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Study of rotation and phase separation in ³He, ⁴He, and mixed ³He/⁴He droplets by X-ray diffraction

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Study of rotation and phase separation in ³He, ⁴He, and mixed ³He/⁴He droplets by X-ray diffraction

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Content i Study of Rotation and Phase Separation in 3 He, 4 He, and Mixed 3 He/ 4 He Droplets by X-ray Diffraction by Sean O’Connell A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (CHEMISTRY) August 2021 ii Acknowledgements My most important acknowledgement belongs to the love of my life, Daisy Lopez. You continue to push me to grow every day. Thank you for helping me through the tough times and for being there to celebrate the good times. All PhD experiences are shaped by advisors, and I was lucky to have Dr. Andrey Vilesov, who helped me tremendously in learning how to develop scientific ideas, conduct complicated experiments, troubleshoot problems, and write and present results. Driven and patient, Dr. Vilesov pushed me beyond what I thought I could do, leading me to contribute to projects at renowned scientific facilities, Stanford Linear Accelerator Center, SLAC and European X-ray Free Electron Laser, EXFEL. Without Dr. Vilesov, who was my sole advisor at the University of Southern California (USC), this thesis would not have been possible. I also thank my committee members: Dr. Jahan Dawlaty, Dr. Armand Tanguay, Dr. Hanna Reisler, and Dr. Karl Kristie. I also had many mentors at SLAC and EXFEL, including Dr. Oliver Gessner (SLAC and EXFEL), Dr. Christoph Bostedt (SLAC), Dr. Peter Walter (SLAC), Dr. Thomas Möller (EXFEL), Dr. Daniela Rupp (EXFEL), Dr. Rico Mayro P. Tanyag (EXFEL), Dr. Yevheniy Ovcharenko (EXFEL), and Dr. Paul Scheier (EXFEL). Note that the results from the EXFEL experiments are still being analyzed and are not presented in this thesis. I particularly thank Drs. Gessner, Walter, Rupp, and Tanyag. Dr. Gessner was involved with me in every beamtime, and we worked together on several papers over the years. Dr. Walter was the sole staff scientist at SLAC during the beamtime that produced the results in Chapter 5; his work at SLAC was invaluable. Dr. Rupp’s insights, ability to direct people, efficiency in delegating tasks, and nuanced analysis, often on the fly, made working for her at the EXFEL a great experience. Dr. Tanyag, who was a senior graduate student in Dr. Vilesov’s lab when I joined the group, taught me the ropes of how to run an iii experiment at USC and how to navigate the different waters of experiments at SLAC. I owe him many thanks. I also acknowledge other former Vilesov group members including Charles Bernando, Curtis Jones, and, especially, Deepak Verma. Deepak and I shared several drives up to SLAC in rented moving trucks, and his presence made the trips much more fun than they would have otherwise been. I also thank Weiwu Pang, an undergraduate student who helped develop the DCDI MATLAB code that made my thesis possible. Last, I thank Alexandra (Allie) Feinberg and Swetha Erukala, current graduate students in the group. In working with many other groups, I met other graduate students and postdocs, whom I thank for their help on beamtimes. Dr. Camila Bacellar, Catherine Saladrigas, Dr. Ben Toulson, and Dr. Mario Borgwardt from Dr. Gessner’s group all helped at various SLAC beamtimes. Catherine, especially, helped tremendously with all X-ray diffraction results presented in this work, and with her work at EXFEL beamtimes. I thank Anatoli Ulmer (from Dr. Möller’s group), a talented experimentalist and productive member of the team at EXFEL experiments. Katharina Kolatzki, Björn Senfftleben, and Julian Zimmerman from Dr. Rupp’s group were wonderful, smart, and supportive. Björn and Katharina were terrific beamtime teammates; Katharina brought out the best in everyone, and Björn patiently helped others with their problems while working on his own analysis. Most of my time was spent at USC, and I thank those who helped me stay sane during the dark days and helped me celebrate the bright moments. Laura Estergreen, Michael Kellogg, Anuj Pennathur, Ryan Hunt, Huy Phan, Dan Kwasniewski, and Bibek Samanta were all great floormates to have in SSC. Drs. Hanah Reisler, Alex Benderskii, Chi Mak, Jahan Dawlaty, and Steve Bradforth, all excellent faculty, were all very generous, providing parts in pinch times. I appreciate iv Dr. Vitaly Kresin being an excellent troubleshooting resource and for helping with various parts that had long lead times. I was also very lucky to have support outside of work. I thank Joe Patrow, with whom I attended the University of Wisconsin – Eau Claire (UWEC), for helping me find my footing in Los Angeles. Eric Driscoll was another important fellow graduate student outside the walls of SSC. We rode our first centuries together (100 miles on a bike in one day), and he, along with Sophie Samson, introduced me to bike infrastructure on the outskirts of LA. He also introduced me to the LA Marathon Crash Ride, which is my favorite yearly tradition in LA. Most importantly, my conversations with Eric convinced me to start seeing a therapist, and my wellness improved tremendously. I also thank Nemal Gobalasingham, Robert Pankow, and Sanket Samal for the many nights we shared together. Matt Bain came to USC late in my PhD tenure, but he is a smart, kind fellow and an excellent friend. Last, I thank Patrick Edwards for all the support, all the nonsense, and for the one particular encore. Our paths should have crossed earlier, but I am so glad that they crossed at all. I also thank the Thiccball crew. Sometimes, the necessary balance in life comes through getting incredibly sore after a pick-up game of kickball. I specifically thank John Niman, Ben Kamerin, Christina Cole, Tom Linker, Brian Weaver, Anna Haynie, James Farmer, and Darian Hartsell. You all have made my life so much better with your presences. Thank you all for staying in touch after the move to Miami. I also thank the crew at Casey’s, especially Aaron, Thomas, Jess, and Gus. Last, I thank everyone at Improv for the People for giving me a place to perform improvisational comedy again. I especially thank Matt Moore, Jaymie Parkkinen, and Mindy Sterling for being such insightful, encouraging teachers. v It took a lot of people to get me to USC, and I thank the faculty at UWEC, especially my undergraduate advisor Dr. James Boulter as well as Drs. Matt Evans, Patricia Cleary, Nathan Miller, George Stecher, Fred King, Jim Rybicki, Elizabeth Glogowski, Tom Lockhart and Erik Hendrickson. I also thank Mr. John Stupak for all his help with research projects and talks about life. I thank the members Backwards Thinkers Society for letting me run the group for so long. I especially thank Alex Brandt, Paul Matthews, Sami Johnson, and Michael Renner for letting me take the reins and allow the group to take itself more seriously. Thank you to Amber Dernbach, Jordan James, and Cullen Ryan for letting us expand off-campus and giving us space in the Eau Claire comedy scene. I acknowledge Eric Johnson for hiring me at Anybody’s Bike Shop and being patient with me while I found the balance between work and “book learning.” Joe Decker, Peter Zernia, and Ian Hall were also excellent co-workers. Thank you, Joe for the continued hospitality at the Nucleus! I also recognize people I met in Eau Claire who become life-long friends: Cody Alft, Alex Brandt, Paul Matthews, Ryan Nichols, Mitch Wood, and Michael Yohn, you all mean so much to me. Before attending UWEC, I never saw myself as a scientist, but thank you to Mary Crowley, Timothy Chase, Kathy Kahn, Peter Grebner, and Colin Marsh for pushing me to try and bring out my best in your classes. Colin, thank you especially for running with the “Safety Duck” bit. Thank you, Makiki Reuvers, for staying in touch for all these years. Thank you, Dave Bole, for giving me my first job out of high school. I learned a lot about how to be an employee while working at the Bicycle Chain. I thank everyone at the Brave New Institute for helping me find my voice and confidence in high school. Special thanks to directors Mike Fotis and Joe (Joe! Joe! Joe!) Bozic and coaches Nate Melcher, Josh Kuehn, Nels Lennes, Jen Scott, and Dave Jennings. Special thanks to my troupe mates Emmet Cowen, Eric Geistfeld, Dena Coffman, Briget Diehl (Travis Fiero), vi Rose Gottlieb, Laura Mulcahy, and William Kennedy, William Schroeder, and Jackson Collins. Jackson has been a long-time inspiration to always be kind, be funny, and be true to myself and others. Thank you, Lucas Forster, Isaak Ridge and my oldest friend, Nick McGrory. I thank my family for continued support throughout the years. Thank you, Uncle Jim, Aunt Karen, Katy, Jimmy, and Ginny. Thank you, Aunt Peggy and Uncle Bob; it was so good to see you one last time. I thank Jim and Martha McGrory and Jim, Siobhan, Nick and Liam Dizio, my non-blood-related families. Last, thank you, Emmet, Mom and Dad for putting up with me for all these years. I love you all, including Kitz, Cali, Tiria, and Mabel, more than words can express. vii Table of Contents Acknowledgements ..................................................................................................................................... ii List of Tables ............................................................................................................................................... xi List of Figures ........................................................................................................................................... xii Abstract ..................................................................................................................................................... xvi Chapter 1 Introduction .......................................................................................................................... 1 1.1 He Droplets .......................................................................................................................................... 1 1.2 Angular Momentum of Superfluid Droplets ........................................................................................ 3 1.3 Non-Superfluid 3 He Droplets and Mixed 3 He/ 4 He Droplets ................................................................ 6 1.4 Projects and Scientific Contributions ................................................................................................ 10 1.5 References .......................................................................................................................................... 12 Chapter 2 Experiments with He Droplets ......................................................................................... 16 2.1 Droplet Production ............................................................................................................................. 16 2.2 Experimental Apparatus at the LCLS Free Electron Laser ............................................................... 26 2.3 USC Experimental Apparatus ............................................................................................................ 29 2.4 References .......................................................................................................................................... 33 Chapter 3 Optical Imaging ................................................................................................................. 37 3.1 Experimental ...................................................................................................................................... 38 3.2 Results ................................................................................................................................................ 40 3.3 Discussion .......................................................................................................................................... 45 3.4 Conclusions ........................................................................................................................................ 46 3.5 References .......................................................................................................................................... 47 Chapter 4 3 He and 4 He Droplet Size Comparisons ........................................................................... 50 4.1 Introduction ........................................................................................................................................ 50 4.2 Experimental ...................................................................................................................................... 51 4.3 Results ................................................................................................................................................ 53 4.4 Discussion .......................................................................................................................................... 57 4.4.1 Droplet size distribution ............................................................................................................. 57 4.4.2 Droplet aspect ratios ................................................................................................................... 59 4.4.3 Average angular momenta and angular velocities of 3 He and 4 He droplets ............................... 60 4.4.4 Formation of rotating droplets in the fluid jet expansion ........................................................... 63 4.5 Conclusions ........................................................................................................................................ 65 4.6 References .......................................................................................................................................... 67 viii Chapter 5 Angular Momentum in Rotating Superfluid Prolate Droplets ...................................... 72 5.1 Introduction ........................................................................................................................................ 72 5.2 Experimental ...................................................................................................................................... 74 5.3 Results ................................................................................................................................................ 78 5.4 Discussion .......................................................................................................................................... 80 5.4.1 Droplet shape and Vortex Configurations .................................................................................. 80 5.4.2 Kinematic Parameters of the Droplets ........................................................................................ 81 5.4.3 Density Functional Calculations ................................................................................................. 86 5.5 Conclusions ........................................................................................................................................ 88 5.6 References .......................................................................................................................................... 89 Chapter 6 Mixed 3 He/ 4 He Droplets .................................................................................................... 93 6.1 Introduction ........................................................................................................................................ 93 6.2 Experimental ...................................................................................................................................... 95 6.3 Results ................................................................................................................................................ 96 6.3.1 Density Reconstructions from Diffraction Patterns .................................................................... 96 6.3.2 Determining 3 He Content of Droplets via Time-of-Flight Spectra ........................................... 100 6.4 Discussion ........................................................................................................................................ 103 6.4.1 Comparison Between Theory and Experiment ......................................................................... 103 6.4.2 Density Reconstruction via the Error Reduction Algorithm. ................................................... 108 6.4.3 Modeling Experimental Results ............................................................................................... 111 6.5 Conclusions ...................................................................................................................................... 116 Chapter 7 Laser-induced Reconstruction of Ag Clusters in Helium Droplets ............................. 122 7.1 Introduction ...................................................................................................................................... 122 7.2 Experimental .................................................................................................................................... 124 7.3 Results .............................................................................................................................................. 128 7.4 Discussion ........................................................................................................................................ 130 7.4.1 Absorption Cross Sections ........................................................................................................ 131 7.4.2 Heat Transfer in Superfluid Ag Droplets .................................................................................. 134 7.4.3 Absorption Bleaching ad Reconstruction of the Ag Clusters Upon Laser Irradiation ............. 137 7.5 Conclusions ...................................................................................................................................... 138 7.6 References ........................................................................................................................................ 139 Chapter 8 Magnetic Circular Dichroism Spectroscopy in He Nanodroplets ............................... 