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X-ray coherent diffractive Imaging of doped quantum fluids

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X-ray coherent diffractive Imaging of doped quantum fluids

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Content X-ray Coherent Diffractive Imaging of Doped Quantum Fluids Alexandra J. Feinberg A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfilment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY (CHEMICAL PHYSICS)) August 2022 Copyright 2022 Alexandra J. Feinberg ii Epigraph “I don’t have to know an answer. I don’t feel frightened by not knowing things, by being lost in the mysterious universe without having any purpose – which is the way it really is, as far as I can tell.” - Richard P. Feynman iii Dedication This thesis is dedicated to my dear friend, Jesse Alexander Spivey. Without you, I never would have pursued this degree in the first place. iv Acknowledgements There are so many people I want to acknowledge because without them, I likely wouldn’t be here today. I suppose I’ll start with the obvious. I must acknowledge my parents for (a) giving birth to me in the first place and (b) supporting me throughout my studies, emotionally and financially. Mom and Dad, I appreciate everything you’ve done and everything you continue to do for me. I love you very much. Next would be my partner and best friend - Kristopher M. Koskela – who is also graduating this year with his Ph.D. in chemistry! Kris, you were my first and closest friend when we first moved here. I was immediately drawn to you in a way that I couldn’t fully comprehend at the time and now, five years later we’re ready to start the next phase of our lives together. Without your emotional support I wouldn’t have finished my degree. Thank you for being there for me and for telling me that I’m smart and strong and competent. You’re everything I could have ever asked for. My next acknowledgement is reserved for my platonic best friend, Jesse Spivey, who I met in my general chemistry sequence in college. Jesse, you were literally the only reason I changed my major to chemistry, so if I hadn’t met you, I would have certainly ended up with a different career path. Thank you for all the laughs, the hours of studying, and exploring Denver with me. I’m sorry I’m graduating before you (!), but you’ll be done before you know it. Along those lines there are some other women I need to thank - Carly Meador from DU, who took advanced physics and math courses with me, and my sisters, Claire, Brooke, and Rachel, for generally being the best. I love all of you! v I have several professors to thank for their kindness and support throughout my studies. Getting to the graduate school finish line was as much of an emotional battle for me as it was academic, so without the help of the following people I wouldn’t be here today. First and foremost, I must thank my advisor Dr. Andrey Vilesov for being a great mentor and introducing me to the Quantum Fluid Clusters community. Andrey, you’re a great person, I’ll cherish the memories we have from our XFEL experiments for the rest of my life. I want to thank Dr. Armand Tanguay for providing me with crucial emotional support at a time when I really needed it. Armand, thanks so much for being understanding and helping me realize that I was capable of finishing not only your course, but my entire Ph.D. I want to thank Dr. Curt Wittig for generally being a friend to me, being easy to talk to, and always providing good life advice. I must also thank my undergraduate advisor Dr. Deborah Gale Mitchell who was my general chemistry instructor and, later, my friend. Debbie, you will be one of those professors that I remember for the rest of my life for how much of an emotional impact you made on me. You’re such a sweet person and a great chemistry instructor. I really appreciate the help you gave to me when I first started general chemistry and was apprehensive about my performance. Without your words of encouragement, I probably wouldn’t have changed my major to chemistry. I want to thank two other professors from DU, both Dr. Kingshuk Ghosh for the number of exciting physics classes he taught me, and Dr. Aysel Erey for making calculus fun, memorable, and not just an overblown algebra course. Both of you are great at what you do, and I wish you the best. vi Some more people to thank would be my group members and numerous collaborators. Most importantly, I need to thank Dr. Sean M.O. O’Connell-Lopez for being my mentor in lab and putting up with me when I wasn’t the most pleasant person to be around. You’re such a great person and I’m so thankful for all the academic support you gave me early on, as well as your continual words of encouragement and sharing of quality memes. I want to thank Ben Kamerin and John Niman of the Vitaly Kresin lab who were, for all intents and purposes, members of my group. I also want to thank Swetha Erukala, who suffered through everything with me. On top of that, I want to thank my key collaborators: Dr. Oliver Gessner and Catherine Saladrigas of Lawrence Berkeley National Laboratory, Rico Mayro P. Tanyag, a previous member of my group now at TU Berlin, Daniela Rupp of MBI Berlin and ETH Zurich, Paul Scheier of Innsbruck, and the staff scientists at SLAC and EXFEL, specifically Peter Walter, Yevheniy Ovcharenko, and Simon Dold. There are probably more people to thank, but the individuals mentioned above were the most crucial in getting where I am today. Thank you for all your kind words and for believing in me. I am forever grateful. vii Table of Contents Epigraph………….………………………………………………………………………………..ii Dedication………..……………………………………………………………………………….iii Acknowledgements……………………………………………………………………………….iv List of Tables……………………………………………………………………………………...x List of Figures…………………………………………………………………………………….xi Abstract………………………………………………………………………………………….xvi List of Projects and Scientific Contributions…………………………………………………….xx Chapter 1 – Introduction…………………………………………………………………...1 1.1 Liquid Helium…………………………………………………………………………1 1.2 Imaging Quantum Vortices in Bulk Liquid Helium…………………………………..8 1.3 Helium Nanodroplets………………………………………………………………….9 1.4 References……………………………………………………………………………11 Chapter 2 – X-ray CDI of Helium Nanodroplets ……………………….…………..14 2.1 – Experimental Setup………………………………………………………………...14 2.2 – X-ray CDI of Helium Nanodroplets……………………………………………….29 2.3 – Diffraction Patterns from Helium Nanodroplets…………………………………...30 2.4 – Sizes and Shapes of Helium Nanodroplets………………………………………...32 2.5 – Xenon Doped Helium Nanodroplets……………………………………………….41 2.6 – References………………………………………………………………………….45 Chapter 3 – Aggregation of Solutes in Bosonic and Fermionic Quantum Fluids………………….………………………………………………………………………..50 3.1 – Abstract…………………………………………………………………………….50 viii 3.2 – Introduction………………………………………………………………………...51 3.3 – Results……………………………………………………………………………...54 3.4 – Discussion………………………………………………………………………….56 3.5 – Materials and Methods……………………………………………………………..58 3.6 – References………………………………………………………………………….65 Chapter 4 – X-ray CDI of Highly Ionized Helium Nanodroplets………………..69 4.1 – Abstract…………………………………………………………………………….69 4.2 – Introduction………………………………………………………………………...69 4.3 – Experimental……………………………………………………………………….71 4.4 – Results……………………………………………………………………………...73 4.5 – Discussion………………………………………………………………………….76 4.6 – References………………………………………………………………………….81 Chapter 5 – X-ray CDI of Helium Nanodroplets Doped with Small Molecules...…………………………………………………………………………………….86 5.1 – Abstract…………………………………………………………………………….86 5.2 – Introduction………………………………………………………………………...86 5.3 – Experimental……………………………………………………………………….87 5.4 – Results……………………………………………………………………………...89 5.5 – Discussion………………………………………………………………………….97 5.6 – References………………………………………………………………………...100 Chapter 6 – Validating the Droplet Coherent Diffractive Imaging (DCDI) Algorithm…………………………………………………………………………………….102 6.1 – Introduction……………………………………………………………………….102 6.2 – Principals of Validating DCDI……………………………………………………103 ix 6.3 – DCDI for diffraction of different intensity……………………………………….106 6.4 – DCDI for extended embedded objects……………………………………………108 6.5 – Conclusions……………………………………………………………………….113 6.6 – References………………………………………………………………………...113 Chapter 7 – Conclusions and Outlooks ……………………….……………………..114 References……………………………………………………….……………………...117 x List of Tables Table 1. Typical absolute pressures maintained in the vacuum chambers for best possible alignment. The pressure reading in the source chamber corresponds to the pressure of helium gas while the pressure readings for the other chambers at 298 K correspond to rest gases. ………………………………………………………………..28 xi List of Figures Abstract Figure 1.0.0. The Vilesov research group at SLAC in 2018……………………………………xxi Chapter 1 – Introduction Figure 1.1.1. Phase diagram of 4 He. Two liquid phases are accessible: helium-I, a normal fluid, and helium-II, a superfluid.…………………………………………………………………1 Figure 1.2.1. Heat capacity of helium-4 at low temperatures.……………………………...........2 Figure 1.1.3. Superfluid film flow. A test tube of liquid helium is immersed in a larger bath of liquid helium. The levels of the two fluids are initially not equal. Due to its ability to flow without friction, superfluid helium will climb the container walls and enter the test tube until the levels of the bath and the tube are equal. ………………………………………..3 Figure 1.1.4. Superfluid and normal fluid fractions in 4 He versus temperature. ……………...…5 Figure 1.1.5. Bosonic 4 He atom’s nucleus versus fermionic 3 He atom’s nucleus. ………………6 Figure 1.1.6. Rotating liquid buckets. Left: Normal fluids and superfluids at rest. Middle: A normal fluid rotating as a rigid body. Right: A superfluid set into rotation. A vortex array can be seen with the vortices aligned parallel to the axis of rotation. Each vortex has a unit of vorticity in the same direction as the entire entity. ………………………………….7 Figure 1.2.1. Imaging quantum vortex configurations in bulk liquid helium-4. Adapted from Packard and coworkers. ……………………………………………………………….........9 Figure 1.3.1. Sizes of small helium clusters versus large helium nanodroplets. ……………………………………………………………………………………………………10 Chapter 2 – X-ray CDI of Helium Nanodroplets Figure 2.1.1. Simplified schematic of the XFEL experiment. The droplets exit the nozzle and enter the adjacent pickup chamber where they pick up dopant atoms such as xenon. The doped droplets continue traveling through the vacuum chambers before entering the interaction region. Here, the doped droplets are irradiated with the focused XFEL beam. Scattered photons are collected on a pnCCD detector behind the interaction region. Droplet fragments are collected via TOF. Further downstream, the droplet flux is analyzed by RGA. ………………………………………………16 Figure 2.1.2. Block diagram of the experiment showing the five vacuum chambers. From top: source chamber, pickup chamber, interaction chamber, detection (pnCCD) chamber, and RGA chamber….…………………………………………………………………17 Figure 2.1.3. From right: Source chamber (DN 200 flange), pickup chamber (DN 300 flange), differential pumping stage, NQS (scattering) chamber at EXFEL. ………………………………………………………………………………….………18 Figure 2.1.4. A view of the doping chamber utilized at EXFEL. Left: gas doping cell. Right: heated cell used to evaporate metal atoms.………………………………………………...19 Figure 2.1.5. pnCCD chamber to the NQS (scattering) chamber at EXFEL. ……………………20 Figure 2.1.6. The 4 He cryocooler, with the nozzle assembly attached to its second xii stage. A temperature sensor is attached to the nozzle assembly to monitor T0 and a resistive heating pad is attached to manipulate T0. The Swagelok fitted copper helium gas line can be seen wrapped around the cryocooler. Cylindrical aluminum shields cover the entire assembly to minimize the effect of thermal fluctuations. Taken with permission from Tanyag et al. ……………………………………………………………....21 Figure 2.1.7. Shows photographs of liquid helium jets produced at 3.5 K and 3, 7, 12, 20, 40 and 60 bar. The jet was imaged at three different regions, covering the total distance of 3.5 mm from the nozzle cap. Each region is about 1.5 mm long and there is ~0.5 mm overlap between these regions. The 100 μm scale for each region is given in the lower right corner in region 1. Large grey spots visible in the same positions in all panels are artefacts of the imaging setup. ……………….….23 Figure 2.1.8. Liquid helium flowing cryostat used to produce 3 He droplets. Non-research grade liquid helium is used to cool the nozzle by flowing from a dewar through a transfer line into the cryostat body. Research grade helium gas (99.9999% pure) is expanded through the nozzle to produce droplets. ……………...…………...26 Figure 2.1.9. 3 He recycling system, built by Sean O’Connell. The system has three components: a compressor (left), liquid nitrogen dewars (right bottom), and the gas manifold (right top)…….. ……………...…………...……………...…………...…………….....27 Figure 2.3.1. Diffraction patterns of pure 4 He droplets shown on a logarithmic color scale as indicated on the left. Images represent the central 660 x 660 detector pixels. Taken with permission from Gomez et al. …………………………………………….….31 Figure 2.3.2. Diffraction patterns of pure 3 He droplets shown on a logarithmic color scale as indicated on the right. Images represent the central 660×660 detector pixels. Corresponding droplet projection half axes (A, C) and their aspect ratios (AR) are displayed on top of each image. ……………………………………….…………32 Figure 2.4.1. Pressure-temperature phase diagrams for 4 He (a) and 3 He (b). The pink SVP curve marks the saturated vapor pressure boundaries. …………………………...……33 Figure 2.4.2. Shapes of classical droplets: (a) spheroidal, (b) elliptical, and (c) caspule. Adapted from Baldwin, Butler, and Hill. ………………………………...…………….35 Figure 2.4.3. Stability diagram for rotating droplets in equilibrium as a function of reduced angular velocity (Ω) and angular momentum (Λ) (see equations 2.1a and 2.1b). The upper branch corresponds to oblate, axisymmetric shapes, whereas the lower branch corresponds to two-lobed shapes. The bifurcation point is located at Λ = 1.2, Ω = 0.56 with AR = 1.48. …………………………………...…………...………………37 Figure 2.4.4. Droplet size A (a) and aspect ratio AR (b) distributions for 3 He (blue) and 4 He (red) isotopes presented in logarithmic scales. The counts for 4 He were multiplied by a factor of 3 for ease of comparison, as the total number of diffraction images obtained for 3 He and 4 He were ∼900 and ∼300, respectively. Squares in panel (c) show average AR as obtained for each bin in panel (a) for the points with AR < 1.4. The results of single measurements with AR > 1.4 are shown by stars in panel (c). The blue line in panel (c) represents a linear fit of the data points (blue squares) for 3 He droplets. …………………………………………………………….…….38 Figure 2.4.5. Red curve: Calculated aspect ratio ar as a function of reduced angular momentum (L) for axially symmetric oblate droplet shapes. Blue curve: stability diagram of rotating droplets in terms of reduced angular velocity (W) and reduced angular momentum (L). The upper branch (dashed blue) corresponds to xiii unstable axially symmetric shapes. The lower branch (dotted blue) is associated with prolate triaxial droplet shapes resembling capsules and dumbbells. The green circle and black cross on the red curve represent the average ⟨ar⟩ for 3 He and 4 He droplets, respectively, obtained in this work (with AR < 1.4). Similar markers on the blue curve indicate the (W, L) values corresponding to 3 He and 4 He droplets.…...…………...41 Figure 2.5.1. A and B) Diffraction images from Xe-doped helium nanodroplets exhibiting Bragg patterns. C) Simulation of how the vortices are oriented in the droplet. Adapted from Gomez et al (16). ………………………………………………………...42 Figure 2.5.2. Schematic of the DCDI algorithm. The algorithm is initiated using a preset helium droplet density, ρ !"#$% . A series of inverse Fourier transforms are performed with iterative reinforcement of constraints. The process rapidly converges to a solution yielding the density of the Xe clusters inside the droplet. Taken with permission from Tanyag et al. …………………………………...…………................................44 Figure 2.5.3. Diffraction patterns from Xe-doped helium nanodroplets of various shapes: (a1) axisymmetric (nearly spherical), (b1) triaxial ellipsoidal, and (c1) capsule shaped. Reconstructions for each diffraction pattern as produced by DCDI (a2-c2) are shown below. The basis vectors of the vortex lattice in (c2) are shown in the upper right corner. Photo taken with permission from O’Connell et al. (32). ………………………………………………………………………………………...45 Chapter 3 – Aggregation of Solutes in Bosonic and Fermionic Quantum Fluids Figure 3.1.1. Outlines (black) and xenon dopant density distributions (blue-red) of superfluid 4 He droplets. Panels a-d show results for four different representative 4 He droplets. The values a and b of the long and short half axis, respectively, of the droplet’s projection onto the detector plane are given in each panel. For visualization, circular contours (magenta) have been superimposed on the droplets with a radius equal to that of the minor half axis. Closer inspection reveals slightly elliptical distortions, most prominent in droplet b. ………………………………52 Figure 3.1.2. Outlines (black) and Xe dopant density distributions (blue-red) of normal fluid 3 He droplets. Panels a-d show results for four different representative 3 He droplets. The values of the long and short half axis of the droplet’s projection onto the detector plane are given in each panel. For visualization, circular contours (magenta) have been superimposed on the droplets with a radius equal to that of the minor half axis. Note the partly significant elliptical distortions of the droplet outlines. ………………………………………………………………………………….53 Figure 3.1.3. Diffraction images from 4 He nanodroplets in a logarithmic color scale. …………..63 Figure 3.1.4. Diffraction images from 3 He nanodroplets in a logarithmic color scale. …………...…………...…………...………………………………….....…………...64 Chapter 4 – X-ray CDI of Highly Ionized Helium Nanodroplets Figure 4.3.1. Schematic of the experiment. Helium droplets are ionized via electron impact and pass-through plane capacitor electrostatic deflectors. Ionized helium droplets are doped with Xe atoms which cluster around the positions of the charges and serve as markers. The droplets are interrogated with the xiv XFEL. Diffraction patterns are recorded on a pnCCD detector and processed using a phase retrieval algorithm to obtain the density profile and charge distributions. The predicted configuration for 18 charges is used as an example to simulate the displayed diffraction pattern. ……………………………………………….…...71 Figure 4.4.1. Results for uncharged, Xe doped 4 He droplets. (1a),(2a): Diffraction patterns showing the central 600 x 600 detector pixels. The vertical streak in the upper half of the patterns is caused by stray light. (1b),(2b): Density reconstructions obtained with the DCDI algorithm………………...…………...……………......73 Figure 4.4.2. Same as in Fig. 2, but for charged, Xe-doped 4 He droplets. For droplets 1 – 3, the electron energy was 40 eV and emission current 30 μA, while for droplet 4, the ionizer was set to 200 eV and 1.3 mA emission current. …………………………………………………………………...……………74 Figure 4.6.1. Our experimental team. …………………………………………………………...81 Chapter 5 – X-ray CDI of Helium Nanodroplets Doped with Small Molecules Figure 5.3.1. (a) Diagram of experimental setup. (b) sample ion TOF spectrum for a neat 4 He droplet………………………………………………………………….…………89 Figure 5.4.1. (1a, 2a) Diffraction patterns from CHF3-doped superfluid 4 He droplets, zoomed into 600 x 600 pixels. The blank horizontal stripe in the diffraction stems from the physical gap between the upper and lower pnCCD detector plates. (1b, 2b) Density reconstructions obtained from the DCDI algorithm. ………………………...…90 Figure 5.4.2. (1a) Diffraction pattern from CF4-doped 4 He droplets, zoomed into 600 x 600 pixels. The blank horizontal stripe in the diffraction stems from the physical gap between the upper and lower pnCCD detector plates. (1b) Density reconstruction obtained from the DCDI algorithm. …………...………………………………………..……….93 Figure 5.4.3. (1a) Diffraction pattern from superfluid SF6-doped 4 He droplets, zoomed into 600 x 600 pixels. The blank horizontal stripe in the diffraction stems from the physical gap between the upper and lower pnCCD detector plates. (1b) Density reconstruction obtained from the DCDI algorithm.……………………………………… ……...94 Figure 5.4.5. Cation TOF spectra for neat 3 He (purple) and 4 He (green) droplets doped with atoms or small molecules. The black tick marks show the mass scale. Taller bars are positioned every 10 mass units. ………………………………………………………....95 Figure 5.5.1. (a) Density plot of xenon clusters aggregating in 4 He nanodroplets (16). (b) Density plot of CHF3 aggregating in 4 He nanodroplets (from Fig. 2(1b) in this text). …………………………………...…………...……………...…………………….….97 Figure 5.5.2. A quantum vortex capturing particles of small (top) and large (bottom) sizes in time. Vortices are displayed in red, and particles are in green. Large particles distort the vortex more than small particles, causing pronounced curvature. Adapted from Giuriato and Krstulovic. …………………...…………...……………...…………...………98 xv Chapter 6 – Validating the Droplet Coherent Diffractive Imaging (DCDI) Algorithm Figure 6.2.1. (a) Experimentally obtained diffraction pattern. (b) The droplet density reconstruction obtained via DCDI. (c) Synthetic input droplet produced from the real and imaginary parts of (b). (d) Synthetic diffraction produced from (c) via FT. The apparent difference in the color scale between (a) and (d) stems from different normalization, which is calculated from the zero-order diffraction maximum but gets cutoff in (a) due to the hole in the detector. ………………………………...…………...……………...…………...……….105 Figure 6.2.2. Schematic of the mask employed for DCDI tests. (a) shows the pnCCD gap in the experimental diffraction pattern. (b) shows the mask employed to cover the gap. The central black stripe and circle are not considered in the DCDI calculations. ……………………………......……………...…………...……………...…………...................106 Figure 6.3.1. Applying a Poisson distribution to a continuous diffraction pattern. (a) shows the continuous pattern. (b) shows the discretized pattern. …………………….……...107 Figure 6.3.2. Density reconstructions produced from discretized synthetic diffraction patterns with varying intensities. The number of photons below each droplet shows the number of photons in the input diffraction with a mask applied, as in Fig 2(b). The minimum number of detected photons needed to produce a valid reconstruction is on the order of 10 5 . …………………………………………….………..107 Figure 6.4.1. Left: Synthetically produced droplet of radius b containing a single spherical cluster of radius b/14. Right: Diffraction pattern obtained by FT of the density in (a). ……………………………...…………...……………...………….............…109 Figure 6.4.2. DCDI reconstructions from clusters of varying sizes. For the initial densities shown, the cluster is in the middle position. The program rotates the droplet in the output, hence why the clusters are on the opposite sides as the input. ……………………………………………………………...…………...............