Shapes of rotating superfluid helium nanodroplets

Rotating superfluid He droplets of approximately 1 μm in diameter were obtained in a free nozzle beam expansion of liquid He in vacuum and were studied by single-shot coherent diffractive imaging using an x-ray free electron laser. The formation of strongly deformed droplets is evidenced by large anisotropies and intensity anomalies (streaks) in the obtained diffraction images. The analysis of the images shows that in addition to previously described axially symmetric oblate shapes, some droplets exhibit prolate shapes. Forward modeling of the diffraction images indicates that the shapes of rotating superfluid droplets are very similar to their classical counterparts, giving direct access to the droplet angular momenta and angular velocities. The analyses of the radial intensity distribution and appearance statistics of the anisotropic images confirm the existence of oblate metastable superfluid droplets with large angular momenta beyond the classical bifurcation threshold.

Figure 1

Diffraction images from He droplets obtained with the rear pn-CCD ($1024\times 1024$ pixels) detector. The logarithmic intensity color scale is shown on the left. Images (f)–(j) in the bottom row contain streaks, whereas images (a)–(e) in the upper row exemplify patterns with large aspect ratios devoid of any streaks. The blank horizontal stripe results from the gap between the upper and lower plates of the pn-CCD detector. Both pn-CCD plates have a rectangular cut next to the gap to accommodate the primary x-ray beam. The vertical stripes in (a), (b), (c), and (h) are caused by imperfect data readout for strong diffraction images. The empty regions near the top and bottom edges of the images, which are particularly noticeable in panels (a)–(c), are the shadows of the front pn-CCD plates. All images were obtained from bare He droplets except in panel (i), which was obtained upon doping with Xe atoms, as described in the text.

Figure 2

Abundance of streaked (red) and large AR (blue) events as a function of their aspect ratios. The data includes measurements with bare droplets as well as weakly Xe-doped droplets $(N_{\mathrm{He}}/N_{\mathrm{Xe}}>1500)$. In total, 3140 diffraction images of bare and Xe-doped He droplets were obtained during the two experimental runs.

Figure 3

Correlation between the decay exponent α of the $q$-dependent scattering intensity and the AR for streaked (red) and large AR images (blue). The solid pink curve corresponds to the expected trend for axisymmetric classical rotating droplets imaged edge-on. The data points measured with weakly Xe doped droplets are marked by open circles.

Figure 4

Correlation between semimajor $b$ and semiminor $a$ axis for images with streaks. The solid line represents a linear fit to the data points, $b=\langle \mathrm{AR}\rangle ·a$, giving $\langle \mathrm{AR}\rangle =1.86\pm 0.05$. The data points in red correspond to $\mathrm{AR}<1.6$ and are not included in the linear fit calculation.

Figure 5

Extended diffraction image derived by combining the rear detector data of Fig. 1 (within the red rectangle) with the appropriately scaled image from the front detector (outside the red rectangle). The logarithmic intensity color scale is shown on the left.

Figure 6

Stability diagram for rotating droplets in equilibrium as a function of the reduced angular velocity, Ω, and the reduced angular momentum, Λ [see Eqs. (1) and (2)]. The upper branch corresponds to oblate axisymmetric shapes, whereas the lower branch to prolate two-lobed shapes. The bifurcation point is located at $\mathrm{\Lambda }=1.2,\mathrm{\Omega }=0.56$ with $\mathrm{AR}=1.48$.

Figure 7

Calculated cross sections of rotating classical droplets for various reduced angular momenta Λ, as indicated. The angular momentum is aligned along the short $a$ axis. The panels correspond to axisymmetric oblate shapes (a), two-lobed prolate shapes in the equatorial plane (b), and in the plane containing the rotational axis and the long axis (c). Note the different scales in (a) and (b), (c). All droplets have the same volume of $4\pi /3$.

Figure 8

(a) Calculated aspect ratio, AR, vs reduced angular momentum Λ for classical axially symmetric oblate (red) and two-lobed prolate (green, blue, and brown) droplet shapes. The color codes indicate the following aspect ratios: red: $b/a$; green: $c/a$; blue: $c/b$; brown: $b/a$. (b) Ratio of the droplet volume to the cube of the $c$ axis for prolate shapes and the $b$ axis for oblate shapes in units of $4\pi /3$. Red: axially symmetric oblate shapes; blue: two-lobed prolate shapes.

Figure 9

Droplet contours obtained by inverse Fourier transformation of the diffraction images in Figs. 1, 1, and 1 are outlined by red squares in (a), (b) and (d), respectively. (c) The inverse Fourier transform of the diffraction image in Fig. 2C of Ref. [18]. Violet curves are calculated contours for axially symmetric shapes with the same aspect ratio as found experimentally. The green contour in (c) is the result of a calculation for a two-lobed shape with $\mathrm{\Lambda }=1.3$ and a 9° tilt of the rotation axis relative to the detector plane, whereas the one in (d) is the result of a calculation for a two-lobed shape with $\mathrm{\Lambda }=1.5$ and a 45° tilt.

Figure 10

(a) The image is reproduced from Fig. 5. (b) A simulated diffraction image obtained from a two-lobe shape in (c), with $\mathrm{\Lambda }=1.5,c=550\mathrm{nm}$, and $a=230\mathrm{nm}$ tilted out of plane around the $a$ axis by $\pi /4$. The x-ray beam enters perpendicular to the plane of the figure. See the text for more details.

Figure 11

Probability to observe streaked images with a certain aspect ratio for randomly orientated prolate droplets with $\mathrm{\Lambda }=1.3,1.4$, or 1.5 (yellow). The three values of Λ have been assigned the same probability. The experimentally observed streaked events (red) are the same as in Fig. 2 but are sorted into larger bin sizes of 0.2.

Figure 12

Calculated values of decay exponent α vs AR for axially symmetric droplets imaged edge-on (pink curve) and for two-lobed shapes imaged along their semi-axes $a$, $b$, and $c$ [see Figs. 7 and 7 for axis orientations]. The calculated two-lobed shapes have $\mathrm{\Lambda }=1.23,1.26.1.3,1.35,1.4,1.5,1.6$, and 1.7 in order of increasing AR. Points corresponding to the same shapes are connected with lines, i.e., each triangle corresponds to one shape imaged along three different axes.