144 8.1 Motivation ........................................................................................................................................ 144 8.2 Experimental .................................................................................................................................... 147 8.3 Future Directions ............................................................................................................................. 150 ix 8.4 References ........................................................................................................................................ 151 Chapter 9 Conclusions and Outlooks ............................................................................................... 154 9.1 Summary .......................................................................................................................................... 154 9.2 Future Outlooks ............................................................................................................................... 158 9.3 References ........................................................................................................................................ 159 Appendix 1: Pickup Cell Design ............................................................................................................. 160 10.1 Beam Deflection via Anisotropic Capture of Atoms ..................................................................... 161 10.2 Radiative Heat Loss Estimates ...................................................................................................... 162 Appendix 2: Overview of 3 He Recycling System ................................................................................... 167 11.1 Collection of Gas ........................................................................................................................... 169 11.2 Purification of Gas ......................................................................................................................... 169 11.3 Pressurization of Gas ..................................................................................................................... 171 11.4 Filling the System .......................................................................................................................... 171 11.5 Storage of Gas ................................................................................................................................ 172 11.6 Operation Parameters and System Components ............................................................................ 172 11.7 Leak Testing the Recycling System at SLAC ............................................................................... 173 11.8 References ...................................................................................................................................... 174 Appendix 3: 3 He Recycling System Operation Manual ....................................................................... 176 12.1 System Introduction ....................................................................................................................... 177 12.2 Evacuating the System ................................................................................................................... 179 12.2.1 Purpose ................................................................................................................................... 179 12.2.2 Introduction ............................................................................................................................. 179 12.2.3 Procedure ................................................................................................................................ 179 12.3 Filling the System .......................................................................................................................... 181 12.3.1 Purpose ................................................................................................................................... 181 12.3.2 Introduction ............................................................................................................................. 181 12.3.3 Procedure ................................................................................................................................ 181 12.4 Normal Recycling Operation ......................................................................................................... 183 12.5 Purpose .......................................................................................................................................... 183 12.6 Introduction .................................................................................................................................... 183 12.7 Normal Run Procedure .................................................................................................................. 183 12.8 Gas Mixing .................................................................................................................................... 183 12.9 Method 1 ........................................................................................................................................ 184 12.10 Method 2 ...................................................................................................................................... 184 12.11 Stopping Recycling and Filling Cylinders ................................................................................... 186 x 12.11.1 Purpose ................................................................................................................................. 186 12.11.2 Introduction ........................................................................................................................... 186 12.11.3 Procedure .............................................................................................................................. 186 12.12 Index ............................................................................................................................................ 188 12.12.1 List of Valves ........................................................................................................................ 188 12.12.2 Compressor Notes ................................................................................................................. 190 12.12.3 Introduction to Priming the Compressor Heads ................................................................... 190 12.12.4 Procedure: ............................................................................................................................. 192 12.12.5 Troubleshooting the Compressor .......................................................................................... 194 xi List of Tables Table 4.1. Nozzle temperatures and corresponding numbers of recorded diffraction images for the results presented in Figure 2 (a), (b) of the main text. All measurements were performed at nozzle stagnation pressure of P0 = 20 bar. ....................................................................................................................................................................... 55 Table 5.1. Kinematic parameters for the droplets in Figure 5.3. ................................................................................. 83 Table 6.1. The size of the major and minor axes, orientation of the major axis with respect to the x-axis of the image, and the X values of the droplets displayed in Figure 6.3. .................................................................................. 103 Table 7.1. Initial He droplet size, size of the obtained Ag clusters, and overall absorption cross section per Ag atom at 355 nm and 532 nm excitations. The values obtained with CW excitation at 532 nm are given in parenthesis. ........................................................................................................................................................................... 126 xii List of Figures Figure 1.1. The stability diagram for classical droplets (red line) shown as reduced angular velocity, Ω, as a function of the reduced angular momentum, Λ. The blue curve shows the solutions for prolate droplets that only acquire angular momentum through capillary waves [29]. The dashed curve indicates the unstable oblate branch. Pictograms from [18] illustrate droplet shapes for RBR rotation at different values of Λ. ................................... 4 Figure 1.2. The phase diagram for mixed 3 He/ 4 He liquid. The content of the 3 He atoms is defined as X = N3/(N3 + N4). The hatched region shows the range of forbidden X as a function of T. The figure is adapted from Reference [21]. ........................................................................................................................................................................ 8 Figure 1.3. Panel a) shows solutions of the Navier-Stokes equations for classical, rotating two-phase droplets [39]. Panel b) shows results from DFT calculations of mixed 3 He/ 4 He droplets (X = 0.8) [40]. All images show cross- sections of the droplets. For each configuration, the outer shell corresponds to the 3 He phase, while the inner core illustrates the 4 He-rich phase. The top left droplet is at rest, with both the core and shell having spherical shapes. The top middle droplet’s shell is an oblate quasi-spheroid, with the spherical 4 He core devoid of vortices. The top right image shows the results at the same Λ as the middle, but with the 4 He phase containing one quantum vortex. The panels in the bottom show the following vortex-free configurations, from left to right: a prolate shell with a peanut-shaped 4 He phase, an oblate shell with a “3-lobed” 4 He phase, and a peanut-shaped shell with a fissioned 4 He core. The reduced angular momentum for each case is given above its respective droplet. The size bar for the DFT results is given on the right side of Panel b). ............................................................................. 10 Figure 2.1. Attachment of the nozzle base to the second stage of the cryocooler. a) Second stage of the cryocooler with anchored electrical wires for the temperature sensors and the resistive heating. Behind the cryocooler are the aluminum shields, which are attached to the cryocooler during normal operation. b) Exploded view of the “Göttingen-style” nozzle assembly. In the lower left corner is a photograph of the actual nozzle assembly. The design of the nozzle assembly is adapted from Toennies, et al. [8-10]. The figure is from Tanyag et al. [2]. ... 18 Figure 2.2. P-T phase diagrams for (a) 4 He and (b) 3 He. The diagrams are based on refs. [21, 22] for 4 He and [23] for 3 He. The pink SVP curves mark the saturated vapor pressure boundaries. ......................................................... 21 Figure 2.3. A schematic diagram of the 3 He recycling system. The black lines indicate tubing connections between valves and other parts of the system. Blue arrows indicate the direction of helium flow during operation. The system can be evacuated before operation to preserve the purity of the gas. Connections to external vacuum pumps are indicated by labels “To Vacuum Pump.” ........................................................................................... 24 Figure 2.4. Chamber pressures as a function of temperature for an alignment optimized at 7 K. The black squares show the source pressure, the red circles indicate the pressures in the first pickup chamber, the blue triangles indicate the pressures in the second pickup chamber, and the green stars indicate the pressures in the QMS (detection) chamber. ............................................................................................................................................ 26 Figure 2.5. The nominal turbo-molecular pumping speeds in each vacuum chamber at the AMO hutch of the LCLS free electron laser facility are given in L/s, and the diameters of the skimmers and the apertures are given in mm. Labels used in the figure: NZ is the cold nozzle; SK1 and SK2 are the skimmers; PC is the pickup chamber with Xe inlet and pressure sensor tubes; PC AP1 and PC AP2 are the pickup cell apertures; IP is the interaction point with the X-ray beam, which enters perpendicular to the plane of the drawing; TOF is the time of flight mass spectrometer; AP3 and AP4 are the downstream apertures; FLAG is the beam shutter; and QMS is the quadrupole mass spectrometer. ............................................................................................................................ 27 Figure 2.6. a) Schematic diagram and b) photograph of the helium droplet beam vacuum apparatus at USC. CH: cold head; NZ: nozzle; SK: skimmer; PC1 and PC2: resistively heatable pickup cells; A1 and A2: 5 and 6 mm diameter apertures, respectively; BB: beam block; GV: gate valve; QMS: quadrupole mass spectrometer; W: Window. The green glow in each chamber results from the scattering of a 532 nm laser as it propagates from the detection chamber to the source chamber. ........................................................................................................... 31 Figure 3.1. Bird’s-eye view of the observation chamber that houses the jet imaging setup. Liquid helium expands in vacuum through the nozzle attached to the second stage of the Sumitomo RDK-408D cryocooler. A cell filled xiii with Coumarin dye is pumped by the third harmonic of an Nd:YAG laser. The fluorescence from the dye is focused by using a lens and illuminates the droplets. The images of the droplet beam are collected with a Navitar ZOOM-6000 microscope attached on a commercial Canon Rebel T1i camera body. ........................................ 39 Figure 3.2. Photographs of a helium droplet beam produced at T0 = 3.5 K and P0 = 20 bars and obtained at different distances from the nozzle. The distance is given with respect to the edge of the nozzle cap on the right of Panel (a). The insets [(i) and (ii)] are zoomed-in portions of the droplet beam. The white dashed circle in (i) indicates the breakup point. ................................................................................................................................................ 41 Figure 3.3. Photographs of liquid helium jets produced at T0 = 3.5 K and P0 = 3 bars, 20 bars, and 60 bars in Panels (a), (b), and (c), respectively. Large gray and black spots consistent in all Panels are artifacts of the imaging setup. .................................................................................................................................................................... 43 Figure 3.4. Photographs of helium droplet beam expansion produced at P0 = 60 bars and T0 = 3.5 K, 4.5 K, and 5.2 K in Panels (a), (b), and (c), respectively. ............................................................................................................... 45 Figure 4.1. 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 AR are displayed at the top of each image. .................................................................................................. 54 Figure 4.2. Droplet size (a) and aspect ratio (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 the ease of comparison, as the total number of diffraction images obtained for 3 He and 4 He were ~900 and ~300, respectively (see Table 4.1). 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 in Panel (c) are shown by stars. The blue line in Panel (c) represents a linear fit of the data points (blue squares) for 3 He droplets. .......................................................................................... 57 Figure 4.3. Red curve: Calculated aspect ratio as a function of reduced angular momentum (Λ) for axially symmetric oblate droplet shapes. Blue curve: stability diagram of rotating droplets in terms of reduced angular velocity (Ω) and reduced angular momentum (Λ). 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 (Ω, Λ) values corresponding to 3 He and 4 He droplets. ............................................................................................................... 63 Figure 5.1. Stability diagram for classical droplets executing RBR. Red solid traces indicate stable shapes for specific combinations of reduced angular velocity ΩRBR and reduced angular momentum ΛRBR [4, 5]. The left branch corresponds to oblate axisymmetric shapes, the right branch to prolate shapes. The unstable portion of the branch is indicated by a dashed curve. The blue curve corresponds to droplets rotating solely through capillary wave motion [6]. Black circles mark the locations of classical droplets with the same aspect ratios as the superfluid droplets studied in this work. ............................................................................................................................... 73 Figure 5.2. (a) Droplet shape reconstruction from inverse Fourier Transform of the square root of the diffraction image in Panel (b). The result of the IFT is shown in a linear color scale. There are two contours of similar intensity with a superimposed red line in between, which marks the droplet contour. The signals inside the droplet are caused by a positive offset, Bragg peaks, and other features of the original diffraction as discussed in the text. ............................................................................................................................................................................. 77 Figure 5.3. Diffraction patterns from Xe-doped droplets with various shapes: (a1) axisymmetric, nearly spherical, (b1) triaxial pseudo-ellipsoidal, and (c1) capsule shaped. The horizontal stripe in (a1)–(c1) results from the gap between the upper and lower detector panels. Panels (a2)–((c2) show column densities retrieved via the DCDI algorithm. The basis vectors of the vortex lattice in (c2) are shown in the upper right corner of the panel. ...... 79 Figure 5.4. Calculated equilibrium density profiles of a deformable 4 He cylinder rotating around its symmetry axis at fixed LSF/NHe = 7.83ℏ for different numbers of vortices. Streamlines are shown in black. The positions of the vortex cores are marked by red dots for visualization. The color bar shows the density in units of Å −3 . ........... 