…110 Figure 6.4.3. Reconstructions for small to medium sized inner clusters of size b/14 to b/6. ……111 Figure 6.4.4. Reconstructions for large inner clusters of size b/4 and b/2. ……………………112 xvi Abstract Liquid helium exhibits fascinating properties macroscopically and on the nanoscale. Its most notable quality is that of superfluidity; a novel state of matter characterized by vanishing viscosity (i.e., fluid flow without the loss of any kinetic energy), very high thermal conductivity, and other bizarre effects like film flow and vortices with quantized circulation (1-3). Many experiments to image quantum vortices in bulk superfluid helium have been attempted (4-7). However, these experiments faced significant difficulties both with achieving low enough temperatures and with the spatial resolution of the optical imaging approaches employed. As such, related experiments pushed to probe superfluids on the nanoscale. Nano-sized droplets of helium present a new possibility for investigation, as they have recently been the focus of many chemical physics groups (8-10). The droplets are versatile in size and can range from few to 10 12 atoms (corresponding to diameters of sub- nm to ~6 micron, respectively) (11-13). Imaging quantum vortices within isolated helium nanodroplets is enticing due to their low temperatures of 0.37 K, and absence of any convection flow, which could not be avoided in bulk experiments (14, 15). The droplets readily pick up atoms or molecules (14, 15), meaning they are perfect for imaging experiments via tagging. The first experiment with tagging of quantum vortices in helium nanodroplets was published in 2012 when large silver clusters, assembled in helium nanodroplets, were deposited onto a substrate (16). The deposits revealed elongated, filament-like silver clusters. It was proposed that inside the droplet, silver clusters are aggregating within the cores of the quantum vortices. This result was the first example of vortex- assisted aggregation in nanoscale helium droplets (16). Recent experiments have pushed to probe helium nanodroplets with x-ray free electron lasers (XFELs) (17-22). Due to short wavelength and their high degree of transverse coherence, xvii XFELs have substantially increased resolution (< 20 nm) as compared to previous optical imaging techniques (23, 24). In such experiments, single droplets are imaged via x-ray diffraction. Results from the first work of its kind revealed distinct Bragg spots in the diffraction images, indicative of a lattice of quantum vortices within the droplet (18). Later, a computer algorithm was developed to reconstruct the internal structure of the doped droplet from the obtained diffraction patterns (24). Application of the algorithm to diffraction patterns of doped 4 He nanodroplets revealed filaments of vortices throughout the droplets volume, or triangular lattice patterns depending on the observer’s viewing angle. The application of XFELs holds promise for other experiments involving helium nanodroplets. This thesis presents a variety of x-ray diffraction experiments with nanodroplets, with both 4 He and 3 He isotopes, as well as highly charged 4 He nanodroplets. The results of such experiments have concluded our series of works regarding the existence of quantum vortices in finite systems. References 1. D. R. Tilley, J. Tilley, Superfluidity and Superconductivity. (Institute of Physics Publishing, Bristol, 1990). 2. J. Wilks, D. S. Betts, An Introduction to Liquid Helium. (Clarendon Press, Oxford, 1987). 3. R. P. Feynman, Application of Quantum Mechanics to Liquid Helium. Progress in Low Temperature Physics 1, 17-53 (1955). 4. G. P. Bewley, D. P. Lathrop, K. R. Sreenivasan, Superfluid helium—visualization of quantized vortices. Nature, 441:588 (2006). xviii 5. G.W. Rayfield, F. Reif, Evidence for the creation and motion of quantized vortex rings in superfluid helium. Phys. Rev. Lett. 11, 305-308 (1963). 6. P. W. Karn, D. R. Starks, W. Zimmerman, Observation of quantization of circulation in rotating superfluid He-4. Phys. Rev. B 21, 797-805 (1980). 7. W. F. Vinen, Detection of single quanta of circulation in liquid helium-II. Proc. R. Soc. A 260:2, 18-36 (1961). 8. M. Y. Choi et al., Infrared spectroscopy of helium nanodroplets: novel methods for physics and chemistry. Int. Revs. in Phys. Chem. 25, 15-75 (2006). 9. J. P. Toennies, A. F. Vilesov, Superfluid helium droplets: A uniquely cold nanomatrix for molecules and molecular complexes. Angew. Chem. Int. Edit. 43, 2622 (2004). 10. D. Verma, R. M. P. Tanyag, S. M. O. O’Connell-Lopez, A. F. Vilesov, Infrared Spectroscopy in Superfluid Helium Droplets. Adv. in Phys. X. 4, (2019). 11. J. P. Toennies, A. F. Vilesov, Spectroscopy of Atoms and Molecules in Liquid Helium. Annu. Rev. Phys. Chem. 49, (1998). 12. J. P. Toennies, A. F. Vilesov, K. B. Whaley, Superfluid Helium Droplets: an Ultracold Nanolaboratory. Phys. Today 2, 31 (2001). 13. J. A. Northby, Experimental studies of helium droplets. J. Chem. Phys. 115, 10065 (2001). 14. R. M. P. Tanyag et al., in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, A. Osterwalder, O. Dulieu, Eds. (Royal Society of Chemistry, Cambridge, 2018). xix 15. R. Hartmann, R. E. Miller, J. P. Toennies, A. F. Vilesov, Rotationally Resolved Spectroscopy of SF6 in Liquid-Helium Clusters - a Molecular Probe of Cluster Temperature. Phys. Rev. Lett. 75, 1566 (1995). 16. L. F. Gomez, E. Loginov, A. F. Vilesov, Traces of Vortices in Superfluid Helium Droplets. Phys. Rev. Lett. 108, 155302 (2012). 17. A. J. Feinberg et al., Aggregation of solutes in bosonic versus fermionic quantum fluids. Science Advances 7, (2021). 18. L. F. Gomez et al., Shapes and vorticities of superfluid helium nanodroplets. Science 345, 906-909 (2014). 19. B. Langbehn et al., Three-dimensional shapes of spinning helium nanodroplets. Phys. Rev. Lett. 121, 255301 (2018). 20. S. M. O. O’Connell et al., Angular momentum in rotating superfluid droplets. Phys. Rev. Lett. 124, 215301 (2020). 21. O. Gessner, A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Droplets. Annu. Rev. Phys. Chem. 70, 173 (2019). 22. C. Bernando et al., Shapes of Rotating Superfluid Helium Nanodroplets. Phys. Rev. B 95, 064510 (2017). 23. C. F. Jones et al., Coupled motion of Xe clusters and quantum vortices in He nanodroplets. Physical Review B 93, 180510 (2016). 24. R. M. P. Tanyag et al., Communication: X-ray coherent diffractive imaging by immersion in nanodroplets. Struct Dynam-Us 2, 051102 (2015). xx List of Projects and Scientific Contributions Below is a list of works that I led or contributed to throughout my Ph.D. Most of my papers came from beam time experiments, of which I have participated in four throughout my studies. My first beam time was conducted at Stanford’s LCLS-I XFEL in 2018. Shortly thereafter, I participated in two beam time experiments at the European XFEL in the summer of 2019, led by Daniela Rupp, and by Rico Mayro P. Tanyag. In 2020, amidst the COVID-19 pandemic, I led a remote beam time at the European XFEL in collaboration with the groups of Paul Scheier, Daniela Rupp, and Oliver Gessner. Besides XFEL-based imaging experiments, I performed in-house infrared spectroscopy experiments, worked on the analysis and publication of some previous experiments from our group, and acted as Scientific Secretary for the 2022 International Conference on Quantum Fluid Clusters in Erice, Sicily, Italy. Completed works 1. Alexandra J. Feinberg, Deepak Verma, Sean M.O. O’Connell-Lopez, Swetha Erukala, Rico Mayro P. Tanyag, Weiwu Pang, Catherine Saladrigas, Benjamin W. Toulson, Mario Borgwardt, Niranjan Shivaram, Ming-Fu Lin, Andre Al-Haddad, Wolfgang Jäger, Christoph Bostedt, Peter Walter, Oliver Gessner, Andrey F. Vilesov. Aggregation of solutes in bosonic versus fermionic quantum fluids. Science Advances 7, abk2247, (2021). In this paper, aggregation of Xe clusters is studied in helium-4 nanodroplets, which contain vortices, and compared to helium-3 nanodroplets, which are devoid of vortices. Results showed that while Xe atoms aggregate in the cores of the quantum vortices in helium-4, they form a ring around the periphery of the helium-3 droplet. Results from these experiments were the first of their xxi kind and validated the existence of quantum vortices in bosonic helium-4. This work was performed at SLAC XFEL in 2018 and was one of my main projects. I was a co-proposer of the beamtime proposal, took part in the experiments, and worked on the data analysis and preparation of the publication. Figure 1.0.0. The Vilesov research group at SLAC in 2018. 2. Alexandra J. Feinberg, Felix Laimer, Rico Mayro P. Tanyag, Björn Senfftleben, Yevheniy Ovcharenko, Simon Dold, Michael Gatchell, Sean M.O. O’Connell-Lopez, Swetha Erukala, Catherine A. Saladrigas, Benjamin W. Toulson, Andreas Hoffman, Ben Kamerin, Rebecca Boll, Alberto De Fanis , Patrik Grychtol, Tommaso Mazza, Jacobo Montano, Kiana Setoodehnia, David xxii Lomidze, Robert Hartmann, Philipp Schmidt, Anatoli Ulmer, Alessandro Colombo, Michael Meyer, Thomas Möller, Daniela Rupp, Oliver Gessner, Paul Scheier, and Andrey F. Vilesov. Diffractive imaging of highly charged helium nanodroplets. Phys. Rev. Res. In press, (2022). In this paper, aggregation of Xe clusters is studied in highly charged helium-4 nanodroplets and compared to results for uncharged. This experiment relates to the Thomson problem, which asks what is the minimum energy configuration of N unit charges on a sphere? Helium nanodroplets can act as experimental realizations of Thomson systems as they are highly spherical and can hold multiple charges. This work was performed at the European XFEL in Hamburg in 2020 and was one of my main projects. I was a co-proposer of the beamtime proposal, took part in the online experiments, worked on the data analysis, and prepared the publication. Leading this experiment was particularly involved due to the timing overlapping with the start of the COVID- 19 pandemic. As a result, the experiments were performed in an online mode. Despite that, we still managed to complete the experiment with the help of our European based team, collaborators, and the staff scientists at EXFEL. 3. Swetha Erukala, Alexandra J. Feinberg, Amandeep Singh, Andrey F. Vilesov. Infrared spectroscopy of carbocations upon electron ionization of ethylene in helium nanodroplets. J. Chem. Phys., 155, 084306 (2021). In this experiment, the infrared spectra of a variety of carbocations originating from ionized ethylene in helium droplets were obtained. This work was performed in the lab at USC. My role involved setting up the machine in the beginning of the day and carrying out measurements. Additionally, I assisted in writing the manuscript, which was primarily prepared by Swetha Erukala and Andrey F. Vilesov. xxiii 4. Swetha Erukala, Alexandra J. Feinberg, Cheol Joo Moon, Myong Yong Choi and Andrey F. Vilesov. Infrared spectroscopy of ions and ionic clusters upon ionization of ethane in helium droplets. J. Chem. Phys. 156, 204306, (2022). In this experiment, infrared spectra of a variety of carbocations originating from ionized ethane molecules were obtained. My role was the same as with the previous experiment involving ethylene. 5. Catherine A. Saladrigas, Alexandra J. Feinberg, Michael P. Ziemkiewicz, Camilla Bacellar, Maximillian Bucher, Charles Bernando, Sebastian Carron, Adam S. Chatterley, Franz-Josef Decker, Ken R. Ferguson, Luis Gomez, Taisia Gorkhover, Nathan A. Helvy, Curtis F. Jones, Justin J. Kwok, Alberto Lutman, Daniela Rupp, Rico Mayro P. Tanyag, Thomas Moëller, Daniel M. Neumark, Christoph Bostedt, Andrey F. Vilesov, and Oliver Gessner. Charging and ion ejection dynamics of large helium nanodroplets exposed to intense femtosecond soft X-ray pulses. Eur. Phys. J. Spec. Top. 230, 4011 (2021). This paper resulted from the previous XFEL collaborative experiments of our group. I performed the analysis of the diffraction images using our phase retrieval algorithm, which yielded droplet sizes and photon fluxes for individual droplets. Additionally, I assisted in writing and editing the manuscript. 6. Rico Mayro P. Tanyag, Alexandra J. Feinberg, Sean M.O. O’Connell, Andrey F. Vilesov. Disintegration of diminutive liquid helium jets in vacuum. J. Chem. Phys. 152, 324306 (2020). This paper resulted from the previous laboratory experiments. I prepared the manuscript using data obtained by Rico Tanyag and facilitated its publication. xxiv 7. 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, Andre Al Haddad, Wolfgang Jaëger, Christoph Bostedt, Peter Walter, Oliver Gessner, and Andrey F. Vilesov. Shapes of rotating normal fluid 3 He versus superfluid 4 He in molecular beams. Phys. Rev. B. 102, 014504 (2020). This work was performed at SLAC in 2018. I was a co-proposer of the beamtime proposal, took part in the experiments, and worked on the data analysis and preparation of the publication. Works in preparation 1. Alexandra J. Feinberg, Catherine A. Saladrigas, Landin Stein, Oliver Gessner, Andrey F. Vilesov. X-ray coherent diffractive imaging of helium nanodroplets doped with small molecules. In this work, aggregation of novel clusters of varying geometry and polarity are studied in helium-4 and helium-3 nanodroplets. The goal of this study was to determine the effect of intermolecular interactions on aggregation. A variety of atoms and small molecules were used, including Xe, CF4, CHF3, CF3Cl, CH3CN, H2O, and SF6. Droplets and clusters were exposed to soft X-ray pulses and studied by X-ray coherent diffractive imaging in addition to single-pulse ion time-of-flight spectroscopy. The data from this experiment came from the same beam time as the Science Advances paper. Thus, my experimental role, and role in post processing, was the same as in the first paragraph. As far as writing goes, I prepared the initial draft and continued the writing process with Andrey F. Vilesov and our collaborators. xxv 2. Alexandra J. Feinberg, Swetha Erukala, Choel Moon, Andrey F. Vilesov. Infrared spectroscopy of carbocations upon electron ionization of propylene in helium nanodroplets. In this experiment, the infrared spectra of a variety of carbocations originating from ionized propylene were obtained. My role involved planning the day-to-day experiments and carrying out measurements. I was assisted by my lab mates, Swetha Erukala and Choel Moon. Swetha aided with instrumentation and lab software, while Choel assisted me in recording spectra and performing calculations for various ions. I prepared the initial draft of the manuscript, which is in prep. 4. Sean M. O. O’Connell, Deepak Verma, Alexandra J. Feinberg, et al. Phase separation in mixed 3 He/ 4 He nanodroplets. The data from this experiment came from the same beam time as the Science Advances paper. Thus, my experimental role was the same as in the first paragraph. I assisted in writing and editing the manuscript, which was primarily prepared by Sean M. O. O’Connell and Andrey F. Vilesov. 5. Anatoli Ulmer, Andrea Heilrath, Björn Senfftleben, Björn Kruse, Lennart Seiffert, Sean Marcus O. O’Connell, Katharina Kolatzki, Bruno Langbehn, Andreas Hoffmann, Thomas Baumann, Rebecca Boll, Adam Chatterley, Alberto de Fanis, Benjamin Erk, Swetha Erukala, Alexandra Feinberg, Thomas Fennel, Patrick Grychtol, Robert Hartmann, Steffen Hauf, Marcus Ilchen, Manuel Izquierdo, Bennet Krebs, Markus Kuster, Tommaso Mazza, Karl-Heinz Meiwes-Broer, Jacobo Montaño, Georg Noffz, Daniel Rivas, Dieter Schlosser, Fabian Seel, Henrik Stapelfeldt, Lothar Strüder, Josef Tiggesbäumker, Andrey Vilesov, Hazem Yousef, Michael Zabel, Pawel xxvi Ziolkowski, Michael Meyer, Yevheniy Ovcharenko, Thomas Möller, Daniela Rupp, and Rico Mayro P. Tanyag. Xenon nanostructures inside superfluid helium droplets. For this project, I participated in the experiments and managed the data entry for variable experimental parameters that changed on a run-to-run basis. Additionally, I assisted in editing the manuscript, which was primarily prepared by Anatoli Ulmer and Rico Mayro P. Tanyag. 1 Chapter 1 – Introduction 1.1 – Liquid helium Helium is a unique element that exhibits fascinating properties unlike any other material in the universe. Its most notable property is that it remains liquid down to absolute zero temperature, only solidifying at high pressures (see Fig. 1.1.1). This effect is quantum mechanical in nature and stems from helium’s large zero-point energy, which is greater than in any other atom (1, 2). At low temperatures, there are two liquid regimes to be considered: liquid helium-I, a normal fluid, and liquid helium-II, also known as superfluid helium. In these regimes, the two liquids exhibit fundamentally different physical properties with distinct behaviors. Figure 1.1.1. Phase diagram of 4 He. Two liquid phases are accessible: helium-I, a normal fluid, and helium-II, a superfluid. 2 Characteristics of superfluid helium (helium-II) The classical fluid that is liquid helium-I undergoes a phase transition to the superfluid state at 2.17 K and pressures below 2.5 MPa. The transition is marked by a dramatic change in heat capacity that can be seen in Fig. 1.1.2. Here, the shape of the heat capacity curve looks like a Greek letter lambda; thus, the superfluid transition is known as the lambda point. The singularity in the heat capacity is characteristic of second order phase transitions, such as with superfluidity and ferromagnetism, and is representative of particles ordering into a single quantum state (3). Figure 1.1.2. Heat capacity of helium-4 at low temperatures. Along with remaining liquid down to 0 K, superfluids exhibit other notable properties such as the loss of all viscosity – meaning that the fluid can flow indefinitely without the loss of kinetic energy. Additionally, superfluids exhibit very high thermal conductivity (30x that of copper) in 3 the low temperature regime. Other unusual effects have been observed in superfluid helium, like the fountain effect (a.k.a. the thermomechanical effect), second sound, and the manifestation of hydrodynamic vortices (to be discussed) (4, 5). Additionally, superfluids exhibit film flow, which refers to a phenomenon in which a test tube of liquid helium is immersed in a larger bath of liquid helium. Once immersed, liquid helium from the bath will flow up the test tube walls like in a syphon until the levels are equalized (see Fig. 1.13) (1-3, 6). Figure 1.1.3. Superfluid film flow. A test tube of liquid helium is immersed in a larger bath of liquid helium. The levels of the two fluids are initially not equal. Due to its ability to flow without friction, superfluid helium will climb the container walls and enter the test tube until the levels of the bath and the tube are equal. 4 Landau and Tisza’s Two Fluid Model The phenomenon of superfluidity can be better understood in terms of the two-fluid model. Tisza put forward an initial model that was later expanded by Landau (7). The model states that a superfluid is composed of a superfluid component, with zero viscosity and zero entropy, and a normal fluid component, with finite viscosity and entropy of values 𝜂 and S. At absolute zero, the fraction of the superfluid component in a liquid helium bath is 100% with zero contributions from the normal fluid component. However, as the temperature is increased from T = 0 K, the fraction of the normal fluid component increases, representing elementary excitations that occur in the liquid as a function of temperature (see Fig. 1.14). There are two main types of excitations in helium: phonons, which are a type of quantized sound wave, and rotons, which are elementary forms of rotational motion (1, 2). These thermally excited states comprise the normal fluid component at finite temperature. As the temperature of the liquid is increased, the concentration of the elementary excitations increases, and the fraction of the normal fluid component increases until reaching 100% at 2.17 K. 5 Figure 1.1.4. Superfluid and normal fluid fractions in 4 He versus temperature. Helium Isotopes and Relation to Bose-Einstein Condensation There are two stable isotopes of helium, 4 He and 3 He, and several unstable radioisotopes. 4 He is the most common, while 3 He is much rarer (only about 1 atom in 10 4 4 He atoms will be 3 He). At first glance, liquid 3 He and 4 He seem very similar, except for their difference in molar volume. However, 3 He and 4 He atoms obey different statistics, so their properties vary dramatically at low temperature. Superfluidity in the isotope of 4 He is related to Bose-Einstein condensation (BEC) (8). A BEC forms when an ensemble of bosonic particles condenses into a single quantum state that can be described by one wavefunction. 4 He atoms are bosonic as they are composed of an even number of subatomic particles – 2 protons, 2 neutrons, and 2 electrons, 6 having a total nuclear spin of I = 0 (see Fig. 1.1.5). Bosonic particles can condense into a single quantum state at low temperature (1, 3, 6, 8). Figure 1.1.5. Bosonic 4 He atom’s nucleus versus fermionic 3 He atom’s nucleus. 3 He atoms cannot become superfluid in the same way as 4 He. 3 He atoms are fermionic, as they are composed of an odd number of subatomic particles – 2 protons, 1 neutron, and 2 elections, having a total nuclear spin of I = ½ (see Fig. 1.1.5). As such, they will only become superfluid by forming a so-called “Cooper pair” at extremely low temperature (~ 1 mK) (9). Here, I = ½ particles will pair and act as a composite entity with spin I = 0 (2). Due to the low temperature of the 3 He superfluid transition, which is inaccessible in our experiments, 3 He exemplifies a low temperature normal fluid devoid of quantum vortices in all our studies. Quantum vortices in superfluid helium Another fundamental hallmark of superfluidity is the manifestation of hydrodynamic vortices. At low temperatures (below the lambda point), when the total wavefunction becomes 7 coherent and orders into a single quantum state, the circulation in the fluid becomes quantized. Quantum vortices can be observed if the fluid is set in rotation, such as in a rotating bucket (see Fig. 1.1.6). Figure 1.1.6. Rotating liquid buckets. Left: Normal fluids and superfluids at rest. Middle: A normal fluid rotating as a rigid body. Right: A superfluid set into rotation. A vortex array can be seen with the vortices aligned parallel to the axis of rotation. Each vortex has a unit of vorticity in the same direction as the entire entity. When set in rotation, liquid helium responds by creating several vortices parallel to the axis of rotation. A rigorous mathematical treatment proving how quantum vortices arise can be found in references (1, 2, 10). It follows that the velocity circulation around the vortex core is quantized in the units of 𝜅 = h/m4 in which m4 is the mass of a helium-4 atom (1, 6). When many straight vortices are present, they will form an array with a specific lattice spacing (11, 12). According to the Feynman relation, the entire vortex array rotates at angular velocity of (𝜅𝑛 & )/2, in which 𝑛 & is the areal density of vortices (1). Superfluid helium is therefore one of the very few instances where one can visualize quantum mechanics on a macroscopic scale. As such, many experiments aimed at imaging quantum vortices in superfluid helium have been attempted. 8 1.2 – Imaging quantum vortices in bulk liquid helium The first experimental work detecting quantum vortices in 4 He was by Vinen (10) and later revisited by Zimmerman and coworkers (13). The experiments involve a vortex attached along a vibrating wire in a rotating container filled with superfluid helium. The presence of the vortex induced a disturbance in the vibrational modes of the wire (10, 13). Subsequent experiments attempted to dope the vortices with impurities such as electrons or hydrogen atoms and observe the drift velocity of such dopants (14). Such measurements provided evidence that ions were attached to the vortex rings within the droplet. These pioneering measurements led to the first and only determination of the radius of a vortex core, which was found to be 1 Å (14, 15). Later, the first two-dimensional images of the geometric arrangement of several vortices rotating in liquid helium-II were obtained by Packard and colleagues with their phosphor screen experiment (16). Electrons were employed as a dopant to image quantum vortices, which were extracted from the bulk liquid along the axis of rotation and accelerated into a phosphor screen that was imaged with a video camera. Results showed the first geometric arrangements of vortices in helium (see Fig.1.21) (16). A quarter of a century later, techniques were developed based on in situ imaging of the spatial distribution of vortices in liquid helium doped with hydrogen clusters used as tracer particles (17). After cooling the container of hydrogen-doped liquid helium below the lambda point, the tracers were observed to form arrays of parallel lines, indicative that tracers are bound to the quantum vortices (17). Recent experiments in the bulk facilitated significant advancements in imaging vorticity in superfluid helium. However, there were several unavoidable drawbacks to bulk techniques, one being that it was difficult to achieve temperatures significantly lower than 1.7 K in the experiment with particle tracers; thus, a large normal fluid fraction is present that participates in convection 9 flow. Additionally, the spatial resolution was limited by the optical imaging approaches utilized. Such limitations set the precedent for expanding experiments to nanoscale droplets of liquid helium with doped vortex cores. The foreign particles are attracted to the vortex cores by hydrodynamic forces (3, 8, 18) and serve as contrast agents for in-situ imaging experiments (19). Figure 1.2.1. Imaging quantum vortex configurations in bulk liquid helium-4. Adapted from Packard and coworkers (16). 