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arXiv:cond-mat/9705070 · Physical review. B, Condensed matter

Density of superfluid helium droplets

Jan Harms, Jan Peter Toennies, F. Dalfovo

Published Aug 1, 1998• 182 citations
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Abstract

The classical integral cross sections of large superfluid ${}^{4}{\mathrm{He}}_{N}$ droplets and the number of atoms in the droplets ${(N=10}^{3}\ensuremath{-}{10}^{4})$ have been measured in molecular beam scattering experiments. These measurements are in good agreement with the cross sections predicted from density functional calculations of the radial density distributions. By using a simple model for the density profile a 10--90 % surface thickness of about 6--8 \AA{} is extracted directly from the data.

DOI
10.1103/physrevb.58.3341

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The classical integral cross sections of large superfluid 4HeN{}^{4}{\mathrm{He}}_{N}4HeN​ droplets and the number of atoms in the droplets (N=103\ensuremath−104){(N=10}^{3}\ensuremath{-}{10}^{4})(N=103\ensuremath−104) have been measured in molecular beam scattering experiments. These measurements are in good agreement with the cross sections predicted from density functional calculations of the radial density distributions. By using a simple model for the density profile a 10--90 % surface thickness of about 6--8 \AA{} is extracted directly from the data.
The integral cross sections of large 3He{}^{3}\mathrm{He}3He droplets and the number of atoms in the corresponding droplets $(\overline{N}=2\ifmmode\times\else\texttimes\fi{}{10}^{3}--2\ifmmode\times\else\texttimes\fi{}{10}^{4}\mathrm{atoms})$ have been measured in molecular-beam scattering experiments. The experimental results are in very good agreement with integral cross sections calculated from the radial density distributions predicted from density-functional theory calculations. The experimentally confirmed theoretical 10--90% surface thicknesses vary between 8.0 \AA{} (N‾=103)(\overline{N}{=10}^{3})(N=103) and 7.6 \AA{} $(\overline{N}=4\ifmmode\times\else\texttimes\fi{}{10}^{4})$ and are about 30% larger than calculated for 4He{}^{4}\mathrm{He}4He droplets of similar sizes.
Below Tλ = 2.17 K, bulk 4He is a superfluid and has a non-zero Bose-Einstein condensate fraction. This work will focus primarily on how phenomena such as superfluidity, Bose condensation and superfluid vortices are manifested in microscopic, inhomogeneous helium systems. Path Integral Monte Carlo is a powerful method for calculating the equilibrium properties of quantum systems at finite temperature. We have achieved linear scaling of computer time with number of particles through the use of neighbor lists, allowing us to simulate systems of several thousand atoms. We have derived a local superfluid estimator and used it to examine the microscopic superfluid response around a molecule rotating in a helium nanodroplet. We found that the first solvation layer is well-described by a two dimensional superfluid, with thermal excitations occuring at a lower temperature than in bulk helium. The effective moment of inertia of a linear impurity in a helium droplet is calculated, and compared with experimental scattering results. In addition, we calculated the vortex formation energy for both pure droplets and droplets doped with linear impurities, and found that the linear impurities had a negligible impact on the formation energy. A possible spectroscopic signature of vortices in doped helium droplets was suggested. After deriving a local estimator, we calculated the condensate fraction throughout the free helium surface of a semi-infinite slab. These results, along with densitydensity correlation functions, were used to characterize the surface excitations and calculate the extent to which ripplons are present. In addition, the ripplon dispersion relation was calculated using imaginary-time correlation functions, and found to be lll in good agreement with experimental results. Finally, we have calculated the Dey be-Waller factor in solid helium for a range of temperatures and densities, and compared the scaling behavior with the predictions of harmonic theory. The first non-Gaussian contribution to the density distribution was calculated.
The first minimum appearing in molecular rotational constants as a function of helium droplet size has been previously associated with the onset of superfluidity in these finite systems. We investigate this relationship by bosonic density functional theory calculations of classical molecular rotors (OCS, N2O, CO, and HCN) interacting with the surrounding helium. The calculated rotational constants are in fair agreement with the existing experimental data, demonstrating the applicability of the theoretical model. Inspection of the spatial evolution of the global phase and density shows the increase in the rotational constant after the first minimum correlates with continuous coverage of the molecule by helium and the appearance of angular phase coherence rather than completion of the first solvent shell. We assign the observed phenomenon to quantum phase transition between a localized state and one-dimensional superfluid, which represents the onset of rotational superfluidity in small helium droplets.
Path Integral Monte Carlo was used to calculate the Bose-Einstein condensate fraction at the surface of a helium film at T = 0:77 K, as a function of density. Moving from the center of the slab to the surface, the condensate fraction was found to initially increase with decreasing density to a maximum value of 0.9, before decreasing. Long wavelength density correlations were observed in the static structure factor at the surface of the slab. A surface dispersion relation was calculated from imaginary-time density-density correlations. Similar calculations of the superfluid density throughout He-4 droplets doped with linear impurities (HCN)(sub n) are presented. After deriving a local estimator for the superfluid density distribution, we find a decreased superfluid response in the first solvation layer. This effective normal fluid exhibits temperature dependence similar to that of a two-dimensional helium system.
Helium droplets spanning a wide size range, N(He) = 10(3)-10(10), were formed in a continuous-nozzle beam expansion at different nozzle temperatures and a constant stagnation pressure of 20 bars. The average sizes of the droplets have been obtained by attenuation of the droplet beam through collisions with argon and helium gases at room temperature. The results obtained are in good agreement with previous measurements in the size range N(He) = 10(5)-10(7). Moreover, the measurements give the average sizes in the previously uncharacterized range of very large droplets of 10(7)-10(10) atoms. The droplet sizes and beam flux increase rapidly at nozzle temperatures below 6 K, which is ascribed to the formation of droplets within the nozzle interior. The mass spectra of the droplet beam upon electron impact ionization have also been obtained. The spectra show a large increase in the intensity of the He(4) (+) signal upon increase of the droplet size, an effect which can be used as a secondary size standard in the droplet size range N(He) = 10(4)-10(9) atoms.