87 Figure 6.1. The phase diagram of 3 He/ 4 He mixtures using data from Chaudhry et al. [4] and Qin et al. [3]. The concentration of 3 He is given as X = N3/(N3+N4). Arrows indicate possible trajectories droplets may take upon evaporative cooling in vacuum. The arrows are tilted to indicate predominant evaporation of 3 He atoms. ....... 94 Figure 6.2. Representative diffraction patterns of mixed 3 He/ 4 He droplets and corresponding traces along the major and minor axes of the diffraction. Panels (a1) and (a2) show the central (400×400 pixels) portion of the diffraction patterns. Continuous traces in Panels b1)-b2) and c1)-c2) show the radial intensity along the xiv corresponding wedges in the upper Panels, which were scaled as described in the text. Dotted traces are calculations based on the reconstructed density, see in the text. The abscissas are in pixel units. One pixel corresponds to the change of X-ray wavevector by 7.76×10 -4 nm -1 . ................................................................... 97 Figure 6.3. Density reconstruction of mixed droplets shown in a linear color scale. Panels i), ii) are reconstructed from the diffraction images in Figure 6.2 (a1) and (a2), respectively. The distance between each tick mark is 100 nm. ............................................................................................................................................................................. 99 Figure 6.4. A pictogram of inner droplet shapes seen in this work. The inner droplets are shown as spheres with a linear color scaling. The outline of the outer droplet is given in black and is based on the droplet in Figure 3 i). ........................................................................................................................................................................... 100 Figure 6.5. Panel a) shows the TOF traces for weak events: one recorded ~4 seconds after the event for Droplet 1 (black trace) and the other recorded ~10 s before the event for Droplet 2 (red trace) in the main text. The peaks in the TOF are indicated with an inverted blue triangle, and their assignments are given with black text. Panel b) shows a graph of time of flight versus the square root of mass per charge used to verify peak assignments in the TOF spectrum shown in a). The blue line shows the linear fit to the data points. ............................................ 102 Figure 6.6. Left column: Reconstructed 2D density for some representative 3 He droplets of different size. Right column: Corresponding density profiles along the minor (blue) and major axis (red) integrated within 60 o wedges. In the left column the abscissa is given in pixel units with a single pixel corresponding to 7.8 nm. Continuous and dotted curves correspond to the spheroid of constant density scaled to give the observed diffraction intensity and result from the reconstruction, respectively. .............................................................. 111 Figure 6.7. (a1)-3): Densities representing 3 models for the morphology of the 4 He rich phase: two lobes, one lobe, and a torus in Panels (a1), a2) and a3), respectively. b1-3): the diffractions from the models in the top row displayed in a logarithmic color scale. The red and blue wedges represent the areas summed to produce the red and blue traces shown in the bottom of the figure. The traces shown compare the two-lobe model and experiment in c1) and d1), the two-lobe and one-lobe models c2) and d2), and the two-lobe and torus models in c3) and d3). The solid trace marks the two-lobe model, and the dashed traces mark the experimental trace in c1) and d1), the one-lobe model in c2) and d2), and the torus model in c3) and d3). One pixel corresponds to the change of wavevector by 7.76×10 -4 nm -1 . .......................................................................................................................... 114 Figure 6.8. Intensity traces from 2-lobe models using representative density ratios of the 4 He-rich and 3 He-rich phases at T = 0.15 K (black traces) and T = 0.5 K (colored traces). The graphs with the red and blue traces are along the short and long axis of the diffraction, respectively. ........................................................................................... 116 Figure 7.1. (a) Schematic of the He droplet beam vacuum apparatus. NZ—5 µm diameter nozzle, SK—1 mm diameter skimmer, PC1 and PC2—upstream and downstream pickup cells, respectively, SH—beam shutter, A1 and A2— 6 mm diameter apertures, GV1 and GV2—gate valves, EI—electron impact ionizer, IB—ion bender, QMS— quadruple mass spectrometer. A pyroelectric detector, PD, was attached to SH. (b) Cross section of the oven inside of the pickup cell: CW—cold water jacket, CR—alumina ceramic crucible, RS—radiation shield, TF— tungsten filament, Ag—metallic silver. (c) Typical depletion dip upon laser excitation at t = 0, as measured at mass M = 8 at T0 = 8 K upon 532 nm excitation. Weak secondary pulse at about t = 7 ms shows the effect of laser pulse heating of the nozzle. ....................................................................................................................... 124 Figure 7.2. Depletion of the M=8 signal with laser fluence obtained for droplets with initial average size of <NAg> = 1.13 × 10 7 , 2.1 × 10 5 , 6700, 1200 produced at T0 = 5.5, 6, 7, and 9 K used to grow Ag clusters of <NAg> = 1.13 × 10 7 , 2.1 × 10 5 , 6700, and 1200 atoms, respectively. Pulsed laser excitation is at 532 nm and 355 nm in a) and b), respectively. .................................................................................................................................................. 129 Figure 7.3. Depletion in M=8 signal with laser irradiance for different Ag cluster sizes as measured upon continuous laser excitation at 532 nm. ................................................................................................................................. 130 Figure 8.1. Energy level diagrams (left) and sample illustrations of MCD spectra (right). For the energy level diagrams, the left side shows the energy states in the absence of a magnetic field, and the right side of the energy diagrams shows how the degenerate levels split in the presence of a magnetic field. The |A⟩ state denotes the ground state while the |J⟩ and |K⟩ states denote excited states. The RCP and LCP absorptions are illustrated as red and blue arrows, respectively. On the right side of the figure, the RCP and LCP signals are shown as red and blue dotted lines, respectively. Their difference gives rise to the black MCD spectra. In i), there is no magnetic field applied, so there is no MCD signal. Panel ii) shows A-term MCD signal in which the excited |J⟩ state splits. B-term MCD signal is shown in iii), in which the |J⟩ and |K⟩ states mix due to being close in energy. Lastly, xv Panel iv) shows the C-term absorption in which the ground level splits and there is a greater population in the lower level, giving rise to more LCP than RCP signal. ..................................................................................... 146 Figure 8.2. Panel a) left: a sketch of the crucible halves in their tungsten heater. Right: a sketch of the two crucible halves showing the opening in the top half to accommodate the He beam. Panel b): The evaporative source inside the vacuum chamber. The He beam traverses from left to right. The red and yellow wire are leads to a type-K thermocouple. The insulation on the leads was only rated to ~900° C, so the insulation was stripped close to the cell. The copper rods connect to a feedthrough mounted to the top of the chamber. There is a stainless-steel plate with a central hole to accommodate the beam shown on the left side of the image. Another such plate is mounted at the right side of the chamber but is not in view of this picture. ..................................................................... 148 Figure 8.3. Top: manufactured holder with 6 magnets inside. Bottom: magnetic field along the axis [21]. ............ 150 Figure 10.1. A plot of vapor pressures of metals versus temperature. Al has10 -8 mm Hg at ~900 K, and Fe is 3 lines to the right of Al. ................................................................................................................................................ 161 Figure 10.2. A sketch of the final heater design. The heater and crucible are suspended in the cup by the feedthroughs attached to the top of the vacuum chamber (omitted in the drawing). The height and tilt of the heat shield can be adjusted via nuts that rest on a stainless-steel plate that spans the chamber. Originally, we planned to use hex nuts, but we opted for knurled knobs (McMaster-Carr part number 91833A112). The heat shield was custom- built by the machine shop. ................................................................................................................................. 164 Figure 10.3. The heat shield assembly with crucible enclosed. The temperature of the crucible is ~1500 K. ......... 166 Figure 11.1. A color-coded schematic of the manifold pictured in Figure 11.2. Blue arrows indicate the direction of helium flow during operation. Blue valves denote connections between parts of the manifold, while green valves denote connections between the gas manifold and other parts of the recycling system. The black lines indicate tubing connections on the board. The dotted green lines illustrate recycling system connections that are off the board. Purple valves and lines denote connections to storage cylinders that are mounted to the manifold. Orange valves connect the gas manifold to the vacuum pump used to evacuate the recycling system. The orange lines denote tubing that is under vacuum while the evacuation pump is turned on. Red valves are currently unused. Pressure gauges, shown as PG, are installed on the lecture bottles and after the LN2 traps. The pressure reducers similarly have pressure gauges, the input being indicated as H and the output indicated as L. ........................ 168 Figure 11.2. A picture of the recycling system compressor (left), LN2 trap (middle) and gas manifold (see Figure S1 for schematic) at the AMO hutch at LCLS in 2017. .......................................................................................... 169 Figure 11.3. Cross-sectional view of the cyrogenic trap. It is filled with zeolite pellets (MDC part number 500003) and immersed in LN2. The filter (Swagelok SS-4F-VCR-2) is used to avoid dust contamination from zeolite in other parts of the recycling system. ................................................................................................................... 170 Figure 11.4. A picture of me in front of the recycling manifold and compressor. Taken at the AMO hutch at LCLS, SLAC, October 2018. ........................................................................................................................................ 174 Figure 12.1. A color-coded schematic of the manifold, copied from Figure 11.1. Blue arrows indicate the direction of helium flow during operation. Blue valves denote connections between parts of the manifold, while green valves denote connections between the gas manifold and other parts of the recycling system. The black lines indicate tubing connections on the board. The dotted green lines illustrate recycling system connections that are off the board. Purple valves and lines denote connections to storage cylinders that are mounted to the manifold. Orange valves connect the gas manifold to the vacuum pump used to evacuate the recycling system. The orange lines denote tubing that is under vacuum while the evacuation pump is turned on. Red valves are currently unused. Pressure gauges, shown as PG, are installed on the lecture bottles and after the LN2 traps. The pressure reducers similarly have pressure gauges, the input being indicated as H and the output indicated as L. ........................ 178 xvi Abstract Helium nanodroplets have been used as spectroscopic matrices for the last few decades. These unique systems can also shed light on more fundamental questions about quantum mechanics and superfluidity. This thesis features the results from several experiments that explored helium droplets as a vehicle to study molecules and clusters, and as a unit of study itself. The following contains excerpts from previously published abstracts [1-4]. The formation of large droplets upon expansion of supercritical fluid remains poorly understood. The phenomenon of liquid jets disintegrating into droplets has attracted the attention of researchers for more than 200 years. Most of these studies considered classical viscous liquid jets issuing into ambient atmospheric gases, such as air. Here, optical shadowgraphy was applied to study the disintegration of a cryogenic liquid 4 He jet produced with a 5 µm diameter nozzle into vacuum. The physical properties of liquid helium, such as its density, surface tension, and viscosity, change dramatically as the fluid moves through the nozzle and evaporatively cools in vacuum, eventually reaching the superfluid state. This study demonstrates that, at different stagnation pressures and temperatures, droplet formation may involve spraying, capillary breakup, jet branching, and/or flashing and cavitation. The average droplet sizes reported in this optical imaging experiment range from 3.4 × 10 12 to 6.5 × 10 12 helium atoms, equating to 6.7–8.3 µm in diameter. Large 4 He droplets have been employed by different groups for at least a decade, but similar 3 He droplets remain to be studied, due in part to the significant cost of 3 He gas. Previous single- pulse extreme ultraviolet and X-ray diffraction studies revealed that superfluid 4 He droplets obtained in a free jet expansion acquire sizable angular momentum, resulting in significant centrifugal distortion. Similar experiments with normal fluid 3 He droplets may help elucidate the xvii origin of the large degree of rotational excitation and highlight similarities and differences of dynamics in normal and superfluid droplets. Another experiment presented in this thesis compares the shapes of isolated 3 He and 4 He droplets following expansion of the corresponding fluids in vacuum at temperatures as low as ∼2 K. Large 3 He and 4 He droplets with average radii of ∼160 and ∼350 nm, respectively, were produced. It is found that most of the shapes of 3 He droplets in the beam correspond to rotating oblate spheroids, in agreement with previous observations for 4 He droplets. The aspect ratio of the droplets is related to the degree of their rotational excitation, which is discussed in terms of reduced angular momenta (Λ) and reduced angular velocities (Ω), the average values of which are found to be similar in both isotopes. This similarity suggests that comparable mechanisms induce rotation regardless of the isotope. It is hypothesized that the observed distribution of droplet sizes and angular momenta originate from processes in the dense region close to the nozzle, where a significant velocity spread and frequent collisions between droplets induce excessive rotation followed by droplet fission. The study of rotation in bulk superfluids is often complicated by interaction with container walls, which could not be well characterized. Helium nanodroplets offer unique opportunities to study quantum systems as they freely rotate in vacuum. The angular momentum of rotating superfluid droplets originates from quantized vortices and capillary waves, the interplay among which remains to be uncovered. Here, the rotation of isolated sub-micrometer superfluid 4 He droplets is studied by ultrafast X-ray diffraction using a free electron laser. The diffraction patterns provide simultaneous access to the morphology of the droplets and the vortex arrays they host. In capsule-shaped droplets, vortices form a distorted triangular lattice, whereas they arrange along elliptical contours in ellipsoidal droplets. The combined action of vortices and xviii capillary waves results in droplet shapes close to those of classical droplets rotating with the same angular velocity. The findings are corroborated by density functional theory calculations describing the velocity fields and shape deformations of a rotating superfluid cylinder. Mixed 3 He/ 4 He droplets present another interesting rotational system. It is well known that a mixture of 3 He and 4 He separates into discrete phases below ~0.9 K. While this phase separation has been characterized in the bulk, so far, the phase separation studies were limited to small quiescent droplets of few nm in diameter. In fact, no experimental studies of rotation in classical or quantum multiphasic droplets could be found. Herein, large mixed droplets of a few hundred nm in diameter were imaged via X-rays from a free electron laser, similar to the previous 3 He and 4 He studies. It was found that droplets containing equal amounts of 3 He and 4 He did not show any signs of phase separation during the ~4 ms time of flight from the nozzle to the point of interaction with the X-rays. However, droplets containing ~75% 3 He and 25% 4 He did undergo phase separation. In these phase-separated droplets, a variety of configurations were seen: i) a centrally located inner droplet of 4 He, ii) an off-center inner 4 He droplet, and iii) two off-center inner 4 He droplets. In each case, these inner 4 He-rich droplets were encased in an outer 3 He droplet. These results are compared to classical and density functional theory calculations of general multiphasic and 3 He/ 4 He droplets, respectively. Heat transfer from embedded species to He droplets is still not fully understood. In another experiment in this thesis, silver clusters were assembled in helium droplets of different sizes ranging from 10 5 to 10 10 atoms. The absorption of the clusters was studied upon laser irradiation at 355 nm and 532 nm, which is close to the plasmon resonance maximum in spherical Ag clusters and in the range of the absorption of the ramified Ag clusters, respectively. The absorption of the pulsed (7 ns) radiation at 532 nm shows some pronounced saturation effects that are absent upon xix continuous irradiation. This phenomenon has been discussed in terms of the melting of the complex Ag clusters at high laser fluence, resulting in a loss of the 532 nm absorption. Estimates of the heat transfer also indicate that a bubble may be formed around the hot cluster at high fluences, which may result in ejection of the cluster from the droplet, or disintegration of the droplet entirely. This thesis also contains plans for a magnetic circular dichroism (MCD) experiment that was interrupted due to the COVID-19 pandemic. Signals derived from MCD spectroscopy generally investigate the energy difference in degenerate states that split in the presence of a magnetic field. While there are several ways for these states to split, the one most advantageous for study using He droplets involves a ground state splitting that is directly proportional to the magnetic field strength and inversely proportional to temperature. While no data was taken, the pickup cell containing zinc phthalocyanine, a common MCD analyte, was aligned and the system was set up for standard depletion absorption spectroscopy measurements. If these standard measurements would have been completed, a magnet tube with a field that rapidly switches between ±0.4 T would have been inserted to perform MCD experiments using zinc phthalocyanine. Upon successful proof-of-concept experiments, the MCD setup may then be adapted to studies of Fe clusters. [1] L. F. Gomez, S. M. O. O’Connell, C. F. Jones, J. Kwok, and A. F. Vilesov, Laser- Induced Reconstruction of Ag Clusters in Helium Droplets. The Journal of Chemical Physics 145, 114304 (2016). [2] S. M. O. O'Connell et al., Angular Momentum in Rotating Superfluid Droplets. Physical Review Letters 124, 215301 (2020). xx [3] R. M. P. Tanyag, A. J. Feinberg, S. M. O. O’Connell, and A. F. Vilesov, Disintegration of Diminutive Liquid Helium Jets in Vacuum. The Journal of Chemical Physics 152, 234306 (2020). [4] D. Verma et al., Shapes of Rotating Normal Fluid 3 He Versus Superfluid 4 He Droplets in Molecular Beams. Physical Review B 102, 014504 (2020). 