1.3 – Helium nanodroplets Nanoscale droplets of superfluid helium present a new opportunity for investigation, as they have recently gained notoriety in the physical chemistry community (20-22). Helium droplets are widely versatile in size and can range from few to 10 12 atoms, corresponding to diameters of sub nm to ~6 micron, respectively (see Fig. 1.3.1) (23). They are produced via continuous free jet 10 expansion of research grade (6.0) helium gas through a pinhole nozzle (24). The droplets evaporatively cool to a temperature of 0.37 K for 4 He and 0.15 K for 3 He and enter an adjacent vacuum chamber where they can readily pick up atoms or molecules that come into their flight path (24, 25). Due to the low temperature of the droplets, they have found applications in recent years as spectroscopic matrices (26), as any encapsulated dopants are ensured to be in the ground electronic and vibrational state (24, 25). The quantum liquid environment exhibits negligible interactions with the dopants and facilitates their free rotation inside the droplets. Finally, the droplets do not absorb light between the far IR and the vacuum UV, making them ideal matrices (24). Figure 1.3.1. Sizes of small helium clusters versus large helium nanodroplets. As helium nanodroplets evaporatively cool to below the lambda point (0.4 K), they are ensured to be in the superfluid state. Additionally, as the droplets rotate in free space after being 11 produced in a free jet expansion, they house quantum vortices and can be used for imaging experiments. The first experiment imaging quantum vortices in helium nanodroplets was performed in 2012. Here, the vortices were tagged with silver clusters that assembled in the nanodroplets and were deposited onto a substrate (27). The deposits showed elongated, filament- like silver clusters. It was proposed that within the droplets, silver clusters aggregated within the cores of the quantum vortices. This result was the first example of vortex-assisted aggregation in nanoscale helium droplets (27). Spectroscopic experiments may have made a name for helium droplets in the chemical physics community, but other types of experiments with droplets are certainly possible, including those that probe fundamental properties of superfluids. These types of experiments make up the bulk of this work. 1.4 - References 1. R. P. Feynman, Application of Quantum Mechanics to Liquid Helium. Prog. in Low Temp. Phys. 1, 17-53 (1955). 2. J. Wilks, D. S. Betts, An Introduction to Liquid Helium. (Clarendon Press, Oxford, 1987). 3. D. R. Tilley, J. Tilley, Superfluidity and Superconductivity. (Institute of Physics Publishing, Bristol, 1990). 4. J. G. Daunt, K. Mendelssohn, Superconductivity and Liquid Helium II. Nature 150, 604 (1942). 12 5. D. H. Liebenberg, Thermally Driven Superfluid-Helium Film Flow. Phys. Rev. Lett. 26, 744 (1971). 6. W. F. Vinen, in The Physics of Superfluid Helium. (2004). 7. L. Landau, On the theory of superfluidity. Phys. Rev. 75, 884 (1949). 8. L. Pitaevskii, S. Stringari, Bose-Einstein Condensation and Superfluididty. (Oxford Univ. Press, Oxford, UK, 2016). 9. E. R. Dobbs, Helium Three. (Oxford University Press, New York, 2000). 10. W. F. Vinen, Detection of single quanta of circulation in liquid helium-II. Proc. R. Soc. A 260:2, 18-36 (1961). 11. A. J. Feinberg et al., Aggregation of solutes in bosonic versus fermionic quantum fluids. Sci. Advan. 7, abk2247, (2021). 12. S. M. O. O’Connell et al., Angular momentum in rotating superfluid droplets. Phys. Rev. Lett. 124, 215301 (2020). 13. P. W. Karn, D. R. Starks, W. Zimmerman, Observation of quantization of circulation in rotating superfluid He-4. Phys. Rev. B 21, 797-805 (1980). 14. G. W. Rayfield, F. Reif, Evidence for the creation and motion of quantized vortex rings in superfluid helium. Phys. Rev. Lett. 11, 305-308 (1963). 15. G. W. Rayfield, F. Reif, Quantized vortex rings in superfluid helium. Phys. Rev. A 136, 1194-1208 (1964). 16. E. J. Yarmchuck, M. V. J. Gordon, R. E. Packard, Observation of stationary vortex arrays in rotating superfluid-helium. Phys. Rev. Lett. 43, 214-217 (1979). 17. G. P. Bewley, D. P. Lathrop, K. R. Sreenivasan, Superfluid helium—visualization of quantized vortices. Nature, 441:588 (2006). 13 18. R. J. Donnelly, C. F. Barenghi, The observed properties of liquid helium at the saturated vapor pressure. J. Phys. Chem. Ref. Data 27, 1217 (1998). 19. O. Gessner, A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Droplets. Annu. Rev. Phys. Chem. 70, 173 (2019). 20. M. Y. Choi et al., Infrared spectroscopy of helium nanodroplets: novel methods for physics and chemistry. Int. Revs. in Phys. Chem. 25, 15-75 (2006). 21. J. P. Toennies, A. F. Vilesov, Superfluid helium droplets: A uniquely cold nanomatrix for molecules and molecular complexes. Angew. Chem. Int. Edit. 43, 2622 (2004). 22. D. Verma, R. M. P. Tanyag, S. M. O. O’Connell-Lopez, A. F. Vilesov, Infrared Spectroscopy in Superfluid Helium Droplets. Adv. in Phys. X 4, (2019). 23. L. F. Gomez, E. Loginov, R. Sliter, A. F. Vilesov, Sizes of Large Helium Droplets. J. Chem. Phys. 135, 154201 (2011). 24. R. M. P. Tanyag et al., in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, A. Osterwalder, O. Dulieu, Eds. (Royal Society of Chemistry, Cambridge, 2018). 25. R. Hartmann, R. E. Miller, J. P. Toennies, A. F. Vilesov, Rotationally Resolved Spectroscopy of SF6 in Liquid-Helium Clusters - a Molecular Probe of Cluster Temperature. Phys. Rev. Lett. 75, 1566 (1995). 26. D. Verma, R. M. P. Tanyag, S. M. O. O’Connell-Lopez, A. F. Vilesov, Infrared spectroscopy in superfluid helium droplets. Advan. in Phys. X 4, 1 (2019). 27. L. F. Gomez, E. Loginov, A. F. Vilesov, Traces of Vortices in Superfluid Helium Droplets. Phys. Rev. Lett. 108, 155302 (2012). 14 Chapter 2 – X-ray coherent diffractive imaging of helium nanodroplets Parts of this chapter are based on the following publications: Deepak Verma, Sean M. O. O’Connell, Alexandra J. Feinberg, Swetha Erukala, Rico M. Tanyag, Charles Bernando, Weiwu Pang, Catherine A. Saladrigas, Benjamin W. Toulson, Mario Borgwardt, Niranjan Shivaram, Ming-Fu Lin, Andre Al Haddad, Wolfgang Jaeger, Christoph Bostedt, Peter Walter, Oliver Gessner and Andrey F. Vilesov. “Shapes of Rotating Normal Fluid 3 He Versus Superfluid 4 He Droplets in Molecular Beams”. Physical Review B. 102 (2020), 014504. Rico Mayro P. Tanyag, Alexandra J. Feinberg, Sean M. O. O’Connell, Andrey F. Vilesov. “Disintegration of Diminutive Liquid Helium Jets in Vacuum”. Journal of Chemical Physics, 152 (2020), 234306. 2.1 - Experimental Setup Our coherent diffractive imaging (CDI) experiments with helium nanodroplets (HNDs) are conducted at XFEL facilities such as Stanford’s Linac coherent light source, LCLS-I, and the European XFEL, affiliated with the Deutches Elektronen-Synchrotron (DESY). The use of XFEL radiation is necessary to obtain diffraction patterns that can be processed with our phase reconstruction algorithm (to be explained later). Figure 2.1.1 shows a simplified schematic of the XFEL experiment, while Fig 2.1.2 shows a block diagram of the experiment in greater detail. To begin, helium nanodroplets are produced in the source chamber via continuous free jet expansion of helium gas through a cryogenic nozzle with an orifice of 5 micron in diameter. A stagnation pressure of P0 = 20 bar and a nozzle temperature T0 = 5 K is typically employed for our experiments. The droplets form a beam which is further collimated by a skimmer placed in the beam’s path. From here, droplets enter the doping chamber, also called the “pickup” chamber, 15 where they can collide with and pick up dopant atoms (represented as stars in the figures). No dopants are added for the interrogation of neat (dopant-free) helium nanodroplets. The droplets continue traveling through the vacuum chambers until they reach the interaction region. Here, they are intersected perpendicularly with the XFEL beam which is typically operated in the soft x-ray regime at 1.5 keV (𝜆 = 0.826 nm). The XFEL beam consists of ultrashort x-ray pulses, containing up to ~10 12 photons/pulse, with a repetition rate of 120 Hz, a pulse energy of 1.5 mJ, and a pulse duration of ~100 fs (FWHM). The small pulse length and large number of photons per pulse enables the instantaneous capture of diffraction patterns from individual droplets. Diffraction images are recorded with a pn-charge-coupled device (pnCCD) detector containing 1024 × 1024 pixels, each 75 × 75 μm 2 in size, which is centered along the XFEL beam axis downstream from the interaction point. The detector consists of two separate panels (1024 × 512 pixels each) located closely above and below the x-ray beam. Both panels have a central, rectangular cutout to accommodate the primary x-ray beam. Due to the extensive ionization by the XFEL beam, the imaged droplets are eventually destroyed. The imaging event, however, takes place on a shorter timescale (< 100 fs) than the droplet disintegration. Droplet fragments are collected by a time-of- flight mass spectrometer aligned parallel to the nanodroplet beam and perpendicular to the x-ray beam. The droplet flux is analyzed via the quadrupole residual gas analyzer (RGA), which is placed in the terminal vacuum chamber. 16 Figure 2.1.1. Simplified schematic of the XFEL experiment. The droplets exit the nozzle and enter the adjacent pickup chamber where they pick up dopant atoms such as xenon. The doped droplets continue traveling through the vacuum chambers before entering the interaction region. Here, the doped droplets are irradiated with the focused XFEL beam. Scattered photons are collected on a pnCCD detector behind the interaction region. Droplet fragments are collected via TOF. Further downstream, the droplet flux is analyzed by RGA. 17 Figure 2.1.2. Block diagram of the experiment showing the five vacuum chambers. From top: source chamber, pickup chamber, interaction chamber, detection (pnCCD) chamber, and RGA chamber. The experimental setup varied depending on the XFEL facility where the experiments were conducted, however, the base principles presented in Figs. 2.1.1 and 2.1.2 are the same. The following figures show snapshots of how parts of the experiment appeared at the European XFEL in Hamburg, Germany. Figure 2.1,3 shows the sequence of vacuum chambers. Starting on the right, one can see the source chamber, which houses the nozzle and is where the droplets originate. Next, we have the pickup chamber, where the droplets are doped with gas. The inside of the pickup chamber can be seen from the side in Fig. 2.1.4. Here, one can see a gas doping cell outfitted with 18 a Swagelok feedthrough fitting on the left, and a solid doping cell on the right (the solid doping cell was left from previous experiments at the XFEL and not used for any of our experiments). Then, we have the differential pumping stage which bridges the pickup chamber and interaction chamber which have vastly different pressures. Figure 2.1.3. From right: Source chamber (DN 200 flange), pickup chamber (DN 300 flange), differential pumping stage, NQS (scattering) chamber at EXFEL. 19 Figure 2.1.4. A view of the doping chamber utilized at EXFEL. Left: gas doping cell. Right: heated cell used to evaporate metal atoms. Finally, we have the interaction chamber, which can be seen from another angle in Fig. 2.1.5. The interaction chamber (also called the Nano Quantum Systems or NQS chamber) is directly adjacent to the pnCCD detector chamber and the beam stop. The XFEL beam enters the NQS chamber along the direction of the gold arrow in Fig. 2.1.5. The imaging event occurs, and scattered photons are collected on the pnCCD detector. Finally, the droplet beam terminates at the beam stop. 20 Figure 2.1.5. pnCCD chamber to the NQS (scattering) chamber at EXFEL. Producing 4 He droplets The 4 He droplet source is cooled using a Sumitomo SRDK 408 cryocooler, with a lowest operational temperature of ~ 3.6 K. A copper nozzle assembly is attached directly to the second stage of the cryocooler (see Fig. 2.1.6). Research grade helium gas is supplied to the nozzle from a high-pressure tank. One meter of stainless-steel gas line is wound around the first stage of the cryocooler to pre-cools the gas down to ~30 K. The cold gas then enters the nozzle and undergoes free jet expansion. The stagnation pressure of the gas, P0, and the temperature of the nozzle, T0, are critical experimental parameters that affect the size distribution of the droplets produced. A 21 temperature sensor and resistive heater are attached to the second stage of the cryocooler to monitor T0. Aluminum shields enclose the second stage of the cryocooler and the nozzle assembly to minimize the effect of thermal fluctuations, caused by either collision with residual helium gas or heating through black body radiation. The nozzle assembly is comprised of three parts: the copper nozzle base, the tip, and the orifice. Detailed schematics can be found in Ref. (1) but the critical parts will be reviewed here. Inside the copper nozzle base, there is a cylindrical reservoir that holds pressurized helium prior to expansion. The nozzle tip is directly attached to the base and sealed with a gasket of indium. The nozzle orifice, which is inside the tip, is a 5 µm diameter electron microscope diaphragm (95%/5% Pt/It alloy). A 0.5 µm filter is placed in between the nozzle base orifice to trap contaminant particles which can clog the nozzle and severely complicate the experiment. 22 Figure 2.1.6. The 4 He cryocooler, with the nozzle assembly attached to its second stage. A temperature sensor is attached to the nozzle assembly to monitor T0 and a resistive heating pad is attached to manipulate T0. The Swagelok fitted copper helium gas line can be seen wrapped around the cryocooler. Cylindrical aluminum shields cover the entire assembly to minimize the effect of thermal fluctuations. Taken with permission from Tanyag et al (1). After the cryocooler/nozzle pair has been assembled, it’s extremely important to measure the flux of helium exiting the nozzle before installing the entire assembly into vacuum. As previously mentioned, multiple steps are taken to prevent contamination (sealing with an indium gasket, using a filter) but particulates still sometimes find their way into the system. Sealing the vacuum chambers, realizing the nozzle is clogged, and having to open the vacuum chambers can eat away at an entire week of experimenting, sometimes longer. Therefore, one should carefully immerse the tip of the nozzle in a reservoir of methanol and observe the flow rate by utilizing a graduated cylinder, upside down and immersed in methanol. For a 5 µm nozzle, P0 = 20 bar, and T0 = room temperature, the flux should be around 0.3 - 0.4 cm 3 bar s -1 . At T0 = 4 - 5 K, the flux is around 3 cm 3 bar s -1 (1). If the flux deviates from these values by more than 20% the nozzle assembly should be changed. After being tested for leaks, the cryocooler and nozzle assembly are installed upright into vacuum on a flange with four movable bellows. The nozzle will experience a periodic displacement by about 0.1 mm due to its attachment to the cryocooler, however, the effect of this oscillation can be minimized through careful alignment of the beam via the bellows. The helium beam is aligned when one observes the highest possible pressure rise in the detection chamber (about 6 x 10 -9 mbar) relative to the beam being blocked, or when the pressure in the source chamber drops by ~50% (1). The pressure rise in the detection chamber is dependent on the beam alignment through the apertures of the system. To speed up aligning, depending on the conditions (T0 < 4.5 K and P0 > 10 bar) one may observe the liquid jet through a microscope. In this regime, 23 the droplets are large enough to cause significant light scattering such that the beam can be seen with the naked eye (see following section). For other conditions, however, one must iteratively align the beam by cooling to T0 = 7 K and manipulating the bellows until the highest pressure is observed in the detection chamber. Then, one can cool further to T0 = 5 K and repeat the process. Below T0 = 4.5 K, the beam becomes strongly collimated and achieving proper alignment is difficult (2). Physical characteristics of a liquid helium-4 jet To learn more about the physical characteristics of low temperature helium droplet jets, we imaged a jet of large helium droplets via optical shadowgraphy (2). We found that at different stagnation pressures and temperatures, the formation of droplets proceeds via the following mechanisms: spraying of the droplets, branching of the liquid jet, and flashing and evaporative instabilities of the jet. Figure 2.1.7 shows photographs of the jets produced at varying P0 and at a constant T0 = 3.5 K. Figure 2.1.7. Shows photographs of liquid helium jets produced at 3.5 K and 3, 7, 12, 20, 40 and 60 bar. The jet was imaged at three different regions, covering the total distance of 3.5 mm from the nozzle cap. Each region is about 1.5 mm long and there is ~0.5 mm overlap between these 24 regions. The 100 μm scale for each region is given in the lower right corner in region 1. Large grey spots visible in the same positions in all panels are artefacts of the imaging setup. At the lowest stagnation pressure of 3 bar, the liquid column jet is not discernable. In fact, the beam seems to vacillate between a jetting mode with a collimated droplet beam, and a spray mode with an opening angle of ~5°. In a series of photographs collected at this condition, the collimated beam only appears about 20% of the time. When a main jet is observed, it is composed of larger droplets forming the centerline of a ~2.5 o cone, evenly filled with smaller droplets. In a complete spray mode, the droplets of different sizes fill the cone and with the larger droplets at the beam axis. Similar vacillations are also observed at farther distances from the nozzle. At 7 bar, a liquid jet becomes distinguishable close to the nozzle but the droplet beam remains rather broad with an opening angle of ~2°. In addition to the main jet, the measurements at 7 bar also reveal some side jets emerging at ~1° above and below the main jet. These side jets consist of smaller droplets with respect to the main jet. At 20 bar, the continuous jet extends up to about 0.5 mm from the nozzle cap. Upon breakup, the jet remains well collimated and contains droplets of similar sizes. At 40 bar, the beam shows a strong main jet with more pronounce side jets than at 20 bar. Finally, at 60 bar, there are four jets visible with the two central ones being the most intense. These two central jets are composed of droplets of similar sizes and appear at same angles in the consecutive images. Close to the nozzle at P0 = 40 bar and 60 bar, the continuous liquid column appears as two nearly parallel dark lines with a light stripe in the middle. Such an image is expected to originate from a liquid cylinder. It is also seen that the side jets originate at about the point of breaking of the column jet and not at the nozzle exit, which is at ~250 μm behind the dark rim on the right most part of the images in the region I. Contrary to our expectation of observing a single train of the droplets upon Rayleigh breakdown of the columnar jet at low T0, we have observed the 25 branching of the jet into multiple droplets' prongs angle between which decreases with the pressure at constant T0. The formation of the multiple side jets which direction may migrate over some minutes complicates an experiment, which generally requires long time stability of the droplet beam with a collimation of better than ~0.5 o , such as in spectroscopic or x-ray scattering experiments. Ideally, the branching should be eliminated upon understanding of its origin. Special consideration for producing 3 He droplets Considering the lower critical point of 3 He (T c = 3.3 K, P c = 1.1 atm) compared to that of 4 He (T c = 5.2 K, P c = 2.3 atm), lower nozzle temperatures are required to obtain 3 He droplets of similar size to of 4 He droplets. For example, for a nozzle stagnation pressure of P 0 = 20 bars, previous experiments demonstrate that 4 He droplets with an average number of atoms ⟨N 4 ⟩ = 10 7 are produced at a nozzle temperature of T 0 = 7K, (3) while T0 = 5K is required to obtain 3 He droplets with the same average number of atoms ⟨N 3 ⟩ = 10 7 (4-6). The temperature difference of 2 K in T 0 correlates well with the corresponding difference in critical temperatures of the two isotopes. Large 4 He droplets can be produced with modern closed-cycle refrigerators that can reach temperatures down to ∼3.5 K. However, to reach the lower temperatures required to produce large 3 He droplets, we instead employed a liquid helium flow cryostat, the LT3 from Advanced Research Systems, with a cooling power of up to 1 W at 1.8 K. Figure 2.1.8 shows how the liquid helium flow cryostat was installed in the system of vacuum chambers. Instead of being mounted on top of the chamber like with the Sumitomo, the flow cryostat is installed parallel to the helium nanodroplet beam. The flow cryostat employs liquid helium (non-research grade) to cool the system, which is pictured in the dewar. Droplets of 3 He and 4 He are produced at constant P 0 = 20 26 bars and varying T 0 , ranging from 2 to 4.5 K. The temperature was measured using a calibrated silicon diode (Lakeshore DT-670- CU) attached to the copper block close to the nozzle. Figure 2.1.8. Liquid helium flowing cryostat used to produce 3 He droplets. Non-research grade liquid helium is used to cool the nozzle by flowing from a dewar through a transfer line into the cryostat body. Research grade helium gas (99.9999% pure) is expanded through the nozzle to produce droplets. Due to the considerable cost of 3 He gas, a recycling system is employed during the experiments (see Fig. 2.1.9). Filling the system requires about 10 liter × bar of room temperature 3 He. For comparison, at standard operating conditions (T0 = 3 K, P0 = 20 bars), the flow rate of the gas is ∼3 cm 3 bars/s and the filling amount of gas would only be sufficient for about 1 hour of operation. During the experiments, 3 He gas is continuously collected from the exhausts of the backing scroll pumps, purified in a liquid nitrogen cooled zeolite trap, pressurized by a metal membrane compressor, and resupplied to the nozzle with minimal losses. Any contaminants are 27 constantly removed from the 3 He sample by the recycling system. The droplet source was stable over several days, indicating the purity of the 3 He remained high throughout the experiment. The 3 He gas used is 99.9% pure with the remaining 0.1% impurity being mostly 4 He. The residual 4 He will be mostly dissolved in the 3 He droplets, considering that its solubility is ∼0.1% at 0.15 K (7). Any possible pockets of a 4 He-rich phase in 3 He droplets are too small to give rise to any measurable effects in the diffraction patterns. Based on the rest gas pressure of less than 10 -7 mbar and a beam path length from the nozzle to the interaction point of about 70 cm, the droplets will capture, on average, fewer than 500 rest gas particles (mostly water molecules), again too small a number to be detected in the diffraction experiments. Figure 2.1.9. 3 He recycling system, built by Sean O’Connell. The system has three components: a compressor (left), liquid nitrogen dewars (right bottom), and the gas manifold (right top). 28 Maintaining vacuum and an oil-free system The types of vacuum pumps employed are determined by the desired base pressures in each chamber. Reference (1) gives a detailed description of ideal vacuum pumping system parameters to produce large helium droplets. Generally, we want a pressure gradient across the chambers such that the source chamber is maintained around 10 -4 mbar and the detection chamber is maintained around 10 -8 mbar. Table 1 shows the absolute pressures measured in different vacuum chambers for the best possible alignment of the helium nanodroplet beam. Nozzle T0 (K) Source chamber Pickup 1 Pickup 2 Detection 3.9 7.0 x 10 -4 8.9 x 10 -4 2.9 x 10 -5 7.5 x 10 -6 5.0 9.7 x 10 -4 1.9 x 10 -4 3.8 x 10 -6 3.1 x 10 -6 7.0 8.6 x 10 -4 3.9 x 10 -6 1.7 x 10 -7 1.6 x 10 -7 11.0 6.1 x 10 -4 2.5 x 10 -6 1.0 x 10 -7 9.5 x 10 -8 298 4.4 x 10 -5 4.7 x 10 -8 1.5 x 10 -8 4.6 x 10 -9 Table 1. Typical absolute pressures in mbar maintained in the vacuum chambers for best possible alignment. The pressure reading in the source chamber corresponds to the pressure of helium gas while the pressure readings for the other chambers at 298 K correspond to rest gases (1). The specific brands and models of the pumps sometimes change when we take the system to beamlines. The most important thing to note is that all pumps should be oil free – any turbomolecular pumps employed should be backed by dry roughing pumps. Oil based pumps introduce the possibility of contaminating the chambers and are not permitted for the beam time experiments. 