiments in which a mixture of N₂: He (1: 1000) was passed through a discharge directed at the surface of liquid helium. Inside the liquid, a snow-like condensate was observed via the $\left({}^{2}D\to {}^{4}S\right)$ emission of the N atom [22]. The group of Gordon in Chernogolovka also pioneered the growth of high aspect ratio metal nanowires inside bulk superfluid helium [23]. Metal nanoobjects grown in helium droplets are discussed in Chap. 11 by F. Lackner in this volume. Several groups used laser ablation of the metal inside the cryostat and investigated the atoms by laser-induced fluorescence [24, 25]. So far, only metal atoms and clusters have been investigated spectroscopically in the bulk liquid and solid. For a review, see [26]. Generally, however, the atomic lines of metals are considerably shifted and broadened as a result of the strong Pauli repulsion between the helium electrons and the unpaired electron of the electronically excited atom.

Bulk helium spectroscopic experiments have the advantage as a spectroscopic matrix over droplets of temperature and pressure flexibility. Consequently, the spectra inside the normal liquid can be compared with that inside the superfluid and solid, [27] further allowing investigations over a wide range of temperatures and pressures. Unfortunately, it still remains a challenge to stabilize closed-shell atoms and molecules in bulk liquid helium. Once inserted, they drift to the walls or coagulate to form large aggregates [28].

Parallel with and independent of the above developments there were a few experiments and theories exploring the physics of large helium droplets formed in gas expansions in vacuum. In 1961 Becker, Klingelhöfer and Lohse reported on the first investigation of the time-of-flight of droplets of helium formed in nozzle expansions [29]. These scientists were interested in possible applications to plasma physics and the experiments have been reviewed by E. W. Becker in 1986 [30]. In 1983 Stephens and King reported distinct magic numbers of the ion signals in the mass spectra of both ³He and ⁴He clusters [31]. The first reported theoretical prediction that finite-sized droplets like nuclei could be superfluid was published in 1978 [32]. In 1983 the nuclear theoretician V. R. Pandharipande and colleagues reported the first Variational Monte Carlo quantum calculations of the ground state properties of helium droplets as a function of the number of atoms up to N = 728 [33]. These pioneering calculations have been largely confirmed by many subsequent theories.

In 1986 our group reported mass spectroscopic investigations of ⁴He clusters beams [34]. Then, in 1990, Scheidemann, Toennies and Northby reported the observation of additional Ne cluster ions in the mass spectrum of the droplet beam after it had been scattered from a Ne atom nozzle beam [35]. A more complete description of some of the early mass spectrometer experiments, can be found in the chapter “Mass Spectroscopy of Pure and Doped Droplets” by A. Schiller, L. Tiefenthaler and F. Laimer in this volume.

Shortly afterwards in 1992 Goyal, Schutt and Scoles observed two unusually sharp infrared lines of a CO₂ laser in the spectrum of SF₆ molecules, which they thought were attached to the helium droplets [36]. Two years later, Fröchtenicht and Vilesov from our laboratory used a tunable diode laser and observed well-resolved rotational lines of the P- and R-branches centered around a sharp Q-branch of SF₆ in droplets of several thousand atoms [37, 38]. This experiement demonstrated for the first time that single molecules could be inserted into droplets and that the molecules were free to rotate. Since then rotationally resolved spectra have found a wide application as reported in Chap. 4 by Gary Douberly.

1.2 Thermodynamic Properties of Helium

As implied earlier helium is a unique element. It is also the best characterized of all the elements in all its phases. The book by Keller from 1969 [39] and the book edited by Bennemann and Ketterson [40] still provide the most comprehensive introduction to the many remarkable thermodynamical and physical properties of helium. The more recent book by Tilley and Tilley is also recommended as a modern source [16].

Figure 1.2 shows the phase diagram of helium in a double logarithmic plot. Helium is the only substance that remains liquid and does not solidify at 0 K under atmospheric pressure. To solidify helium, high pressures of about P = 29 bar at below 1.73 K are required. Its critical point is at a pressure of P_c.p. = 2.27 bar and T_c.p. = 5.2 K. At atmospheric pressure below about 4.2 K helium becomes a liquid, denoted He I. Then, on further cooling to 2.17 K it undergoes a second-order transition into the new liquid superfluid phase denoted He II (Fig. 1.2). In place of the triple point, where in all other substances the gas, liquid and solid states coexist, it has a unique point at which the gas, liquid He I and superfluid He II coexist, called the λ-point. Furthermore, instead of a phase line marking the transition from the liquid to solid, as in all other substances, it exhibits a λ-line, which marks the transition from the normal liquid He I to superfluid He II. The name derives from the sharp spike in the specific heat which approaches infinity at the superfluid transition. At the λ-line, helium undergoes many dramatic changes in its properties. For example, as a result of the superfluidity of He II the viscosity drops by 6 orders of magnitude to less than 10^–11 poise and the heat conductivity jumps by almost 7 orders of magnitude.

Fig. 1.2

Pressure–temperature phase diagram of ⁴He in a double logarithmic diagram. The dashed horizontal line marks atmospheric pressure

Table 1.1 summarizes some of the thermodynamic properties of ⁴He and the rare ³He isotope. Because ³He is much lighter than ⁴He and is a fermion and not a boson as ⁴He, it has much different physical properties than the more abundant ⁴He. Below about 2.5 mK, ³He also becomes a superfluid by the pairing of two fermions to produce a type of boson. Because of the many different magnetic phases ³He is presently a hot topic in the low temperature community which regularly meet at the biannual conferences Quantum Fluids and Solids (QFS).

Table 1.1 Thermodynamic and physical properties of ${}^{4}\mathrm{He}$ and ³He

1.3 Formation and Characterization of Helium Nanodroplets

1.3.1 Production of Nanodroplets in Free Jet Expansions

Beams containing helium clusters and droplets are readily produced by expanding the high purity gas or the liquid at a stagnation source pressure ${P}_{0} \, \mathrm{between} \, 1.0 \,\mathrm{ and } \,80 \, {\mathrm{bar}}$ and low temperatures ${T}_{0} \, \mathrm{between} \, 3 \,\mathrm{and} \, 40 \,\mathrm{K}$ into vacuum. A thin-walled orifice with diameter $d=5-10 \, \upmu$ is commonly used as the source [41, 42]. Since the flow is not guided, as compared to a Laval-shaped relatively long nozzle, the corresponding jet is termed “free jet”. For preliminary collimation and to reduce the gas load on the vacuum system, the orifice is followed by a conical skimmer [43]. In the ensuing radial expansion of the gas, the particle density $n$ falls off with the inverse square of the distance $z$ measured from the orifice, $n\propto {z}^{-2}$. For a review on free jet expansions, reference is made to the article by Miller [44]. Pulsed helium droplet sources were first successfully operated by Uzi Even and colleagues [45] and by Slipchenko et al. [46] and further improved by Pentlehner et al. [47] Pulsed sources of helium droplets at cryogenic temperatures were first developed by Ghazarian and colleagues [48]. Typically pulsed sources have two orders of magnitude larger orifices of about 0.5 mm dia. Since the short pulses of about 100 µs have a repeat frequency of about 50 Hz the load on the vacuum system is much the same as with a continuous source.

In both types of sources, changes in the thermodynamic state of helium as it flows through the orifice can be approximated by assuming an adiabatic isentropic process in which the gas or liquid is always in thermodynamic equilibrium. The gas or liquid can adjust adiabatically since it takes only less than about a hundred collisions (≈10^–12 s) to adapt to a new pressure and temperature. Thus, it is possible to describe the expansion by following the isentropes in the phase diagram shown in Fig. 1.2. The thermodynamic properties of helium and their dependence on pressure and temperature, as well as the isentropes, are available in tabular form from extensive calculations [49, 50] and measurements [51].

