Three-Dimensional Shapes of Spinning Helium Nanodroplets

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.

Figure 1

Evolution of spinning helium nanodroplet shapes. Experimental data (a)–(e), model shapes (f)–(j), and corresponding simulations (k)–(o); see text for details. We can classify our data into five groups (I)–(V), with a transition from spherical (f) to oblate (g) and prolate (h)–(j) shapes.

Figure 2

Origin of characteristic asymmetric features in the droplets’ diffraction patterns. Any given droplet orientation can be characterized by the tilt angles $\alpha$ [rotation around $y$ axis, i.e., $\alpha =\angle (\widehat{x},\widehat{a})$] and $\gamma$ [rotation around droplet’s long principal axis, i.e., $\gamma =\angle (\widehat{y},\widehat{c})$], where the hat denotes the unit vector of the respective axes. (a) Oblate (biaxial) droplet, $\gamma =0$. (b) Oblate (biaxial) droplet. Tilting the short principal semiaxis $c$ out of the scattering plane by $\gamma >0$ leads to a characteristic one-sided asymmetry in the diffraction pattern (indicated by a dotted line). (c) Prolate (triaxial) droplet. Tilting the long principal semiaxis $a$ out of the scattering plane by $\alpha >0$ leads to a bending of the diffraction pattern (see dotted line). An additional asymmetry along the bending of the scattering pattern occurs when tilting the droplet by $\gamma$ (here exemplified as two maxima versus five maxima until the detector edge). Note that rotation of the droplet around the optical axis $\widehat{z}$ will only rotate the diffraction pattern.

Figure 3

Ratios of the principal semiaxis lengths $a$, $b$, $c$, and the droplets’ volume $V$. The dimensionless ratio $b^3/V$ is plotted versus the aspect ratio $a/c$. Dashed line: Analytical model for axisymmetric equilibrium shapes of rotating droplets [2]. Squares: Numerical model for classical droplet shapes of spinning droplets [6]. Triangles: This experiment. Our data follow the oblate (axisymmetric) branch up to an aspect ratio of $a/c\approx 1.5$ and then evolve along the prolate (triaxial) branch, with pill-shaped droplets starting at $a/c>1.8$ and dumbbell-like shapes when $a/c>2.5$. The stability limit for classical axisymmetric droplets is indicated by the filled dot. To visualize the droplet shape evolution, model shapes from Fig. 1 are reproduced.