29 2.2 - X-ray coherent diffractive imaging of helium nanodroplets Imaging of individual helium nanodroplets in situ has recently become possible with the advent of XFELs. Linear particle accelerators such as XFELs are known to produce extremely short (≤ 100 fs) pulses of x-ray radiation, thus having the ability to resolve features as small as ≤ 20 nm, like vortices (8, 9). Additionally, XFELs have a high degree of transverse coherence, and a large enough flux (≥10 18 photons cm 2 ) to produce a substantial number of scattered photons in diffraction experiments from single droplets (10, 11). Light scattering from helium nanodroplets Light scattering from helium nanodroplets is well described by Mie theory which presents a complete mathematical description of electromagnetic scattering by homogenous spherical particles of all sizes relative to the wavelength of light. When hit with radiation, helium droplets respond by either absorbing or scattering the incident photons. The total scattering cross section of a helium droplet of radius RD is given by (12, 13): 𝜎 '()**+,-./ ,1+ (𝜆) = 23 ! 4 " # 5 $ |𝑛(𝜆)−1| 6 (2.2a) Here, 𝑛(𝜆) is the refractive index of liquid helium and 𝜆 is the wavelength of the XFEL. The refractive index is given by (14, 15): 𝑛(𝜆) = 1− . %& 4 & 5 $ 63 [𝑓 7,1+ 8 (𝜆)−𝑖𝑓 6,1+ 8 (𝜆)] (2.2b) where 𝑅 + is 2.82×10 -6 nm, the classical radius of an electron, and 𝑓 7,1+ 8 (𝜆) and 𝑓 6,1+ 8 (𝜆) are the atomic scattering factors for helium, which equal 2.0 and 2.4×10 -3 at ℎ𝜈 = 1.5 keV (𝜆 = 0.83 nm), 30 respectively (15). The total number of scattered photons then can be calculated from the scattering cross section and the flux of incident photons per laser pulse in the XFEL focus region, 𝜙 8,-.(-9+.* : 𝐼 '()**+,-./ (𝜆) = 𝜎 '()**+,-./ ,1+ (𝜆) 𝜙 8,-.(-9+.* (2.2c) As previously mentioned, in addition to scattering, many photons are absorbed by the droplet. The absorption cross section of a helium nanodroplet can be calculated via (14): 𝜎 ):';,<*-;. (𝜆) = 23 = 𝑛 1+ 𝑅 + 𝑓 6,1+ 8 (𝜆) 𝜆 𝑅 > = (2.2d) Whereas the total number of absorbed photons can be calculated via: 𝐼 ):';,<*-;. (𝜆) = 𝜎 ):';,<*-;. (𝜆) 𝜙 8,-.(-9+.* (2.2e) For a droplet with RD = 500 nm, and assuming a photon flux of 4×10 4 photons/nm 2 , that 𝐼 '()**+,-./ (𝜆) is around 2×10 8 photons, and 𝐼 ):';,<*-;. (𝜆) is roughly 5×10 6 photons. Helium atoms have a very high ionization potential (24.6 eV) but will inevitably become ionized by the intense interaction with the FEL, leading to the production of ~1.5 keV electrons. Many these electrons will escape, but some will become trapped by a Coulomb potential of the ionized droplet and contribute to secondary ionization processes which remains to be fully elucidated. Complete destruction and disintegration of the droplet occurs shortly after the interaction with the laser pulse, yielding free Hen + ions. 2.3 – Diffraction patterns from helium nanodroplets Diffraction patterns from neat 4 He (Fig. 2.3.1) droplets are characterized by sets of concentric contours. The images in Fig. 2.3.1(a, b) exhibit circular and elliptical contours, 31 respectively. Fig. 2.3.1(c), however, shows an elongated diffraction contour with pronounced streaks radiating away from the center. These diffraction patterns are characteristic of distinct droplet shapes, as will be discussed in the following section. Distorted droplet shapes come from rotating droplets with high angular momentum. Figure 2.3.1. Diffraction patterns of pure 4 He droplets shown on a logarithmic color scale as indicated on the left. Images represent the central 660 x 660 detector pixels. Taken with permission from Gomez et al (16). Figure 2.3.2 shows several diffraction patterns from pure 3 He droplets. As with 4 He, images in Figs. 2.3.2(a) and 2.3.2(b) exhibit a series of circular and elliptical contours, respectively, with different spacings between their respective rings. Figure 2.3.2(c) shows an elongated diffraction contour like that of 4 He. Thus, we observe the similar shapes for droplets of both helium isotopes. 32 Figure 2.3.2. Diffraction patterns of pure 3 He droplets shown on a logarithmic color scale as indicated on the right. Images represent the central 660×660 detector pixels. Corresponding droplet projection half axes (A, C) and their aspect ratios (AR) are displayed on top of each image. 2.4 – Sizes and shapes of helium nanodroplets Superfluid helium nanodroplets are characterized by their overall size and shapes. Despite helium’s weak interatomic interactions, stable clusters are formed for a wide range of sizes (3). For N < 10 3 atoms, the moieties are referred to as clusters, whereas they are called droplets for N > 10 3 . For 4 He, very small clusters containing two and more atoms are possible. However, for 3 He, clusters with N < 30 are unstable (17). A helium nanodroplet’s radius can be calculated from the number of helium atoms using the equations R4He= 2.22(N 1/3 ) Å for 4 He and R3He = 2.44(N 1/3 ) Å for 3 He (17). Droplets of different average size can be produced by varying the expansion conditions; the stagnation pressure, P0 and nozzle temperature, T0. In our experiments, we usually keep the value of P0 constant at 20 bar. The expansion of the gas can be described using the pressure-temperature phase diagram for 4 He (a) and 3 He (b) (see Fig. 2.4.1) (3, 18, 19). 33 Figure 2.4.1. Pressure-temperature phase diagrams for 4 He (a) and 3 He (b). The pink SVP curve marks the saturated vapor pressure boundaries. The adiabatic expansion proceeds along the line of constant entropy – i.e., an isentropes, indicated by dashed lines, which starts at an initial condition defined by T0, P0, and ends at a set of final values Tf, Pf on the saturated vapor pressure (SVP) curve. Depending on how the isentrope approaches the SVP curve, the expansion is defined as being subcritical or supercritical. If the expansion starts at P0 = 20 bar and T0 > 10 K, the isentrope will cut the evaporation curve from the gas side at ~3.7 K and the expansion will be in the sub-critical regime. Here, small droplets (N < 10 5 ) will be formed via gas condensation. However, at T0 = 6 K, the isentropic line cuts the evaporation curve from the liquid side at 4.5 K and the expansion is in the supercritical regime (3, 19). Here, large (N > 10 5 ) droplets are formed from the breakup of liquid. At T0 < 4 K, a liquid jet emanates from the nozzle followed by a Rayleigh breakup into very large droplets, whose diameter is comparable to that of the nozzle (2, 20). In vacuum, the temperature of the droplets further decreases via evaporative cooling down to 0.15 K and 0.38 K for 3 He (21) and 4 He (22), respectively. Below 2.17 K, boiling of superfluid 4 He ceases, which may lend further stability to the droplets. The superfluid transition temperature for 3 He is < 1 mK, far below the temperature of the droplets in any of our experiments (therefore, 3 He droplets remain normal fluids in all our 34 experiments). The droplets assume a uniform density with a definitive outer edge as in the liquid drop model. The particle density in the center of 4 He droplet is approximately that of the bulk phase value of 0.022 atoms/Å 3 . However, droplet theory predicts that in the surface regions, the particle density drops from 90% to 10% of its bulk phase value within less than 1 nm (23, 24). Obtaining droplet shapes from diffraction patterns A droplet’s shape is characterized by its aspect ratio (ar, major axis/minor axis) and ranges from close to spherical to distorted capsule-like shapes, see Fig. 2.4.3. A nanodroplet’s shape is described by the distances between the center and the surface of the droplet in three mutually perpendicular directions: a > b > c. For oblate axisymmetric droplets (also called spheroidal), a = b > c, with c along the rotation axis, whereas a > b > c in the case of prolate, triaxial (capsule) shapes with c along the rotation axis. Two-lobed “dumbbell” shaped droplets can also be formed; however, they are unstable and subject to fission. For dumbbell shapes, a > b ~ c, see Fig. 2.4.2 (16, 25). 35 Figure 2.4.2. Shapes of classical droplets: (a) spheroidal, (b) elliptical, and (c) caspule. Adapted from Baldwin, Butler, and Hill (26). The obtained diffraction patterns do not provide direct access to the actual values of a, b, and c, due to the droplets’ unknown orientations with respect to the x-ray beam. Instead, the images are characterized by the two half axes of the projection of a droplet onto the detector plane, which will be referred to as A and C (A > C), corresponding to a projection aspect ratio, AR = A/C. For an axisymmetric oblate droplet having an unknown orientation with respect to the x-ray beam, the value of A corresponds to the a axis, whereas the value of C only constitutes an upper bound for the c axis. In the case of a triaxial droplet, the value of A gives a lower bound for the a axis, whereas the value of C gives a lower bound for the b axis and an upper bound for the c axis. The values of A and C are obtained from the diffraction patterns as described in supplemental material in Ref. (25). 36 The aspect ratio of a droplet measures the extent to which it is distorted from a perfect sphere. Thus, it provides information on the angular momentum and angular velocities of the droplets. Most droplets are close to spherical with AR ~1.0. However, large AR events up to 2.0 have been observed, bigger than the largest predicted stable aspect ratio for classical axially symmetric droplets of ar = 1.47 (27, 28). The shapes of rotating submicron helium nanodroplets change with increasing AR and closely follow the predicted shapes for classical rotating liquid droplets (25). The equilibrium shape of a droplet rotating as a rigid body is defined by the balance between capillary forces from surface tension and centrifugal forces (25). A stability diagram and corresponding representative shapes are shown in Fig. 2.4.2 for the axially symmetric and two- lobed branches (26-30). Figure 2.4.3 relates the reduced angular momentum of the droplet, L, and the reduced angular velocity, W, given by the following equations: Λ = 7 ?2∗A∗B∗4 ' ∗𝐿 (2.4a) Ω = ? B∗ 4 ! 2∗A ∗𝜔 (2.4b) Here, L and 𝜔 are the angular momentum and angular velocity, in absolute units, respectively, 𝜎 is the surface tension of the liquid, 𝜌 is the density of the liquid, and R is the radius of the spherical droplet with the same volume as the distorted droplet. One can see from the graph that upon increasing the droplet’s angular momentum, the shape progresses from spherical to oblate axially symmetric. At large angular momentum, the shapes appear flattened. Beyond Λ = 1.2, axially symmetric shapes become unstable, and the curve is instead described by a lower branch representative of prolate, triaxial shapes which look like elongated pills between Λ = 1.2 – 1.6 and dumbbells from Λ = 1.6 – 2.0. Beyond Λ = 2.0, all shapes are unstable and are subject to breakup. 37 Figure 2.4.3. Stability diagram for rotating droplets in equilibrium as a function of reduced angular velocity (Ω) and angular momentum (Λ) (see equations 2.1a and 2.1b). The upper branch corresponds to oblate, axisymmetric shapes, whereas the lower branch corresponds to two-lobed shapes. The bifurcation point is located at Λ = 1.2, Ω = 0.56 with AR = 1.48. Size and shape distribution of helium nanodroplets Figure 2.4.4(a) displays the measured distribution of the droplet’s major half axis, A, for 3 He and 4 He droplets, as represented by blue and red bars, respectively. The average value of A in 4 He droplets is approximately a factor of 2 larger than that of 3 He droplets. The values for the 3 He droplets vary between A = 52 nm and A = 796 nm, whereas 4 He droplets exhibit a larger spread, ranging from A = 55 nm to A = 1250 nm. Figure 2.44(b) shows the AR distribution for 3 He and 38 4 He droplets. The largest ARs are 1.99 for 3 He and 1.72 for 3 He. Figures 2.4.4(a) and (b) show that both the values of A and (AR − 1) follow exponential distributions. Square symbols in panel (c) of Fig. 2.4.4 show the average aspect ratio for each of the bins in panel (a) for droplets with AR < 1.4, which correspond to oblate, pseudo-spheroidal shapes as discussed in the following. The results of single measurements with AR > 1.4 in panel (c) are shown by stars. It is seen that in 3 He droplets, the average aspect ratio increases linearly from ∼1.03 in small droplets with A < 100 nm to ∼1.15 in larger droplets with A ∼ 600 nm. Corresponding points for 4 He droplets also follow linear dependence with a somewhat smaller slope and have ⟨AR⟩ ∼ 1.12 at A > 600 nm. 39 Figure 2.4.4. Droplet size in terms of A (a) and aspect ratio AR (b) distributions for 3 He (blue) and 4 He (red) isotopes presented in logarithmic scales. The counts for 4 He were multiplied by a factor of 3 for ease of comparison, as the total number of diffraction images obtained for 3 He and 4 He were ∼900 and ∼300, respectively. Squares in panel (c) show average AR as obtained for each bin in panel (a) for the points with AR < 1.4. The results of single measurements with AR > 1.4 are shown by stars in panel (c). The blue line in panel (c) represents a linear fit of the data points (blue squares) for 3 He droplets. Angular momentum and angular velocities of helium droplets As previously described for 4 He droplets (16, 25, 31-33), we ascribe the shape deformation in 3 He droplets to centrifugal distortion. It has been reported that the shapes of rotating 4 He droplets closely follow the equilibrium shapes of classical droplets having the same values of angular momentum (25, 31, 32, 34, 35). This pattern is also expected to be the case of 3 He droplets, which at the temperature of these experiments (∼0.15 K) (21) should behave classically due to the high viscosity of about 200 μP and small mean free path (a few nanometers) of elementary excitations at this temperature (7). In recent density functional calculations, the shapes of rotating 3 He droplets were found to be very close to those predicted for classical droplets (36). The blue curves in Fig. 2.45 show the stability diagram of the classical droplets in terms of the reduced angular momentum (L) and reduced angular velocity (W) which are given by equations 2.4(a) and 2.4(b). For liquid 4 He and 3 He at low temperature, the surface tensions are σ4 = 3.54 × 10 -4 N/m (37) and σ3 = 1.55 × 10 -4 N/m (38), respectively, while their densities are ρ4 = 145 kg/m 3 (37) and ρ3 = 82 kg/m 3 (7). With increasing L, the droplet’s equilibrium shape transitions from spherical to oblate axially symmetric, which is shown by the solid blue curve. At Λ = 1.2, Ω = 0.56, ar = 1.48, the stability curve bifurcates into two branches: an unstable upper branch (dashed blue curve) representing axially symmetric droplets and a stable lower branch (dotted blue curve) representing prolate triaxial droplets. The stable prolate branch represents triaxial ellipsoidal and capsule-shaped 40 droplets with 1.2 < L < 1.6, and dumbbell-shaped droplets at L > 1.6 (25, 27, 28, 31). For L > 2, droplets become unstable and break up. Also shown in Fig. 2.4.5 is the ar of droplets along the axisymmetric branch as a function of L, which is represented by the red curve. Using an exponential distribution of the ar values, 𝑃(𝑎𝑟−1) = 7 〈), E7〉 exp (− ), E7 〈), E7〉 ) (see Fig. 2.4.4(b)) and the functions of L(ar ) and W(ar) in Fig. 2.4.5, integration over ar gives the average L and W for 3 He and 4 He droplets to be ⟨L 3 ⟩ = 0.47, ⟨W3⟩ = 0.27 and ⟨L 4 ⟩ = 0.51, ⟨W4⟩ = 0.29. Those values are indicated in Fig. 2.4.5 as green circles and black crosses, respectively. Very similar values were obtained for 3 He by integrating a double exponential distribution of the form 𝑃(𝑎𝑟−1,𝐴) = 7 〈), (H)E7〉 exp (− ), E7 〈), (H)E7〉 ) 7 〈H〉 exp (− H 〈H〉 ) where the values of ar(A) were obtained from the blue line in Fig. 2.4.4(c) multiplied by 1.5. Although the values of ⟨AR⟩ versus A in Fig. 2.4.4(c) lie somewhat lower for 4 He than for 3 He, the corresponding average values for 4 He are larger due to a larger prevalence of large 4 He droplets. From the values of ⟨L3,4⟩ and ⟨W3,4⟩ and using equations 2.1a and 2.1b, the angular momentum is obtained as L 3 =1.5×10 9 and L 4 =6.9×10 10 ℏ for the average-sized 3 He and 4 He droplets, respectively. Next, L per atom of the droplet is obtained as 5.7 ℏ and 19.3 ℏ for 3 He and 4 He droplets, respectively. Lastly, ω was calculated as 1.6x10 7 and 5.9x10 6 rad/s for 3 He and 4 He respectively. Although the 4 He droplets and 3 He droplets have similar ⟨L⟩, 4 He droplets have about a factor of 3 larger L per atom. Mathematically, this effect stems from the different factors of σρR 7 (Eq. 2.1a) in 3 He and 4 He droplets. 41 Figure 2.4.5. Red curve: Calculated aspect ratio ar as a function of reduced angular momentum (L) for axially symmetric oblate droplet shapes. Blue curve: stability diagram of rotating droplets in terms of reduced angular velocity (W) and reduced angular momentum (L). The upper branch (dashed blue) corresponds to unstable axially symmetric shapes. The lower branch (dotted blue) is associated with prolate triaxial droplet shapes resembling capsules and dumbbells. The green circle and black cross on the red curve represent the average ⟨ar⟩ for 3 He and 4 He droplets, respectively, obtained in this work (with AR < 1.4). Similar markers on the blue curve indicate the (W, L) values corresponding to 3 He and 4 He droplets. 2.5 – Xenon doped helium nanodroplets Diffraction patterns from neat helium droplets are used to obtain information on their sizes and shapes. However, patterns from helium droplets doped with xenon atoms can reveal additional information about the internal structure of the droplets, i.e., their quantum vortices. The addition of xenon creates interference in the scattering event due to the differences in indices of refraction 42 between helium and xenon. Patterns from xenon doped helium nanodroplets were often found to exhibit Bragg spots that lie on a line crossing the image center, or in triangular lattice configurations, see Fig. 2.5.1, consistent with periodic structures within the droplets (16). It was concluded that the Bragg patterns are likely due to the presence of quantum vortices, which are consistent with previous observations of vortex structures in both nanodroplets via ex situ tracing experiments (39) and in rotating containers of bulk liquid helium (40). Figure 2.5.1. A and B) Diffraction images from Xe-doped helium nanodroplets exhibiting Bragg patterns. C) Simulation of how the vortices are oriented in the droplet. Adapted from Gomez et al (16). Reconstructing density of clusters with Droplet Coherent Diffractive Imaging (DCDI) Reconstructing a nonperiodic object from its scattering pattern is a technique known as coherent diffractive imaging (CDI). For small scattering angles and optically thin objects just as a helium nanodroplet, the complex amplitude of the scattered photons corresponds to the Fourier transform (FT) of the projection of the droplet’s density onto a detector plane (41). Reconstructing the object is, in principle, possible by applying an inverse FT (IFT). However, experimentally, one cannot obtain all the information required to apply a IFT as the pnCCD detector only collects 43 information on absolute photon intensity, but not phase information. This issue is known as the phase problem of CDI, of which there are a variety of techniques to circumvent (42, 43). One of the most common techniques is to apply a trial phase (a best guess at the initial phase) and solve multiple iterations of transforms to obtain a final convergence. Tanyag et al worked to develop an iterative phase retrieval algorithm of such nature with increased accuracy (9) which is based on a modified version of the method in Ref. (44). The algorithm, known as Droplet Coherent Diffractive Imaging (DCDI), exploits the fact that diffraction off pure helium nanodroplets have a known phase, and that the diffraction amplitude is the sum of the known, real amplitude from the droplet and the unknown amplitude from the dopants. A schematic of the processes in DCDI is shown in Fig. 2.5.2. The process is initiated with an experimental diffraction image, from which an analytical droplet density 𝜌 -.<J* , corresponding to a pure helium droplet is extracted. From here, a FT is performed. Once in phase space the amplitude is replaced with the square root of the measured diffraction intensity coming from both the dopants and the helium droplet itself. Then, an IFT is performed, and various constraints are applied; that the dopants must lie inside the predetermined droplet boundary and that the real and imaginary part of the dopants' density is positive. Then, a new cycle of FT/IFT is initiated until the modulus of the calculated scattering pattern has converged to the square root of the measured intensity. From the final object density output, a FT is performed to obtain a calculated diffraction pattern which should agree well with the experimental. 44 Figure 2.5.2. Schematic of the DCDI algorithm. The algorithm is initiated using a preset helium droplet density, ρ !"#$% . A series of inverse Fourier transforms are performed with iterative reinforcement of constraints. The process rapidly converges to a solution yielding the density of the Xe clusters inside the droplet. Taken with permission from Tanyag et al (9). Reconstructions of Xe-doped helium nanodroplets Reconstructions of xenon doped helium nanodroplets have been performed for a variety of sizes and shapes, shown in Fig. 2.5.3 (32). Xe clusters are depicted by red/yellow colors and the droplet density is depicted in blue. One can see the internal structure varies depending on the observer’s view angle. In Fig. 2.5.3(a2), the vortices are viewed from the side and appear as elongated filaments. In this instance, the imaging event occurs at some angle to the direction of angular momentum of the droplet. However, in Fig. 2.5.3(b2) and (c2), the vortices are viewed head on, going into the detector. For these cases, the imaging event occurs approximately parallel 45 to the direction of angular momentum of the droplets, resulting in a lattice-type array of compact dots in the reconstruction. The probability to observe such an event is low as several conditions must first be satisfied: one, the droplet must be large enough to be imaged, two, the droplet must be within the focus of the XFEL beam, and three, the droplet must be oriented along the direction of propagation of the beam. As such, reconstructions showing lattice arrays are fortuitous hits. Figure 2.5.3. Diffraction patterns from Xe-doped helium nanodroplets of various shapes: (a1) axisymmetric (nearly spherical), (b1) triaxial ellipsoidal, and (c1) capsule shaped. Reconstructions for each diffraction pattern as produced by DCDI (a2 - c2) are shown below. The basis vectors of the vortex lattice in (c2) are shown in the upper right corner. Photo taken with permission from O’Connell et al. (32). 2.6 - References 1. R. M. P. Tanyag et al., in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, A. Osterwalder, O. Dulieu, Eds. (Royal Society of Chemistry, Cambridge, 2018). 46 2. R. M. P. Tanyag, A. J. Feinberg, S. M. O. O’Connell-Lopez, A. F. Vilesov, Disintegration of diminutive liquid helium jets in vacuum. J. Chem. Phys. 152, 234306 (2020). 3. L. F. Gomez, E. Loginov, R. Sliter, A. F. Vilesov, Sizes of Large Helium Droplets. J. Chem. Phys. 135, 154201 (2011). 4. M. Farnik, U. Henne, B. Samelin, J. P. 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Commun. 8, 493 (2017). 34. F. Ancilotto, M. Barranco, M. Pi, Spinning superfluid He-4 nanodroplets. Phys. Rev. B 97, 184515 (2018). 35. F. Ancilotto, M. Pi, M. Barranco, Vortex arrays in nanoscopic superfluid helium droplets. Phys. Rev. B 91, 100503 (2015). 49 36. M. Pi, F. Ancilotto, M. Barranco, Rotating He droplets. J. Chem. Phys. 152, 184111 (2020). 37. R. J. Donnelly, C. F. Barenghi, The observed properties of liquid helium at the saturated vapor pressure. J. Phys. Chem. Ref. Data 27, 1217 (1998). 38. M. Iino, M. Suzuki, A. J. Ikushima, Y. Okuda, Surface tension of liquid 3He down to 0.3 K. J. Low Temp. Phys. 59, 291 (1985). 39. L. F. Gomez, E. Loginov, A. F. Vilesov, Traces of Vortices in Superfluid Helium Droplets. Phys. Rev. Lett. 108, 155302 (2012). 40. E. J. Yarmchuck, M. V. J. Gordon, R. E. Packard, Observation of stationary vortex arrays in rotating superfluid-helium. Phys. Rev. Lett. 43, 214-217 (1979). 41. O. Gessner, A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Droplets. Annu. Rev. Phys. Chem. 70, 173 (2019). 42. J. Miao, P. Charalambous, J. Kirz, D. Sayre, Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens. . Nature 400, 342 (1999). 43. H. N. Chapman et al., High-resolution ab initio three-dimensional x-ray diffraction microscopy. J. Opt. Soc. Am. A. 23, 1179 (2006). 44. J. R. Fienup, Phase retrieval algorithms: a comparison. Appl. Opt. 21, 2758 (1982). 