Helium droplets are usually generated by a supersonic expansion of helium through a continuous-flow nozzle with a diameter of 2-10 µm into a vacuum chamber. Typical sta-gnation conditions are a high pressure of 10 to 100 bar and low temperatures between 4 and 30 K. Depending on the stagnation conditions such as stagnation pressure p0 and nozzle temperature T0 droplets of different sizes are formed. The size distribution of a droplet beam cannot be measured by means of standard mass selective techniques, because the ionization prerequisite of most mass filters destroys the nascent size distri-bution of the neutral droplets. [BKN+90] Instead, the size distribution and the scattering cross section were measured by Toennies and coworkers for different expansion condi-tions by deflecting the droplet beam through the impact of particles from a secondary beam of Ar, Kr or SF6 and detecting the angular distribution. [LST93, HTD98, TV04]
This technique was applied for droplets up to a number of N = 104 He atoms per dro-plet. For larger droplets the deflection angle is too small. Thus, their size distribution was obtained by charging the droplets with single electrons and deflecting them in static electric fields. [JN92, KH99, TV04] With this method droplet sizes of 105 <N<108 can be probed. Using a combination of scattering techniques (deflection and attenuation) determining the size and the total scattering cross section, the radial distribution of the particle density in the droplets was determined. [TV04, HTD98] Evaluating the data with density functional calculations revealed a density close to the bulk-liquid value of 0.022 atoms ˚A−3 in the center. In the surface region the density drops from 90 % to 10 % within about 6 ˚A. [TV04, HTD98] Droplets are expected to be spherical and assuming a uniform density and a sharp outer edge (as in the liquid-drop model) the radius R0 of a droplet is related to the number of He atoms per droplet N by R0 = 2.22N1/3˚A. [BS90]
In Fig. 2.2 the mean droplet size and the diameter of the droplets are plotted versus the nozzle temperature for different stagnation pressures.
Droplets produced via adiabatic expansion of helium can be formed in three different ways, namely by condensation of gas, fragmentation of liquid, and Rayleigh break-up of liquid. [TV04] The thermodynamic equilibrium states occurring in the adiabatic expan-sion lie on isentropes starting from the initial stagnation conditions (fig.2.1). Droplets are formed whenever an isentrope hits the phase transition between liquid and gaseous phase. Depending on the stagnation conditions, the isentropes are separated into three
Fig. 2.2: Mean droplet size N4 (4He atoms per droplet) and droplet diameters D0 versus nozzle temperature T0 for various stagnation pressures p0. Droplets are formed by different processes. (figure taken from [TV04])
regimes as shown in fig. 2.1.
In regime I (subcritical expansion) the expansion starts from the gas phase. When the isentrope cuts the gas-liquid phase boundary droplets are formed by condensation of the gas. The formation of droplets takes place only if the helium density is high enough to warrant for three-body collisions, i.e. close to the nozzle orifice. Further downstream the droplets stop growing and instead start cooling by evaporation of He atoms (evaporative cooling, see below). The average size of the droplets depends on how far the expansion has progressed before the phase interface is hit. In particular, for a certain stagnation pressure, smaller droplets are formed for higher nozzle temperatures as can be seen from fig. 2.2 (lower part). If an isentrope does not reach the phase transition between gaseous and liquid He, e.g. if the nozzle temperature is too high, the gas only cools down but does not condense. Droplets formed by condensation of helium gas consist of about 102 -104 He atoms per droplet. The obtained droplet size distribution can be fitted with a log-normal distribution [TV98, LST93]
f(N) = 1 N σ√
2πe
−(lnN−µ)2
2σ2 (2.1)
wherein N is the droplet size (number of He atoms per droplet) and µ and σ are para-meters. The mean droplet size is given by
N =eµ+σ
2
2 (2.2)
and the full width at half maximum of the distribution is sN =Np
eσ2 −1 (2.3)
which turns out experimentally to be comparable to the most probable size. The dro-plet size distribution is plotted in fig. 2.3 for different expansion conditions. The drodro-plet size distribution for the supercritical expansion is also reflected in the lineshape of elec-tronic [DS01] and vibrational (IR) [HPS+99] transitions, as will be discussed in more detail in chapter 3.2.3.3. The velocity of the droplets depends on the expansion conditi-ons and covers the range between 200 (smaller droplets) and 400 m/s (larger droplets) with a sharp velocity distribution of ∆v/v ≈ 0.01-0.03 or a speed ratio of v/∆v ≈ 50, respectively. [BKN+90]