1.3.2 The 4 Regimes of Isentropic Expansions

The free jet expansions of helium depend on the nature of the corresponding isentropes. These can be divided into four flow regimes, which are denoted as Regimes I through IV. Figure 1.3 shows a number of isentropes as dashed colored lines starting from a fixed stagnation pressure of ${P}_{0}=20 \, \rm{bar}$ and for a range of source temperatures ${T}_{0}$ from 0 to 40 K. Regime I expansions are frequently referred to as supercritical since the temperature range lies above the critical temperature. Consequently, the Regime III expansions below the critical point are termed subcritical. In Regime II expansions the isentropes pass to or near the critical point. Whereas in Regimes I, II and III normal He I is expanded, in Regime IV a jet of superfluid He II liquid is formed. Each regime has different operation conditions and leads to clusters or nanodroplets with different sizes and size distributions.

Fig. 1.3

Pressure-temperature phase diagram for ${}^{4}He$ with isentropes (----) for free jet expansions starting from a stagnation pressure of ${P}_{0}=20 \, bar$ and a range of temperatures ${T}_{0}$ [52]. As discussed in the text, qualitatively different behavior is expected for the four regimes indicated at the top of the diagram; Regime I:${T}_{0}\ge 12K, \,{\text{Regime II}:T}_{0}\approx 8-12K, \,{\text{Regime III}:T}_{0}\approx 2.8-8K$, and $\text{Regime IV}:<2.2K$. The isentropes are based on data in [50, 51] for different temperatures ${T}_{0}$ and a single initial pressure of ${P}_{0}=20 \, {\mathrm{ bar }}$ [52]. To obtain the state of the gas or liquid at the orifice, the velocity has been calculated from the change in enthalpy during the expansion (the lines of equal enthalpy are not shown). The sonic condition requires that the flow velocity equals the speed of sound at the narrowest point of the orifice and at this point the flow changes from subsonic to supersonic. Where this occurs is indicated by the dotted curves in Fig. 1.3 [52]. Reproduced from [52] with the permission of AIP Publishing

1.3.2.1 Isentropes: Regime I

In Regime I the source temperatures are well above the critical temperature. Accordingly, the isentropes are nearly those of an ideal gas and accordingly follow straight lines, and cross the phase line from the gas into the liquid phase. Early on in the expansion, the flow reaches the sonic condition and the expansion continues along the isentrope beyond the orifice. For an ideal gas, the changes in state (T,P) for adiabatic isentropic flow can be calculated from the following equation

$${T}^{\gamma }{P}^{1-\gamma }={T}_{0}^{\gamma }{P}_{0}^{1-\gamma }$$

(3.1)

It follows then that for an ideal gas the temperature at some point downstream from the nozzle $z$ is given by

$$T\left(z\right)={T}_{0}{\left(\frac{n\left(z\right)}{{n}_{0}}\right)}^{\gamma -1},$$

(3.2)

where $\gamma$ is the ratio of the specific heats ($\gamma =1.67$ for atoms). The sharp drop in the density $n\left(z\right)$ beyond the orifice leads to a rapid decrease in the temperature. Stein has calculated the gas cooling rates of H₂, He, Ar, and N₂ as a function of orifice diameters at a source temperature of 100 K and higher [53]. The temperature drops continuously as long as the density is sufficient that collisions still occur in significant numbers to assure local equilibrium. Immediately, after crossing from the gas phase into the liquid He I or superfluid He II phase region the system will, before it can become liquid, continue on for some distance in a metastable supercooled gaseous state. At some point belonging to the Wilson line [54] condensation begins. The heat released will raise the temperature and bring the system trajectory back to the saturated vapor phase curve as the droplets continue to grow in size. This continues until collisions cease to occur (sudden freeze). Beyond the point of sudden freeze, the temperature of the droplets is determined by evaporation (see Sect. 1.4.4).

1.3.2.2 Isentropes: Regime II

In Regime II the source conditions correspond to isentropes that converge on or in the vicinity of the critical point. Table 1.2 lists the source temperatures T_CI for several source pressures which correspond to critical isentropes passing exactly through the critical point. In expansions which converge at and near the critical point unusual behavior is expected. Here, the correlation length becomes macroscopically large, corresponding to large fluctuations in density. Light scattering experiments under similar steady state conditions reveal a critical opalescence which indicates the presence of large clusters under static conditions [55]. Also the speed of sound is minimized and the surface tension vanishes [56]. The transition from Regime I to III is, therefore, not sharply defined and occurs over a range of isentropes and source temperatures.

Table 1.2 Source temperatures for critical isentropes at four different source pressures

Regime II was accessed in the first experiment demonstrating that He clusters could capture foreign particles, in this case Ne atoms [35]. The capture probability was found to be largest for expansions in this regime.

1.3.2.3 Isentropes: Regime III

In Regime III at source pressures of 20 bar and temperatures less than about 8 K, below the range of Regime II, but higher than the temperature corresponding to the superfluid transition of helium, the isentropes deviate greatly from the straight lines of the ideal gas. This is consistent with the increasing fraction of liquid and the large increase in heat capacity at the transition from He I to superfluid He II as the isentrope approaches the λ-line. Thus, the isentropes all bend downwards, and cross the liquid–gas phase line from the liquid side. Since the sonic point is at the phase line, the liquid flashes and disintegrates into large clusters after leaving the orifice. Related processes occur in the fragmentation of brittle materials, laser ablation, nuclear collisions, and in the big bang. Holian and Grady among others have simulated this phase transition in a molecular dynamics simulation which provided an expression for the distribution of cluster sizes after the break-up of a liquid [57,58,59].

1.3.2.4 Isentropes: Regime IV

At even lower temperatures in superfluid He II, the isentropes are vertical lines. The liquid leaves the orifice initially as a flowing cylinder with approximately the orifice diameter [60]. After a few millimeters, capillary instabilities inherent in the passage through the orifice lead to a Rayleigh break-up into a sequence of large, equally-sized droplets about 1.89 times the nozzle diameter. Depending on the orifice diameter, droplets produced from liquid helium jet breakup could contain about ${10}^{10}$ atoms. Flashing has not been observed in these expansions [60].

1.3.3 Droplet Sizes and Size Distributions in Regimes I, II, III, and IV

Figure 1.4 presents an overview of droplet sizes obtained with 5 µ orifices as a function of source pressure and temperatures for each of the four regimes. Different nanodroplet sizes over a range of more than 10 orders of magnitude are obtained in the four different regimes. Correspondingly different methods have been found to be suitable for characterizing the sizes, size distributions and velocities. Some of the methods employed rely on the fact that the velocity of the helium cluster and droplet beams have a sharp velocity distribution.

Fig. 1.4

Overview of the average number sizes $\overline{N }$ and liquid droplet diameters D in Å of ${}^{4}\mathrm{He}$ clusters and droplets as a function of source temperature and pressures in the four expansion regimes (Fig. 1.2). In all regimes the sizes increase with decreasing source temperature. Small clusters and droplets are obtained in supercritical expansions (Regime I). Larger droplets are produced by expanding liquid helium in critical (Regime II) and subcritical (Regime III) expansions. Within Regime III at the lowest temperatures of about 3 K liquid He I leaves the orifice in a cylindrical column which breaks up into a series of nearly uniformly-sized droplets with about 10¹⁰–10¹² atoms. Similar phenomena are observed in the expansion of the superfluid He II liquid in Regime IV. Adapted from [64]

The final cluster sizes are determined not only by growth processes, such as droplet coagulation, but also by subsequent cooling by evaporation with a correspondingly large loss of atoms. As discussed in Sect. 1.4.4, see also Fig. 1.14, the droplets lose about half of their atoms resulting from the extensive evaporation. The evaporation is very rapid and the internal temperatures drop with a time constant of about ${10}^{-11}$ – 10^–9 s until the droplet gets so cold that the rate of evaporation becomes negligible. After travelling typical apparatus distances of one meter, the final droplet temperatures have been calculated to be $0.38K{\left({}^{4}He\right)}$ [61, 62] and $0.15K{\left({}^{3}He\right)}$ [61,62,63] (see Fig. 1.14).