50 Chapter 3 - Aggregation of solutes in bosonic versus fermionic quantum fluids This chapter is based on the following publication: Alexandra J. Feinberg, Deepak Verma, Sean M.O. O’Connell-Lopez, Swetha Erukala, Rico Mayro P. Tanyag, Weiwu Pang, Catherine Saladrigas, Benjamin W. Toulson, Mario Borgwardt, Niranjan Shivaram, Ming-Fu Lin, Andre Al-Haddad, Wolfgang Jäger, Christoph Bostedt, Peter Walter, Oliver Gessner, Andrey F. Vilesov. Aggregation of solutes in bosonic versus fermionic quantum fluids. Science Advances 7, abk2247, (2021). Abstract Quantum fluid droplets made of helium-3 ( 3 He) or helium-4 ( 4 He) isotopes have long been considered as ideal cryogenic nanolabs, enabling unique ultracold chemistry and spectroscopy applications. The droplets were believed to provide a homogeneous environment in which dopant atoms and molecules could move and react almost as in free space, but at temperatures close to absolute zero. Here, we report ultrafast X-ray diffraction experiments on xenon-doped 3 He and 4 He nanodroplets, demonstrating that the unavoidable rotational excitation of isolated droplets leads to highly anisotropic and inhomogeneous interactions between the host matrix and enclosed dopants. Superfluid 4 He droplets are laced with quantum vortices that trap the embedded particles, leading to the formation of filament-shaped clusters. In comparison, dopants in 3 He droplets gather in diffuse, ring-shaped structures along the equator. The distorted shapes of droplets carrying both filaments and rings are direct evidence that rotational excitation is the root cause for the inhomogeneous dopant distributions. 51 3.1 - Introduction Quantum fluid nanodroplets made of liquid helium are exceptional hosts for isolated cryogenic matrix applications (1-5). The droplets readily pick up atoms and molecules (6), providing unique opportunities to study the formation of molecular complexes close to absolute zero temperatures. Indeed, the large degree of quantum mechanical delocalization in helium enables unique matrix configurations around the dopants, giving rise to a perfectly tailored void around each particular molecule (3). Previously, small 4 He droplets containing less than ~10 4 atoms, roughly 10 nm in diameter, were used for the spectroscopic interrogation of molecules and molecular complexes at a temperature of about 0.4 K (1-5). It was long believed that, unlike immobilized dopant molecules in solid matrices, dopants in helium nanodroplets could move unhindered and stochastically (3, 7). Recent ultrafast X-ray coherent diffractive imaging (CDI) experiments with large xenon-doped superfluid 4 He droplets, a few hundreds of nm in diameter, have revealed a dramatically different scenario (8-10). Instead of forming the once proposed ramified entities (7), dopant atoms were found to aggregate in arrays of elongated filament-shaped clusters (9, 10). This effect was assigned to inhomogeneities within the droplets due to the presence of quantum vortices, which attract dopant particles (11-14). The vortices were found to originate from an unavoidable rotational excitation of free helium droplets in the beam (8, 15-17), implying that the superfluid nature of 4 He enhances the inhomogeneity of matrix-dopant interactions. To provide unequivocal proof for the link between inhomogeneous dopant distributions, the superfluid nature of 4 He droplets, and their rotational excitation, comparative measurements are required on fermionic 3 He and superfluid, bosonic 4 He droplets. It is important to note that 3 He can also enter the superfluid state but does so at much lower temperatures (T ~ 1 mK) (18, 19) than 52 are present in our experiment (T ~ 0.15 K) (20). Thus, 3 He droplets act as a normal fluid under our experimental conditions and serve as a reference droplet devoid of vortices. Here, we present a comparative study of the aggregation of xenon atoms in sub micrometer sized 3 He and 4 He droplets. Our results show that, in fact, dopants are subject to a high degree of spatial confinement within both 3 He and 4 He nanodroplets, with each isotope giving rise to dramatically different dopant morphologies. 53 Fig. 3.1.1. Outlines (black) and xenon dopant density distributions (blue-red) of superfluid 4 He droplets. Panels a-d show results for four different representative 4 He droplets. The values a and b of the long and short half axis, respectively, of the droplet's projection onto the detector plane are given in each panel. For visualization, circular contours (magenta) have been superimposed on the droplets with a radius equal to that of the minor half axis. Closer inspection reveals slightly elliptical distortions, most prominent in droplet b. Fig. 3.1.2. Outlines (black) and Xe dopant density distributions (blue-red) of normal fluid 3 He droplets. Panels a-d show results for four different representative 3 He droplets. The values of the long and short half axis of the droplet's projection onto the detector plane are given in each panel. 54 For visualization, circular contours (magenta) have been superimposed on the droplets with a radius equal to that of the minor half axis. Note the partly significant elliptical distortions of the droplet outlines. 3.2 - Results Figures 3.1.1 and 3.1.2 show plane projections of 4 He and 3 He droplets, respectively, with their reconstructed Xe dopant density distributions for a variety of representative droplets (10). The details on the reconstruction of density from diffraction images and the description of the results are described in the Materials and Methods (MM) section. The 3 He and 4 He droplets studied in this work have similar diameters in the range of 400-600 nm, containing on the order of 10 9 helium atoms per droplet. Corresponding diffraction images are presented in the Supplementary Materials (SM). Most outlines are ellipses, consistent with spheroidal, rotating droplets (8, 15, 17, 21, 22). In previous studies, it was found that a cryogenic fluid expansion into vacuum readily produces rotating 4 He and 3 He droplets (15, 17). It was also found that droplets of different isotopes have very similar average aspect ratios of about 1.05 for their projections on the detector plane (17). We hypothesized that during the passage of fluid helium through the nozzle, the fluid interacts with the nozzle channel walls and acquires vorticity, which is eventually transferred to the droplets. Figure 1 illustrates several 4 He droplets and their dopant density distributions. As previously demonstrated (9, 10), the droplets contain several strongly aligned tracks of high density, which are assigned to xenon atoms aggregating inside the cores of quantum vortices. Vortices in Figs. 3.1.1(a-c) are viewed from the side, while vortices in Fig. 3.1.1(d) point toward the viewer, revealing their arrangement in a triangular lattice configuration that closely resembles the 55 arrangements of vortices observed in rotating cylinders filled with 4 He (11, 14) and in trapped Bose-Einstein condensates (23). Results are dramatically different for xenon-doped 3 He droplets, as illustrated in Fig. 3.1.2. Here, xenon clusters either appear as a stripe (Fig. 3.1.2(a)) or as elliptical structures (Fig. 3.1.2(b- d)) that are aligned along the droplets’ long axes. In Fig. 3.1.2(d), xenon atoms form a loose ring of clusters on the droplet's periphery. During the imaging event, the X-ray beam forms an arbitrary angle with the droplet's figure axis; therefore, their real aspect ratios are larger than indicated by their outlines in Figs. 3.1.1 and 3.1.2, which correspond to projections of the droplets on the detector plane. The images are characterized by the two half axes of the droplet’s projection, referred to as a and b (a > b), corresponding to a projection aspect ratio, AR = a/b. The formation of rings is observed in 3 He droplets having AR = 1.04 to AR = 1.2 (Fig. 3.1.2). A smaller amount of data was obtained for 4 He droplets during the same experimental run. Most of the intense, reconstructable 4 He images have a small aspect ratio (AR < 1.05). However, the results obtained during our previous studies show the formation of vortex arrays in droplets having up to AR = 2.4 (9). Thus, we observe confinement of dopants across a wide range of aspect ratios. The lower boundaries for the droplet’s angular velocity, estimated from their aspect ratios (17), are ≈1.1∙10 7 rad/s and ≈1.5∙10 6 rad/s for the droplets in Figs. 3.1.2(c) and 3.1.2(d), respectively. In comparison, the angular velocity of the 4 He droplet in Fig. 3.1.1(d) is estimated to be ≈1.7∙10 6 rad/s based on the areal density of the vortices and using the Feynman relation (24). The pronounced alignment of the xenon cluster contours along the long axes of the 3 He droplets strongly suggests that the xenon dopants form rings in the droplets' equatorial planes, with their apparent ellipticity determined by the viewing angle. 56 Statistically, there is a large difference between the shapes of the xenon density distributions within 3 He and 4 He droplets. No aligned filaments, which are characteristic for superfluid 4 He droplets, are observed in 3 He droplets. Instead, these fermionic droplets contain diffuse ring-shaped structures. It is unlikely that the ring structures could be attributed to any impurities. The 3 He gas used was 99.9% pure with the remaining 0.1% being mostly 4 He. Considering that 4 He’s solubility in 3 He is ∼0.1% at 0.15 K, any residual 4 He will likely be dissolved in the 3 He droplets. Even if any pockets of a 4 He-rich phase were formed, they would be too small to give rise to any measurable effects in the diffraction patterns (17). 3.3 - Discussion It is immediately apparent from the dopant density distributions presented in Figs. 3.1.1 and 3.1.2 that helium nanodroplets are not homogenous nanolabs. In both isotopes, dopants are subject to unavoidable, high degrees of spatial confinement due to the droplets’ rotation. In 4 He, vortex-bound xenon is aligned along the minor axis of the droplets, as discussed in more detail elsewhere (9), whereas in 3 He, xenon is confined along the droplet equator. The direct relation between the direction of the 3 He droplet’s long axis and the concentration of xenon along the equator is visually apparent in Fig. 3.1.2. The distorted shapes of 3 He and 4 He droplets carrying dopant rings and filaments, respectively, are direct evidence that rotational excitation is indeed the root cause for the inhomogeneous dopant distributions. Clusters formed in fermionic 3 He and bosonic 4 He droplets exhibit distinctly different structures. Thus, nuclear spin, which has no impact on any property of ordinary solvents at higher temperatures, plays a crucial role in determining the aggregation dynamics of dopants at low temperatures. We propose that the mechanism for cluster formation in large helium droplets differs between superfluid 4 He versus normal fluid 3 He. In 4 He, single xenon atoms are picked up by the 57 droplet, rapidly thermalize, and begin to move freely within the confines of the droplet’s boundaries (3). Atoms form clusters upon collisions. At the same time, xenon atoms and small clusters are attracted to the cores of the vortices by hydrodynamic forces (11-14) and form large, filament-shaped aggregates. In comparison, in a 3 He droplet devoid of vortices, xenon clusters will likely form throughout the entire droplet volume, followed by coalescence into larger globular cluster-cluster aggregates. However, due to the high viscosity of 3 He droplets, dopants assume the same angular velocity as the host and congregate close to the droplet's surface along the equator, i.e., in a plane perpendicular to the direction of the angular momentum. The positions of the dopants are defined by a balance between centrifugal forces and the dopants' solvation potential (25). In principle, similar ring-shaped clusters are expected to be formed in classical rotating droplets (e.g., water droplets with heavy colloidal clusters), however we are unaware of such studies. The ring-shaped xenon structures appear to consist of separate, small (~50 nm) clusters, some of which exhibit branched shapes. The structures are likely defined during their formation and remain frozen at the low droplet temperature. The clusters appear to be separated and do not collapse into larger cluster-cluster aggregates, indicating that some mechanism stabilizes the porous network. Previously, it was proposed that some weakly interacting atoms (e.g., Magnesium) may form a so-called foam (26 - 28), where the atoms stay at sub-nm distance because of the shell of surrounding helium atoms. Whereas X-ray diffraction could be a useful technique for identifying the foam state, the resolution of the current small-angle soft X-ray scattering experiment of about 20 nm is insufficient to resolve spatial features on this level of detail. It is conceivable that the clusters have some interlinks that are too thin to be detected. The smallest compact cluster that can be detected in this work contains ~1000 xenon atoms and will appear in an image as approximately 3 × 3 pixels in size. This limit is set by the threshold of the phase 58 retrieval algorithm and the spatial resolution of the measurements (10). Future high-resolution experiments may shed more light on the atomic structure of aggregates obtained at temperatures close to 0 K. The few 100-nm sized droplets in this study, which are produced from fragmentation of the supercritical fluid in the cryogenic nozzle, are marked by large angular velocities of 10 6 -10 7 rad/s. This contrasts with the results for small droplets of few nm in diameter produced via aggregation of helium atoms. For example, extensive spectroscopy experiments on molecules in small (a few nm) 4 He droplets did not indicate any presence of quantum vortices (29). On the other hand, centrifugal displacement of molecules from the droplet’s center was discussed (30). The locations of molecules in small droplets could not be identified in the previous spectroscopy studies on either 3 He or 4 He, and the dopants are often assumed to reside close to the droplet's center (3, 4). We observe that vortices in 4 He are typically separated by distances of 100 - 200 nm, thus smaller droplets of 150 - 200 nm in diameter may contain just a single vortex. This shows that smaller 4 He droplets between 50 - 100 nm in diameter may be devoid of vortices. Some other techniques of producing helium droplets at small velocity, other than in a molecular beam, may be considered to produce 4 He droplets devoid of vortices. 3.4 - Materials and Methods Production and doping of 3 He and 4 He droplets Large nanodroplets are produced by expanding pressurized 4 He (99.9999%) or 3 He (99.9%) fluid through a cryogenic nozzle into vacuum with a stagnation pressure of P0 = 20 bar and a nozzle temperature T0 = 5 K (3, 8, 17, 31). At these expansion conditions, droplets with 59 average radii of ~160 nm and ~350 nm were produced for 3 He and 4 He, respectively (17). Once in vacuum, the droplets evaporatively cool to respective temperatures of 0.15 K for 3 He (20) and 0.38 K for 4 He (32). The droplets exit the source chamber with an average velocity of about 160 m/s for 3 He and 190 m/s for 4 He and subsequently enter the pickup chamber, which is filled with xenon (99.9%) gas. The droplets collide with, and pick up, several xenon atoms, evaporating off ~750 3 He or ~250 4 He atoms with the pickup of each xenon atom. The amount of xenon added is measured by monitoring the relative depletion of the mass M = 8 signal for 4 He (or M = 6 for 3 He), representative of He2 + ions, in a quadrupole mass spectrometer installed in the terminal vacuum chamber (8). The droplets in Figs. 3.1.1 and 3.1.2 contain ~10 9 helium atoms and between 10 5 - 10 6 xenon atoms. The 3 He gas was collected, purified, and recirculated by a gas-recycling system as described elsewhere (17). X-ray diffraction from Xe-doped 3 He and 4 He droplets Xenon-doped droplets are irradiated by a focused X-ray Free-Electron Laser (XFEL) beam operated at 1.5 keV (𝜆 = 0.826 nm) (8). The FEL beam consists of ultrashort X-ray pulses, containing up to ~10 12 photons/pulse, with a repetition rate of 120 Hz, a pulse energy of 1.5 mJ, and a pulse duration of ~100 fs (FWHM). The small pulse length and large number of photons per pulse enables the instantaneous capture of the shapes of individual droplets. Diffraction images are recorded with a pn-charge-coupled device (pnCCD) detector containing 1024 × 1024 pixels, each 75 × 75 μm 2 in size, which is centered along the FEL beam axis ∼735 mm downstream from the interaction point. The detector consists of two separate panels (1024 × 512 pixels each) located closely above and below the x-ray beam. Both panels have a central, rectangular cutout to accommodate the primary x-ray beam. 60 Density retrieval, size, and shape determination The diffraction patterns are recorded at small scattering angles and, thus, predominantly contain information on the column density of the droplets in the direction perpendicular to the detector plane. During the measurements, roughly 550 diffraction patterns from Xe doped 3 He nanodroplets were obtained, whereas 200 patterns were obtained as a reference for Xe doped 4 He droplets. Among them, only the brightest images containing more than ~10 5 detected photons were selected for reconstruction (10). Ten brightest hits were selected from the 4 He data, whereas 15 hits were selected from the 3 He data. Using an iterative phase retrieval algorithm, termed Droplet Coherent Diffractive Imaging (DCDI), the density profiles of the Xe clusters inside the droplets are reconstructed and the sizes and shapes are determined (10). Similar 3 He and 4 He droplet reconstructions are compared based on size, aspect ratio, and overall number of photons detected. Helium droplet shapes are described by the distances between the center and the surface in three mutually perpendicular directions: A > B > C. The observed diffraction patterns do not provide direct access to the actual values of A, B, and C, due to the droplets’ unknown orientations with respect to the X-ray beam. Instead, the images are characterized by the two half axes of the projection of a droplet onto the detector plane, which are referred to as a and b (a > b), corresponding to a projection aspect ratio, AR = a/b. The majority (99%) of helium droplets are close to spherical with AR < 1.4 corresponding to oblate, axially symmetric shapes. For those shapes with AR < 1.4, the average aspect ratios for each isotope are similar, with AR = 1.049 ± 0.003 for 3 He and 1.059 ± 0.005 for 4 He (17). 61 Acknowledgements This material is based upon work supported by the National Science Foundation under Grant No. DMR-1701077 and CHE-1664990 (A.F.V.). C.S., B.W.T., M.B. and O.G. are supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division, through Contract No. DE-AC02-05CH11231. M.B. acknowledges support by the Alexander von Humboldt foundation. Portions of this research were carried out at the Linac Coherent Light Source (LCLS) at the SLAC National Accelerator Laboratory. SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02- 76SF00515. Additionally, we acknowledge the Natural Sciences and Engineering Research Council of Canada. Author Contributions A.F.V., O.G. and C.B. conceived the experiment, A.J.F., D.V., S.M.O.O’C., S.E., C.A.S., B.W.T., M.B., N.S., M.-F.L., P.W., O.G. and A.F.V. prepared and carried out the experiment. A.J.F., D.V. and A.F.V. performed the data analysis. R.M.T. and W.P. contributed to the development of the phase retrieval algorithm and took part in the preliminary experiments. A.A.H. and C.B. provided support for online data analysis. S.M.O.O’C., W.J. and A.F.V. constructed the recycling apparatus for 3 He gas. A.J.F., O.G. and A.F.V. wrote the manuscript. Competing Interests Statement The authors declare no competing financial interests. 62 Data Availability Statement Raw data were generated at the Linac Coherent Light Source (LCLS) large-scale facility. Derived data supporting the findings of this study are available from the corresponding author upon request. Supplementary Information (SI) Diffraction Images Figures S1(a-d) and S2(a-d) show the diffraction images for Xe-doped 4 He and 3 He droplets, respectively, from which the densities in Fig. 3.1.1(a-d) and 3.1.2(a-d) of the main text were obtained. The images are plotted in the indicated logarithmic color scale and the patterns are cropped to the central 600 × 600 pixels of the 1024 × 1024 pixels detector. The diffraction images from 4 He droplets in Fig. S1 exhibit circular ring structures close to the center and speckled patterns in the outer region due to scattering off the embedded Xe clusters. Figure S1(d) shows Bragg spots in a hexagonal arrangement, indicating that a highly ordered structure is contained inside the droplet. As with diffraction from 4 He, the patterns from 3 He droplets in Fig. S2 exhibit circular ring structures close to the center and speckled patterns in the outer region. However, in comparison to patterns from 4 He droplets, the speckles are extended and seem to be arranged more closely along the rings. The different appearances of the diffraction patterns indicate different density distributions of Xe atoms inside 4 He compared to 3 He droplets, which are presented in Figs. 3.1.1 and 3.1.2, respectively, in the main text. 63 Figure 3.1.3. Diffraction images from 4 He nanodroplets in a logarithmic color scale. 64 Figure 3.1.4. Diffraction images from 3 He nanodroplets in a logarithmic color scale. 65 References 1. M. Hartmann, R. E. Miller, J. P. Toennies, A. F. Vilesov, High-resolution molecular spectroscopy of van der Waals clusters in liquid helium droplets. Science 272, 1631-1634 (1996). 2. K. K. Lehmann, G. Scoles, Superfluid helium - The ultimate spectroscopic matrix? Science 279, 2065-2066 (1998). 3. J. P. Toennies, A. F. Vilesov, Superfluid helium droplets: A uniquely cold nanomatrix for molecules and molecular complexes. Angew. Chem. Int. 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Tanyag et al., Communication: X-ray coherent diffractive imaging by immersion in nanodroplets. Struct. Dynam. 2, 051102 (2015). 11. G. Bewley, D. Lanthrop, K. Sreenivasan, Visualization of quantized vortices. Nature 441, 588 (2006). 12. F. Coppens, F. Ancilotto, M. Barranco, N. Halberstat, M. Pi, Dynamics of impurity clustering in superfluid He4 nanodroplets. Phys. Chem. Chem. Phys. 31, 17423-17432 (2019). 13. W. Guo, M. La Manita, D. Lathrop, S. Sciver, Visualization of two-fluid flows of superfluid helium-4. PNAS 111, 4653-4658 (2014). 14. E. J. Yarmchuck, M. V. J. Gordon, R. E. Packard, Observation of stationary vortex arrays in rotating superfluid-helium. Phys. Rev. Lett. 43, 214-217 (1979). 15. B. Langbehn et al., Three-dimensional shapes of spinning helium nanodroplets. Phys. Rev. Lett. 121, 255301 (2018). 16. D. Rupp et al., Coherent diffractive imaging of single helium nanodroplets with a high harmonic generation source. Nat. Commun. 8, 493 (2017). 17. D. 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Bernando, A. F. Vilesov, Kinematics of the doped quantum vortices in superfluid helium droplets. J. Low Temp. Phys. 191, 242-256 (2018). 26. E. B. Gordon, Impurity condensation in liquid and solid helium. J. Low Temp. Phys. 30, 756-762 (2004). 27. S. Gode, R. Irsig, J. Tiggesbaumker, K. H. Meiwes-Broer, Time-resolved studies on the collapse of magnesium atom foam in helium nanodroplets. New J. of Phys. 15, 015026 (2013). 28. J. Eloranta, Theoretical Study of Quantum Gel Formation in Superfluid He-4. J. Low Temp. Phys. 162, 718-723 (2011). 29. K. K. Lehmann, R. Schmied, Energetics and possible decay mechanisms of vortices in helium nanodroplets. Phys. Rev. B 68, 224520 (2003). 30. K. K. Lehmann, Potential of a neutral impurity in a large helium droplet. Molec. Phys. 97, 645-666 (1999). 31. O. Bünermann et al., Calcium atoms attached to mixed helium droplets: A probe for the 3He - 4He interface. Phys. Rev. B 79, 214511 (2009). 68 32. S. Grebenev, J. P. Toennies, A. F. Vilesov, Superfluidity within a small helium-4 cluster: The microscopic Andronikashvili experiment. Science 279, 2083-2086 (1998). 69 Chapter 4 - X-ray diffractive imaging of highly ionized helium nanodroplets This chapter is based on the following accepted manuscript: Alexandra J. Feinberg, Felix Laimer, Rico Mayro P. Tanyag, Björn Senfftleben, Yevheniy Ovcharenko, Simon Dold, Michael Gatchell, Sean M.O. O’Connell-Lopez, Swetha Erukala, Catherine A. Saladrigas, Benjamin W. Toulson, Andreas Hoffman, Ben Kamerin, Rebecca Boll, Alberto De Fanis , Patrik Grychtol, Tommaso Mazza, Jacobo Montano, Kiana Setoodehnia, David Lomidze, Robert Hartmann, Philipp Schmidt, Anatoli Ulmer, Alessandro Colombo, Michael Meyer, Thomas Möller, Daniela Rupp, Oliver Gessner, Paul Scheier, and Andrey F. Vilesov. Diffractive imaging of highly charged helium nanodroplets. Phys. Rev. Res. In press (2022). 