1 Introduction He Droplets Helium is a unique substance in that it remains liquid down to absolute zero. Furthermore, 4 He, by far the most abundant isotope of helium, becomes superfluid below 2.17 K [1, 2]. A superfluid is a unique state of matter that can be characterized by a macroscopic wavefunction and has several interesting properties including zero viscosity and quantized vortices. Superfluidity in 4 He was discovered in 1938 [3, 4]. Since its discovery, studies of bulk superfluid helium have been prolific in the low temperature physics community, investigating its thermodynamic, rotational, and other basic properties [2, 5-8]. More recently, droplets of superfluid helium have become the focus of many physical chemistry research groups. These droplets range in number of atoms from NHe ~2,000 to 10 12 , and corresponding radii, RD, from ~3 nm to ~3 µm [9]. The droplets are created by expansion of cold helium gas or fluid into vacuum, where they cool evaporatively to a temperature of 0.37 K [10, 11]. Aggregates can also form inside droplets upon capture of multiple atoms or molecules. Droplets lack viscosity (allowing for free movement of particles inside) and are transparent to light over a wide range between the far infrared and the vacuum ultraviolet [11]. Thus, droplets have proven to be useful spectroscopic matrices and have been employed by many groups [12-14]. Helium droplets are also interesting objects in their own right. In 2012, an experiment was published that focused on the deposition on a substrate of large silver clusters assembled in helium droplets ~1 µm in diameter [15]. The deposits showed elongated clusters, marking the first experimental evidence of vortex-assisted aggregation in the droplets. Subsequently, experimentalists turned to X-ray diffraction as an in-situ technique to study droplets and the 2 vortices therein [11, 16-18]. Results of X-ray diffraction experiments showed string-like clusters, indicating that vortices were present in large droplets and dominated cluster aggregation [17]. These X-ray imaging experiments have recently been reviewed [16]. Previous X-ray [18] and extreme ultraviolet (XUV) [19] experiments were aimed at comparing rotating superfluid droplets to their classical, viscous counterparts. These experiments established that, similar to classical droplets, superfluid droplets exist in two classes: axisymmetric droplets, such as oblate pseudo-spheroids, and tri-axial prolate droplets, such as pseudo-ellipsoids, capsule, and dumbbell shapes. However, these experiments were only able to evaluate the shapes and aspect ratios of droplets, whereas a full kinematic analysis requires simultaneous determination of a droplet’s shape, angular momentum, and angular velocity. Determining the latter two quantities requires observation of the vortex configuration inside a droplet. The work described herein continues X-ray diffraction studies on rotating superfluid droplets and provides the first measurement of angular momenta and angular velocities in droplets of oblate and prolate shapes [20]. This thesis also presents a study of phase separation in mixed 3 He/ 4 He droplets by X-ray diffraction. Isotopic mixtures of 3 He and 4 He are well-characterized in the bulk [21]. It was shown that, below ~1 K, the two isotopes separate into a non-superfluid 3 He phase and a superfluid 4 He- rich phase that contains a small amount of 3 He, as discussed in detail later. The 3 He phase remains in a normal fluid state until < 1 mK. Furthermore, it is less dense than its superfluid counterpart and so was shown to sit atop the 4 He-rich phase in bulk experiments. Bulk studies yield ensemble- averaged results, so non-equilibrium systems are rarely observed [22-24]. However, droplets cool up to 8 orders of magnitude faster than liquid in a dewar [11]. Thus, He droplets provide a unique vessel for phase separation studies. Moreover, phase separation has not yet been studied in rotating 3 systems. In this thesis, X-ray diffraction results are presented that shed light on the topological structure of mixed 3 He/ 4 He rotating droplets for the first time. Angular Momentum of Superfluid Droplets The study of rotation of superfluid droplets is a developing field. In classical, viscous droplets, rotation is achieved through rigid body rotation (RBR), wherein each part of the droplet is at rest in the rotating frame [25-28]. Figure 1.1 shows a plot, often referred to as the droplet stability diagram, of reduced angular velocity, Ω, as a function of the reduced angular momentum, Λ, for classical droplets. The reduced values are unitless and are used to compare droplets of different sizes all on one curve. The equations to find these values are as follows: 𝛬 = ! "#∙%∙&∙' ! ∙𝐿 (1.1) 𝛺 = ' &∙' " #∙% ∙𝜔 (1.2) in which σ is the surface tension of the liquid, ρ is the density of the liquid, R is the radius of a spherical droplet isochoric to the droplet of interest, L is the absolute angular momentum of the droplet, and ω is the absolute angular velocity. A spinning droplet becomes more oblate as its angular momentum increases. However, at Λ ≈ 1.2, the curve bifurcates into an unstable oblate branch (dashed curve) and a stable prolate branch (solid curve). In the prolate branch, droplets take the shapes of triaxial ellipsoids, capsule shapes, and eventually dumbbells before fissioning beyond Λ ≈ 2. The blue curve represents the solutions for droplets containing only quadrupolar capillary wave excitations that deform droplets into prolate, two-lobe shapes [29]. These excitations decay rapidly to RBR in viscous droplets, meaning that the blue curve represents only metastable solutions for classical droplets. The curves in Figure 1.1 represent solutions for rotating droplet shapes that minimize total energy, which is 4 the sum of the kinetic energy of a rotating solid shape and the energy due to surface tension at fixed angular momentum. The pictograms in Figure 1.1 from Bernando et al. [18] illustrate some stable RBR shapes at different Λ. Rigid body rotation is forbidden in superfluids. A superfluid is defined by a wavefunction, which necessarily has a phase component. The velocity at each point is proportional to the gradient of this phase. In RBR, the velocity increases linearly along the radial direction away from the rotational axis. In a superfluid, the RBR velocity distribution would imply a high gradient of phase and would thus correspond to a high kinetic energy. Therefore, in superfluids, angular momentum must only arise through the creation of capillary waves (which are sustained in superfluid droplets lacking viscosity) and/or quantum vortices [20, 29]. Figure 1.1. The stability diagram for classical droplets (red line) shown as reduced angular velocity, Ω, as a function of the reduced angular momentum, Λ. The blue curve shows the solutions for prolate droplets 5 that only acquire angular momentum through capillary waves [29]. The dashed curve indicates the unstable oblate branch. Pictograms from [18] illustrate droplet shapes for RBR rotation at different values of Λ. The shapes of rotating classical and superfluid droplets have been compared in several experimental and theoretical studies [17-19, 29]. The angular momentum or angular velocity of the droplets could not be easily derived from X-ray or XUV experiments studying neat droplets, which deliver only their shapes. Therefore, most experimental studies [19, 30] have tacitly assumed that the RBR stability curve holds for superfluid droplets and that the angular momentum and angular velocity terms are then derived from a classical RBR treatment. The validity of this assumption will be evaluated in this thesis using configurations of quantum vortices to obtain estimates for Ω and Λ in rotating superfluid droplets. Most of the previous experimental studies [17, 18, 31] were focused on oblate droplets, the angular momentum of which is solely due to vortices and make up more than 98% of all droplets obtained from nozzle beam expansion. The prolate branch, however, is different in that angular momentum can be stored in capillary waves and/or quantum vortices [18]. In this work, we quantify the angular momentum of superfluid, rotating droplets and find that their shapes are very close to those of their rotating classical counterparts. Thus, we conjecture that, in prolate superfluid droplets, the angular momentum is partitioned between vortices and capillary waves to minimize the energy of the droplet and to give shapes that are similar to that of classical droplets. The question of how superfluid droplets acquire their angular momentum is still unresolved. One conjecture is that vorticity stems from uneven flow of liquid 4 He in the nozzle [15, 30, 32]. Another explanation may be that droplets are driven along the stability diagram at constant ω from collisions with He atoms in the high-pressure region immediately outside the nozzle [32]. A recent X-ray imaging experiment studied the shapes of superfluid 4 He and viscous 3 He droplets under the same expansion conditions [32]. It was found that the average reduced 6 angular momenta and angular velocities of the two kinds of droplets were very similar. This result implies that the mechanism that delivers angular momentum to the droplets operates independently of the He isotope. Furthering the understanding of rotation in superfluids may help us understand the origin of angular momentum in He droplet experiments, making this nanoscopic tool more powerful. Non-Superfluid 3 He Droplets and Mixed 3 He/ 4 He Droplets Helium-3, 3 He, is also a stable isotope of He, with a nucleus consisting of two protons and one neutron. This configuration means that 3 He is a fermion, while 4 He is a boson. The difference between isotopes is manifested in their critical points (T3 = 3.3 K and P3 = 1.1 bar, T4 = 5.2 K and P4 = 2.3 bar) and liquid number densities (n3 = 1.62 × 10 28 kg m -3 , n4 = 2.18 × 10 28 kg m -3 ). Helium- 3 also undergoes a superfluid transition, but only at T3 ≲ 1 mK [33], orders of magnitude lower than T4 = 2.17 K for 4 He [34]. One property that these two isotopes share is that they are both liquid down to absolute zero. Therefore, 3 He acts as a viscous, normal fluid at temperatures when 4 He is a superfluid, which allows for directs comparisons between viscous fluids and superfluids at temperatures close to absolute zero [32]. Mixed 3 He/ 4 He liquids have been thought of as effective vehicles to study quantum nucleation [35], and many experiments have investigated bulk solutions of 3 He and 4 He [21]. Figure 1.2 illustrates the phase diagram, in which the vertical axis gives the temperature and the horizontal axis the fraction of 3 He atoms in the mixture, X = N3/(N3 + N4). At T >~ 0.9 K, the superfluid and normal phases are miscible. At small X, at T > 0.87 K the mixture is superfluid, and becomes normal fluid at some threshold as X increases, indicated by the lambda line. Pure 4 He has a lambda transition at 2.2 K; however, the temperature of the transition decreases with an increase of X as T ~ (1 - X) 2/3 . This behavior is qualitatively similar to that of an ideal mixture of Bose and Fermi 7 gases [21]. Such a model also predicts miscibility of the two gases down to T = 0 K. However, at T < ~ 0.87 K and X ≈ 0.67, the 3 He/ 4 He mixture separates into two phases: a normal-fluid 3 He phase containing small amounts of 4 He, and a superfluid 4 He-rich phase. As the temperature of the system decreases, so does the content of the minor isotopes in the two phases. At T = 0 K, there is a virtually pure 3 He phase and the superfluid 4 He-rich phase containing ~6% 3 He [21]. The hatched region indicates the “forbidden region” with X-values that have proven elusive in experiments. Some experiments have been able to observe supersaturation of ~1% [22-24] in the 4 He rich phase, but larger amounts of supersaturation predicted by theory have yet to be observed [35]. The mechanism of phase separation has been an intriguing question. Two mechanisms of phase separation have been suggested to date. The first theory is that a 4 He-rich nucleus forms via the coalescence of a critical nucleus by quantum mechanical tunneling [35, 36]. According to the other theory, the cores of 4 He quantum vortices serve as nucleation centers for 3 He phase [37, 38]. In bulk experiments, the walls of the containers may also serve as nucleation sites. 8 Figure 1.2. The phase diagram for mixed 3 He/ 4 He liquid. The content of the 3 He atoms is defined as X = N 3/(N 3 + N 4). The hatched region shows the range of forbidden X as a function of T. The figure is adapted from Reference [21]. Recently, the shapes of the two-phase classical and quantum 3 He/ 4 He rotating droplets have been investigated via numerical calculations [39, 40]. The results are shown in Figure 1.3. In classical Navier-Stokes-based calculations using densities as well as surface and interphase tensions that re the same as in a 3 He / 4 He phase separated system at T = 0 K [39], it was found that the inner phase took on a similar shape as the outer shell in oblate rotating droplets, as seen in Panel (a) in Figure 1.3. 9 Panel b) of Figure 1.3 displays cross-sections of rotating, mixed 3 He/ 4 He droplets, containing N3 = 6,000 and N4 = 1,500 from density functional theory (DFT) calculations [40]. In droplets, the 4 He-rich phase resides on the inside of the droplet, while the 3 He phase forms a shell around the other phase [40, 41]. The reduced angular momentum for each result is given directly above its respective picture. Droplets without vortices have an oblate 3 He-rich shell and spherical 4 He-rich core. However, droplets with vortices contain a torus of 4 He-rich density. There is also a case in which the 4 He-rich density forms a tri-lobed shape. DFT calculations also show that the inner 4 He phase remains single-lobed until the entire droplet becomes very prolate, or the 4 He core hosts a vortex [40]. When a vortex is present in the 4 He core (as in Panel b), upper right side), the phase takes on a toroidal shape. 