1.3.3.1 Droplet Sizes Regime I

The initial growth from the gas phase of small helium clusters, which serve as nuclei for further growth, is accelerated due to a quantum resonance enhanced cross section resulting from the weak 1 mK binding energy of the dimer [65, 66]. For a detailed description of the helium dimer see the chapter by M. Kunitski, Small helium clusters studied by Coulomb explosion imaging, in this volume. The resonance has been calculated to increase the cross section from 30 Å² at room temperature to 259,000 Å² as T $\to 0 \, \mathrm{K}$ [67]. The large increase in the quantum cross sections and resulting increase in collision frequency can also explain the narrow velocity distributions of $\frac{\Delta v}{v}\le 0.01$ of helium atom beams at T₀ = 80 – 300 K [66, 68]. Under conditions where condensation does not take place, this narrow velocity distribution corresponds to an ambient temperature in the moving frame of the beam of only about T_amb $\cong$ 10^–3 K. In mild cryogenic expansions (T₀ = 30 K, P₀ = 5 bar), the heat released in the coagulation to small clusters increases the velocity half width to only about $\frac{\Delta v}{v}=0.015$ (T_amb = 2 × 10^–3 K). In an expansion at higher source pressures (T₀ = 30 K, P₀ = 80 bar), the velocity half width increases further to $\frac{\Delta v}{v}=0.06$ (T_amb = 3 × 10^–2 K) [69]. Compared to expansions with heavier rare gases, however, this resolution is still very good.

The growth processes leading to small clusters at T₀ = 6, 12 and 30 K, P₀ $\le 1.5, 7.0$ and 80 bar, respectively, have been analyzed with a detailed kinetic theoretical model which takes account of three body recombination and break-up processes [69]. The formation of small clusters He₂, He₃ and He₄ from a 5 µ diameter orifice starts already several nozzle diameters from the source and continue to grow up to a distance of about 220 nozzle diameters (${\approx}1 \mathrm{mm})$ [69]. There the expansion undergoes a “sudden freeze” which implies that collisions essentially cease and the clusters continue their forward motion in vacuum without further encounters.

In Regime I, ⁴He dimers, trimers, small clusters, and droplets have been produced with sizes up to about 10⁴ atoms. The size distributions of small clusters up to about N ≈ 100 have been measured with matter-wave diffraction from nanoscopic transmission gratings [69]. Since the diffraction pattern is made up of a superposition of many coherent de Broglie particle waves, which pass through the slots of the grating without interaction with the slits, the method is non-destructive. Matter-wave diffraction from nanoscopic transmission gratings is discussed in more detail in the chapter Small Helium Clusters studied by Coulomb Explosion Imaging by M. Kunitski in this volume.

In other matter-wave diffraction experiments from transmission nano-gratings at 1.25 bar and 6.7 K, the number size distributions peaked at about 10 atoms with an exponential fall-off extending up to cluster size of about 100 atoms [70, 71]. The results for helium clusters with N atoms at T₀ = 6.7 K have been shown to be well fitted by the following distribution: $P\left(N\right)={AN}^{a}{e}^{-bN}$ derived from a theory which accounts for all the recombination and break-up rate constants [72, 73].

For a review of experimental techniques to determine the sizes of larger clusters and droplets up to 1997 the reader is referred to the article by Knuth [74]. The size distributions of larger clusters and droplets of helium were first measured with mass spectroscopy [52, 75]. The ion intensity distributions are severely affected by the heat released (2.2 eV) in the recombination of the initially formed He⁺ with a nearby He atom to form He₂⁺, which is sufficient to evaporate about 3500 atoms [76]. For this reason, the method is not very suitable in Regime I. For the latest applications of mass spectroscopy see the chapter Helium Droplet Mass Spectroscopy by Schiller, Laimer and Tiefenthaler in this volume.

A more suitable and gentle method, which was pioneered by Gspann in 1981, consists of deflecting the helium droplet beam by collisions with a secondary beam of heavy atoms [77,78,79]. The method was later used in our group for studying the size distributions in Regime I by scattering from a secondary beam of heavy particles such as krypton or SF₆ [80, 81]. A careful analysis of the deflected droplets, taking account of the velocity spreads of both beams, indicated that the Kr atoms were fully captured by the droplets so that the entire momentum was imparted to the droplets. From the angular distribution of deflected droplets at angles as small as ${10}^{-3}\mathrm{ radians},$ the momentum distribution of the incident droplets could be measured. Moreover, since the velocity of the droplet beam is sufficiently sharp the droplet mass and number size distributions could be determined. The number sizes were found to be well fitted by a log-normal distribution shown in Fig. 1.5 and given in (3.3) [80].

Fig. 1.5

Droplet size distributions in Regime I for three different source temperatures and a source pressure of 80 bar (5-μ diameter orifice). The distributions were obtained from the angular distribution of the deflected droplets after collisions with a beam of SF₆ molecules [80]. The curves are least squares fits to a log-normal distribution. Reproduced from [80] with the permission of AIP Publishing

$$P\left(N\right)=\frac{1}{\sqrt{2\pi N\delta }}exp\left[-\frac{{\left(lnN-\mu \right)}^{2}}{{2\delta }^{2}}\right],$$

(3.3)

where the mean number of atoms $\overline{N}$ is

$$\overline{N}=exp\left(\mu +\frac{{\delta }^{2}}{2}\right),$$

(3.4)

and the width of the distribution (FWHM) is

$${\Delta N}_{1/2}=exp\left(\mu -{\delta }^{2}+\delta \sqrt{2ln2}\right)-exp\left(\mu -{\delta }^{2}-\delta \sqrt{2ln2}\right),$$

(3.5)

Typically, the FWHM is very similar in size as the mean $\overline{N }$. Best fit parameters $\mu \, \mathrm{and} \, \delta$ are tabulated in [81].

The mean sizes in Regime I over a wide range of temperatures and pressures have also been determined from the size dependent attenuation of the droplet beam by an electron beam [82] Based on these measurements and beam velocities, Knuth et al. have derived expressions for estimating the fraction of the beam which is condensed as liquid [83].

Mean sizes can also be estimated from kinetic and thermodynamic scaling parameters first introduced by Hagena for a wide variety of substances including metals [84]. Knuth found these scaling laws to be unsatisfactory for helium since they considered the clusters to solidify [74]. Knuth introduced kinetic and thermodynamic scaling parameters which took account of the fluid nature of helium clusters and droplets to reliably predict the number sizes of droplets produced in Regime I.

The sizes of droplets from pulsed sources operating at cryogenic temperatures are much larger than droplets from the continuous sources. To investigate the distributions in pulsed beams, Slipchenko et al. [46] doped the droplet bursts with the dye phthalocyanine. The droplet sizes were estimated from the suppression of the laser-induced fluorescence (LIF) signal by scattering from argon gas. The beam attenuation provided information on the collision cross section, which in turn depends on the number size. Slipchenko et al. reported mean droplet sizes as a function of temperature from T₀ = 10 to 26 K at P₀ = 6, 20 and 40 bar. In comparison with continuous droplet sources under the same conditions, they found an increase in droplet sizes between a factor 50–100 at comparable source temperatures. In Regime I Kuma and Azuma also reported a similar increase in sizes [85]. In a more recent experiment from the Vilesov group, the peak flux from a pulsed source was found to be a factor 10³ greater than with a continuous source [86].