4.1 - Abstract Finding the lowest energy configuration of N unit charges on a sphere, known as Thomson’s problem, is a long-standing query which has only been studied via numerical simulations. We present its physical realization using multiply charged He nanodroplets. The charge positions are determined by x-ray coherent diffractive imaging with Xe as a contrast agent. In neutral droplets, filaments resulting from Xe atoms condensing on quantum vortices are observed. Unique to charged droplets, however, Xe clusters that condense on charges are distributed on the surface in lattice-like structures, introducing He droplets as experimental model systems for the study of Thomson’s problem. 4.2 – Introduction In 1904, J.J. Thomson sought to find the configuration of N unit charges on a sphere that minimizes the overall Coulombic energy [1]. Thomson’s problem has since appeared in the research of highly ordered finite systems such as viral morphology [2], crystallography [3] and 70 molecular structure [4]. Generally, Thomson’s problem remains unsolved, though numerical simulations have yielded a variety of configurations depending on the number of charges. Locally, calculations yield charge distributions characterized by triangular lattice configurations with ~6 nearest neighbors per charge [5]. Surprisingly, it has been shown that the minimum energy configuration isn’t always the one that places the charges at furthest distance from each other, nor the configuration with greatest symmetry [5-9]. It isn’t immediately clear if an idealized Thomson problem that ignores the roles of thermal or quantum mechanical kinetic energies has any practical significance. However, charged helium droplets present an extraordinary experimental realization [10] because they remain liquid down to absolute zero temperature and can hold multiple charges, which are effectively confined and pushed to the droplet’s surface due to their mutual Coulomb repulsion. Surface liquid helium acts as a suitable support to study the structure and collective properties of two-dimensional Coulomb systems, i.e., those made of electrons or cations. Recent examples include the observation of multielectron nanobubbles in 4 He [11] (which have been predicted to exhibit several exciting low- temperature properties like Wigner crystallization, quantum melting, and electron localization by ripplon-polaron lattice [12,13]), or the study of the spatial distribution of cations on the surface of liquid 4 He when placed in an inhomogeneous electric field [14]. Recently, Laimer et al. demonstrated that multiply charged 4 He droplets can be produced via electron impact [15]. Here, the arrangement of charges in helium droplets is studied via scattering of radiation from an x-ray free electron laser (XFEL). Visualization of charges is achieved by doping the droplets with xenon (Xe) atoms, which cluster around the charges and serve as contrast agents. The cluster positions are obtained from diffraction images using an iterative phase retrieval algorithm [16]. Distinctly different from neutral helium droplets, aggregation within charged helium droplets leads to 71 fractured dot-like patterns with compact spots of Xe clusters throughout the droplets. We assign the compact dots to charged Xe clusters near the droplet surface. 4.3 – Experimental Figure 4.3.1. Schematic of the experiment. Helium droplets are ionized via electron impact and pass-through plane capacitor electrostatic deflectors. Ionized helium droplets are doped with Xe atoms which cluster around the positions of the charges and serve as markers. The droplets are interrogated with the XFEL. Diffraction patterns are recorded on a pnCCD detector and processed using a phase retrieval algorithm to obtain the density profile and charge distributions. The predicted configuration for 18 charges is used as an example to simulate the displayed diffraction pattern. The experiments are performed at the Small Quantum Systems (SQS) instrument of the European XFEL [17,18] using the Nano-sized Quantum Systems (NQS) end station. A schematic is shown in Fig. 4.3.1. Neutral helium droplets are produced via cryogenic nozzle beam expansion 72 of pressurized helium gas through a 5 μm nozzle into vacuum [19,20]. The droplets are ionized by an electron gun operating at 40 – 100 eV acceleration voltage and ionization currents between 5 μA and 1 mA [15]. Charged droplets pass through two parallel plate capacitors (each containing 2 × 2 cm 2 plates placed 2 cm apart from each other), used to verify the charging through electrostatic beam deflection. The ionized droplets are doped with Xe atoms (see Fig. 4.3.1) in a pickup cell. The focused XFEL beam (~3.5 µm full-width-at-half-maximum, FWHM focal diameter) intersects the Xe doped helium droplets ~950 mm downstream from the droplet source. The XFEL is operated at 10 Hz with 1 FEL pulse per bunch train, a photon energy of 1.5 keV, a mean pulse energy of 3 mJ, and a nominal pulse duration of ~25 fs. Diffraction images are recorded with a 1- Mpixel pnCCD detector (1024 × 1024 pixels, 75 × 75 μm 2 each) [21], centered along the XFEL beam axis 542 mm behind the interaction point. The detector consists of two separate panels (1024 × 512 pixels each), located above and below the x-ray beam with a central, rectangular cutout to accommodate the primary beam [19]. Each diffraction pattern is used to reconstruct the density distribution of Xe clusters within a single, isolated doped droplet (see Fig. 4.3.1) by applying an iterative phase retrieval algorithm termed Droplet Coherent Diffractive Imaging (DCDI) [16]. Diffraction images of neutral droplets are recorded for the initial droplet beam characterization and confirm that the droplets have an average radius of 250 nm (roughly 10 9 atoms) and an average aspect ratio (AR, major diffraction axis/minor diffraction axis) of 1.04, like previous conditions [22]. The non-sphericity can be assigned to centrifugal deformation of rotating droplets [20,22-24]. At low angular momenta and aspect ratios of 1< AR < 1.1, these can be described as spheroids. The flux of atoms carried by the droplets is monitored by the pressure rise in the beam dump chamber [20]. Ionization of the droplets did not lead to any measurable decrease 73 (<10%) in flux. However, the application ~100 V/cm to the parallel plate capacitors downstream from the ionizer completely extinguishes the droplet flux, demonstrating effective charging. 4.4 – Results Figure 4.4.1. Results for uncharged, Xe doped 4 He droplets. (1a), (2a): Diffraction patterns showing the central 600 x 600 detector pixels. The vertical streak in the upper half of the patterns is caused by stray light. (1b), (2b): Density reconstructions obtained with the DCDI algorithm. Initial measurements are performed on neutral Xe doped droplets, see Fig. 4.4.1. The diffraction images, Fig. 4.4.1(1a) and Fig. 4.4.1(2a), exhibit rings close to the center and speckles in the outer region due to scattering off Xe clusters. Fig. 4.4.1(1b) and Fig. 4.4.1(2b) show the corresponding density reconstructions (helium is depicted in blue and Xe clusters in red/yellow) as obtained via DCDI. The Xe clusters form filaments aligned along a common direction. They are the result of Xe atoms condensing on the cores of quantum vortices, as documented in our previous works [16,20,24-27]. 74 Figure 4.4.2. Same as in Fig. 4.4.1, but for charged, Xe-doped 4 He droplets. For droplets 1 – 3, the electron energy was 40 eV and emission current 30 μA, while for droplet 4, the ionizer was set to 200 eV and 1.3 mA emission current. Figure 4.4.2 shows diffraction patterns and density reconstructions from charged, Xe doped 4 He nanodroplets. The diffraction images Fig. 4.4.2(1a) – Fig. 4.4.2(3a) resemble those for uncharged droplets. Figure 4.4.2(4a) is noteworthy as it contains six Bragg spots, arranged hexagonally, which have been marked by white rings. Density reconstructions are shown in Fig. 4.4.2(1b) – Fig. 4.4.2(4b). In distinction to neutral droplets, aggregation within charged helium droplets appears unique in that it leads to fractured dot-like patterns with compact spots of Xe clusters. The comparison of Xe densities in charged droplets (Fig. 4.4.2) with neutral droplets (Fig. 4.4.1), obtained in this work as well as in previous [16,20,24-27] indicates important differences. Xe clusters in neutral droplets are seen as filaments. However, it’s possible that a droplet’s direction of angular momentum can be aligned with the x-ray beam such that the cylindrical 75 vortices appear as dots on the detector plane. The coexistence, however, of dot-like features and filaments has never been observed in neutral doped droplets [16,20,24-27], indicating that the dots aren’t the result of a particular viewing angle. Thus, we assign the dots to the location of charged Xe clusters near the droplets’ surface. Figure 4.4.2(1b) showcases a droplet (AR = 1.02, a = 199 nm, N = ~7×10 8 helium atoms) containing two filaments and at least 16 dot-shaped clusters. The filaments primarily occupy the central volume of the droplet whereas the dots are scattered throughout the image. Figure 4.4.2(2b) shows a droplet (AR = 1.04 and a = 201 nm, N = ~7×10 8 helium atoms) with a pattern of small irregularly shaped dots. Twelve approximately equidistant Xe clusters are arranged into an ordered, oval-shaped structure. Figure 4.4.2(3b) shows a much larger droplet (AR = 1.01, a = 550 nm, N = ~10 10 helium atoms) with several small clusters arranged in the center. During the experiments, droplets 1, 2, and 3 from Fig. 4.4.2 had a constant doping level, set by evaporation of ~20% of the constituent helium atoms. Considering that the pickup of one Xe atom leads to evaporation of ~250 helium atoms [20,26], the number of doped Xe atoms are estimated to be ~7×10 5 Xe atoms for the droplets in Fig. 4.4.2(1b) and Fig. 4.4.2(2b), and ~10 7 Xe atoms for Fig. 4.4.2(3b) (or about 10 -3 per helium atom). Figure 4.4.2(4b) exemplifies aggressive doping and charging conditions in larger droplets. Here, droplets were produced at nozzle settings of 60 bar, 4 K and with the ionizer set to 200 eV and 1.3 mA emission current. The droplet (AR = 1.01, a = 337 nm, N = ~4×10 9 He atoms) in Fig. 4.4.2(4b) was doped with ~1.6×10 7 Xe atoms, leading to evaporation of ~50% of helium atoms. The reconstruction shows a very dense pattern of small clusters. The details cannot be fully resolved. Note that the resolution limit of the experiment is ~20 nm [16] or three pixels in Fig. 4.4.2(4b), whereas the average distance between the Xe clusters is estimated to be ~40 nm. 76 4.5 – Discussion Significant differences in the Xe density distributions upon doping neutral versus charged droplets are readily apparent. While neutrals exhibit exclusively filament-shaped Xe structures, charged droplets contain compact, dot-shaped Xe clusters often in greater numbers than the filaments. Evidently, the presence of charges influences the formation of Xe clusters. Charges in helium droplets were previously proposed to serve as nucleation centers for embedded atoms [15], catalyzing the formation of clusters such as in Fig. 4.4.2(1b). The charge acquired by the droplet is estimated from the current density in the experiment (~1 A/m 2 ), time of flight of the droplet through the ionizer (~25 μs), and assuming the ionization cross section of the droplet corresponds to its geometric cross section. The estimated number of ~30 charges can be compared with the observed 16 dot-shaped clusters in Fig. 4.4.2(1b). The agreement is reasonable considering the coarse estimate and the fact that upon creation, some charges leave the droplet in the form of Hen + clusters [28]. Calculations [10] show that charges in helium droplets reside close to the surface. For 20 charges, they’re submerged by about 3% of the droplet radius, and move closer to the surface with increasing charge [10]. Xe clusters forming around the charges increases the solvation energy, pulling the charged clusters deeper inside. Other calculations [29] show that the potential energy of Xe clusters is flat in the droplet's interior but increases significantly ~10 nm from the surface. Thus, we expect charges encapsulated in Xe clusters will reside close to the surface. The distance will likely be comparable to the resolution of the present experiments of ~20 nm. Upon doping helium droplets, vortices and surface charges compete to attract the dopants. Like other particles, some charges will be captured by vortices, while others will remain free [28,30,31]. The partitioning depends on the droplet size, number of vortices, and the binding 77 energy of He + to vortices. The capture impact parameter for Xe atoms by a vortex has been estimated to be ~0.5 nm [32], and the capture cross section of a 200 nm long filament is ~200 nm 2 . This can be compared to the cross section for the capture of a Xe atom by a charge, which is estimated to be ~ 1 nm 2 based on the atom's polarizability. Therefore, Xe atoms are mostly attracted to vortices. The situation in Fig. 4.4.2(1b) appears fortunate as only two vortices are present. However, the interpretation of Fig. 4.4.2(1b) is not straightforward, as the filaments likely contain multiple charges, the locations of which cannot be determined. Figure 4.4.2(2b) shows an elliptic constellation of Xe clusters around the droplet center. No such structure has previously been observed in neutral droplets. The corresponding 3D structure is not obvious. For example, the figure may correspond to charged clusters attached to vortices arranged in a circle and viewed sideways, as was observed previously [26]. The other Xe clusters may correspond to charges on the droplet's surface that are not attached to vortices. Most images obtained in this work resemble that of Fig. 4.4.2(3b), exhibiting a constellation of clusters in the middle that lacks recognizable symmetry, and a small density on the periphery. Figure 4.4.2(4b) represents the extreme case of intensive charge exposure (200 eV at 1.3 mA emission current) coupled with heavy Xe doping. Although cluster configurations in Fig. 4.4.2(4b) cannot be resolved, some conclusions can be drawn from the observation of hexagonal Bragg spot arrangements in Fig. 4.4.2(4a). This pattern may originate from a system having short-range order, but lacking long-range order, such as a 2D liquid. From the spot scattering angle of about 0.021 rad, the average distance between the scattering centers is estimated to be ~40 nm. The distribution of Xe clusters at a comparable level of doping in neutral droplets (~40% of helium atoms evaporated) was previously obtained [33] and displays different features 78 such as a network of interconnected filaments. Therefore, we assign the presence of excess Xe clusters in Fig. 4.4.2(4b) to droplet charging. Previously observed Bragg spots in neutral droplets were assigned to lattices of quantum vortices. However, the smallest observed distance between vortices was about ~150 nm, [20] much larger than the estimated distance between scattering centers in Fig. 4.4.2(4b). A droplet having such a tight distance between vortices will rotate at an angular velocity about two times higher than the stability threshold for droplet fission [20,22,23]. Assuming the scattering centers are distributed evenly (with an average distance of 40 nm) on the surface of an R = 340 nm droplet, then the number of charges is estimated to ~1000. In comparison, the Rayleigh criterion [34] predicts a maximum charge of 1800. It’s likely that the diffraction in Fig. 4.4.2(4a) corresponds to a droplet containing a few vortices, but the formation of filaments is suppressed by aggressive charging and doping. Small angle diffraction from a 3D object is well described by the 2D Fourier transform of the column density along the light propagation direction. For a lattice on a sphere, the projection will accurately reflect the distances between the clusters near the droplet center, but peripheral distances will appear smaller. The combined effect will be a rapid decrease in the intensity of the Bragg spots in the radial direction. This agrees with the observation of the first order Bragg spots in Fig. 4.4.2(4a). Therefore, the results in Fig. 4.4.2(4a) and Fig. 4.4.2(4b) are consistent with the Xe clusters evenly filling the surface of the droplet. The location of charges in a spherical helium droplet was studied via Monte Carlo calculations [10], where at low T similar configurations were found as in previous calculations minimizing the potential energy [5-9]. Melting of the charge lattice at higher temperature is characterized by a sharp increase in the number of dislocations, which in the limit of large N can 79 be associated with the Kosterlitz–Thouless transition. The melting point corresponds to temperatures T* = 0.025 and 0.01 for N = 32 and 92 charges, respectively. Here, the dimensionless temperature, T*, is defined as the ratio of the absolute temperature to the Coulomb energy of two elementary charges at a distance equal to the radius of the droplet. In 4 He droplets T* ≈ 5×10 -3 at R = 200 nm, suggesting that a lattice could be observed in helium droplets relevant to this work. 4.6 – Conclusions In summary, charged helium droplets are a promising experimental realization of Thomson's problem for determining the minimum energy configuration of charges on a sphere. This work shows that droplets of a few hundreds of nm in radius and containing up to several hundreds of charges can be utilized as Thomson systems. While Thomson-type surface lattices of charge sites were observed, it was also discovered that quantum vortices can coexist with the charge lattice and act as dominant scavengers of Xe atoms. Nevertheless, our results indicate that quasi-free charges occupy positions near the droplet surface, forming lattice structures consistent with numerical solutions of Thomson’s problem. Future experiments with 3 He droplets could provide opportunities to advance these studies as they are devoid of quantum vortices above ~0.15 K. Data Availability Statement The data that support the findings of this study are stored under a proper DOI available from the corresponding authors upon reasonable request at Ref. [35]. Acknowledgements 80 We acknowledge European XFEL in Schenefeld, Germany, for provision of x-ray free electron laser beam time at the SQS instrument and would like to thank the staff for their assistance. A.J.F., S.M.O.C., S.E. and A.F.V. were supported by the NSF Grant No. DMR-1701077. B.S., D.R., and A.H. acknowledge funding from the Leibniz-Gemeinschaft via grant No. SAW-2017- MBI-4. D.R. and A.C. acknowledge funding from the Swiss National Science Foundation via grant No. 200021E_193642 and the NCCR MUST. C.A.S., B.W.T. and O.G. were supported by the Atomic, Molecular, and Optical Sciences Program of the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division, through Contract No. DE-AC02-05CH11231. F.L. and P.S. were supported by the Austrian Science Fund, FWF, projects I4130 and P31149. M.G. was supported by the Swedish Research Council (contract numbers 2016-06625 and 2020-03104). T.M. was supported by DFG, project numbers 719/13 and 719/14. T.M., R.M.P.T. and A.U. acknowledge funding from the Bundesministerium für Bildung und Forschung via grant No. 05K16KT3, within the BMBF Forschungsschwerpunkt Freie-Elektronen-Laser FSP-302. This article is based upon work from COST Action CA18212 - Molecular Dynamics in the GAS phase (MD-GAS), supported by COST (European Cooperation in Science and Technology). An Aside on this Experiment This experiment was originally scheduled to be performed in March 2020, right when the COVID-19 pandemic began. I had a plane ticket to fly to Hamburg and was meant to stay there for a month to set up the experiment and conduct measurements. One week before my flight was meant to depart, a no-fly order was issued, and we went on lockdown. Through some greater power we were eventually able to conduct the experiment in a hybrid manner. EU citizens could come to 81 the EXFEL site, but US participants were banned. We managed to pull off the experiment via ZOOM working closely with our European collaborators. I want to thank all the staff scientists at EXFEL for being patient and helping us finish our project. It was a miracle that despite these issues we still managed to take decent data. Figure 4.6.1 shows a snapshot of our ZOOM room right after finishing the experiment. Figure 4.6.1. Our experimental team. 4.7 - References 1. J. J. Thomson, On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure. 82 The London, Edinburgh, and Dublin and Philosophical Magazine and Journal of Science 6, 237-265 (1904). 2. R. Zandi, D. Reguera, R. F. Bruinsma, W. M. Gelbart, J. Ruddnick, Origin of icosahedral symmetry in viruses. PNAS 101, 155556-115560 (2004). 3. H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, R. E. Smalley, C60: Buckminsterfullerene. Nature 318, 162-163 (1985). 4. T. Liu, E. Diemann, L. Huilin, A. W. M. Dress, A. Müller, Self-assembly in aqueous solution of wheel-shaped Mo-154 oxide clusters into vesicles. Nature 426, 59-62 (2003). 5. A. P. Garrido, M. J. W. Dodgson, M. A. Moore, Influence of dislocations in Thomson’s problem. Phys. Rev. B 56, 3640-3643 (1996). 6. A. P. Garrido, M. A. Moore, Symmetric patterns of dislocations in Thomson’s problem. Phys. Rev. B 60, 15628-15631 (1999). 7. E. L. Altschuler et al., Possible Global Minimum Lattice Configurations for Thomson's Problem of Charges on a Sphere. Phys. Rev. Lett. 78, 2681-2685 (1997). 8. E. L. Altschuler, T. J. WIlliams, E. R. Rather, F. Dowla, F. Wooten, Method of constrained global optimization. Phys. Rev. Lett. 72, 2671-2674 (1994). 9. E. L. Altschuler, A. P. Garrido, Global minimum for Thomson’s problem of charges on a sphere. Phys. Rev. E 71, 047703 (2005). 10. A. M. Livshts, Y. E. Lozovik, Crystallization and Melting of a System of Charges in a Liquid Helium Cluster. J. of Exp. and Theo. Phys. 105, 571-586 (2007). 11. N. Yadav, P. Sen, A. Ghosh, Bubbles in superfluid helium containing six and eight electrons: Soft, quantum nanomaterial. Sci. Advan. 7, eabi7128 (2021). 83 12. P. Leiderer, Electrons at the surface of quantum systems. J. Low. Temp. Phys. 87, 242- 287 (1992). 13. J. Tempere, I. F. Silvera, J. T. Devreese, Multielectron bubbles in helium as a paradigm for styudying electrons on surfaces with curvature. Surf. Sci. Rep. 62, 159-217 (2007). 14. T. B. Möller, P. Moroshkin, K. Kono, E. Scheer, P. Leiderer, Critically Charged Superfluid 4He Surface in Inhomogeneous Electric Fields. J. Low. Temp. Phys. 202, 431- 443 (2020). 15. F. Laimer et al., Highly Charged Droplets of Superfluid Helium. Phys. Rev. Lett. 123, 165301 (2019). 16. R. M. P. Tanyag et al., Communication: X-ray coherent diffractive imaging by immersion in nanodroplets. Struct. Dynam. 2, 051102 (2015). 17. T. Tschentscher et al., Photon Beam Transport and Scientific Instruments at the European XFEL. Appl. Sci. 7, 592 (2017). 18. W. Decking et al., A MHz-repetition-rate hard X-ray free-electron laser driven by a superconducting linear accelerator. Nat. Photon. 14, 391-397 (2020). 19. R. M. P. Tanyag, C. Jones, C. Bernando, S. M. O. O’Connell, A. F. Vilesov, in Cold Chemistry: Molecular Scattering and Reactivity Near Absolute Zero. (2018). 20. L. F. Gomez et al., Shapes and vorticities of superfluid helium nanodroplets. Science 345, 906-909 (2014). 21. M. Kuster et al., The 1-Megapixel pnCCD detector for the Small Quantum Systems Instrument at the European XFEL: system and operation aspects. J. Synch. Radiat. 28, 576-587 (2021). 84 22. D. Verma et al., Shapes of rotating normal fluid He-3 versus superfluid He-4 droplets in molecular beams. Phys. Rev. B 102, 014504 (2020). 23. C. Bernando et al., Shapes of rotating superfluid helium nanodroplets. Phys. Rev. B 95, 064510 (2017). 24. A. J. Feinberg et al., Aggregation of solutes in bosonic versus fermionic quantum fluids. Sci. Advan. 7, eabk2247 (2021). 25. O. Gessner, A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Nanodroplets. Ann. Rev. Phys. Chem. 70, 173-198 (2019). 26. C. F. Jones et al., Coupled motion of Xe clusters and quantum vortices in He nanodroplets. Phys. Rev. B 93, 180510 (2016). 27. S. M. O. O’Connell et al., Angular momentum in rotating superfluid droplets. Phys. Rev. Lett. 124, 215301 (2020). 28. T. González-Lezana et al., Solvation of ions in helium. Int. Rev. in Phys. Chem. 39, 456- 516 (2020). 29. C. Bernando, A. F. Vilesov, Kinematics of the doped quantum vortices in superfluid helium droplets. J. Low. Temp. Phys. 191, 242-256 (2018). 