10 Figure 1.3. Panel (a) shows solutions of the Navier-Stokes equations for classical, rotating two-phase droplets [39]. Panel b) shows results from DFT calculations of mixed 3 He/ 4 He droplets (X = 0.8) [40]. All images show cross-sections of the droplets. For each configuration, the outer shell corresponds to the 3 He phase, while the inner core illustrates the 4 He-rich phase. The top left droplet is at rest, with both the core and shell having spherical shapes. The top middle droplet’s shell is an oblate quasi-spheroid, with the spherical 4 He core devoid of vortices. The top right image shows the results at the same Λ as the middle, but with the 4 He phase containing one quantum vortex. The panels in the bottom show the following vortex- free configurations, from left to right: a prolate shell with a peanut-shaped 4 He phase, an oblate shell with a “3-lobed” 4 He phase, and a peanut-shaped shell with a fissioned 4 He core. The reduced angular momentum for each case is given above its respective droplet. The size bar for the DFT results is given on the right side of Panel b). Due to the ~34% difference in the number densities between the 3 He rich and 4 He rich phases, the phase separation in the mixed 3 He/ 4 He droplets can be studied by X-ray diffraction experiments as discussed below. The results showed that the mixed droplets contain two discrete phases, with the 4 He-rich phase forming two separate lobes of density within the 3 He shell. The experimental results are at variance with the theoretical results highlighted in Figure 3. Projects and Scientific Contributions In my work at USC, I contributed to two reviews and several papers. The first review is a comprehensive discussion of the 4 He work that the Vilesov lab does at USC and at XFEL facilities. Tanyag, Rico Mayro P., Curtis F. Jones, Charles Bernando, Sean M. O. O’Connell, Deepak Verma, and Andrey F. Vilesov. "Experiments with Large Superfluid Helium Nanodroplets." In Cold Chemistry, (2017) 389-443. The other review reported on progress made in the field of infrared spectroscopy of molecules and clusters in He droplets. Deepak Verma, Rico Mayro P. Tanyag, Sean M. O. O'Connell, and Andrey F. Vilesov. “Infrared spectroscopy in superfluid helium droplets.” Advances in Physics-X 4 (2019). I co-authored the following papers and manuscripts. The results from some of them will be summarized in this thesis.: Sean M. O. O’Connell, Rico Mayro P. Tanyag, Deepak Verma, Charles Bernando, Weiwu Pang, Camila Bacellar, Catherine A. Saladrigas, Johannes Mahl, Benjamin W. Toulson, Yoshiaki Kumagi, 11 Peter Walter, Francesco Ancilotto, Manuel Barranco, Marti Pi, Christoph Bostedt, Oliver Gessner, and Andrey F. Vilesov. “Angular Momentum in Rotating Superfluid Droplets.” Physical Review Letters, 124 (2020), 215301. [20] Rico Mayro P. Tanyag, Alexandra J. Feinberg, Sean M. O. O’Connell, and Andrey F. Vilesov. “Disintegration of Diminutive Liquid Helium Jets in Vacuum”. Journal of Chemical Physics, 152 (2020), 234306. [42] 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 Jäger, 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. [32] Luis F. Gomez, Sean M. O. O'Connell, Curtis F. Jones, Justin Kwok, and Andrey F. Vilesov. (2016). “Laser-induced Reconstruction of Ag Clusters in Helium Droplets”. The Journal of Chemical Physics, 145 (11), 114304. [43] Camila Bacellar, Adam S. Chatterley, Florian Lackner, C. D. Pemmaraju, Rico Mayro P. Tankag, Deepak Verma, Charles Bernando, Sean M. O. O’Connell, Maximilian Bucher, Ken R. Ferguson, Tais Gorkhover, Ryan N. Coffee, Giacomo Coslovich, Dipanwita Ray, Timur Osipov, Daniel M. Neumark, Christoph Bostedt, Andrey F. Vilesov, and Oliver Gessner. “Anisotropic Surface Softening and Core Depletion During the Expansion of a Strong-Field Induced Nanoplasma.” Nature Physics, under review (2021). Alexandra J. Feinberg, Deepak Verma, Sean M.O. O’Connell, Swetha Erukala, Rico M. Tanyag, Weiwu Pang, Catherine A. Saladrigas, Benjamin W. Toulson, Mario Borgwardt, Niranjan Shivaram, Ming-Fu Lin, Andre Al Haddad, Wolfgang Jäger, Christoph Bostedt, Peter Walter, Oliver Gessner and Andrey F. Vilesov. “Aggregation of Atoms in Bosonic Versus Fermionic Helium Nanodroplets.” Nature Communications, submitted (2021). Sean M. O. O’Connell, Deepak Verma, Alexandra J. Feinberg, Swetha Erukala, Catherine A. Saladrigas, Benjamin W. Toulson, Mario Borgwardt, Niranjan Shivaram, Ming-Fu Lin, Andre Al Haddad, Wolfgang Jäger, Christoph Bostedt, Peter Walter, Oliver Gessner and Andrey F. Vilesov. “Phase Separation in Freely Rotating 3 He/ 4 He Nanodroplets.” in preparation. Finally, I describe attempts to perform magnetic circular dichroism studies in He nanodroplets at USC. This project was stalled by the COVID-19 pandemic, and I was unable to complete the measurements. The fundamental ideas of the study are elaborated in the text, while my notes and ideas for moving forward are included in the appendix. This work was supported by the following NSF grants: DMR-1701077 and CHE-1664990. 12 References [1] L. Landau, The Theory of Superfluidity of Helium Ii. Journal of Physics-Ussr 5, 71 (1941). [2] R. P. Feynman, Chapter Ii Application of Quantum Mechanics to Liquid Helium, C. J. Gorter Ed., North-Holland Publishing Company: Amsterdam, 1955: Vol. 2, p. 17. [3] J. F. Allen and A. Misener, Flow of Liquid Helium Ii. Nature 141, 75 (1938). [4] P. Kapitza, Viscosity of Liquid Helium Below the Lambda-Point. Nature 141, 74 (1938). [5] L. Landau, Theory of the Superfluidity of Helium Ii. Physical Review 60, 356 (1941). [6] D. G. Henshaw and A. D. B. Woods, Modes of Atomic Motions in Liquid Helium by Inelastic Scattering of Neutrons. Physical Review 121, 1266 (1961). [7] A. L. Fetter, Vortex Nucleation in Deformed Rotating Cylinders. Journal of Low Temperature Physics 16, 533 (1974). [8] E. L. Andronikashvili, A Direct Observation of Two Kinds of Motion in Helium Ii. J. Phys. USSR 10, (1946). [9] L. F. Gomez, E. Loginov, R. Sliter, and A. F. Vilesov, Sizes of Large He Droplets. Journal of Chemical Physics 135, 154201 (2011). [10] M. Hartmann, R. E. Miller, J. P. Toennies, and A. Vilesov, Rotationally Resolved Spectroscopy of Sf6 in Liquid-Helium Clusters - a Molecular Probe of Cluster Temperature. Physical Review Letters 75, 1566 (1995). [11] R. M. P. Tanyag, C. F. Jones, C. Bernando, D. Verma, S. M. O. O'Connell, and A. F. Vilesov, in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, edited by A. Osterwalder, and O. Dulieu (Royal Society of Chemistry, Cambridge, 2018), p. 389. [12] J. P. Toennies and A. F. Vilesov, Superfluid Helium Droplets: A Uniquely Cold Nanomatrix for Molecules and Molecular Complexes. Angewandte Chemie-International Edition 43, 2622 (2004). 13 [13] M. Y. Choi, G. E. Douberly, T. M. Falconer, W. K. Lewis, C. M. Lindsay, J. M. Merritt, P. L. Stiles, and R. E. Miller, Infrared Spectroscopy of Helium Nanodroplets: Novel Methods for Physics and Chemistry. International Reviews in Physical Chemistry 25, 15 (2006). [14] D. Verma, R. M. P. Tanyag, S. M. O. O'Connell, and A. F. Vilesov, Infrared Spectroscopy in Superfluid Helium Droplets. Advances in Physics-X 4, (2019). [15] L. F. Gomez, E. Loginov, and A. F. Vilesov, Traces of Vortices in Superfluid Helium Droplets. Physical Review Letters 108, 155302 (2012). [16] O. Gessner and A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Droplets. Annual Review of Physical Chemistry 70, 173 (2019). [17] L. F. Gomez et al., Shapes and Vorticities of Superfluid Helium Nanodroplets. Science 345, 906 (2014). [18] C. Bernando et al., Shapes of Rotating Superfluid Helium Nanodroplets. Physical Review B 95, 064510 (2017). [19] B. Langbehn et al., Three-Dimensional Shapes of Spinning Helium Nanodroplets. Physical Review Letters 121, 255301 (2018). [20] S. M. O. O'Connell et al., Angular Momentum in Rotating Superfluid Droplets. Physical Review Letters 124, 215301 (2020). [21] D. O. Edwards and M. S. Pettersen, Lectures on the Properties of Liquid and Solid He-3- He-4 Mixtures at Low-Temperatures. Journal of Low Temperature Physics 87, 473 (1992). [22] V. A. Maidanov, V. A. Mikheev, N. P. Mikhin, N. F. Omelaenko, E. Y. Rudavski, V. K. Chagovets, and G. A. Sheshin, Supersaturation of Superfluid He-3-He-4 Solutions in the Region of Phase Stratification. Fizika Nizkikh Temperatur 18, 943 (1992). [23] V. A. Mikheev, E. Y. Rudavsky, V. K. Chagovets, and G. A. Sheshin, Kinetics of Phase- Separation in Superfluid He-3-He-4 Solutions. Fizika Nizkikh Temperatur 18, 1091 (1992). [24] T. Satoh, M. Morishita, M. Ogata, and S. Katoh, Critical Supersaturation of He-3-He-4 Liquid-Mixtures - Decay of Metastable States at Ultralow Temperatures. Physical Review Letters 69, 335 (1992). 14 [25] S. Chandrasekhar, The Stability of a Rotating Liquid Drop. Proc. Roy. Soc. London A 186, 1 (1965). [26] S. Cohen, F. Plasil, and W. J. Swiatecki, Equilibrium Configurations of Rotating Charged or Gravitating Liquid Masses with Surface-Tension .2. Annals of Physics 82, 557 (1974). [27] R. A. Brown and L. E. Scriven, The Shape and Stability of Rotating Liquid-Drops. Proceedings of the Royal Society of London Series A 371, 331 (1980). [28] S. L. Butler, M. R. Stauffer, G. Sinha, A. Lilly, and R. J. Spiteri, The Shape Distribution of Splash-Form Tektites Predicted by Numerical Simulations of Rotating Fluid Drops. Journal of Fluid Mechanics 667, 358 (2011). [29] F. Ancilotto, M. Barranco, and M. Pi, Spinning Superfluid He-4 Nanodroplets. Physical Review B 97, 184515 (2018). [30] C. Bernando and A. F. Vilesov, Kinematics of the Doped Quantum Vortices in Superfluid Helium Droplets. Journal of Low Temperature Physics 191, 242 (2018). [31] D. Rupp et al., Coherent Diffractive Imaging of Single Helium Nanodroplets with a High Harmonic Generation Source. Nature Communications 8, 493 (2017). [32] D. Verma et al., Shapes of Rotating Normal Fluid 3 He Versus Superfluid 4 He Droplets in Molecular Beams. Physical Review B 102, 014504 (2020). [33] E. R. Dobbs, Helium Three (Oxford University Press, New York, 2000). [34] R. J. Donnelly, Quantized Vortices in Helium Ii (Cambridge University Press, 1991), Vol. 2. [35] M. Barranco, M. Guilleumas, D. M. Jezek, R. J. Lombard, J. Navarro, and M. Pi, Nucleation in Dilute He-3-He-4 Liquid Mixtures at Low Temperatures. Journal of Low Temperature Physics 117, 81 (1999). [36] D. M. Jezek, M. Guilleumas, M. Pi, and M. Barranco, Critical Supersaturation of 3 He- 4 He Liquid Mixtures at Low Temperatures. Physical Review B 51, 11981 (1995). 15 [37] E. A. Pashitskii, V. N. Mal'nev, and R. A. Naryshkin, Vortex Nucleation in the Process of Phase Separation of a Supersaturated He-3-He-4 Mixture. Low Temperature Physics 31, 105 (2005). [38] S. N. Burmistrov and L. B. Dubovskii, Nucleation at Quantized Vortices and the Heterogeneous Phase Separation in Supersaturated Superfluid He-3-He-4 Liquid Mixtures. Low Temperature Physics 44, 985 (2018). [39] S. L. Butler, Equilibrium Shapes of Two-Phase Rotating Fluid Drops with Surface Tension. Physics of Fluids 32, 012115 (2020). [40] F. A. M. Pi, J. M. Escartin, R. Mayol, M. Barranco, Rotating Mixed 3He-4He Nanodroplets. arxiv:2003.03306 (2020). [41] S. Grebenev, B. G. Sartakov, J. P. Toennies, and A. F. Vilesov, The Structure of the Ocs- H-2 Van Der Waals Complex Embedded inside He-4/He-3 Droplets. Journal of Chemical Physics 114, 617 (2001). [42] R. M. P. Tanyag, A. J. Feinberg, S. M. O. O’Connell, and A. F. Vilesov, Disintegration of Diminutive Liquid Helium Jets in Vacuum. The Journal of Chemical Physics 152, 234306 (2020). [43] L. F. Gomez, S. M. O. O’Connell, C. F. Jones, J. Kwok, and A. F. Vilesov, Laser- Induced Reconstruction of Ag Clusters in Helium Droplets. The Journal of Chemical Physics 145, 114304 (2016). 16 Experiments with He Droplets In this section, two He droplet beam sources are described that are based on a closed-cycle refrigerator that was used for measurements with 4 He droplets and a helium flowing cryostat employed to reach lower temperatures required for 3 He. Then, the XFEL-imaging apparatus employed at the Atomic, Molecular, and Optical (AMO) end-station of the Linac Coherent Light Source (LCLS) is detailed, followed by the spectroscopy and optical imaging setup at the University of Southern California (USC). This section is based on the following publications: Luis F. Gomez, Ken R. Ferguson, James P. Cryan, Camila Bacellar, Rico Mayro P. Tanyag, Curtis Jones, Sebastian Schorb, Denis Anielski, Ali Belkacem, Charles Bernando, Rebecca Boll, John Bozek, Sebastian Carron, Gang Chen, Tjark Delmas, Lars Englert, Sascha W. Epp, Benjamin Erk, Lutz Foucar, Robert Hartmann, Alexander Hexemer, Martin Huth, Justin Kwok, Stephen R. Leone, Jonathan H. S. Ma, Filipe R. N. C. Maia, Erik Malmerberg, Stefano Marchesini, Daniel M. Neumark, Billy Poon, James Prell, Daniel Rolles, Benedikt Rudek, Artem Rudenko, Martin Seifrid, Katrin R. Siefermann, Felix P. Sturm, Michele Swiggers, Joachim Ullrich, Fabian Weise, Petrus Zwart, Christoph Bostedt, Oliver Gessner, and Andrey F. Vilesov. “Shapes and Vorticities of Superfluid Helium Nanodroplets”. Science 345.6199 (2014): 906-909. [1] Tanyag, Rico Mayro P., Curtis F. Jones, Charles Bernando, Sean M. O. O’Connell, Deepak Verma, and Andrey F. Vilesov. “Experiments with Large Superfluid Helium Nanodroplets.” In Cold Chemistry, (2017) 389-443. [2] 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 Jäger, 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. [3] Droplet Production Helium droplets are produced in the source chamber from the continuous free jet expansion of research grade helium gas (6.0 purity) through a nozzle (NZ) with a nominal diameter of 5 μm [2, 4]. As the beam travels downstream, it is collimated by a skimmer, SK, which is about 17 10-15 mm downstream from the nozzle. As shown in Figure 2.1 (a) [2], an oxygen-free copper nozzle base is attached directly to the second stage of a Sumitomo SRDK 408 cryocooler, which employs Gifford-McMahon closed-cycle refrigeration [5, 6]. Despite working for more than ten years, one of our cryocoolers can still cool down to T0 ≈ 3.6 K without any maintenance. The cryocooler is installed upright and mounted on a flange with bellows and four threaded fasteners. The cryocooler assembly rests on a Viton O-ring and can glide along the horizontal plane for alignment. This mounting allows for the alignment of the height and tilt of the nozzle in the X-Y- Z directions. The fasteners and alignment hardware can be seen on the left-hand side of Figure 2.6 b). A thin layer of silicone grease is applied between the nozzle base and the cryocooler for better thermal contact. As a consequence of its attachment to the cryocooler, the nozzle experiences periodic displacement by about 0.1 mm, corresponding to the 1.2 Hz cooling strokes [7]. The effect of this oscillation on the beam intensity can be minimized through careful alignment of the nozzle. Helium from a high-pressure tank is maintained at P0 ≈ 20 bar by a mechanical regulator and supplied to the nozzle assembly through a Swagelok feedthrough fitting. The stagnation pressure P0 is a critical experimental parameter, which is measured with a Wika Type-S general purpose pressure transmitter. On the vacuum side, about one meter of a high-pressure stainless- steel gas line (Swagelok 1/8” outer diameter) is wound around and anchored to the first stage of the cryocooler. This construction serves to pre-cool the helium gas down to about 30 K before being cooled further at the nozzle base. Figure 2.1 (a) shows the second stage of the cryocooler with the nozzle assembly and helium gas line attached. The electrical wiring around the second stage connects the silicon diode sensors and the resistive heater to the temperature controller. Aluminum shields, shown behind the cryocooler in Figure 2.1 (a), are attached during normal 18 operation and enclose the second stage and the nozzle assembly to minimize heating through black body radiation and collisions with the residual helium gas in the source chamber. Figure 2.1. Attachment of the nozzle base to the second stage of the cryocooler. (a) Second stage of the cryocooler with anchored electrical wires for the temperature sensors and the resistive heating. Behind the 19 cryocooler are the aluminum shields, which are attached to the cryocooler during normal operation. b) Exploded view of the “Göttingen-style” nozzle assembly. In the lower left corner is a photograph of the actual nozzle assembly. The design of the nozzle assembly is adapted from Toennies, et al. [8-10]. The figure is from Tanyag et al. [2]. The temperature of the nozzle T0 is measured using a silicon diode sensor (Lakeshore DT- 670B-CU) and is regulated by resistive heating, which is controlled via a proportional-integral- derivative (PID) controller (CryoCon 32B). The diode is calibrated to a standard voltage- temperature curve programmed directly into the controller. Like the stagnation pressure, the temperature of the nozzle T0 is a second critical experimental parameter. Accordingly, the temperature sensor should be reliably attached to the copper nozzle base to reflect the temperature of helium before expanding out of the nozzle orifice; see Figure 2.1. The nozzle assembly, as shown in Figure 2.1 b), includes the following major parts: the nozzle base, nozzle tip, and nozzle orifice [8-10]. The nozzle base is made from oxygen-free copper. Inside is a cylindrical reservoir that holds the pressurized helium prior to expansion. The nozzle tip is attached on top of this reservoir, and the junction between them is sealed with an indium gasket. To prevent clogging of the nozzle orifice due to dust particulates, a 0.5 μm sintered filter is placed inside the nozzle tip. Finally, the nozzle tip is secured to the body of the nozzle base with a nozzle tip fixing screw. The nozzle orifice is a commercial 5 μm diameter (95/5% Pt/Ir alloy) electron microscope diaphragm (Plano A0200P), which seats on top of a gold (or copper) gasket at the end of the nozzle tip. These two parts are then secured in place with a nozzle plate cover cap and a nozzle cap. Both under- and over-tightening of the nozzle cap can cause problems while fixing the nozzle orifice in place. Under-tightening will cause leaks in the gap between the nozzle plate cover cap and the nozzle tip, which can be detected by immersing the whole pressurized nozzle assembly into a beaker filled with methanol. A properly tightened nozzle gives an intense fan of bubbles emanating from the nozzle opening with no bubbles escaping from the 20 hole on the body of the nozzle cap; see Figure 2.1 b). On the other hand, if the cap is over-tightened, the orifice will be deformed and can reduce the discharge rate and quality of the helium beam. Therefore, caution must then be taken in how much torque is applied when tightening the nozzle cap in place, by tightening only until no bubbles escape from the hole on the body of the nozzle cap. It is advisable to measure the helium discharge rate drNZ through the nozzle right after installation. For a nominal 5 μm diameter nozzle at P0 = 20 bar and at room temperature, drNZ is usually in the range of 0.3-0.4 cm 3 s -1 ; while at T0 = 4-5 K, drNZ is ~3 cm 3 s -1 . The effective diameter of the nozzle can be estimated at room temperature as: 𝑑 ()) = ' !