Yang and Ellis observed that the heavier droplets with 10⁵ atoms from a pulsed source have significantly longer flight times than the lighter droplets with 4 × 10³ atoms [87]. In several pulsed studies using dopants, which are ionized inside the droplet, the mass distribution of the host droplets could be measured from the intensity of the ions which passed a retarding potential grid [88]. For a similar recent experiment see the chapter by Zhang et al. entitled Electron diffraction of molecules and clusters in superfluid helium in this volume. In a very recent experiment from the same group, Pandey et al. measured the flight time of large droplets which had been doped with cations of the laser dye Rhodamine 6G from an electrospray source [89]. This technique opens up the possibility to select the sizes of droplets containing an ion.

1.3.3.2 Droplet Sizes: Regime II

In Regime II the source conditions correspond to isentropes that converge on the critical point. As mentioned earlier, since large fluctuations occur in the medium at the critical point the transition from Regime I to III is expected to occur over a range of temperatures near the critical isentrope. Gomez et al. carried out a careful study of the sizes in going from Regime I through Regime II to Regime III [90]. They found that there was only a small increase in in the droplet sizes in going from about 9 to 7 K at 20 bar and that a sharp increase by several orders of magnitude occurred below 7 K. As described in the next section, the same large increase could be found at about the same temperature with negatively charged droplets [91]. With pulsed beams, Verma and Vilesov observed a similar plateau in droplet sizes from about 9 to 7.5 K at source pressures of 5 and 10 bar [86].

Only a few studies have been carried at exactly or close to the critical expansion which at 20 bar is at T_o = 9.2 K. (Table 1.2) As mentioned earlier, a large enhancement of the pick-up-probability was observed in expansions in which the isentropes converged on the critical point [35].

Only one experiment has been reported on the properties of beams emerging from the source at conditions corresponding to the critical point (T₀ = T _c.p. = 5.2 K, and P₀ = P _c.p. = 2.3 bar) [92]. As the critical pressure was approached, the velocity of the cluster beam dropped from about 200 m/s to only 50 m/s, whereas the velocity of the atom beam component remained surprisingly unchanged. In a subsequent investigation, the droplets were found to have sizes of N = 1.5 × 10⁹ atoms and a velocity half width of 5% [93]. Under similar source conditions, ³He droplets were found to have slightly smaller sizes of 10⁸ atoms [93]. Since the droplets have velocities below the Landau critical velocity of 58 m/s, they were used to demonstrate the superfluid transmission of He⁴ and He³ atoms through the droplets [93].

1.3.3.3 Droplet Sizes: Regime III

In Regime III, liquid He I flows from the orifice and then once in vacuum cavitates and flashes into smaller droplets and clusters. In 1986, in one of the first experiments in our group, the mass spectrometer signal at critical point conditions jumped-up by several orders of magnitude [34]. As mentioned above, at temperatures well below the transition to Regime III, the droplet sizes increase by about five orders of magnitude.

The first size measurements in this region made use of mass spectroscopy [52]. Jiang et al. [94] tagged the droplets by attaching electrons to their surface with very small binding energies of between 4 × 10^–6 and 1.8 × 10^–4 eV [91] so that fragmentation was minimal. The negatively charged droplets were either retarded or deflected in an electric field to determine the mass distribution [94, 95]. At P₀ = 20.7 bar they observed an increase from N = 4.2 × 10⁴ at 10 K to 5.5 × 10⁵ at 6.5 K. Similar measurements were later carried out by Henne and Toennies [91]. As seen in Fig. 1.6 the distribution of negative ions and positive ions are similar for large droplet sizes with 10⁵ or more atoms.

Fig. 1.6

Reproduced from [91] with permission of AIP Publishing

Mean number sizes of droplets in Regime III measured by deflecting the negatively and positively charged beams in an homogeneous electric field directed perpendicular to the beam direction. Despite the large loss of 3500 atoms upon ionization the cation sizes agree with anion sizes at sizes $N\ge$ 10⁵.

Fig. 1.7

Reproduced from [97] with the permission of AIP Publishing

Typical number size distributions of ionized droplets produced at P₀ = 20 bar. The distribution of the positive ion current is measured by deflecting the charged droplets in an electric field. a The distribution in Regime I at 10 K shows in addition to the expected log-normal distribution N_ln a much smaller exponential tail N_e extending out to 10⁶ atoms, which was overlooked in earlier scattering deflection measurements. b In Regime III at 8 K a pure exponential distribution is observed with a most probable size two orders of magnitude larger.

Figure 1.7 shows the distribution of sizes measured at 20 bar from the deflection of positive ions at 10 K and 8 K on both sides of the critical isotherms at 9.2 K [91]. At 10 K (Fig. 1.7a) in Regime I in addition to the log-normal distribution an additional long tail extending out to larger sizes was found. Since the tail is much less intense than the log-normal distribution it was overlooked in the collisional deflection experiments described in Sect. 1.3.3.1. At 8 K the distribution displays a pure exponential shape (Fig. 1.7b). Confirmation comes from measurements in which the size distribution of very large droplets with more than 10⁹ atoms was determined by analyzing the amplitude of He₂⁺ ion pulses following ionization by electrons of 100 eV. At 5.4 K (20 bar) an exponential size distribution of droplets was determined from the amplitude distribution of the He₂⁺ ion pulses [96]. A combined theoretical and experimental study shows that an exponential distribution is also consistent with theory and is given by

$${P}_{N}=\frac{{e}^{-\frac{N}{\overline{N}}}}{N }$$

(3.6)

where $\overline{N }$ is the average number of atoms in the droplet [97]. This distribution is similar to that obtained in the theory of the fragmentation of brittle materials [98].

At very low temperatures of 3.5 K in Regime III, it was recently discovered by optical imaging that under these extreme conditions the liquid issues from the orifice as a liquid column before breaking up into larger entities [99]. As seen in Fig. 1.8. after a fraction of a mm from the orifice the liquid first breaks up into long ligaments and then, after about one millimeter further downstream, into small droplets. The droplets were measured to be 6.7–8.3 $\upmu$ in diameter and larger than the source orifice of 5 µ. This is consistent with the theory which goes back to a classic paper by Rayleigh (1879). The disturbances which lead to Rayleigh breakup have a length, which converted to the spherical droplets are estimated to have a diameter 1.89 times the nozzle diameter [100,101,102]. For a 5 µ nozzle, this translates to 9.45 µ droplets. These ultra large droplets have found applications as low-Z targets in high energy collisions in storage rings [103]. There, the interest is only in the alpha particle nucleus of the helium atom. A discussion of droplet sizes and size distributions and droplet properties in Regime III can be found in the review by Tanyag et al. [104].

Fig. 1.8

Adapted from [99]

Photographs of droplets in the early stages of a Regime III expansion at P₀ = 20 bars and T₀ = 3.5 K at distances from the orifice at the left. The dashed white oval at 0.5 mm indicates the point at which the liquid cylinder breaks up first into ligaments and then after 1.3 mm into discrete droplets.