Fig. 2.3: Droplet size distribution for stagnation pressures p0 and source temperatures T0 using equ. 2.1. Values for the parametersµ and σ are taken from ref. [Sch93]
In regime III (supercritical expansion) the isentropes approach the gas-liquid phase boundary from the liquid side. (Fig.2.1) Thus, droplets are formed by fragmentation of liquid HeI yielding a bimodal size distribution consisting of large droplets with N
> 104 (fig.2.2) and smaller droplets with N ≈ 104. The smaller droplets are found on-ly for isentropes close to regime II and may be formed by recondensation of helium atoms evaporated from the large droplets. [JN92, BKN+90] For the large droplets an exponentially decreasing size distribution was found. The droplets formed via supercri-tical expansion have a much lower speed of 50-100 m/s with the larger droplet being slower. [JN92, KH99, BKN+90, HTK97]
The boundary between regimes I and III is defined as regime II (fig.2.1). Isentropes belon-ging to this regime pass the phase boundary between liquid and gaseous phase near the critical point (Tc = 5.2 K, pc = 2.3 bar) where the formation process changes. [HTK97]
Even larger droplets with N > 1010-1012 can be produced by expanding liquid HeI or
superfluid HeII at nozzle temperatures below the boiling point of He through a larger orifice into vacuum. Under these conditions, the droplets are formed via Rayleigh break-up of the liquid. [TV04] (Fig. 2.2)
Once droplets are formed they cool down by evaporative cooling, i.e. by the evaporation of He atoms from the surface. The binding energy per He atom was calculated to be about 1.3×10−3 K for the dimer, but reaches the bulk value of 7.2 K for droplets consisting of more than 104 He atoms. [TV04] The evaporation rate is initially very high (≈ 1010 K s−1) but decreases exponentially with time and is predicted by theory to level out at a final temperature of about 0.4 K within 10−4 s. [BS90] This equilibrium temperature agrees well with the experimental value of 0.37 K. Due to the very high initial cooling rate it is not possible to significantly increase the droplet temperature and the droplets can serve as a practically perfect thermostat. [TV04]
The temperature of the helium droplets was determined from the rotational fine struc-ture in the IR-spectra of OCS and SF6 to 0.37 ± 0.02 K [GHH+00] and 0.38 ± 0.01 K [HHTV97], respectively, assuming thermal equilibrium between the helium and the dopants. Rotationally resolved spectra reveal the ability of the molecules to rotate freely which is interpreted as microscopic indication of the superfluidity of the dro-plets. [HMTV95, GTV98] The internal pressure of the helium droplets pi can be esti-mated from the droplet radius r and the known surface tension γ of bulk liquid helium.
It is given by pi = 2γ/r and amounts to about 1.8 bar for a droplet radius of r = 36 ˚A (2600 atoms). [LST95] Due to their size, the phase diagram of the bulk helium (fig.2.1) is expected to be applicable also to the droplets. Thus, the internal pressure together with the low temperature is a further evidence for the superfluidity of the droplets. [LST95]
Finally, the phonon wing of the electronic excitation of glyoxal could be simulated using the dispersion curve that is characteristic for superfluid helium. [HMTV96](cf. chapter 4) Droplets can also be formed of the fermionic isotope 3He (nuclear spin I = 1/2). The superfluid phase of bulk 3He is found at much lower temperature (3×10−3 K at atmo-spheric pressure) than for the bosonic isotope 4He (total spin I = 0) with the phase transition at 2.17 K at atmospheric pressure. The droplets formed of pure 3He have a lower temperature of 0.15 K due to the lower binding energy per 3He atom of 2.7 K (bulk value) and thus are not superfluid. The lower binding energy is due to the larger zero-point energy of3He. Due to the non-superfluidity of pure3He droplets and the high price of 3He, pure 3He droplets are usually not used as a host system for spectroscopy.
Experiments with mixed droplets of 3He and 4He revealed that a 4He core is formed inside the 3He droplet and is cooled to 0.15 K. Foreign species doped into these mixed droplets are located in the4He core which was found to be superfluid if it consists of at least 64 atoms. [HHTV97, GTV98] Thus, spectroscopy in superfluid helium droplets is also possible at 0.15 K.