* # +,- $% . & [𝜇m] (2.1) in which C = 2.249×10 -5 bar s cm -1 [11] and drNZ is a third critical experimental parameter, which should also be periodically checked when the nozzle is in a cold state by monitoring the exhaust of the fore-vacuum pump. Once a noticeable change in the measured flow rate (> 20%) is observed, the nozzle orifice should be replaced, since this change usually indicates partial clogging of the nozzle. However, this is a very rare event, as the nozzle orifice can operate for years without any noticeable deterioration. Helium droplets of different average size can be produced by varying P0 and T0 . In our experiments, we usually kept 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, as illustrated in Figure 2.2 [2-4, 12, 13]. The adiabatic expansion proceeds along isentropes, i.e., lines of constant entropy, as indicated by dashed lines, starting at an initial condition defined by the nozzle temperature, T0, and a stagnation pressure, P0, and ending 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 21 is defined as being subcritical or supercritical (shown in Figure 2.2) [2]. In the Rayleigh regime at T < 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 [14-17]. In vacuum, the temperature of the droplets further decreases via evaporative cooling down to 0.15 K and 0.38 K for 3 He [18] and 4 He [19], respectively. Below 2.17 K, boiling of superfluid 4 He ceases, which may lend further stability to the incipient 4 He droplets. The superfluid transition temperature for 3 He is < 1 mK, which is far below the temperature of the droplets in these experiments. Therefore, 3 He droplets remain normal fluids in these experiments. Due to fast evaporative cooling [2, 20], 4 He droplets become superfluid close to the nozzle; however, the location of the superfluid transition and the kinetics of the droplet cooling upon expansion remain unknown. Figure 2.2. P-T phase diagrams for (a) 4 He and (b) 3 He. The diagrams are based on refs. [21, 22] for 4 He and [23] for 3 He. The pink SVP curves mark the saturated vapor pressure boundaries. Considering the lower critical point of 3 He (TC = 3.3 K, PC = 1.1 atm) compared to that of 4 He (TC = 5.2 K, PC = 2.3 atm), lower nozzle temperatures are required to obtain 3 He droplets of the same sizes as those consisting of 4 He. For example, for a nozzle stagnation pressure of P0 = 20 bar, previous experiments demonstrated that 4 He droplets with an average number of atoms 22 <N4> = 10 7 are produced at a nozzle temperature of T0 = 7 K, [4] while T0 = 5 K is required to obtain 3 He droplets with the same average number of atoms <N3> = 10 7 [24-27]. The temperature difference of 2 K in T0 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. Droplets of 3 He and 4 He are produced at constant P0 = 20 bar and varying T0, 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. Due to the considerable cost of 3 He gas, a recycling system was employed during the 2017 and 2018 SLAC beamtimes that studied 3 He and mixed 3 He/ 4 He droplets. Filling the system requires about 10 L×bar of room temperature 3 He. For comparison, at standard operating conditions (T0 = 3 K, P0 = 20 bar), the flow rate of the He gas is ~3 cm 3 ×bar/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 constantly removed from the 3 He sample by the recycling system. The droplet source was stable over several days, indicating that the purity of the 3 He remained high throughout the experiment. The design of the gas recycling system was inspired by a similar system used for experiments with 3 He droplets [26-28] and for 3 He gas circulation systems in dilution refrigerators [29]. The 3 He recycling system fulfills the following functions: (1) collection of recycled gas, (2) 23 cleaning of recycled gas, (3) pressurization of clean gas, and (4) storage of clean gas. Figure 2.4 shows a schematic diagram of the recycling system. The two inward-facing triangles denote valves. Blue arrows in Figure 2.4 indicate the direction of the helium flow during operation. The gas exiting the cryogenic nozzle is pumped by turbo pumps backed by scroll pumps (Leybold SC 30D and Anest Iwata ISP 250C). An Adixen DFT-25 microfiber-based dust filter is installed at the exit of the scroll pump to stop debris from entering the system. Gas is collected from the output of the scroll pump and impurity gases are frozen out on LN2-cooled zeolite traps. The purified helium gas is compressed by a Fluitron S1-20/150 compressor and resupplied to the cryogenic nozzle. When not in operation, gas can be stored in the cylinders shown in the upper right portion of Figure 2.4. The entire system can be evacuated by vacuum pumps before operation. The corresponding connections are noted in Figure 2.4. A more comprehensive figure, photos, and operating instructions are provided in Appendices 2 and 3. 24 Figure 2.3. A schematic diagram of the 3 He recycling system. The black lines indicate tubing connections between valves and other parts of the system. Blue arrows indicate the direction of helium flow during operation. The system can be evacuated before operation to preserve the purity of the gas. Connections to external vacuum pumps are indicated by labels “To Vacuum Pump.” 25 As a reference for future beam alignments, I include a graph of chamber pressures at different beam conditions for comparison and alignment checks. For the following table, the source chamber is pumped by a ~3000 L⸱s -1 turbo pump, the pickup chambers are each pumped by 900 L⸱s -1 turbo pumps, and the magnetic/differential chamber and QMS chambers are both pumped by a 450 L⸱s -1 turbo pumps. The second pickup chamber and magnetic/differential cambers were connected by the full flange instead of the ~6 mm orifice used in other experiments. The turbo pumps were back by oil-free scroll and screw pumps, but the screw pumps needed to have open ballast ports to effectively pump He gas. The QMS chamber was backed by its own Osaka DSP- 250 screw pump with an open ballast port. For all of the following, P0 = 20 bar. The pressures as read from the controller are given in a logarithmic scale and are plotted versus T0 in Figure 2.4. At low T0 < 5.5 K, the pressures fluctuate, as discussed in previous works [2, 4], so the mean values are given. During the measurements, the alignment of the nozzle was optimized for T0 = 7 K. The alignment at low T around 5 K is difficult to achieve due to high collimation and some pointing instability of the beam [30]. Therefore, the pressure rise in the QMS is low at T0 < 4.5 K. Another noticeable effect is the pressure drop in the source chamber at T0 < 5 K. This drop is related to the substantial transport of He through the skimmer in the form of a collimated droplet beam. 26 Figure 2.4. Chamber pressures as a function of temperature for an alignment optimized at 7 K. The black squares show the source pressure, the red circles indicate the pressures in the first pickup chamber, the blue triangles indicate the pressures in the second pickup chamber, and the green stars indicate the pressures in the QMS (detection) chamber. Experimental Apparatus at the LCLS Free Electron Laser The same source of the droplet beam was used during the experiments at USC and LCLS. The discussion below describes the apparatus used at LCLS, and then details the spectroscopy and optical imaging setup at USC. Figure 2.5 shows a schematic diagram of the vacuum apparatus employed at LCLS in 2012. Similar setups were used for experiments in 2014, 2015, 2017, and 2018, the last three in which I participated. For all of these studies, helium nanodroplets form upon expansion of high purity 3 He 27 or 4 He into vacuum through a nozzle with a 5 μm nominal diameter and a 2 μm nominal channel length, at a temperature of 2-5 K and a backing pressure of 20 bar. Average droplet sizes can be determined by titrating the droplets with Xe atoms [4]. For the 2012 experiment, the central part of the He beam expansion was directed through a 0.5 mm diameter skimmer (SK) and the resulting droplet beam was doped with Xe atoms in a 10 cm long pickup cell (PC) placed 15 cm from the droplet source. A similar gas cell was used in later experiments, but was positioned further away from the source. The degree of Xe doping was controlled by varying the Xe pressure in the pickup cell, which was measured in absolute units by a membrane manometer. Figure 2.5. The nominal turbo-molecular pumping speeds in each vacuum chamber at the AMO hutch of the LCLS free electron laser facility are given in L/s, and the diameters of the skimmers and the apertures are given in mm. Labels used in the figure: NZ is the cold nozzle; SK1 and SK2 are the skimmers; PC is the pickup chamber with Xe inlet and pressure sensor tubes; PC AP1 and PC AP2 are the pickup cell apertures; IP is the interaction point with the X-ray beam, which enters perpendicular to the plane of the drawing; TOF is the time of flight mass spectrometer; AP3 and AP4 are the downstream apertures; FLAG is the beam shutter; and QMS is the quadrupole mass spectrometer. The droplet beam then crossed the ~20 µm 2 focus of the X-ray beam from the free-electron laser (FEL) at the interaction point (IP) 640 mm away from the He nozzle. The XFEL specifications for the various experiments are given in their respective chambers. During most of 28 the experiments, the XFEL was operated at hυ = 1.5 keV (λ= 0.827 nm) and delivered approximately 10 12 photons per ~100 fs pulse at a repetition rate of 120 Hz. Early experiments were performed using the CFEL-ASG Multi-Purpose (CAMP) instrument at the Atomic, Molecular, and Optical Science (AMO) beamline of the Linac Coherent Light Source (LCLS); for details, see Refs. [31-33]. Later experiments employed the LAMP chamber [34-36]. Scattered X- rays were detected at small scattering angles (< 0.05 rad) using a cooled pnCCD detector with approximately 10 6 pixels (75×75 μm 2 /pixel). The detector consists of two panels (75 × 37 mm 2 each) separated by about 2 mm and placed perpendicular to the X-ray beam at distances of 564 mm and 567 mm behind the scattering center (see Figure S2). Both panels have semicircular cuts, in the case of early experiments, or rectangular cuts, in later experiments, to accommodate the primary X-ray beam, which crossed the detector approximately in its geometric center. One detected X-ray photon results in ~32 units of intensity, as can be figured out using the following formula: 𝐼 /01 = 1270𝑎𝐸 (2.2) in which Isig is the intensity in ADU (detector units), E is the energy in keV, and a is the amplification, which can range from 1 to 1/128. The probability for a droplet to reside in the FEL focal volume during a laser pulse was estimated to be less than ~10 -3 , using the measured droplet flux [3, 4]. This estimate agrees with the observed hit rate. Therefore, the probability of finding two droplets in the detection volume is negligible (≤ 10 -6 ) and each diffraction image originates from a single droplet irradiated by a single FEL shot. In addition to scattering from He and Xe atoms, many X-ray photons within a single FEL shot are absorbed, leading to ionization and, ultimately, disintegration of the droplets via Coulomb explosion. The increase in the radius of a droplet due to Coulomb explosion [37] during 29 an FEL pulse was estimated to be less than 1%, which is within the uncertainty of droplet size determinations in this work. For each hit, the resulting ion distribution was measured using a time- of-flight (TOF) mass spectrometer, confirming the purity of the droplets. Therefore, the obtained diffraction images represent the shapes of intact, single He droplets in the beam. After passing through the scattering chamber, the droplet beam entered the beam dump vacuum chamber, where the beam intensity was monitored via the partial He pressure, PHe. The average number of Xe atoms captured per He droplet, 〈𝑁 2( 〉, as well as the average droplet size before doping, 〈𝑁 3( 〉, were estimated by measuring the attenuation of the droplet beam due to dopant-induced evaporation of He atoms [4]. Upon repeated capture of Xe atoms and concomitant evaporation of He atoms, NHe decreases, as monitored by a reduction in the average partial pressure of He in the dump chamber, ΔPHe. The energy absorbed by the droplet upon capture of Xe atoms primarily arises from the binding energy released by the formation of XeN clusters and the kinetic energy of room-temperature Xe atoms in the pickup cell. Thus 〈𝑁 2( 〉 can be estimated by: 〈𝑁 2( 〉 = ∆. '( 〈6 )( 〉 . '( 8 '( 8 )( (2.3) in which EHe is the 0.6 meV binding energy of He atoms to the droplet [20], NHe is the initial average size of the He droplet, and EXe ≈ 0.15 eV is the average total energy released upon capture and condensation of one Xe atom. After passing through the interaction point, the beam terminates in a beam dump chamber that is equipped with the ion gauge and a residual gas analyzer. USC Experimental Apparatus The apparatus at USC was recently rebuilt to be oil-free, which allows for the imaging of the droplet beam. The oil-free design of the system also facilitates the use of the He recycling system. Figure 2.6 shows a schematic diagram and a photograph of the vacuum apparatus for experiments with large helium droplets at USC. The source chamber, which is the first chamber 30 from the left in Figure 2.6, accommodates the helium cryostat and nozzle. This chamber is pumped with a ~3000 L⸱s -1 Osaka kineTiG Series magnetically levitated turbomolecular pump. There is a 0.5 mm skimmer separating the source chamber from the rest of the vacuum apparatus. Next are two 8” ConFlat-flange Six-way crosses, which serve as the pickup chambers for droplet doping, and each is pumped with a 900 L-s -1 Osaka TG-MCAB Series magnetically levitated turbomolecular pump. There is a 5 mm aperture separating the two pickup chambers. An intermediate chamber, which can either be used as a differential pumping stage or for metal cluster deposition experiments, is installed between the second pickup chamber and the detection chamber, where the helium droplet beam terminates. A quadrupole mass spectrometer (Extrel Max-500 with a cross-beam deflector ionizer) in the detection chamber serves to detect changes in the helium droplet beam intensity for various spectroscopic and size measurements. The differential and detection chambers are each pumped by a 420 L⸱s -1 Osaka TG-MCAB Series magnetically levitated turbomolecular pump. To keep the entire system oil-free, all turbomolecular pumps are backed by dry roughing pumps. The pressure in each of the chambers is measured using hot ion gauges (MKS (Bayard-Alpert Mini-Ion Gauge)). A more sensitive extraction-type hot ion gauge (Leybold IONIVAC IE 514) with an X-ray limit of ~1×10 -12 mbar is installed in the detection chamber. 31 Figure 2.6. (a) Schematic diagram and b) photograph of the helium droplet beam vacuum apparatus at USC. CH: cold head; NZ: nozzle; SK: skimmer; PC1 and PC2: resistively heatable pickup cells; A1 and A2: 5 and 6 mm diameter apertures, respectively; BB: beam block; GV: gate valve; QMS: quadrupole mass spectrometer; W: Window. The green glow in each chamber results from the scattering of a 532 nm laser as it propagates from the detection chamber to the source chamber. 32 After expansion through the nozzle in the source chamber, the central part of the beam is selected by a skimmer, SK. The resulting beam traverses the pick-up chambers and differential pumping chamber until it terminates in the detection chamber. To achieve the highest flux in the detection chamber, the beam needs to be properly aligned through the skimmer (SK) and apertures A1, and A2. Alignment involves vertical and horizontal translation as well as tilting of the cold head. The initial beam alignment is usually performed at T0 = 7 K, where the beam has a rather broad angular distribution. When properly aligned, the beam causes a noticeable pressure rise (~6 ×10 -9 mbar) in the detection chamber and can be used to measure the He beam flux. Since the effusive beam contributes considerably to the total pressure in the detection chamber, the pressure rise is obtained as a difference between pressure measurements with the beam unblocked and blocked. A flag serves as a beam block and is installed in one of the pick-up chambers (BB in Figure 2.6 (a)). After the alignment was optimized at T0 = 7 K, the nozzle temperature can be decreased down to 5 K and the process of alignment is continued. The best alignment corresponds to the highest pressure rise possible over the entire operation range of T0. Below 4.5 K, the beam becomes strongly collimated, and the alignment at low temperatures is often not optimal at 7 K. While the iterative adjustment of vertical, horizontal and tilting positions can be time consuming and frustrating, this process ultimately gives rise to an optimized pressure rise in the detection chamber that can persist for months without further adjustment. Alignment only needs to be repeated upon the removal of the cold head for various reasons, such as nozzle orifice replacement. Observing the liquid jet through a microscope expedites the process of alignment and is discussed in detail in Chapter 3. However, using a microscope is only possible at T0 < 4.5 K and P0 > 10 bar, when the droplets are large enough to cause significant light scattering so that the beam can be seen even with bare eyes. Proper alignment of the jet at T0 = 4.0 K will cause the 33 pressure in the source chamber to drop by about 44%, as is seen in Figure 2.4. This drop signifies that a larger portion of the liquid helium beam is transported through the skimmer down to the detection chamber in the form of a highly collimated beam. Aside from the decrease of pressure in the source chamber, the lowest achievable temperature of the nozzle can drop by up to 0.5 K when the jet is properly aligned. This difference is due to the smaller base pressure of helium in the source chamber and the concomitant reduced heat conductivity through the residual helium gas. Further details on the imaging and spectroscopy setups are given below in Sections 3 and 8, respectively. References [1] L. F. Gomez et al., Shapes and Vorticities of Superfluid Helium Nanodroplets. Science 345, 906 (2014). [2] R. M. P. Tanyag, C. F. Jones, C. Bernando, D. Verma, S. M. O. O'Connell, and A. F. Vilesov, in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, edited by A. Osterwalder, and O. Dulieu (Royal Society of Chemistry, Cambridge, 2018), p. 389. [3] D. Verma et al., Shapes of Rotating Normal Fluid 3 He Versus Superfluid 4 He Droplets in Molecular Beams. Physical Review B 102, 014504 (2020). [4] L. F. Gomez, E. Loginov, R. Sliter, and A. F. Vilesov, Sizes of Large He Droplets. Journal of Chemical Physics 135, 154201 (2011). [5] R. Radebaugh, Cryocoolers: The State of the Art and Recent Developments. Journal of Physics-Condensed Matter 21, (2009). [6] A. T. A. M. de Waele, Basic Operation of Cryocoolers and Related Thermal Machines. Journal of Low Temperature Physics 164, 179 (2011). [7] L. F. Gomez, E. Loginov, and A. F. Vilesov, Traces of Vortices in Superfluid Helium Droplets. Physical Review Letters 108, 155302 (2012). 