These large droplets have been found to be filled with quantum vortices with interesting quantum properties. These are discussed in Chap. 7 entitled X-ray Imaging of Helium Droplets by Tanyag et al. in this volume. As discussed there, diffraction of X-rays provides detailed information on the sizes and even the shapes, which often deviate from spherical and can be either prolate or oblate. See also [105]

1.3.3.4 Droplet Sizes: Regime IV

Source conditions in the liquid superfluid phase have been explored in only one experiment [60]. A 2 µ thin-walled orifice and also for comparison a convergent large aspect-ratio silica pipette with the same sized opening were used. The transition from the superfluid phase He II to the normal phase He I was explored at temperatures of T₀ = 1.5–2.6 K (P₀ = 2.0 – 20 bar). In both phases shortly after emerging from the orifice the intact beam disintegrated into a sequence of highly collimated droplets similar to those shown in Fig. 1.8. The liquid beam had a sharp velocity distribution of $\frac{\Delta v}{v}=0.01$ both below and above the phase transition at velocities as low as 15 m/s [60]. Because of the high directionality a high brilliance of about ${10}^{22}$ atoms/s sr, which is 2–3 orders of magnitude more than possible when flashing occurs, is observed as in Regime III. At the second order superfluid λ-line phase transition, it appears that some turbulence was observed since the relative velocity spread increased from 1 × 10^–2 to 4 × 10^–2 and the beam doubled in the angular width. A similar and possibly related effect has been observed in the bulk liquid and claimed to be analogous to the formation of cosmic strings in cosmology [106, 107]. Presently, it is not clear how the droplet beams produced by expanding the superfluid differ from the beams from normal He I. Possibly, since the superfluid is a many body coherent system interesting interference effects might be observed in the scattering of two superfluid beams prior to break-up. Recently the break-up of jets of normal and superfluid liquid helium, issuing into the saturated vapor, where an entirely different behavior is expected, has been reported [108].

1.3.4 Velocities of Nanodroplets

In Regime I the velocities of small clusters with sizes up to about 6 atoms have been analyzed with a combination of matter wave diffraction and time-of-flight spectroscopy [69]. With increasing P₀ (0–80 bar at T₀ = 30 K) the velocities decrease from about 560 m/s by about 2% at 30 bar and then increase to 570 m/s at higher pressures. The increase is a result of the heat released in the condensation to clusters. At low pressures the velocities agree qualitatively with calculations based on the conservation of enthalpy, v $=\sqrt{\frac{{2h}_{0}}{m}}$, where h₀ is the enthalpy per atom at P₀ and T_o. As a result of condensation heating, the velocity resolution at P₀ = 80 bar and T₀ = 30 K decreased to $\frac{\Delta v}{v}\cong 6 \times {10}^{-2}$ (T_amb = 3 × 10^–2 K) [69] from the very sharp velocity resolution of beams free of clusters with $\frac{\Delta v}{v}\cong{10}^{-2}$. Thus, in general beams of small clusters of helium have reasonable sharp velocities compared to cluster beams with heavier particles.

Figure 1.9 shows the velocities of nanodroplets as a function of the source temperature for 4 different source pressures encompassing Regimes I to III. As expected, the velocities increase linearly with the source temperature and do not reflect the sharp rise in the droplet sizes following the transition region, Regime II. In the transition the linear increase shows only a small perturbation.

Fig. 1.9

Experimental velocities of droplets as a function of source temperature for four different source pressures. The orifice used has a diameter of 5 μ [109]

In Regime IV in expansions of He II very low velocities were achieved. With the pipette (see the previous section) in He II at P₀ = 3 bar the velocity was 30 m/s. At P₀ = 0.5 bar a velocity of 15 m/s was found [60]. Because of the low velocities in the apparatus great care had to be taken to avoid losing the beam by the downward deflection by gravity.

1.4 Physical Properties of Nanodroplets

The physical properties of pure helium clusters and droplets come mostly from theory. They have been calculated by many methods starting from a simple variational calculation in 1965 [110]. The methods used since then include the Green’s Function Monte-Carlo method (GFMC), [33, 111] the Variational Monte-Carlo method (VMC), [33, 111] the Diffusion Monte-Carlo (DMC) method, [112, 113] the hypernetted-chain/Euler–Lagrange theory (HNC/EL) [114] and the Density Functional (DF) method using the Orsay-Trento (OT) finite range density functional [115]. All these methods are based on the assumption that the droplets have a thermodynamic temperature of 0 K. The Path Integral Monte-Carlo Method (PIMC) perfected by David Ceperley to simulate many of the temperature dependent properties of bulk helium [116] is one of the few temperature dependent methods.

1.4.1 Total Energies

Figure 1.10 shows an example of the many calculations of the total ground state energies per atom of ${}^{4}\mathrm{He}$-nanodroplets as a function of the number of atoms for $N\ge 70$ from the HNC/EL and DMC calculations performed by Chin and Krotscheck [114]. The chemical potential given by $\mu \left(N\right)=E\left(N\right)-E\left(N-1\right)$ is also shown in Fig. 1.10 and was determined from the energy per atom. $\mu \left(N\right)$ is also a measure of the evaporation energy per atom. As shown in Fig. 1.10, the ground state energy and evaporation energy for clusters and droplets increases from about 2–3 to 7.21 K with increasing size.

Fig. 1.10

Reproduced from [117] with permission of AIP Publishing

The top curve shows the results for the total ground state energy per atom from HNC/EL (+ symbols) and DMC ( black squares) calculations as a function of N^−1/3 [117]. The crosses (x) show the results for the chemical potential $\mu$ from a generalized Hartree function. The lower straight line is obtained by differentiating the mass formula fit to the upper curve.

The calculated energies are customarily fitted to a mass formula as a function of powers of N^−1/3. Equation (4.1) is a fit to the calculations of Chin and Krotscheck, where the coefficients are in K [114],

$$ \frac{E\left( N \right)}{N} = - 7.21 + 17.71\,N^{ - 1/3} - 5.95N^{ - 2/3} , $$

(4.1)

The first term on r.h.s. in (4.1) is the evaporation energy in the bulk which is obtained by extrapolating to N = $\infty$. The coefficient of the second term a_s on the r.h.s. in (4.1) accounts for the energy of the droplet surface and is calculated from ${a}_{S}=4\pi {r}_{0}^{2}{\tau }_{4}$, where ${r}_{0}$ is the bulk unit radius defined by $\frac{4}{3}\pi {r}_{0}^{3}{\rho }_{0}=1$ and ${\tau }_{4}$ is the surface tension which is about ${\tau }_{4}=0.274{K\AA }^{-2}.$ The next coefficient is referred to as the curvature term. As discussed in the review by Barranco et al. [62] very similar results are also obtained with the following methods: GFMC MEDHE-2 [33, 111], DMC [114] OTDF [115, 118], VMC [33, 111].

1.4.2 Excited State Energies

The first attempts to account for the excited states of helium droplets were based on the liquid drop model which was introduced for dealing with the vibrations of nuclei [61, 118, 119]. The relevant normal modes of He clusters and droplets within the liquid drop model are ripplons, which are quantized capillary surface waves, and phonons, which are quantized bulk volume compressional waves. Krotscheck and colleagues have carried out both DMC and HNC/EL method calculations of the collective excitations of droplets with sizes from N = 20 up to 1000 atoms [113, 114, 117]. Instead of the usual angular momentum quantum number $l$ to account for the surface ripplon modes, these authors have introduced an effective wave number $k$ defined by

$$k=\sqrt{l(l+1})/R$$

(4.2)

where $R$ is an equivalent hard sphere droplet radius given by $R=\sqrt{\frac{5}{3}}$ r_ms and r_ms is the root mean square radius. The lowest compressional volume mode, with $l=0$ in the limit of large $N\left(N>300\right)$ can be approximated by

$${\hslash \omega }_{0}\cong {21N}^{{-1}/{3}}$$

(4.3)

Their latest calculations for both types of excitations are displayed in a plot of the energy and the effective wave number $k$ in Fig. 1.11c for two particle sizes $N=40$ and $N=200$ [117]. The horizontal lines at ${\hslash \omega }_{0}=3.9 \mathrm{K }\left(N=40\right)$ and $5.2 \mathrm{K} \, \left(N=200\right)$ mark the corresponding chemical potentials discussed in the previous section. The results contain both the actual quantum states and the virtual states lying above the chemical potential. In the $N=200$ dispersion curve the remnant of the bulk maxon-roton dispersion curve of bulk He II (Fig. 1.1) can be clearly seen in the virtual states.