30. K. R. Atkins, Ions in Liquid Helium. Physical Review 116, 1339-1343 (1959). 31. W. I. Glaberson, W. Johnson, Impurity ions in liquid helium. J. Low. Temp. Phys. 20, 313-338 (1975). 32. I. A. Pshenichnyuk, N. A. Berloff, Inelastic scattering of xenon atoms by quantized vortices in superfluids. Phys. Rev. B 94, 184505 (2016). 33. R. M. P. Tanyag, University of Southern California, Los Angeles (2018). 85 34. L. Rayleigh, On the equilibrium of liquid conducting masses charged with electricity. The London, Edinburgh, and Dublin and Philosophical Magazine and Journal of Science 14, 184-186 (1882). 35. A. F. Vilesov, P. Scheier, D. Rupp, O. Gessner, E. XFEL, Ed. (European XFEL, 2020). 86 Chapter 5 – X-ray coherent diffractive imaging of helium nanodroplets doped with small molecules 5.1 - Abstract Previous coherent diffractive imaging (CDI) studies of the doped helium droplets were limited to using xenon atoms. Here we expand the studies to molecular dopants such as CF4, CHF3, CH3CN, and SF6. We report the results of the single shot x-ray CDI in conjunction with time-of-flight (TOF) spectroscopy. X-ray CDI provides information on the droplet size, photon flux, and dopant distribution inside the droplet, whereas TOF spectra provide information about the relative abundance of different ionic species produced in the x-ray – cluster interaction, as well as the ion kinetic energies. It was found that in distinction to atomic dopants, molecular dopants show unique aggregation patterns that appear more compact. 5.2 - Introduction Quantum fluid helium nanodroplets are exceptional hosts for cryogenic matrix isolation (1-5). The droplets readily pick up atoms and molecules and enable “trapping” of the dopants in the lowest possible energy configuration due to the droplets’ extremely low temperature (T < 0.38 K) (6). Previous experiments on 4 He droplets doped with Xe atoms have revealed that Xe aggregates along the cores of 4 He’s quantum vortices as the atoms are attracted to the vortices by hydrodynamic forces (7-9). The aggregates span the length of the vortex and appear as elongated filaments or threads, sometimes featuring curvature (8, 9). Such filament-shaped clusters were previously observed in Xe-doped 4 He nanodroplets via x-ray coherent diffractive imaging (CDI). Concomitant experiments with molecules of varying size, shape, and polarity can deepen our 87 understanding of the self-assembly of atoms and molecules at ultra-low temperatures. Molecular self-assembly is a process that is dictated by thermodynamics. At ambient temperatures, aggregation will evolve toward a structure with the smallest free energy. Here, we explore the regime close to absolute zero temperature which will allow us to “freeze out” unstable structures. In this regime, interparticle interactions should be dominant whereas entropic effects due to temperature will be negligible. Weak intermolecular forces such as dipole-dipole interactions and hydrodynamic attraction by quantum vortices should play a significant role in the resulting structures, which will likely be atypical such as anatomically thin chains or fractals. Here, we expand our experiments to small molecules of different geometries and dipole moments to explore their aggregation at low temperatures. The dopants used for this experiment are Xe, CF4, CHF3, CH3CN, and SF6. The dopants are embedded into 4 He droplets and interrogated with x-ray CDI. In addition to x-ray scattering measurements, simultaneous acquisition of TOF spectra provide information about the relative abundance of different ionic species produced in the x-ray – cluster interaction, as well as the ion kinetic energies (10). It was found that ~38% of the collected diffraction patterns exhibited Bragg spotting, a significant increase relative to previous experiments with Xe-doped helium nanodroplets, where patterns with Bragg spots were only observed ~10% of the time. 5.3 - Experimental Figure 5.3.1 shows the experimental scheme. Large nanodroplets are produced by expanding pressurized 4 He (99.9999%) or 3 He (99.9%) fluid through a cryogenic nozzle into vacuum with a stagnation pressure of P0 = 20 bar and a nozzle temperature T0 = 2-5 K (3, 7, 11, 12). Once in vacuum, the droplets evaporatively cool to 0.38 K and 0.15 K for 4 He and 3 He 88 respectively (6). The droplets exit the source chamber and subsequently enter the pickup chamber, which is filled with dopant gas. The droplets collide with and pick up a large number (~10 5 -10 6 ) dopant molecules, evaporating off several hundreds of helium atoms with the pickup of each molecule. The amount of dopant added is measured by monitoring the relative depletion of the mass M = 8 (6) signals for 4 He ( 3 He) (representative of He2 + ions) in a quadrupole mass spectrometer installed in the terminal vacuum chamber (7). Doped droplets are irradiated by a focused X-ray Free-Electron Laser (XFEL) beam operated at 1.5 keV (𝜆 = 0.826 nm) (7). The FEL beam consists of ultrashort x-ray pulses, containing up to ~10 12 photons/pulse, with a repetition rate of 120 Hz, a pulse energy of 1.5 mJ, and a pulse duration of ~100 fs (FWHM), focused to about 5 µm. The small pulse length and large photon flux enables the instantaneous capture of the diffraction image from the doped droplets. Diffraction images are recorded with a pn-charge-coupled device (pnCCD) detector containing 1024 × 1024 pixels, each 75 × 75 μm 2 in size, which is centered along the FEL beam axis ∼735 mm downstream from the interaction point. The detector consists of two separate panels (1024 × 512 pixels each) located closely above and below the X-ray beam. Both panels have a central, rectangular cutout to accommodate the primary X-ray beam. A single-field Wiley-McLaren ion time-of-flight (TOF) spectrometer (13) is aligned with the constant extraction field oriented perpendicular to both the X-ray and the droplet beams, i.e., vertically in the laboratory frame. The spectrometer consists of a 40 mm long acceleration region between a positively charged repeller (+800 V) and a negatively charged extractor (-800 V) electrode, and a 40 mm long, field-free drift region that is terminated by a single microchannel plate (MCP) detector. Both the repeller and extractor are round and 40 mm in diameter. The 89 interaction volume is shifted by ~4.4 mm from the center of the acceleration region toward the repeller due to mechanical restrictions in the experimental chamber. The extractor electrode is equipped with a 2 mm wide, 15 mm long slit aligned perpendicular to the X-ray beam. Along the beam propagation direction, the slit ensures that only ions created within ≈20% of the Rayleigh range (2xr ≈ 11 mm) around the focal point of the x-ray beam are detected. Figure 5.3.1. a) Diagram of experimental setup. b) sample ion TOF spectrum for a neat 4 He droplet. 5.4 – Results Due to experimental limitations, reconstructable images could only be obtained for three molecules: CHF3, CF4, and SF6. The similarity in the densities amongst these molecules indicates that the reconstructions for CH3CN would likely follow the same trend. TOF spectra, however, could be collected for all four molecules. 90 Figure 5.4.1. 1a, 2a: Diffraction patterns from CHF3-doped superfluid 4 He droplets, zoomed into 600 x 600 pixels. The blank horizontal stripe in the diffraction stems from the physical gap between the upper and lower pnCCD detector plates. 1b, 2b: Density reconstructions obtained from the DCDI algorithm. 91 Figure 5.4.1(1a) and (2a) show diffraction patterns for CHF3-doped 4 He droplets in a logarithmic color scale. Figure 5.4.1(1b) and (2b) show the corresponding density reconstructions as obtained via DCDI. In the reconstructions, helium is depicted as blue and embedded CHF3 clusters are depicted as red/yellow. These droplets contain on average ~10 9 helium atoms and between 10 5 - 10 6 dopant particles. The diffraction pattern in Fig. 5.4.1(1a) exhibits characteristic features, namely, a circular ring structure close to the center that extends outwards. Six Bragg spots arranged along a line are visible in the image, indicative of a periodic structure inside the droplet. The spots have decreasing intensity in going outwards from the center of the image. The reconstruction in Fig. 5.4.1(1b) shows four filaments arranged perpendicularly to the droplet’s long axis. These filaments exhibit no curvature. They appear to be slightly shorter than the droplet diameter, indicating that the droplet’s rotational axis is slightly tilted with respect to the X-ray beam. Figure 5.4.1(2a) shows another diffraction pattern for a CHF3-doped 4 He droplet. This pattern also exhibits a circular ring structure close to the center, however some speckling/intensity modulation is observed in the rings further from the center, which is due to scattering interference with CHF3 clusters. A diagonal streak through the image containing Bragg spots can be observed in Fig. 5.4.1(2a) like that in (1a). However, the spots in (2a) have some elongated shapes and are not as clearly decipherable as in (1a). Fig. 5.4.1(2b) shows the density reconstruction. Seven filaments arranged perpendicularly to the droplet’s long axis can be seen. In comparison to (1b), the filaments are not as compact or defined but there is a more homogenous distribution throughout the droplet. 92 The droplets in Fig. 5.4.1(1a) and (1b), along with the remaining presented droplets, exhibit spheroidal shapes. The shape of a droplet is characterized by the distance between its center and surface in three mutually perpendicular directions: a = b ≥ c, for spheroids. Due to an unknown orientation of the droplets with respect to the x-ray beam, the diffraction patterns only allow us to calculate two semi-axes of the projection of a droplet onto the detector plane: the major half axis, A, which is the axis perpendicular to the direction of the angular momentum, and an upper bound for the minor half axis, C, which is the shape’s symmetry axis. (7, 14). The images are characterized by these projection axes and correspond to a projection aspect ratio, AR = A/C. The aspect ratio provides a measure of how distorted the droplet is from a perfect sphere, which can tell about its angular momentum (14, 15). Most droplets will have AR close to 1, which we can assign to rotating droplets which acquire spheroidal shapes because of centrifugal deformation (14, 15). However, larger AR events have been observed, corresponding to strongly distorted shapes with high angular momenta (8). The droplet corresponding to Fig. 5.4.1(1a, 1b) was found to have a major half axis of A = 370 nm and an aspect ratio of AR = 1.06, while the droplet corresponding to Fig. 5.41(2a, 2b) was found to have a major half axis of A = 350 nm and an aspect ratio of AR = 1.09. These shapes correspond well with what we have previously observed for xenon doped droplets (7, 8, 12). 93 Figure 5.4.2. 1a: Diffraction pattern from CF4-doped 4 He droplets, zoomed into 600 x 600 pixels. The blank horizontal stripe in the diffraction stems from the physical gap between the upper and lower pnCCD detector plates. 1b: Density reconstruction obtained from the DCDI algorithm. Figure 5.4.2(1a) shows a diffraction pattern for a CF4-doped 4 He droplet. The diffraction pattern exhibits some Bragg streaks like CHF3, but with lower contrast. The reconstruction Fig. 5.4.2(1b) shows some diffuse filaments arranged perpendicularly to the droplet’s long axis. Some regions of density break this pattern and appear clustered along the droplet’s periphery. Due to the low contrast of the diffraction pattern itself, it’s difficult to tell if these clusters are real or artefacts of the reconstruction algorithm. The droplet corresponding to Fig. 5.4.2 was found to have a major half axis of A = 330 and an aspect ratio of AR = 1.04. 94 Figure 5.4.3. 1a: Diffraction pattern from superfluid SF6-doped 4 He droplets, zoomed into 600 x 600 pixels. The blank horizontal stripe in the diffraction stems from the physical gap between the upper and lower pnCCD detector plates. 1b: Density reconstruction obtained from the DCDI algorithm. Figure 5.4.3(1a) shows a diffraction pattern for a SF6-doped 4 He droplet which shows similar features to the previous three diffraction patterns, albeit with a significant amount of intensity modulation. The speckling can be attributed to scattering interference between helium atoms and SF6 which have different refractive indices. A Bragg streak can be seen in the diffraction like that of CHF3 and CF4. Figure 5.4.3(2b) shows the droplet reconstruction which exhibits at least six filaments aligned perpendicularly to the droplet’s long axis. The filaments appear to have a diffuse distribution like with CF4. The droplet corresponding to Fig. 5.4.3 was found to have a major half axis of A = 390 and an aspect ratio of AR = 1.02. 95 Figure 5.4.5. Cation TOF spectra for neat 3 He (purple) and 4 He (green) droplets doped with atoms or small molecules. The black tick marks show the mass scale. Taller bars are positioned every 10 mass units. 96 Cation TOF spectra provide information about the relative abundance of different ionic species produced in the X-ray – cluster interaction and the kinetic energies of the ions. An example TOF spectrum for pure 4 He droplets is shown in Fig. 5.3.1(b). The leftmost (i.e., shortest TOF) feature corresponds to the singly charged He + atomic ions, with the features following at longer times of flight corresponding to Hen + clusters with n >1 (10). Figure 5.4.5 shows some typical TOF spectra obtained upon X-ray ionization for neat (undoped) 3 He (purple) and 4 He (green) droplets doped with small molecules. Figure 5.4.5(a) shows the spectrum for neat helium droplets. Here, the leftmost TOF feature corresponds to singly charged 3 He + and 4 He + atomic ions, whereas the features at longer times of flight corresponding to x Hen + clusters with x = 3 or 4 and n > 1. In both cases, the prominent peaks can be seen for the singly ionized case of He + extending out to He5 + . Figure 5.4.5(b) shows the TOF spectra for Xe- doped 3 He and 4 He droplets. As with Fig. 5.4.5(a), a progression of Hen + peaks can be seen at shorter TOF. At later TOF, around m/z = 130 (TOF = 3.2 x 10 -6 s), one can see peaks corresponding to Xe + ions. The Xe peak has a higher intensity in 4 He than in 3 He. Figure 5.4.5(c) shows the TOF spectra for CF4-doped 3 He and 4 He droplets. The two initial peaks correspond to 4 He + and 4 He2 + . Around m/z = 12 (TOF = 1.4 x 10 -6 s), a peak corresponding to carbon ions is observed, followed by fluorine ions shortly thereafter at m/z = 18 (TOF = 1.6 x 10 -6 s). Figure 5.4.5(d) shows the cation TOF spectra for CHF3-doped 3 He and 4 He droplets. An initial peak can be seen for both isotopes corresponding to H + . Two prominent peaks can be seen directly adjacent corresponding to 3 He + , 3 He2 + , 4 He + , and 4 He2 + . Carbon and fluorine peaks are observed at the same m/z and TOF as in Fig. 5.4.5(c), as to be expected. Figure 5.4.5(e) shows the TOF spectra for CH3CN-doped 3 He and 4 He droplets. A sequence of peaks can be seen 97 corresponding to H + , 3 He + , 4 He + , 3 He2 + , 4 He2 + , C + , and N + , however, C + and N + are likely not resolved. Finally, Fig. 5.4.5(f) shows the cation TOF spectra for SF6-doped 3 He (purple) and 4 He (green) droplets. Two prominent peaks for both isotopes can be seen initially corresponding to 3 He + , 3 He2 + , 4 He + , and 4 He2 + . At m/z = 18 (TOF = 1.6 x 10 -6 s), the fluorine peak is observed, followed by a smaller sulfur peak at m/z = 32 (TOF = 1.8 x 10 -6 s). 5.5 – Discussion We will begin by comparing the droplet densities as obtained via DCDI for atomic xenon versus molecular clusters. Previously obtained densities for xenon doped 4 He nanodroplets show that xenon aggregates in the filaments in accordance with what was observed in this work (8, 9, 16). However, the appearance of xenon-doped filaments deviates dramatically from that of molecular clusters, see Fig. 5.5.1. Filaments doped with xenon have been observed to exhibit curvature appearing slightly askew to profoundly distorted (16). Sometimes, the filaments are observed to curl over each other. Figure 5.51(a) shows a pronounced example of this type of curling behavior. Figure 5.5.1. (a) Density plot of xenon clusters aggregating in 4 He nanodroplets (16). (b) Density plot of CHF3 aggregating in 4 He nanodroplets (from Fig. 2(1b) in this text). 98 For the case of CHF3 in 4 He, as in Fig. 5.5.1(b), the filaments appear compact and needle- like. There is little to no curvature in any of the vortex structures present. The curvature of a quantum vortex doped with particles has previously been studied via 3D Gross-Pitaevskii equation to theoretically explore the attractive interaction between particles and vortices (17). It was found that particles of all sizes induce Kelvin waves in the vortex upon their capture. The amplitude of these Kelvin waves is strongly dependent on the side of the particle captured; thus, small particles will induce waves with small amplitude, whereas large particles will induce waves with large amplitude (see Fig. 7). These waves deform the vortex and cause a deviation from a perfectly straight line. Figure 5.5.2. A quantum vortex capturing particles of small (top) and large (bottom) sizes in time. Vortices are displayed in red, and particles are in green. Large particles tend to distort the vortex 99 more than small particles, causing pronounced curvature. Adapted from Giuriato and Krstulovic (17). It is interesting to consider the contribution to Kelvin wave generation from an atomic xenon cluster versus a molecular cluster such as CHF3. The vortices in the case of the CHF3 doped droplet (Fig. 5.5.1 (b)) appear extremely straight, meaning they must not be contributing to vortex deformation to the same extent as with xenon clusters (see Fig. 5.5.1(a)). This could be due to the difference in intermolecular interactions both substances experience. As xenon has a full electron shell, it primarily forms clusters via Van der Waals interactions. CHF3, however, exhibits dipole- dipole interactions. The difference in intermolecular interactions experienced by the two substances could be related to the degree of rigidity of the formed clusters, underlying why we see a larger fraction of diffraction patterns exhibiting Bragg features for molecularly-doped droplets relative to those doped with xenon. Another important observation is that molecular clusters in both 4 He and 3 He irradiated by X-rays fragment very extensively into constituent ions. Previously, it was proposed that the helium environment may decrease the fragmentation of molecules upon X-ray excitation. Our experiments show this is not the case, and that molecules fragment to atomic ions when irradiated with the XFEL. Additionally, we did not observe any noticeable difference in the TOF patterns from 4 He and 3 He droplets. This likely indicates that typical energies of the ions produced upon ionization is very large, much larger than the interaction energy between helium atoms, so that the effects of nuclear spin statistics play no significant role. It is remarkable that ionization of doped helium droplets by X-rays does not produce any noticeable number of molecular ions or molecular ionic clusters, indicating that the energy released is sufficient for the complete atomization of the embedded molecules. In comparison, molecular splitter ions and ionic clusters are readily 100 produced upon ionization of helium droplets in a typical laboratory experiment with a 100 eV electron beam, such as in our recent work with ethylene (18). More work is required to understand the mechanism of ionization of molecular clusters in helium droplets by X-rays. 5.6 - References 1. M. Hartmann, R. E. Miller, J. P. Toennies, A. F. Vilesov, High-resolution molecular spectroscopy of van der Waals clusters in liquid helium droplets. Science 272, 1631-1634 (1996). 2. K. K. Lehmann, G. Scoles, Superfluid helium - The ultimate spectroscopic matrix? Science 279, 2065-2066 (1998). 3. J. P. Toennies, A. F. Vilesov, Superfluid helium droplets: A uniquely cold nanomatrix for molecules and molecular complexes. Angew. Chem. Int. Ed. 43, 2622-2648 (2004). 4. M. Y. Choi et al., Infrared spectroscopy of helium nanodroplets: novel methods for physics and chemistry. Int. Revs. in Phys. Chem. 25, 15-75 (2006). 5. A. Mauracher et al., Cold physics and chemistry: collisions, ionization and reactions inside helium nanodroplets close to zero K. Phys. Reports 751, 1-90 (2018). 6. S. Grebenev, J. P. Toennies, A. F. Vilesov, Superfluidity within a small helium-4 cluster: The microscopic Andronikashvili experiment. Science 279, 2083-2086 (1998). 7. L. F. Gomez et al., Shapes and vorticities of superfluid helium nanodroplets. Science 345, 906-909 (2014). 8. S. M. O. O’Connell et al., Angular momentum in rotating superfluid droplets. Phys. Rev. Lett. 124, 215301 (2020). 101 9. R. M. P. Tanyag et al., Communication: X-ray coherent diffractive imaging by immersion in nanodroplets. Struct. Dynam. 2, 051102 (2015). 10. C. A. Saladrigas et al., Charging and ion ejection dynamics of large helium nanodroplets exposed to intense femtosecond soft X-ray pulses. Euro. Phys. J. Special Topics 230, 4011-4023 (2021). 11. O. Bünermann et al., Calcium atoms attached to mixed helium droplets: A probe for the 3He – 4He interface. Phys. Rev. B 79, 214511 (2009). 12. D. Verma et al., Shapes of rotating normal fluid He-3 versus superfluid He-4 droplets in molecular beams. Phys. Rev. B 102, 014504 (2020). 13. L. Strüder et al., Large-format, high-speed, X-ray pnCCDs combined with electron and ion imaging spectrometers in a multipurpose chamber for experiments at 4 th generation light sources. Nuclear Instruments and Methods in Physics Research A 614, 483 (2010). 14. D. Verma et al., Shapes of rotating normal fluid He-3 versus superfluid He-4 droplets in molecular beams. ArXiv:2003.07466. . 2020. 15. C. Bernando et al., Shapes of rotating superfluid helium nanodroplets. Phys. Rev. B 95, (2017). 16. A. J. Feinberg et al., Aggregation of solutes in bosonic versus fermionic quantum fluids. Sci. Advan. 7, (2021). 17. U. Giuriato, G. Krstulovic, Interaction between active particles and quantum vortices leading to Kelvin wave generation. Sci. Rep. 9, 4839 (2019). 18. S. Erukala, A. J. Feinberg, A. Singh, A. F. Vilesov, Infrared spectroscopy of carbocations upon electron ionization of ethylene in helium nanodroplets. J. Chem. Phys. 155, 084306 (2021). 102 Chapter 6 – Validating the droplet coherent diffractive imaging (DCDI) algorithm 6.1 - Introduction With the recent advent of XFELs, quantum vortices in helium nanodroplets could be visualized for the first time via scattering of XFEL radiation, a technique known as coherent diffractive imaging (CDI) (1). X-ray CDI is a lensless microscopy technique used to obtain the spatial distribution of electron density from an object’s diffraction image (2, 3). For far field coherent diffraction, the diffraction pattern is essentially the Fourier transform (FT) of the density, albeit without any phase information. The missing phase information is known as the “phase problem” of CDI; experimentally, only the amplitude information of the incident photons is retained, thus applying a simple inverse FT is insufficient due to the missing phase information. One can circumvent this problem by immersing the objects in a helium nanodroplet that acts as a matrix/physical host for what is being studied. The droplet limits the amount of space in which the object resides, providing a so-called "support" for the phase reconstruction. The scattering phase off a helium droplet can be used as an approximate trial phase for calculating the reconstruction, thus simplifying the solution of the phase problem. The application of a trial phase significantly reduces computational time and cost as compared to other algorithms that begin calculations assuming a random phase. This technique and its associated algorithm are known as Droplet Coherent Diffractive Imaging (DCDI) (4). DCDI was originally written in 2015 by members of the Vilesov research group at University of Southern California. Since its conception, it has been used to reconstruct Xe-doped 4 He droplets, revealing the configurations of quantum vortices (5, 6). It was observed that weaker 103 diffraction images often result in low contrast reconstructions, with some perceived parts of the density missing. Thus, it is important to establish the total number of detected photons in the diffraction required for a valid reconstruction. Another question is related to the size of the object that can be reconstructed. Xenon clusters inside helium droplets aggregate in vortices, making either compact clusters or thin filaments. Clusters with small dimensions leads to a particular shape in the diffraction from the clusters, in which the zero-order diffraction maximum extends to relatively large scattering angles beyond the central hole in the detector. Questions on the validity of the algorithm for the reconstruction of extended embedded objects inside the droplets remains, such as in the case of mixed 3 He/ 4 He droplets showing phase separation. The purpose of this chapter is to determine the minimum number of photons required to produce a reconstruction and to model the reconstruction of objects of different sizes inside helium droplets. 