34 [8] H. Buchenau, E. L. Knuth, J. Northby, J. P. Toennies, and C. Winkler, Mass-Spectra and Time-of-Flight Distributions of Helium Cluster Beams. Journal of Chemical Physics 92, 6875 (1990). [9] H. Buchenau, J. P. Toennies, and J. A. Northby, Excitation and Ionization of He-4 Clusters by Electrons. Journal of Chemical Physics 95, 8134 (1991). [10] R. G. H. Buchenau, Scheidemann A. and J. P. Toennies, Conference Proceedings of the 15th International Symposium on Rarefied Gas Dynamics(1986). [11] J. H. Moore, C. C. Davis, and M. A. Coplan, Building Scientific Apparatus: A Practical Guide to Design and Construction (Westview Press, Cambridge, Massachusetts, 2003). [12] J. P. Toennies and A. F. Vilesov, Superfluid Helium Droplets: A Uniquely Cold Nanomatrix for Molecules and Molecular Complexes. Angewandte Chemie-International Edition 43, 2622 (2004). [13] D. Verma, R. M. P. Tanyag, S. M. O. O'Connell, and A. F. Vilesov, Infrared Spectroscopy in Superfluid Helium Droplets. Advances in Physics-X 4, (2019). [14] R. E. Grisenti and J. P. Toennies, Cryogenic Microjet Source for Orthotropic Beams of Ultralarge Superfluid Helium Droplets. Physical Review Letters 90, (2003). [15] R. E. Grisenti, R. A. C. Fraga, N. Petridis, R. Dorner, and J. Deppe, Cryogenic Microjet for Exploration of Superfluidity in Highly Supercooled Molecular Hydrogen. Europhysics Letters 73, 540 (2006). [16] M. Kuhnel, N. Petridis, D. F. A. Winters, U. Popp, R. Dorner, T. Stohlker, and R. E. Grisenti, Low-Z Internal Target from a Cryogenically Cooled Liquid Microjet Source. Nuclear Instruments & Methods in Physics Research Section a-Accelerators Spectrometers Detectors and Associated Equipment 602, 311 (2009). [17] M. Kuhnel, J. M. Fernandez, G. Tejeda, A. Kalinin, S. Montero, and R. E. Grisenti, Time- Resolved Study of Crystallization in Deeply Cooled Liquid Parahydrogen. Physical Review Letters 106, (2011). [18] B. G. Sartakov, J. P. Toennies, and A. F. Vilesov, Infrared Spectroscopy of Carbonyl Sulfide inside a Pure He-3 Droplet. Journal of Chemical Physics 136, 134316 (2012). 35 [19] S. Grebenev, J. P. Toennies, and A. F. Vilesov, Superfluidity within a Small Helium-4 Cluster: The Microscopic Andronikashvili Experiment. Science 279, 2083 (1998). [20] D. M. Brink and S. Stringari, Density of States and Evaporation Rate of Helium Clusters. Zeitschrift Fur Physik D-Atoms Molecules and Clusters 15, 257 (1990). [21] M. J. McCarthy and N. A. Molloy, Review of Stability of Liquid Jets and the Influence of Nozzle Design. The Chemical Engineering Journal 7, 1 (1974). [22] R. J. Donnelly and C. F. Barenghi, The Observed Properties of Liquid Helium at the Saturated Vapor Pressure. Journal of Physical and Chemical Reference Data 27, 1217 (1998). [23] R. M. Gibbons and D. I. Nathan, Thermodynamic Data of Helium-3, Technical Report AFML-TR-67-175, Allentown, PA (1967). [24] M. Farnik, U. Henne, B. Samelin, and J. P. Toennies, Differences in the Detachment of Electron Bubbles from Superfluid He-4 Droplets Versus Nonsuperfluid He-3 Droplets. Physical Review Letters 81, 3892 (1998). [25] B. Samelin, Lebensdauer Und Neutralisation Metastabiler, Negative Geladener Helium- Mikrotropfen, University of Göttingen, 1998. [26] J. Harms, M. Hartmann, B. Sartakov, J. P. Toennies, and A. F. Vilesov, High Resolution Infrared Spectroscopy of Single Sf6 Molecules in Helium Droplets. Ii. The Effect of Small Amounts of He-4 in Large He-3 Droplets. Journal of Chemical Physics 110, 5124 (1999). [27] J. Harms, J. P. Toennies, M. Barranco, and M. Pi, Experimental and Theoretical Study of the Radial Density Distributions of Large He-3 Droplets. Physical Review B 63, (2001). [28] M. Hartmann, University of Göttingen, 1997. [29] F. Pobell, Matter and Methods at Low Temperatures (Springer, Berlin, 2007), 3rd edn., 461. [30] R. M. P. Tanyag, A. J. Feinberg, S. M. O. O’Connell, and A. F. Vilesov, Disintegration of Diminutive Liquid Helium Jets in Vacuum. The Journal of Chemical Physics 152, 234306 (2020). 36 [31] L. Struder et al., Large-Format, High-Speed, X-Ray Pnccds Combined with Electron and Ion Imaging Spectrometers in a Multipurpose Chamber for Experiments at 4th Generation Light Sources. Nuclear Instruments & Methods in Physics Research Section a-Accelerators Spectrometers Detectors and Associated Equipment 614, 483 (2010). [32] T. Gorkhover et al., Nanoplasma Dynamics of Single Large Xenon Clusters Irradiated with Superintense X-Ray Pulses from the Linac Coherent Light Source Free-Electron Laser. Physical Review Letters 108, (2012). [33] C. Bostedt et al., Ultra-Fast and Ultra-Intense X-Ray Sciences: First Results from the Linac Coherent Light Source Free-Electron Laser. Journal of Physics B-Atomic Molecular and Optical Physics 46, (2013). [34] K. R. Ferguson et al., The Atomic, Molecular and Optical Science Instrument at the Linac Coherent Light Source. Journal of Synchrotron Radiation 22, 492 (2015). [35] O. Gessner and A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Droplets. Annual Review of Physical Chemistry 70, 173 (2019). [36] S. M. O. O'Connell et al., Angular Momentum in Rotating Superfluid Droplets. Physical Review Letters 124, 215301 (2020). [37] M. Grech, R. Nuter, A. Mikaberidze, P. Di Cintio, L. Gremillet, E. Lefebvre, U. Saalmann, J. M. Rost, and S. Skupin, Coulomb Explosion of Uniformly Charged Spheroids. Physical Review E 84, (2011). 37 Optical Imaging The following section uses previously published text and figures from our paper: 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. [1] Small droplets (NHe < 10 5 ) are typically produced upon the condensation of cold (T0 = 10– 20 K), pressurized helium gas during the course of beam expansion from the nozzle [2, 3]. Larger droplets are formed upon the expansion (flashing) of supercritical helium fluid at lower temperatures (T0 = 5–10 K). At even lower temperatures, liquid helium’s violent boiling ceases and forms a columnar liquid jet after the nozzle exit [4-6]. Here, the jet breaks into very large droplets (with diameters ~1.9 times the nozzle diameter) via capillary instability [7-9]. First reports of helium droplet jets date back to 1961 when Becker and co-workers reported producing beams of condensed helium upon expanding the liquid into vacuum at a nozzle temperature of T0 ≈ 4.2 K and stagnation pressure of P0 ≈ 1 atm [10, 11]. In subsequent works, Gspann et al. [12, 13] and Toennies et al. [2, 14] studied the ionization of helium droplets produced in the flashing regime. The average droplet sizes in this operation were in the range of 10–1000 nm in diameter, depending on T0 and P0 [3, 14, 15]. Helium droplets in the jet capillary breakup regime, which is often called the Rayleigh regime, were observed by optical microscopy [4-6]. An effective use of large helium droplets requires knowledge of the average droplet sizes and droplet densities at a large distance from the nozzle (up to about 1 m) where, for example, the droplet beam is interrogated by an X-ray laser. However, the formation of large droplets from liquid helium jets in the flashing and capillary breakup regimes remains poorly studied. Here, we 38 performed a systematic study of the behavior of helium droplet beams in these regimes by optical microscopy. This study addresses droplets larger than about 2 μm in diameter as determined by the optical resolution limit. Experimental In this work, the liquid helium jet was produced upon the expansion of pressurized helium (99.9999%) into vacuum through a 5 µm diameter nozzle at temperatures T0 < 5 K and stagnation pressures P0 up to 60 bars. Similar experimental arrangements have previously been used for producing helium droplets with different average sizes at higher T0 [3, 16, 17]. The nozzle assembly was cooled by using a Sumitomo RDK-408D closed-cycle refrigerator, yielding a low temperature of about 3.5 K. The strokes of the cryocooler lead to periodic (~1.1 Hz) changes of the nozzle temperature by a few tenths of a degree. These small temperature fluctuations are disregarded in this work. The imaging setup is presented in Figure 3.1. The cryocooler holding the nozzle is installed upright inside a vacuum chamber. It can be moved along the X, Y, and Z directions and tilted along the roll, pitch and yaw axes. Upon traversing through the observation chamber, the jet terminates downstream in a beam dump chamber. The observation chamber is pumped by a turbo-molecular pump with a nominal speed of 3000 L s −1 , while the beam dump chamber has a 900 L s −1 pump. 39 Figure 3.1. Bird’s-eye view of the observation chamber that houses the jet imaging setup. Liquid helium expands in vacuum through the nozzle attached to the second stage of the Sumitomo RDK-408D cryocooler. A cell filled with Coumarin dye is pumped by the third harmonic of an Nd:YAG laser. The fluorescence from the dye is focused by using a lens and illuminates the droplets. The images of the droplet beam are collected with a Navitar ZOOM-6000 microscope attached on a commercial Canon Rebel T1i camera body. The imaging system consists of a laser, a dye cell, a microscope, and a commercial digital single-lens reflex (DSLR) camera. Different portions of the jet are imaged by moving the cryocooler with the attached nozzle along the beam axis by up to 50 mm. The light source is the 355 nm third harmonic of a Nd:YAG laser (Exspla NT-242) with a pulse energy of ~2.4 mJ and a pulse width of ~7 ns. The laser beam of about 9 mm in diameter illuminates a 10 mm thick quartz cuvette filled with a ~14 ppm solution of coumarin 120 in methanol. The dye fluoresces at ~440 nm with a duration comparable to that of the pump pulse. The fluorescent light is collected using a lens, and the helium droplet beam is illuminated. A Navitar Zoom 6000 microscope is employed with a 1.5× lens attachment, a 6.5× zoom prime lens, and a 2.0× adapter. The setup has a focal length of ~50 mm, a magnification of 13.50×, a numerical aperture of 0.106, a depth of field of 0.04 mm, and a nominal resolution limit of 2.5 µm at 440 nm. The images are collected by using 40 a Canon Rebel T1i camera equipped with a CMOS image sensor. A single pixel of the image corresponds to 0.35 µm pixel width. More details on the imaging technique and image processing are presented in Section S1 of the supplementary materials (SM) of Reference [1]. Results Figure 3.2 shows images of the jet obtained at different distances from the nozzle. The nozzle cap edge appears as the black stripe on the right-hand side of Panel (a). The nozzle exit is indented by about 250 µm from the edge of the nozzle cap and is not visible in any of the images. More details are available in Figure S1 of the SM of Reference [1]. Each individual droplet is discernable in the images, and shows a dark periphery with a bright spot in the middle. The bright spots originate from some focusing effect by the spheroids. Light passing through the droplet is refracted and makes the droplet appear dark. The unrefracted light, on the other hand, produces a uniformly illuminated background that appears bright on the image sensor. 41 Figure 3.2. Photographs of a helium droplet beam produced at T 0 = 3.5 K and P 0 = 20 bars and obtained at different distances from the nozzle. The distance is given with respect to the edge of the nozzle cap on the right of Panel (a). The insets [(i) and (ii)] are zoomed-in portions of the droplet beam. The white dashed circle in (i) indicates the breakup point. The droplet diameter can be determined from the outer boundaries of the dark region. In the inset of Figure3.2(c), the bigger droplet has a diameter of about 24 pixels, while the diameter of the smaller one is about 20 pixels, with corresponding diameters of 8.4 µm and 7.0 µm, respectively. Due to limited resolution, the sharp droplet boundary is blurred in the image by about 2.5 µm. Model calculations indicate that if the edge of the droplet is determined at 50% of the optical density, the images give the actual diameters only for droplets larger than 4 µm. Figure 3.2(a) displays a portion of the jet close to the nozzle, and the insets show two zoomed-in regions, one very close to the nozzle (i) and another farther away from it (ii). Very close to the nozzle, the jet is a liquid column with periodic intensity modulation, suggesting the presence of some axial disturbances, which could grow and eventually lead to jet breakup. The breakup 42 point is marked in Inset (i) by a white dashed circle and corresponds to a distance of ~500 µm from the nozzle exit. The breakup length, L, is defined as the length of the continuous portion of the jet from the nozzle exit until the point where it breaks into droplets. The breakup point is measured as the first point from the nozzle edge where the optical density drops by more than 50% along the center line of the jet. At longer distances, such as in Inset (ii), the jet consists of ligaments of about 30–60 µm long. The ligaments are distorted in a peristaltic fashion, characterized by alternating areas of protrusions and depressions. The protruding areas likely indicate places where droplets are formed, and the depressed areas are regions where the droplets probably pinch. When a ligament breaks further, it does not necessarily produce droplets right away. A shorter ligament can be produced in addition to some droplets, whose partitioning depends on the growth of capillary waves on the surface of the ligaments [18]. Several ligaments, such as those around 0.5 mm, exhibit large vertical undulations and become shorter, displaying fewer deformities farther downstream. Finally, beyond 1.3 mm, the image mostly shows single droplets with an average spacing of about 12 µm. The droplet beam near the nozzle follows a straight line. At farther distances, however, the droplets distribute themselves in a cone with an opening angle of ~15 mrad. This spread must result from the fluctuations in the transversal velocity of the droplets. A more detailed set of images and discussion are presented in Section S2 of the SM of Reference [1]. The appearance of the jet depends on the stagnation conditions, T0 and P0. Previous studies [5, 15] have demonstrated that the liquid helium jet velocity can be estimated using Bernoulli’s equation, 𝑣 9(: ≈ ' ;. & & * (3.1) 43 In which ρl is the density of liquid helium at stagnation conditions [19]. Due to the finite compressibility of liquid helium, it cools upon passing from T0 inside the nozzle into vacuum. The expansion isentropes indicate that a temperature drop can be as large as ~2 K for an expansion starting at P0 = 60 bars and T0 = 5 K, but will be smaller at lower P0 and T0 [5]. Figure 3.3. Photographs of liquid helium jets produced at T 0 = 3.5 K and P 0 = 3 bars, 20 bars, and 60 bars in Panels (a), (b), and (c), respectively. Large gray and black spots consistent in all Panels are artifacts of the imaging setup. Figure 3.3 shows photographs of the jets in the vicinity of the nozzle produced at varying P0 and constant T0 = 3.5 K. Images at some intermediate pressures and encompassing longer distances from the nozzle, up to 3.5 mm, are presented in Figure S6 of the SM of Reference [1]. At the lowest P0 = 3 bars, the liquid column jet is not discernable. The beam appears as a spray in which droplets of different sizes fill a ~4° cone with the largest droplets forming the centerline of the beam axis. In a series of photographs collected at this condition, the beam vacillates between a jetting mode and a spraying mode. The jetting mode, however, only appears about 20% of the 44 time. At P0 = 20 bars, a persistent 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. In addition to the main jet, Figure 3.3(b) reveals some side jets of smaller droplets emerging at ~1° above and below the main jet. At P0 = 60 bars in Panel (c), four jets are visible with the two central jets being the most intense. Here, the continuous liquid column close to the nozzle appears as two nearly parallel dark lines with a light stripe in the middle. Due to lower contrast of the images close to the nozzle, it is difficult to accurately determine the origin of the side jets. In some fortuitous images, extrapolations following the side jets indicate that they originate behind the nozzle cap and close to the nozzle exit. Figure 3.4 shows jet images at varying T0 and a constant P0 = 60 bars. Images at some intermediate temperatures and encompassing longer distances from the nozzle, up to 3.5 mm, are presented in Figure S7 of the SM of Reference [1]. At T0 = 4.5 K, the jet forks into five sub-jets: two strong jets in the center and three weaker jets on the periphery with the top side jet being very faint. The forks at T0 > 4.5 K converge to the same point at or very close to the nozzle exit. The separation angle between the two outermost forks of the jet [of ~1° in panel (a)] is referred to as the splay angle. In panel (b), multiple ruptures in the helium droplet beam are observed. These ruptures are about 12–24 µm long and are characterized by concave shapes, as if a bubble had just exploded. The appearance of the gaps suggests cavitation within the columnar jet close to the nozzle. At higher temperatures, the splay angle increases with temperature and is about a factor of five times larger in Panel (c) at T0 = 5.2 K. In addition, no continuous jet column is seen in the images, and the droplets are smaller and indistinguishable. Finally, we observed that the pointing of the jets changes with time. Within minutes, the pointing can change by a few tenths of a degree. 