Fig. 1.11

a The experimental size distribution of small clusters corrected for the drop off in the average size distribution. Adapted from [71] b The grey area below the chemical potential (grey dotted curve) shows the region of real stable energies bounded by the chemical potential µ (dashed curve) from theory. Adapted from [71] The nearly horizontal line curves mark the ripplon states (0,l) and the only stable compressional state (1,0), indicated by the curve which extends from 1.5 K at N = 10 to 3.2 K at N = 50 [71]. The vertical dotted lines connect the maxima in (a) with the locations where the states cross the chemical potential. The arrows on the sides of part (b) mark the energies of the states (0,2), (0,3), and (0,4) for N = 40, indicated by the red dots, which are in agreement with the experiment in (a) and the theory of [71]. c Calculated density of states for a N = 40 and a N = 200 cluster. The arrows mark the energies of the same states in part (b) from the theory in [117]. Reproduced from [117] with permission of AIP Publishing

Experimental confirmation for some of these collective states has been found in matter wave diffraction measurements of helium clusters [70, 71]. As described in Sect. 1.3.3.1, the resolution could be increased to analyze the relative intensities of clusters with sizes up to about $N=45$. Surprisingly, the intensities showed peaks at $N=\mathrm{8,14,25}$ and 41 [70, 71]. Because of the quantum liquid nature of helium clusters this observation was completely unexpected [120]. Calculations of the quantized collective excitations revealed that these magic numbers occur at exactly the threshold sizes at which an additional excited state became bound leading to a greater structural stability [71]. In Fig. 1.11a the grey area below the dashed line curve depicting the chemical potential shows the bound stable area. The nearly horizontal lines in the grey stable region mark the ripplon states $\left(0,l\right)$ and the only stable compressional state $\left(\mathrm{1,0}\right)$. The arrows in Fig. 1.11a for (0, l) = (0, 2), (0, 3), and (0, 4) at N = 40 agree remarkably well with the calculations in part (b) of the same figure. As shown by the horizontal arrows in (c) which mark the same state-energies, highlighted by red dots in (b), the two theories are both in agreement and agree with the experiment in (a). The dynamics behind the magic numbers have recently been attributed to Auger inelastic processes inside the droplet [121].

1.4.3 Radial Distributions

Most of what is known about the particle number densities and radial distributions of droplets is also from theory. In most of the theories, a spherical shape and a temperature of 0 K are assumed. A typical result obtained using the OT-DF method is shown in Fig. 1.12a [62]. For all clusters with $N\ge 50$, the central density is about equal to the bulk atom density ${\rho }_{B}=0.021{\AA }^{-3}.$ For smaller clusters, the central density increases from about $0.005 {\AA }^{-3}$ for $N=3$ up to close to ${\rho }_{B}$ for $N\approx 20$[71]. Figure 1.12b compares the outer surface density fall-off region for N = 50 and higher. In each case, the surface region has been referenced to the radius at which each of the surface densities has fallen to ${\rho }_{B}/2$. Practically all the large clusters have the same outer shape. The small oscillations seen in the outer regions of radial distributions were found in one of the earliest calculations [33] and in several theories since. Evidence for the existence of oscillations is discussed at lengths in Dalfovo et al. [115]. At present, since the same structures appear in all the recent calculations they are considered to be significant. The authors of the 2006 review by Barranco et al. [62], where Fig. 1.12 is from, conjecture “One could think that such oscillations reflect an underlying quasi-solid structure”.

Fig. 1.12

Reproduced with permission from [62]. © copyright Springer Nature. All rights reserved

a Radial density distributions $\rho \left(R\right)$ calculated with the OT-DF method for ${}^{4}{He}_{N}$ nanodroplets. The red curves correspond to steps of N = 10 up to 50. The blue curves correspond to steps of N = 100 up to 500 and the green curves are for N $\approx$ 1000 and 2000. b The shape of the outer density profiles from 50, 100, 500 and 2000 are compared by shifting the radius to where the density is at ${\rho }_{B/2}$, where ${\rho }_{B}$ is the bulk density.

The surface thickness is customarily defined as the difference between the densities at 10% and 90% of the internal density. The surface thickness of the calculations in Fig. 1.12b is 5.2 Å for all droplet sizes. It is about the same as the surface of films on polished silicon wafers and smaller than the surface thickness of the bulk liquid which has been measured with X-ray scattering to be about 9.2 Å [122].

The surface thickness and central densities have also been measured in a deflection scattering experiment discussed already in connection with droplet sizes and their distribution (see Sect. 1.3.3.1) In these deflection experiments the nanodroplets were deflected by free jet beams of Ar or Kr which crossed the droplet beam at an angle of 40 degrees [81]. As in the earlier size measurements [80], the mean number of atoms in the droplets was determined from the measured angular deflection pattern. Simultaneously, the attenuation of the droplet beam was also measured at a low angular resolution (implied by the large momentum of the droplets) which provides the “classical cross section”. This is equivalent to the hard sphere area of the droplet. Assuming a spherical structure, an effective volume is determined. The density obtained from the mass and effective volume was found to be less than the bulk density. With a realistically assumed surface drop-off in density, the decrease compared to the bulk density could be attributed to the drop-off region. The results for 15 different sizes ranging from N = 700 to 13,000 resulted in an average surface thickness of 6.4 $\pm 1.3 \AA$ [81]. This result is in good agreement with five theories similar to the OT-DF method of Fig. 1.12 which gave values between 6 and 7 Å [81]. As shown in Fig. 1.13 the surface region occupies a significant volume of the droplets with less than about 10⁴ atoms.

Fig. 1.13

The fraction of atoms in the surface region, denoted by t in the inset, is shown as a function of the number of atoms in the nanodroplet

Very large He droplets with up to 10¹⁰ atoms have been imaged individually with X-ray diffraction [105]. The images of single droplets reveal prolate and oblate as well as spherical shapes with typically average diameters of 300–2000 nm [123]. The shapes with large aspect ratios up to 3.0, indicate that the droplets are rotating with considerable angular momentum with a rotational frequency up to $\cong {10}^{7}$ rad s⁻¹ [105, 123] The method and results are the subject of the chapter entitled “X-ray and XUV Imaging of Helium Nanodroplets” by R. M. P Tanyag, B. Langbehn, D. Rupp, and T. Möller in this volume.

1.4.4 Internal Temperatures of Nanodroplets

Jürgen Gspann in 1982 was the first to predict that the temperature of helium droplets is less than 1 K [78]. He based his estimate on the correlation between the electron diffraction measured temperature of the heavier rare gas clusters and the potential depth of the corresponding dimers suggested by Farges et al. [124] Then in 1990, Brink and Stringari calculated the temperature from the rate of evaporation from the surface of droplets. For their theory, they estimated the state density of excited states, the energies and chemical potentials of the droplets [61]. They reported that after 10^–3 s the temperature approached 0.3 K and also predicted the temperature of ³He droplets to be about 0.15 K. Similar results for ³He droplets have also been calculated by Guirao et al. [63].

The results of a more recent calculation, shown in Fig. 1.14 taken from the review by Barranco et al., illustrates the time dependent evaporation induced decrease in droplet sizes and temperatures [62]. Prior to the evaporation they assumed that the ⁴He droplet had grown to ${10}^{3}$ atoms and initially had a temperature of 4 K [62]. The evaporation is very rapid and the internal temperatures drop with a time constant of about ${10}^{-11}$–10^–9 s until the droplet is so cold that the rate of evaporation becomes negligible [61]. After travelling typical apparatus distances of one meter, corresponding to a flight time of about ${10}^{-3}\mathrm{s}$, the final droplet temperatures have been calculated to be $0.38K{\left({}^{4}He\right)}$ [61, 62] and $0.15K{\left({}^{3}He\right)}$ [61,62,63] (see Fig. 1.14) in good agreement with experiment [38, 125,126,127,128].