6.2 - Principles of validating DCDI The most straightforward way to validate DCDI requires obtaining a diffraction pattern from a known reference object, which is, however, still experimentally challenging. Therefore, we must resort to a modeling approach. Our method will be to create a synthetic input droplet based on some known reference density distributions for helium droplets that have been obtained experimentally. From the synthetic input droplet, diffraction patterns containing a different number of detected photons will be calculated and used as an input into DCDI. If the algorithm works, it will give back the reference density. We will limit our study to 2D objects, which appears to be a good approximation in view of the small scattering angle of <5°. The objects studied represent the 2D column density of the droplets and clusters, which are 3D objects. Any deviations from a valid reconstruction could be related to possible flaws of the DCDI algorithm itself, or incompleteness 104 of the diffraction measurement itself, such as a limited number of detected photons, or the missing central part of the data. Our calculations started from an exemplary experimentally obtained diffraction pattern for a 4 He droplet doped with xenon, see Fig. 6.2.1(a). This diffraction pattern was run through DCDI to produce the reference density, shown in Fig. 6.2.1(b), which provides a good example of the vortex features typically found in helium nanodroplets (the obtained reference density corresponds to a droplet with R = 300 nm). The synthetic input droplet was then produced from DCDI outputs. One gets four main files back from the program, two of which are the real and imaginary parts of the last density output before the program applies constraints. The synthetic input droplet, shown in Fig. 6.2.1(c), was produced as a sum of these outputs, both the droplet and cluster densities. The synthetic diffraction is obtained as a modulus squared of the discrete Fourier transform (FT) of the synthetic input droplet on a 1024 x 1024 matrix, shown in Fig. 6.2.1(d). The synthetic diffraction pattern produced from the reference density agrees well with the experimentally obtained diffraction pattern. 105 Figure 6.2.1. (a) Experimentally obtained diffraction pattern. (b) The droplet density reconstruction obtained via DCDI. (c) Synthetic input droplet produced from the real and imaginary parts of (b). (d) Synthetic diffraction produced from (c) via FT. The apparent difference in the color scale between (a) and (d) stems from different normalization, which is calculated from the zero-order diffraction maximum but gets cut off in (a) due to the hole in the detector. In all calculations using the synthetic diffraction pattern, a mask is first applied that omits certain data (see Fig. 6.2.2(b)). The mask simulates the effect of the missing photon information in the experimental diffraction pattern; here, the central part of the detector and the region between 106 the upper and lower detector plates have a gap (Fig. 6.2.2(a)) which is required for the exit of the intense XFEL beam. Figure 6.2.2. Diagram of the mask employed for DCDI tests. (a) Shows the pnCCD gap in the experimental diffraction pattern. (b) Shows the mask employed to cover the gap. The central black stripe and circle are not considered in the DCDI calculations. 6.3 - DCDI for diffraction of different intensities An important question regarding DCDI is the number of photons needed to produce a valid reconstruction. The calculated diffraction pattern, such as in Fig. 6.2.1(d), is described by a continuous function; however, our experimentally obtained data is discrete in terms of the number of photons. Applying a Poisson distribution to the continuous diffraction pattern discretizes the signal, thereby making it more representative of the data we obtain experimentally; see Fig. 6.2.2(a). In this way, diffraction patterns containing different numbers of scattered photons could be calculated. 107 Figure 6.3.1. Applying a Poisson distribution to a continuous diffraction pattern. (a) Shows the continuous pattern. (b) Shows the discretized pattern. To determine the minimum number of photons required to produce a valid reconstruction, synthetic, discretized diffraction patterns with varying numbers of photons were used as inputs to the DCDI code. Figure 6.3.1(b) represents the synthetic, discretized diffraction pattern with original intensity. However, a factor was applied to produce four more patterns with x10, x100, /10, and /100 numbers of photons relative to the original. These diffraction patterns of varying intensity were used as inputs to the DCDI code with a mask applied as in Fig. 6.2.2(b). Figure 6.3.2. Density reconstructions produced from discretized synthetic diffraction patterns with varying intensities. The number of photons below each droplet shows the number of photons 108 in the input diffraction pattern with a mask applied, as in Fig. 6.2.2(b). The minimum number of detected photons needed to produce a valid reconstruction is on the order of 5x10 5 . Figure 6.3.2 shows the density reconstructions for diffraction patterns of varying intensities. The reconstruction for the unadulterated diffraction pattern with original intensity is shown in the center, whereas factors of /10, /100, x10, and x100 are shown on the left and right sides, respectively. The original reconstruction (with ~5 × 10 5 photons as in the experiment) is valid in that it accurately reconstructs the internal vortex structures. Decreasing the intensity below 10 5 photons as with the /10 and /100 factors will not yield an adequate reconstruction. For the /10 factor, some incomplete structures are visible, and for the /100 factor little to no structural information can be discerned. Increasing the intensity to 5 × 10 6 photons yields significantly increased resolution and decreased noise with respect to the original reconstruction. However, this trend does not continue indefinitely; further increasing the intensity to 5 × 10 7 photons leads to decreased resolution and substantial background noise. Therefore, it was found that for the case of the synthetic discretized diffraction pattern, the minimum number of detected photons required to produce a valid reconstruction is on the order of 5 × 10 5 . 6.4 - DCDI for extended embedded objects So far, the DCDI reconstructions were applied to small objects, such as compact clusters or thin filaments embedded into the droplets. Such objects produce a diffraction pattern that is characterized by a zero-order maximum and a large radius extending beyond the central hole in the detector. A large object comparable with the radius of the droplet will cause closely spaced diffraction rings. This may pose a problem for the DCDI algorithm because the applied modulation 109 is not accounted for by the zero-order phase guess determined by the diffraction from the droplet itself. It is important to understand how large objects in a helium droplet are reconstructed as it applies to systems under study, such as mixed phase droplets of 3 He and 4 He. Figure 6.4.1. (a) Synthetically produced droplet of radius b containing a single spherical cluster of radius b/14. (b) Diffraction pattern obtained by FT of the density in (a). (c) Other positions considered for the inner cluster. To explore how the size of the inner cluster influences the quality of the reconstruction, we created a synthetic input droplet of radius b containing a single spherical cluster of radius b/x, see Fig. 6.4.1(a). Multiple sizes are studied from b/14 to b/2. Size b/14 is representative of small clusters (like those obtained on vortices), whereas size b/2 is representative of larger clusters (like 3 He/ 4 He mixed phase droplets). Additionally, the location of the cluster was varied such that we 110 could see how the quality of the reconstruction changes as a function of cluster location. Three locations were considered, the center, the periphery, and an intermediate regime between the two (see Fig. 6.4.1(c)). These synthetic input droplets containing a single cluster were used to produce synthetic diffraction patterns as shown in Fig. 6.4.1(b). The synthetic diffraction patterns with masks applied as in Fig. 6.2.2(b) were used as DCDI inputs to produce the reconstructions shown in Figs. 6.4.2, 6.4.3, and 6.4.4. Figure 6.4.2 shows some highlighted reconstructions for cluster sizes b/14, b/6, and b/2. For the highlighted densities shown, the inner cluster is in the middle position. The middle and peripheral positions were chosen as they were found to exhibit the least number of artefacts when producing reconstructions. Clusters placed in the center sometimes produce noise that is related to masking of the central hole. For Fig. 6.4.2, the reconstruction is fair when the embedded object is 1/6 th of the total droplet size. However, when one reaches an embedded object that is 1/2 of the total droplet size, the inner cluster in the reconstruction shows a pronounced hole, which is a failure. 111 Figure 6.4.2. DCDI reconstructions from clusters of varying sizes. For the initial densities shown, the cluster is in the middle position. The program rotates the droplet in the output, hence why the clusters are on the opposite sides as the input. Figure 6.4.3 shows all positions; central, middle, and peripheral, for small to medium inner cluster sizes. For size b/14, the program adequately reconstructs the droplet and inner cluster for all three positions. Size b/8 represents the initial point where the code shows issues performing reconstructions, although only for the central location. Once the inner cluster size reaches 1/6 th of the total droplet size, the program begins producing substantial artefacts for the central position and some slight noise for the middle and peripheral positions. Figure 6.4.3. Reconstructions for small to medium sized inner clusters of sizes b/14 to b/6. 112 Large clusters of sizes b/4 and b/2 face significant issues with being reconstructed for all positions, see Fig. 6.4.4. When the inner cluster is placed in the middle and peripheral positions, the reconstruction exhibits less artifacts, albeit not by much. The reconstructions for size b/2 show that when the inner cluster is placed in the center of the droplet, two separate crescent-shape regions of high density appear in the reconstruction. Reconstructions for the middle and peripheral locations do not exhibit the crescent behavior. However, information is missing, specifically in the central part of the inner cluster. From these tests, it can be concluded that DCDI experiences problems with returning the density of large, embedded clusters. This seems to be related to the missing information due to the central hole in the detector. The diffraction patterns produced from test droplets with embedded objects of varying sizes could be reasonably well reconstructed if no mask was applied. However, these issues must be rectified before using the code to reconstruct experimentally obtained diffraction patterns, as the mask must be employed to cover the missing information in the detector gap (see Fig. 6.2.2(b)). Figure 6.4.4. Reconstructions for large inner clusters of size b/4 and b/2. 113 6.5 - Conclusions We have determined that for the studied droplet of R = 300 nm, DCDI successfully reconstructs the diffraction patterns from the reference objects (with root mean square deviation of the density of less than 10%) in the case when more than 10 5 – 10 6 photons are present. With respect to extended embedded objects, for an R = 300 nm droplet, an inner cluster up to r = 50 nm (1/6 th of the total droplet size) could be reconstructed. Clusters larger than 1/6 th of the total droplet size yield artifacts in the reconstruction. Therefore, we conclude that DCDI needs to be studied more extensively before applying it to extended embedded objects such as mixed phase 3 He/ 4 He droplets. 6.6 - References 1. L. F. Gomez et al., Shapes and vorticities of superfluid helium nanodroplets. Science 345, 906-909 (2014). 2. J. Miao, R. Sandberg, C. Song, Coherent X-Ray Diffraction Imaging. IEEE Journal of Selected Topics in Quantum Electronics 18, 399 (2011). 3. Q. Shen, I. Bazarov, P. Thibault, Diffractive imaging of nonperiodic materials with future X-ray sources. J. Synch. Rad. 11, 432 (2004). 4. R. M. P. Tanyag et al., Communication: X-ray coherent diffractive imaging by immersion in nanodroplets. Struct. Dynam. 2, 051102 (2015). 5. A. J. Feinberg et al., Aggregation of solutes in bosonic versus fermionic quantum fluids. Sci. Advan. 7, (2021). 6. S. M. O. O’Connell et al., Angular momentum in rotating superfluid droplets. Phys. Rev. Lett. 124, 215301 (2020). 114 Chapter 7 – Summary and Future Outlooks To summarize, I performed a variety of X-ray CDI experiments with doped superfluid helium nanodroplets. My group and I were able to, for the first time, image the aggregation of xenon atoms in bosonic versus fermionic quantum fluids. We showed that in bosonic 4 He, aggregation occurs in the cores of the quantum vortices, whereas for fermionic 3 He, aggregation occurs in a stripe along the droplet’s periphery. The differences in aggregation mechanisms stems from the different spin statistics each element obeys, Bose-Einstein for 4 He, and Fermi-Dirac for 3 He. This experiment proved the validity of our previous work imaging quantum vortices in 4 He and concluded our series of experiments regarding the existence of quantum vortices in finite systems. My group and I also performed a first experimental attempt at solving the Thomson problem. So far, the problem had only been studied via numeric calculations. Superfluid 4 He nanodroplets were charged and doped with xenon atoms that aggregate around the positions of the charges. We were able to observe, for the first time, the co-existence of charge-assisted and vortex- assisted aggregation in 4 He nanodroplets. Obtaining coordinates for the charge locations was complicated by the presence of quantum vortices. Going forward, it would be imperative to redo the experiment with 3 He nanodroplets as they are devoid of quantum vortices within our experimental temperature regime. Additionally, I led our groups invesitgation into the low temperature aggregation of dopant atoms versus small molecules within the cores of 4 He’s quantum vortices. It was found that atomic aggregates form more diffuse, floppy structures than their molecular counterparts, evident by the shapes of the vortex structures after doping. Xe atoms in 4 He, which experience van der Waals interatomic interactions, form diffuse structures that cause the vortex to exhibit pronounced 115 curvature. However, for small molecules in 4 He, which experience dipole-dipole intermolecular interactions, more rigid structures are formed that appear as compact, needle-like lines of density. The compactness does not allow the vortex to deform to the same extent as with the atomic clusters due to the stronger intermolecular force experienced by the molecules. My final contribution was related to testing our phase retreival algorithm, DCDI. Questions on the validity of the algorithm for the reconstruction of large, embedded objects (such as in the case of phase separated 3 He/ 4 He droplets) need to be addressed before applying the algorithm to any general experiment. Two questions were considered for the tests: (1) what is the minimum number of photons required to produce a reconstruction, and (2) how large of a dopant cluster can we place in a helium nanodroplet before DCDI breaks down? By calculating a series of diffraction patterns with different numbers of photons, it was found that roughly 10 5 – 10 6 photons are required in the raw diffraction pattern to produce a reconstruction. Additionally, it was found that when a dopant cluster is larger than 1/6 th of the total droplet size, the code breaks down and fails to perform an adequate reconstruction. These findings imply that the code works well for reconstructing nano-sized structures, such as vortices, that are typically less than 1/14 th of the total droplet size. However, more developments should be made before applying the algorithm to phase separated 3 He/ 4 He droplets. Many exciting experiments involving helium nanodroplets have yet to be performed. My work, along with the work of my group, has concluded our fundamental series of experiments involving the existence of quantum vortices in finite systems. Applied experiments involving helium nanodroplets present a new frontier, such as with cryogenic matrix isolation spectroscopy, or the study of photoionized helium nanoplasmas in droplets. Additionally, helium nanodroplets 116 have recently been reported to act as a promising physical support in the synthesis of core-shell metallic nanoparticles, opening a new avenue of research. 117 References For ease of reading, references have been added to the end of each chapter. They can be found on the following pages: Chapter 1 References………….…………………………………………………………………11 Chapter 2 References ……………………….……………………………………………..…….45 Chapter 3 References………………….…………………………………………………………65 Chapter 4 References ………………..…………………………………………………………..81 Chapter 5 References......…………………………………………………………………..…...100 Chapter 6 References…………………………………………………………………………...113

Abstract (if available)

Abstract Liquid helium exhibits fascinating properties macroscopically and on the nanoscale. Its most notable quality is that of superfluidity; a novel state of matter characterized by vanishing viscosity (i.e., fluid flow without the loss of any kinetic energy), very high thermal conductivity, and other bizarre effects like film flow and vortices with quantized circulation (1-3). Many experiments to image quantum vortices in bulk superfluid helium have been attempted (4-7). However, these experiments faced significant difficulties both with achieving low enough temperatures and with the spatial resolution of the optical imaging approaches employed. As such, related experiments pushed to probe superfluids on the nanoscale.
Nano-sized droplets of helium present a new possibility for investigation, as they have recently been the focus of many chemical physics groups (8-10). The droplets are versatile in size and can range from few to 1012 atoms (corresponding to diameters of sub- nm to ~6 micron, respectively) (11-13). Imaging quantum vortices within isolated helium nanodroplets is enticing due to their low temperatures of 0.37 K, and absence of any convection flow, which could not be avoided in bulk experiments (14, 15). The droplets readily pick up atoms or molecules (14, 15), meaning they are perfect for imaging experiments via tagging. The first experiment with tagging of quantum vortices in helium nanodroplets was published in 2012 when large silver clusters, assembled in helium nanodroplets, were deposited onto a substrate (16). The deposits revealed elongated, filament-like silver clusters. It was proposed that inside the droplet, silver clusters are aggregating within the cores of the quantum vortices. This result was the first example of vortex-assisted aggregation in nanoscale helium droplets (16).
Recent experiments have pushed to probe helium nanodroplets with x-ray free electron lasers (XFELs) (17-22). Due to short wavelength and their high degree of transverse coherence, XFELs have substantially increased resolution (< 20 nm) as compared to previous optical imaging techniques (23, 24). In such experiments, single droplets are imaged via x-ray diffraction. Results from the first work of its kind revealed distinct Bragg spots in the diffraction images, indicative of a lattice of quantum vortices within the droplet (18). Later, a computer algorithm was developed to reconstruct the internal structure of the doped droplet from the obtained diffraction patterns (24). Application of the algorithm to diffraction patterns of doped 4He nanodroplets revealed filaments of vortices throughout the droplets volume, or triangular lattice patterns depending on the observer’s viewing angle.
The application of XFELs holds promise for other experiments involving helium nanodroplets. This thesis presents a variety of x-ray diffraction experiments with nanodroplets, with both 4He and 3He isotopes, as well as highly charged 4He nanodroplets. The results of such experiments have concluded our series of works regarding the existence of quantum vortices in finite systems.

References
1. D. R. Tilley, J. Tilley, Superfluidity and Superconductivity. (Institute of Physics Publishing, Bristol, 1990).
2. J. Wilks, D. S. Betts, An Introduction to Liquid Helium. (Clarendon Press, Oxford, 1987).
3. R. P. Feynman, Application of Quantum Mechanics to Liquid Helium. Progress in Low Temperature Physics 1, 17-53 (1955).
4. G. P. Bewley, D. P. Lathrop, K. R. Sreenivasan, Superfluid helium—visualization of quantized vortices. Nature, 441:588 (2006).
5. G.W. Rayfield, F. Reif, Evidence for the creation and motion of quantized vortex rings in superfluid helium. Phys. Rev. Lett. 11, 305-308 (1963).
6. P. W. Karn, D. R. Starks, W. Zimmerman, Observation of quantization of circulation in rotating superfluid He-4. Phys. Rev. B 21, 797-805 (1980).
7. W. F. Vinen, Detection of single quanta of circulation in liquid helium-II. Proc. R. Soc. A 260:2, 18-36 (1961).
8. M. Y. Choi et al., Infrared spectroscopy of helium nanodroplets: novel methods for physics and chemistry. Int. Revs. in Phys. Chem. 25, 15-75 (2006).
9. J. P. Toennies, A. F. Vilesov, Superfluid helium droplets: A uniquely cold nanomatrix for molecules and molecular complexes. Angew. Chem. Int. Edit. 43, 2622 (2004).
10. D. Verma, R. M. P. Tanyag, S. M. O. O’Connell-Lopez, A. F. Vilesov, Infrared Spectroscopy in Superfluid Helium Droplets. Adv. in Phys. X. 4, (2019).
11. J. P. Toennies, A. F. Vilesov, Spectroscopy of Atoms and Molecules in Liquid Helium. Annu. Rev. Phys. Chem. 49, (1998).
12. J. P. Toennies, A. F. Vilesov, K. B. Whaley, Superfluid Helium Droplets: an Ultracold Nanolaboratory. Phys. Today 2, 31 (2001).
13. J. A. Northby, Experimental studies of helium droplets. J. Chem. Phys. 115, 10065 (2001).
14. R. M. P. Tanyag et al., in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, A. Osterwalder, O. Dulieu, Eds. (Royal Society of Chemistry, Cambridge, 2018).
15. R. Hartmann, R. E. Miller, J. P. Toennies, A. F. Vilesov, Rotationally Resolved Spectroscopy of SF6 in Liquid-Helium Clusters - a Molecular Probe of Cluster Temperature. Phys. Rev. Lett. 75, 1566 (1995).
16. L. F. Gomez, E. Loginov, A. F. Vilesov, Traces of Vortices in Superfluid Helium Droplets. Phys. Rev. Lett. 108, 155302 (2012).
17. A. J. Feinberg et al., Aggregation of solutes in bosonic versus fermionic quantum fluids. Science Advances 7, (2021).
18. L. F. Gomez et al., Shapes and vorticities of superfluid helium nanodroplets. Science 345, 906-909 (2014).
19. B. Langbehn et al., Three-dimensional shapes of spinning helium nanodroplets. Phys. Rev. Lett. 121, 255301 (2018).
20. S. M. O. O’Connell et al., Angular momentum in rotating superfluid droplets. Phys. Rev. Lett. 124, 215301 (2020).
21. O. Gessner, A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Droplets. Annu. Rev. Phys. Chem. 70, 173 (2019).
22. C. Bernando et al., Shapes of Rotating Superfluid Helium Nanodroplets. Phys. Rev. B 95, 064510 (2017).
23. C. F. Jones et al., Coupled motion of Xe clusters and quantum vortices in He nanodroplets. Physical Review B 93, 180510 (2016).
24. R. M. P. Tanyag et al., Communication: X-ray coherent diffractive imaging by immersion in nanodroplets. Struct Dynam-Us 2, 051102 (2015).

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Creator Feinberg, Alexandra Joan (author)

Core Title X-ray coherent diffractive Imaging of doped quantum fluids

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry (Chemical Physics)

Degree Conferral Date 2022-08

Publication Date 07/22/2022

Defense Date 07/22/2022

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Advisor Vilesov, Andrey F. (committee chair), Takahashi, Susumu (committee member), Tanguay, Armand (committee member)

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