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arXiv:2003.07466 · Physical review. B./Physical review. B

Shapes of rotating normal fluid He3 versus superfluid He4 droplets in molecular beams

Deepak Verma, Sean M. O. O’Connell, Alexandra J. Feinberg, Swetha Erukala, Rico Mayro P. Tanyag, Charles Bernando, Weiwu Pang, Catherine A. Saladrigas, Benjamin W. Toulson, Mario Borgwardt, Niranjan Shivaram, Ming‐Fu Lin, André Al Haddad, Wolfgang Jäger, Christoph Bostedt, Peter Walter, Oliver Geßner, Andrey F. Vilesov

Published Jul 8, 2020• 19 citations
arXivHTMLPDF

Abstract

Previous single-pulse extreme ultraviolet and x-ray coherent diffraction studies revealed that superfluid $^{4}\mathrm{He}$ droplets obtained in a free jet expansion acquire sizable angular momentum, resulting in significant centrifugal distortion. Similar experiments with normal fluid $^{3}\mathrm{He}$ droplets may help elucidate the origin of the large degree of rotational excitation and highlight similarities and differences of dynamics in normal and superfluid droplets. Here, we present a comparison of the shapes of isolated $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ droplets following expansion of the corresponding fluids in vacuum at temperatures as low as \ensuremath{\sim}2 K. Large $^{3}\mathrm{He}$ and $^{4}\mathrm{He}$ droplets with average radii of \ensuremath{\sim}160 and \ensuremath{\sim}350 nm, respectively, were produced. We find that the majority of the shapes of $^{3}\mathrm{He}$ droplets in the beam correspond to rotating oblate spheroids, in agreement with previous observations for $^{4}\mathrm{He}$ droplets. The aspect ratio of the droplets is related to the degree of their rotational excitation, which is discussed in terms of reduced angular momenta (\ensuremath{\Lambda}) and reduced angular velocities (\ensuremath{\Omega}), the average values of which are found to be similar in both isotopes. This similarity suggests that comparable mechanisms induce rotation regardless of the isotope. We hypothesize that the observed distribution of droplet sizes and angular momenta originates from processes in the dense region close to the nozzle, where a significant velocity spread and frequent collisions between droplets induces excessive rotation followed by droplet fission.

DOI
10.1103/physrevb.102.014504

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Previous single-pulse extreme ultraviolet and x-ray coherent diffraction studies revealed that superfluid 4He^{4}\mathrm{He}4He droplets obtained in a free jet expansion acquire sizable angular momentum, resulting in significant centrifugal distortion. Similar experiments with normal fluid 3He^{3}\mathrm{He}3He droplets may help elucidate the origin of the large degree of rotational excitation and highlight similarities and differences of dynamics in normal and superfluid droplets. Here, we present a comparison of the shapes of isolated 3He^{3}\mathrm{He}3He and 4He^{4}\mathrm{He}4He droplets following expansion of the corresponding fluids in vacuum at temperatures as low as \ensuremath{\sim}2 K. Large 3He^{3}\mathrm{He}3He and 4He^{4}\mathrm{He}4He droplets with average radii of \ensuremath{\sim}160 and \ensuremath{\sim}350 nm, respectively, were produced. We find that the majority of the shapes of 3He^{3}\mathrm{He}3He droplets in the beam correspond to rotating oblate spheroids, in agreement with previous observations for 4He^{4}\mathrm{He}4He droplets. The aspect ratio of the droplets is related to the degree of their rotational excitation, which is discussed in terms of reduced angular momenta (\ensuremath{\Lambda}) and reduced angular velocities (\ensuremath{\Omega}), the average values of which are found to be similar in both isotopes. This similarity suggests that comparable mechanisms induce rotation regardless of the isotope. We hypothesize that the observed distribution of droplet sizes and angular momenta originates from processes in the dense region close to the nozzle, where a significant velocity spread and frequent collisions between droplets induces excessive rotation followed by droplet fission.
Helium nanodroplets are considered ideal model systems to explore quantum hydrodynamics in self-contained, isolated superfluids. Rotation of a superfluid droplet is manifested as a collection of quantum vortices. Here, the authors use single-shot femtosecond X-ray coherent diffractive imaging to investigate the shapes and vorticity of rotating superfluid 4{}^{4}4He droplets. They find that most of the diffraction contours exhibit noticeable ellipticity. A smaller but still significant number of distorted diffraction contours are marked by regions of high intensity concentrated along the direction of the long axis in the diffraction image, indicating the presence of extremely deformed 4{}^{4}4He droplets. Forward modeling of the diffraction images shows that the shapes of superfluid 4{}^{4}4He droplets are very similar to their classical counterparts. In particular, the existence of both axially symmetric (oblate) and, for higher angular momenta, two-lobed (triaxial prolate) droplets is revealed. The latter shapes are strongly deformed compared to the spheroidal bodies that are expected for smaller rotational speeds. The observed shapes provide direct access to the droplet angular momenta, angular velocities, and estimated number of quantum vortices inside the droplets.
The angular momentum of rotating superfluid droplets originates from quantized vortices and capillary waves, the interplay between which remains to be uncovered. Here, the rotation of isolated submicrometer superfluid ^{4}He droplets is studied by ultrafast x-ray diffraction using a free electron laser. The diffraction patterns provide simultaneous access to the morphology of the droplets and the vortex arrays they host. In capsule-shaped droplets, vortices form a distorted triangular lattice, whereas they arrange along elliptical contours in ellipsoidal droplets. The combined action of vortices and capillary waves results in droplet shapes close to those of classical droplets rotating with the same angular velocity. The findings are corroborated by density functional theory calculations describing the velocity fields and shape deformations of a rotating superfluid cylinder.
A significant fraction of superfluid helium nanodroplets produced in a free-jet expansion has been observed to gain high angular momentum resulting in large centrifugal deformation. We measured single-shot diffraction patterns of individual rotating helium nanodroplets up to large scattering angles using intense extreme ultraviolet light pulses from the FERMI free-electron laser. Distinct asymmetric features in the wide-angle diffraction patterns enable the unique and systematic identification of the three-dimensional droplet shapes. The analysis of a large data set allows us to follow the evolution from axisymmetric oblate to triaxial prolate and two-lobed droplets. We find that the shapes of spinning superfluid helium droplets exhibit the same stages as classical rotating droplets while the previously reported metastable, oblate shapes of quantum droplets are not observed. Our three-dimensional analysis represents a valuable landmark for clarifying the interrelation between morphology and superfluidity on the nanometer scale.
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Motivated by recent experiments, we study normal-phase rotating 3He droplets within density functional theory in a semi-classical approach. The sequence of rotating droplet shapes as a function of angular momentum is found to agree with that of rotating classical droplets, evolving from axisymmetric oblate to triaxial prolate to two-lobed shapes as the angular momentum of the droplet increases. Our results, which are obtained for droplets of nanoscopic size, are rescaled to the mesoscopic size characterizing ongoing experimental measurements, allowing for a direct comparison of shapes. The stability curve in the angular velocity–angular momentum plane shows small deviations from the classical rotating drop model predictions, whose magnitude increases with angular momentum. We attribute these deviations to effects not included in the simplified classical model description of a rotating fluid held together by surface tension, i.e., to surface diffuseness, curvature, and finite compressibility, and to quantum effects associated with deformation of the 3He Fermi surface. The influence of all these effects is expected to diminish as the droplet size increases, making the classical rotating droplet model a quite accurate representation of 3He rotation.
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