Fig. 1.14

Reproduced with permission from [62] © copyright Springer Nature. All rights reserved

Calculated time evolution of the mean number sizes and temperatures of ³He and ⁴He droplets after they have grown to 10³ atoms. It is assumed that the ⁴He droplets have initially a temperature of 4.0 K and the ³He droplets 0.8 K. Concomitant with the large evaporative loss the temperatures decrease by about an order of magnitude to below 0.3 K (⁴He) and 0.1 K (³He) [62, 63].

Calculations by Tanyag et al. [104] for N = 10⁷, 10¹⁰, and 10¹³ show that after 10^–3 s the temperature versus time curves are virtually the same independent of the size. Interestingly they find that for these large droplets the reduction in sizes is always about 40% independent of the droplet size.

The first experimental evidence for the internal temperatures of droplets came from the Boltzmann distribution of rotational line intensities of the completely resolved spectra of SF₆ [37, 38]. Shortly afterwards in 1998, the rotational spectrum of linear carbonyl sulfide (O³²CS) molecule inside ⁴He droplets showed an even sharper well-resolved rotational fine structure [126, 128]. Figure 1.15 compares the spectrum of OCS measured in ⁴He droplets with that of the free molecules in a seeded beam [129] and with the spectrum inside non-superfluid ³He droplets [127]. In addition, the spectrum in the colder (0.15 K) inner ⁴He core of mixed ⁴He/³He droplets is shown [126].

Fig. 1.15

Reproduced from [127] with the permission of AIP Publishing

a The OC³²S rotational IR absorption spectrum of the free molecules in an Ar seeded beam [129] is compared with b the depletion IR spectrum in pure ⁴He droplets ($\overline{N}_4 \approx 10^3$) [126, 128] and c with the depletion spectrum in the ⁴He core of a mixed ⁴He/³He droplet ($\overline{N}_4 \approx 10^3 ,\,\overline{N}_3 \approx 10^4$) [126, 127] and (d) with the depletion spectrum in pure ³He droplets ($\overline{N}_3 \approx 10^4$) [126, 127]. The P-branch corresponds to $j\to j-1$ and R-branch to $j\to j+1$ transitions and are labelled by the initial j value. The corresponding transitions are coupled to the transition of the OCS asymmetric stretch vibration mode. Note the Q-branch is missing (position indicated by red arrows) both for the free molecule and in the mixed droplets as expected for a linear molecule. The reduced spacing of the lines in the ⁴He droplet is due to the larger effective moment of inertia of the molecules with effectively several attached helium atoms. The smaller shift in the band origin in the ³He droplets (Fig. 1.15d) can be explained by the lower density compared to ⁴He.

Since the OCS rotational line intensities closely follow a Boltzmann distribution the rotational temperature could be determined to be 0.37 K in excellent agreement with the earlier result obtained for SF₆ in ⁴He droplets and with the theory mentioned above. The agreement with theoretical predictions of the droplet temperatures from evaporation cooling indicates, moreover, that despite the weak coupling with the surrounding bath the dopant is thermally equilibrated in the vibrational and rotational degrees of freedom. The droplet temperature is much less than the temperature of 5 K in a seeded beam (Fig. 1.15a). In seeded beams the vibrational temperatures often lag behind the rotational temperature.

Several differences and similarities between the spectrum in ⁴He droplets and that of the free molecule are noteworthy. For one, the line widths in the droplets are only slightly broadened compared to the free molecule. The well-resolved spectrum and the narrow lines indicate that the molecule rotates practically freely despite the dense liquid environment. Secondly, the vibrational band origins are only slightly red shifted by 0.557 cm⁻¹ (⁴He) and 0.450 cm⁻¹ (³He). The biggest effect seen in ⁴He droplets, however, is the greatly reduced line spacing in the droplets by about a factor 2.8, indicating that the embedded molecule has a larger effective moment of inertia (MOI) by the same amount.

Several physical models have been proposed to explain the increase in the MOI and are discussed in [125, 128, 130]. Several theories have also been published; see for example [131]. Most theories assume that a small number of He atoms are attached to the molecule and rotate with it. The increase in the MOI has been observed in many molecules. The review by Callegari et al. contains a list with about 50 small molecules all of which have a greater MOI in helium droplets than in the gas phase [132]. The infrared spectroscopy of radicals, carbenes and ions is the subject of Chap. 4 in this volume by Gary Douberly.

Also of significance is that the Q-branch is missing in both the ⁴He and the mixed ⁴He/³He droplet spectra (Fig. 1.15) as is the case in the free molecule. The Q-branch, which corresponds to transitions in which j does not change, is forbidden for free linear rotor type molecules but is allowed for symmetric top molecules. Thus, the absence of a Q-branch in the droplet spectrum indicates that the helium environment has no apparent effect also on the symmetry of the rotating chromophore.

Mixed droplets produced by expanding mixtures of ${}^{4}\mathrm{He}$ and ${}^{3}\mathrm{He}$ have been found experimentally [126, 133] and theoretically [134, 135] to consist of an inner core of nearly pure ${}^{4}\mathrm{He}$ atoms and an outer shell of ${}^{3}\mathrm{He}$ atoms. The latter, being on the outside, serve for evaporative cooling and thereby determine the temperature of the molecules inside the central ⁴He core predicted to be about 0.1–0.15 K [61, 63] and experimentally determined at 0.15 K [126, 127, 133]. Pure ³He droplets have even a lower experimentally determined temperature of 0.07 K [127]. As seen in Fig. 1.15, the rotational line widths in the colder mixed droplets are less than in the pure ⁴He droplets and approach those of the free molecule.

At the time when these spectra were first measured, it was speculated that since bulk helium is superfluid below 2.17 K the droplets might be superfluid. Alternatively, it was argued that perhaps the low temperature and the inertness of He might account for the free rotation. This interpretation could be excluded by the broad practically structureless spectrum in the ³He droplets (Fig. 1.15d). Since ³He is lighter and has a weaker interaction with dopants the spectrum would be even sharper. Moreover since ³He is a fermion, superfluidity in the bulk occurs at much lower temperatures of 3 × 10^–3 K and can be excluded even at the low experimentally determined temperature of 0.07 K [127]. Thus, the structureless spectrum in the ³He droplets is evidence that the bosonic nature of the ⁴He droplets explains the sharp rotational features. The spectra in Fig. 1.15 and the earlier rotational resolved IR spectra of SF₆ [36, 37] provided the first evidence that the droplets might be superfluid. The phenomenon of free rotations has been designated as a manifestation of molecular superfluidity [126].

Confirmation that indeed the droplets are superfluid came shortly afterwards in 1989 from theory [136] and the experiments described in the next section.

1.5 Evidence for Superfluidity in Finite-Sized Helium Nanodroplets

Prior to the theory from 1989 [136] and the advent of the spectroscopic experiments in helium nanodroplets described above, it was not clear if finite-sized objects could support superfluidity. Up to this time all the evidence for superfluidity was based on the macroscopic properties of bulk He II, such as the viscous-less flow through narrow channels, the fountain effect, film flow and the enormous heat conductivity compared to He I [16]. The only confirmation that superfluidity might occur at the microscopic level came from a 1977 experiment in which the drift velocity of electrons was measured in the bulk at high pressures close to freezing [137]. In this experiment the electron drift velocity was found to satisfy the upper limit of about 58 m/s specified by Landau and thereby provided confirmation for Landau’s prediction of frictionless motion in a superfluid (Fig. 1.1) [137].

The first theoretical evidence that small clusters of ${}^{4}He$ with 64 and 128 atoms were superfluid came from the 1989 Path-Integral Monte Carlo calculations of Sindzingre et al. [136]. The superfluidity was determined by simulating a slow rotation of the entire cluster and calculating the reduction of the moment of inertia of the cluster from its classical value. The transition was found to be smeared as expected for a finite-sized system. The superfluid fraction increased from about 50% at about 1.5 K and approached 100% below 1 K. The same effect was at the basis of the macroscopic experiment of Andronikashvili which provided the first evidence for the temperature dependence of the bulk superfluid fraction (two fluid model) below the 2.2 K λ-transition [141].

Previously in 1988 Lewart, Pandharipande and Pieper had calculated the Bose condensate fraction for a ⁴He droplet of 70 atoms [140]. Later, in related variational Monte Carlo calculations, Chen, McMahon and Whaley calculated the Bose fraction in smaller clusters with 7 and 40 atoms shown in Fig. 1.16 [139]. Further evidence for superfluidity came from Rama Krishna and Whaley in 1990 [119]. They calculated the dispersion curves for the $l=0$ compression mode which in the spherical droplet is equivalent to the phonon dispersion curve of the bulk. In their T = 0 K calculations, they found for $N=240$ cluster a visible equivalent of a roton which for $N=70$ was weaker and disappeared at $N=20$ [119]. See also Fig. 1.11 and the related discussion. For a more detailed theoretical discussion of the superfluidity of helium droplets, we call attention to the early review by Whaley [142] and the reviews by Dalfovo and Stringari [15] and by Szalewicz [143], which are also listed in Appendix A.

Fig. 1.16

Variational Monte-Carlo calculations of the radial total density distribution ${\rho }_{tot}$ of small clusters of ⁴He. The blue area shows the radial distribution of the Bose–Einstein condensate ${\rho }_{0}$. f₀ is the overall cluster condensate fraction for angular momentum l = 0. Note that in the low density on the periphery the local condensate fraction approaches unity as in BEC of alkali gases [138]. The results for N = 7 and 40 are from Cheng et al. [139] and those for N = 70 are from [140]. Note also that in the bulk the condensate fraction is only about 10%. Adapted from [139] and [140]

The first direct unequivocal experimental evidence that ⁴He droplets are indeed superfluid came in 1996 from electronic excitation spectra of the ${S}_{1}\leftarrow {S}_{0}$ transition of the glyoxal (C₂H₂O₂) molecule embedded in a 5,000 atom droplet [144]. In addition to a sharp zero phonon line (ZPL) at higher frequencies an unusual clearly peaked phonon wing (PW) was found to be well separated from the ZPL as shown in Fig. 1.17. Broad PWs are well known for species trapped in low-temperature classical matrices where they are found to be proportional to the density of the available phonon states. While usually the PW is broad and largely merged with the ZPL, the glyoxal PW is sharply peaked and well separated from the ZPL. More remarkable is that the exact shape reflects the Landau dispersion curve with a large maximum at 6 cm⁻¹ (8.4 K) equal to the energy of the roton and another peak at the energy of the maxon (Fig. 1.1). The distinct PW and the coincidence of its energy distribution with the density of states associated with the roton minimum provides direct evidence that the ⁴He droplets are indeed superfluid in the microscopic sense. Phonon wings have been found in other systems but few have been found to be so sharp as in glyoxal. For further details the reader is referred to Chap. 5 entitled “Electronic Spectroscopy in Superfluid Helium Droplets” by F. Schlaghäufer, J. Fischer and A. Slenczka.

Fig. 1.17

a Depletion spectrum of the band origin of the S₁ ← S₀ in glyoxal (C₂H₂O₂) in a ⁴He droplet with ${\overline{N} }_{4}=5 x 10$ ³ [147]. b The fine structure of the ${0}_{0}^{0}$ line reveals a rotationally resolved zero phonon line (ZPL) followed by a structured phonon wing (PW). c The density of states corresponding to the sharp dispersion curves predicted by Landau for superfluid ⁴He. d A simulation of the phonon wing (red curve) based on the position of the roton and maxon agrees with the shape of the PW

The above interpretation is confirmed by the spectrum of glyoxal in pure non-superfluid ³He droplets [125, 145]. There the sharply peaked phonon wing is replaced by a gradually falling intensity starting at the ZPL and extending to higher energies. This is consistent with the fact that in ³He the density of states is smeared out and partly concealed by particle-hole pair excitations at the Fermi level [145, 146].

Another equally important experiment is the observation of the frictionless ejection of atoms from the droplets with a velocity of about 58 m/s as predicted by Landau. In the 2013 experiment of Brauer et al., $Ag{}^{2}{P}_{1/2}$ and five different molecules, including NO, the large organic molecules such as trimethylamine(TMA),1,4-diazabicyclo[2.2.2]octane(DABCO), and 1-azabicyclo [2.2.2]octane (ABCO) were electronically photo-excited by a pulsed laser [148]. Previously, it had been established that certain electronically excited atoms and molecules are efficiently ejected from droplets [149]. The velocities of the ejected particles were measured by Brauner et al. with a velocity map imaging spectrometer. In all cases, the most probable speeds were in the range of 40 – 60 m/s. With adequate corrections the velocity of ejected excited $Ag{}^{2}{P}_{1/2}$ atoms was found to be 56 m/s in excellent agreement with the Landau velocity [148]. A wide range of different sized droplets were investigated and the limiting velocities were all quite similar. A small dependence on the masses of the 5 particles investigated was roughly in accord with theory [148]. For a more detailed discussion of these photoejection experiments see Chap. 5 entitled “Electronic Spectroscopy in Superfluid Helium Droplets “ by F. Schlaghäufer, J. Fischer and A. Slenczka.

An equally direct experimental evidence that very large droplets with between 10⁸–10¹¹ atoms are superfluid comes from the observation of quantum vortices in the diffraction images from pulsed coherent X-ray scattering [105]. The droplet shapes were found to be either oblate or prolate [150]. Ancillotto et al. on the basis of DFT calculations conclude that the shapes observed experimentally can only be attributed to the presence of quantum vortices inside the large droplets [151]. By doping the droplets with Xe atoms, which are known to be localized at the vortex cores and which are strong scatterers it was possible to directly observe the vortices in the diffraction image. The Fourier-transformed images revealed a regular array with up to 170 vortices [152]. This is considered as evidence that very large droplets, which are close to macroscopic are also superfluid. In an earlier famous experiment, similar regular vortex arrays were found in a rotating cylinder containing superfluid bulk helium [153]. A related vortex lattice was seen much earlier at the surface of superconductors and named after the Nobel Prize laureate Aleksei Abrikosov [154]. The method of X-ray imaging of superfluid droplets and the experimental results are dealt with in detail in the chapter entitled “X-ray and XUV Imaging of Helium Nanodroplets” by. R. M. P. Tanyag, B. Lengbehn, D. Rupp, and T. Möller in this volume.

There are several additional experiments that can only be fully explained when the superfluidity of the droplets is invoked. These include the field detachment of electrons from droplets [155] and the transmission of ³He atoms through large ⁴He droplets [93].

At the present time the superfluidity of helium droplets is well established. An interesting, but not fully solved question is “How many atoms are required for superfluidity?” or to put it differently “How does superfluidity manifest itself in the limit of few atoms?”. Related experiments can be found in the [126] and [156, 157].

This introduction covers only a small selection of the vast literature on the properties of helium droplets and the methods that have been developed to create clusters with tailored made properties. Also, the various techniques which are used in the spectroscopic investigations have hardly been touched upon. For more information the reader is referred to the chapters in this book and the reviews in the Appendix.