45 Reference [1] further discusses the size, aspect ratio, and tilt distributions of the imaged droplets, but these further details are omitted in this work. Figure 3.4. Photographs of helium droplet beam expansion produced at P0 = 60 bars and T 0 = 3.5 K, 4.5 K, and 5.2 K in Panels (a), (b), and (c), respectively. Discussion The disintegration of liquid jets has been studied extensively, particularly by engineers [7, 9, 19-21]. Jet atomization into small and possibly uniform droplets is beneficial for a wide range of applications such as fuel injection in internal combustion engines, painting, and dispersing pesticides. However, in most previous studies, the systems under consideration were that of liquid jets emanating into gaseous media, such as water jets impinging into air at atmospheric pressure. Such studies established several parameters affecting jet fragmentation, most notably the incipient jet velocity. At low velocities, the Rayleigh mechanism dominates. As the jet velocity increases, the interactions between the liquid and the surrounding gas start to play a dominant role in the 46 breakup mechanism. This interaction induces the growth of short wavelength perturbations on the surface of the jet, giving rise to small droplets. A jet’s visual appearance, as well as its breakup length, is often used to categorize different regimes of operation. With the increase in the jet velocity, five different regimes have been identified. They are known as the dripping, Rayleigh, first-wind induced, second-wind induced, and atomization regimes, respectively [7, 9, 19-21]. In the Rayleigh regime, the jet exists as a column whose length increases approximately linearly with the jet velocity and may reach up to ~1000 nozzle diameters. The breakup length slightly decreases for the first wind-induced regime, where the continuous jet starts showing noticeable peristaltic oscillations. The droplets obtained in both regimes have diameters comparable to that of the jet. In the second wind-induced regime, the breakup length increases slightly again. However, in this regime, small droplets start to break off from the sides of the jet even close to the nozzle exit. Finally, in the atomization regime, the visible portion of the jet nearly disappears and instead takes the form of a dense droplet spray. In the last two regimes, the droplets are much smaller than the jet diameter. Conclusions This study has demonstrated the multifaceted behavior of the liquid helium jet breakup. By optical microscopy, it was shown that, at different stagnation pressures and temperatures, droplet formation may involve spraying, capillary breakup, jet branching, and/or flashing and cavitation. Spraying was observed at lower stagnation pressures. Further improvement in the production of the helium droplets, such as the use of a continuous flow cryostat, a nozzle channel with a known surface roughness, or possibly a smooth glass capillary, increasing the purity of the helium used for the expansion, or reduction in the jet velocity are needed to achieve a more robust and reproducible operation as needed in experiments. Our work calls for more studies to elucidate the 47 mechanism of the liquid helium jet breakup and to optimize droplet production for future experiments requiring well collimated, evenly sized and spaced droplets. For example, data acquisition in X-ray coherent diffractive imaging experiments could be greatly improved if better collimated beams with pointing stability were used. Eliminating the jet branching is critical to these experiments. In the future, obtaining images of the jet in the immediate vicinity of the nozzle should also be attempted. Obtaining images of the entire length of the jet breakup is imperative to elucidating the branching phenomenon and should be investigated further. Furthermore, studying the dynamics of jets produced upon expanding of a superfluid at lower temperatures should be attempted to gauge the effect of the normal fluid core–superfluid shell structure on the jet breakup as in the present work. References [1] R. M. P. Tanyag, A. J. Feinberg, S. M. O. O’Connell, and A. F. Vilesov, Disintegration of Diminutive Liquid Helium Jets in Vacuum. The Journal of Chemical Physics 152, 234306 (2020). [2] H. Buchenau, E. L. Knuth, J. Northby, J. P. Toennies, and C. Winkler, Mass-Spectra and Time-of-Flight Distributions of Helium Cluster Beams. Journal of Chemical Physics 92, 6875 (1990). [3] J. P. Toennies and A. F. Vilesov, Superfluid Helium Droplets: A Uniquely Cold Nanomatrix for Molecules and Molecular Complexes. Angewandte Chemie-International Edition 43, 2622 (2004). [4] C. C. Tsao, J. D. Lobo, M. Okumura, and S. Y. Lo, Generation of Charged Droplets by Field Ionization of Liquid Helium. Journal of Physics D-Applied Physics 31, 2195 (1998). [5] R. E. Grisenti and J. P. Toennies, Cryogenic Microjet Source for Orthotropic Beams of Ultralarge Superfluid Helium Droplets. Physical Review Letters 90, (2003). [6] M. Kuhnel, N. Petridis, D. F. A. Winters, U. Popp, R. Dorner, T. Stohlker, and R. E. Grisenti, Low-Z Internal Target from a Cryogenically Cooled Liquid Microjet Source. Nuclear 48 Instruments & Methods in Physics Research Section a-Accelerators Spectrometers Detectors and Associated Equipment 602, 311 (2009). [7] S. P. Lin, Breakup of Liquid Sheets and Jets (Cambridge University Press, Cambridge, 2003). [8] N. Ashgriz and A. L. Yarin, in Handbook of Atomization and Sprays: Theory and Applications, edited by N. Ashgriz (Springer US, Boston, MA, 2011), p. 3. [9] A. H. Lefebvre and V. G. McDonell, Atomization and Sprays (Taylor & Francis, CRC Press, Boca Raton, FL, 2017), 2nd edn. [10] E. W. Becker, R. Klingelhofer, and P. Lohse, Strahlen Aus Kondensiertem Helium Im Hochvakuum. Zeitschrift Fur Naturforschung Part a-Astrophysik Physik Und Physikalische Chemie 16, 1259 (1961). [11] E. W. Becker, P. Lohse, and R. Klingelhofer, Strahlen Aus Kondensiertem Wasserstoff, Kondensiertem Helium Und Kondensiertem Stickstoff Im Hochvakuum. Zeitschrift Fur Naturforschung Part a-Astrophysik Physik Und Physikalische Chemie A 17, 432 (1962). [12] J. Gspann and H. Vollmar, Metastable Excitations of Large Clusters of He-3, He-4 or Ne Atoms. Journal of Chemical Physics 73, 1657 (1980). [13] J. Gspann and H. Vollmar, Ejection of Positive Cluster Ions from Large Electron- Bombarded He-3 or He-4 Clusters. Journal of Low Temperature Physics 45, 343 (1981). [14] U. Henne and J. P. Toennies, Electron Capture by Large Helium Droplets. Journal of Chemical Physics 108, 9327 (1998). [15] L. F. Gomez, E. Loginov, R. Sliter, and A. F. Vilesov, Sizes of Large He Droplets. Journal of Chemical Physics 135, 154201 (2011). [16] C. Callegari and W. E. Ernst, in Handbook of High Resolution Spectroscopy, edited by F. Merkt, and M. Quack (John Wiley & Sons, Chichester, 2011), p. 1551. [17] R. M. P. Tanyag, C. F. Jones, C. Bernando, D. Verma, S. M. O. O'Connell, and A. F. Vilesov, in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, edited by A. Osterwalder, and O. Dulieu (Royal Society of Chemistry, Cambridge, 2018), p. 389. 49 [18] P. Marmottant and E. Villermaux, Fragmentation of Stretched Liquid Ligaments. Physics of Fluids 16, 2732 (2004). [19] M. J. McCarthy and N. A. Molloy, Review of Stability of Liquid Jets and the Influence of Nozzle Design. The Chemical Engineering Journal 7, 1 (1974). [20] C. Dumouchel, On the Experimental Investigation on Primary Atomization of Liquid Streams. Experiments in Fluids 45, 371 (2008). [21] J. Eggers and E. Villermaux, Physics of Liquid Jets. Reports on Progress in Physics 71, (2008). 50 3 He and 4 He Droplet Size Comparisons This section is based on our published paper: 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 Jäger, 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. [1] Introduction A number of recent experiments in our group and elsewhere focused on the study of quantum rotation in superfluid 4 He droplets containing up to ~10 11 atoms using coherent scattering of femtosecond X-ray and XUV pulses from free electron lasers (FEL) and intense, laboratory- based high-order harmonic sources. It was found that large 4 He droplets can have sizable angular momentum and can be subject to considerable centrifugal distortion [2-6]. Rotation of superfluid 4 He droplets is associated with the creation of quantum vortices, a physical manifestation of quantized angular momentum in the bosonic species [7-10]. Molecular beam laboratory experiments involving droplets of the rare fermionic helium-3 isotope ( 3 He) have also been performed [11-20]. While 3 He may exist as a superfluid under temperatures T ≈ 1 mK [21, 22], it is a normal fluid at typical molecular beam temperatures of ~0.15 K [20]. Recent density functional calculations show that the rotating 3 He droplets should follow corresponding classical shapes [23]. This work expands X-ray imaging experiments to rotating 3 He droplets to enable a direct comparison of droplet shapes and rotational properties of the two quantum fluids. Herein, we report on the characterization of 3 He droplets produced upon nozzle beam expansion into vacuum at temperatures as low as ~2 K. Using X-ray scattering at an XFEL, the properties of individual, free 3 He and 4 He droplets are analyzed and their size, shape and angular 51 momenta are compared. A wide range of 3 He and 4 He droplet sizes are obtained with average radii of 162 nm and 355 nm, respectively. An overwhelming majority of the droplets have pseudo- spheroidal shapes, characterized by the AR of the major and minor axes. From the aspect ratio, the reduced angular momenta and angular velocities are obtained, using the corresponding stability diagram for classical viscous droplets [3, 24, 25]. The aspect ratios of droplets from both isotopes are found to have average values of 1.074 for 3 He and 1.088 for 4 He. Accordingly, the reduced angular momentum and reduced angular velocity in 3 He and 4 He droplets are similar. Comparison of the results obtained with 4 He and 3 He at different expansion conditions may help to gain a better understanding of the mechanism underlying the production of rotating droplets in free nozzle beam expansion sources. Experimental Droplets are produced via expansion of pressurized He through a nozzle, as described in Chapter 2. In vacuum, the temperature of the droplets further decreases via evaporative cooling down to 0.15 K and 0.38 K for 3 He [20] and 4 He [15], respectively. Considering the lower critical point of 3 He (TC = 3.3 K, PC = 1.1 atm) compared to that of 4 He (TC = 5.2 K, PC = 2.3 atm), lower nozzle temperatures are required to obtain 3 He droplets of the same sizes as those consisting of 4 He. For example, for a nozzle stagnation pressure of P0 = 20 bar, previous experiments demonstrated that 4 He droplets with an average number of atoms <N4> = 10 7 are produced at a nozzle temperature of T0 = 7 K, [26] while T0 = 5 K is required to obtain 3 He droplets with the same average number of atoms <N3> = 10 7 [14, 16, 18, 27]. The temperature difference of 2 K in T0 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 52 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, as described in Chapter 2. Droplets of 3 He and 4 He are produced at constant P0 = 20 bar and varying T0, 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. Due to the considerable cost of 3 He gas, a recycling system is employed during the experiments as described in Chapter 2 of this thesis. Filling the system requires about 10 L×bar of room temperature 3 He. For comparison, at standard operating conditions (T0 = 3 K, P0 = 20 bar), the flow rate of the He gas is ~3 cm 3 ×bar/s and the filling amount of gas would only be sufficient for about 1 hour of operation. 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, taking into account that its solubility is ~0.1% at 0.15K [21]. Any possible pockets of 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. The experiments are performed using the LAMP end station at the Atomic, Molecular and Optical (AMO) instrument of the Linac Coherent Light Source (LCLS) XFEL [28, 29]. The focused XFEL beam (~2 μm full-width-at-half-maximum, FWHM) intersects the He droplet beam ~70 cm downstream from the nozzle. The XFEL is operated at 120 Hz, a photon energy of 1.5 keV (λ = 0.826 nm), 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 (~10 12 ) enables the instantaneous capture of the shapes of individual droplets. Diffraction images are recorded with a pn-charge-coupled device 53 (pnCCD) detector containing 1024×1024 pixels, each 75×75 μm 2 in size, which is centered along the XFEL 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 also have a central, rectangular section cut-out to accommodate the primary X-ray beam. 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. Results Figure 1 shows several diffraction patterns from pure 3 He droplets. The images are characterized by sets of concentric contours. Images in Figures 1(a) and (b) exhibit a series of circular and elliptical contours, respectively, with different spacing between their respective rings. Figure 1(c), however, shows an elongated diffraction contour with pronounced streaks radiating away from the center. These diffraction patterns are characteristic of spherical (Figure 1(a)), and spheroidal (oblate) or capsule (prolate) (Figures 4.1(b) and(c)) droplet shapes, as previously observed in 4 He droplets [30-33]. Spheroidal and prolate shapes result from centrifugal deformation of droplets with considerable angular momentum. The obtained diffraction patterns are characteristic of spherical, spheroidal (oblate) or capsule (prolate) droplet shapes, as previously observed in 4 He droplets [2, 3, 5, 6]. Spheroidal and prolate shapes result from centrifugal deformation of droplets with considerable angular momentum. The droplet shapes are characterized by the distances between the center and the surface in three mutually perpendicular directions: a ≥ b ≥ c. For an oblate axisymmetric droplet, a = b > c, with c along the rotation axis, whereas a > b > c in the case of triaxial prolate shapes with c along the rotation axis [2, 3]. The observed diffraction patterns do not provide direct access 54 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. In this section, we discuss the experimental results in terms of the apparent A, C and AR values, from which the average actual sizes of the axisymmetric droplets are obtained. The values of A and C are obtained from the diffraction patterns as described elsewhere (SM in Reference [3]). Figure 4.1. 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 AR are displayed at the top of each image. During the measurements, approximately 900 intense diffraction images from pure 3 He droplets were obtained, each providing a unique set of A and C values. Similar measurements are performed for 4 He droplets, providing ~300 patterns as an independent reference for comparison. The measurements for a given isotopic fluid do not exhibit any systematic variation with nozzle 55 temperature. Thus, the results obtained at different temperatures are combined to improve statistics. Table 4.1 lists the nozzle temperatures and corresponding numbers of recorded diffraction images for all experimental runs. He isotope Nozzle temperature, K Number of images obtained 4 He 4.36 68 3.95 131 2.59 120 3 He 2.58 92 2.67 132 2.91 229 2.03 86 2.10 119 2.07 132 2.12 73 Table 4.1. Nozzle temperatures and corresponding numbers of recorded diffraction images for the results presented in Figure 2 (a), (b) of the main text. All measurements were performed at nozzle stagnation pressure of P 0 = 20 bar. Figure 4.2(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 of 4 He droplets is approximately a factor of two times 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 56 spread, ranging from A = 55 nm to A = 1250 nm. Figure 2(b) shows the AR distribution for 3 He and 4 He droplets. The largest ARs are 1.99 for 3 He and 1.72 for 4 He. Panels (a) and (b) in Figure 4.2 show that both the values of A and (AR-1) follow exponential distributions. Panel (c) in Figure 4.2 shows 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. The AR distribution in each bin from Figure 4.2(a) is found to be close to exponential. In contrast to the temperature-independent droplet sizes reported here, previous measurements on 4 He droplets found continuous increases in sizes with decreasing temperature [26]. At T0 < 4 K and P0 = 20 bar, 4 He expansion leads to the formation of a jet that breaks up into micron-sized droplets due to Rayleigh instability [26, 34, 35]. This mechanism gives rise to an extremely collimated beam of droplets, the occurrence of which was not observed during this work with either 3 He or 4 He. We conclude that, most likely, the flow through the nozzle in this work was affected by imperfections such as microscopic damage or partial obstruction by some solid impurities. Previous experiments with 4 He droplets in our group demonstrated that, under such conditions, decreasing the nozzle temperature below a certain value does not result in any increase in average droplet size [36], which is in agreement with the observations in this work. 57 Figure 4.2. Droplet size (a) and aspect ratio (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 the ease of comparison, as the total number of diffraction images obtained for 3 He and 4 He were ~900 and ~300, respectively (see Table 4.1). 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 in Panel (c) are shown by stars. The blue line in Panel (c) represents a linear fit of the data points (blue squares) for 3 He droplets. Discussion 4.4.1 Droplet size distribution Droplet size distributions are usually discussed in terms of the number of atoms per droplet, owing to the detection technique, which is often based on mass spectroscopy [37]. For an oblate droplet the number of He atoms is given by 𝑁 = <=>? + @ A , where n is the number density of liquid 58 3 He or 4 He at low temperature, with values of 1.62×10 28 m -3 [22] and 2.18×10 28 m -3 [38], respectively. Here, we approximate the true values of a and c for each droplet by the measured projection values of A = a and C ≤ c. Since, as discussed in the following section, the average aspect ratio of the droplets is close to unity, this approximation will overestimate the droplet number sizes by a few percent, which is comparable to the statistical error. Accordingly, average number sizes for droplets with aspect ratios of less than 1.4 are <N3> = (5.6 ± 0.1) ´ 10 8 and <N4> = (1.1 ± 0.1)´10 10 . Figure 4.3 shows the number size distribution for 3 He droplets on a natural logarithmic scale. For comparison, the red line indicates an exponential distribution 𝑃(𝑁 A ) = B C6 " D exp (− 6 " C6 " D ), with S being the total number of detected droplets. This approximation is in good agreement with the experimental data for sizes N3 ≤ 3×10 9 . An exponentially declining size distribution was also found in a recent study of 4 He droplets obtained from a pulsed nozzle at the FERMI FEL [32]. In comparison, the size distribution of smaller 3 He droplets with N3 ≤ 10 7 obtained at P0 = 20 bar and T0 ≥ 5 K was found to be close to log-normal [27]. 59 Figure 4.3. Measured size distribution of 3 He droplets (blue squares). The red line represents ln(P) = 4.7 – N 3/<N 3> (blue squares) with <N 3> = 5.6´10 8 , see the text for details. The figure does not show 9 droplets in the range N 3 = 7×10 9 -2×10 10 , which are off scale. 4.4.2 Droplet aspect ratios The aspect ratios provide access to the angular momentum and angular velocity of the droplets. One can obtain the average actual aspect ratio áarñ from the average apparent aspect ratio áARñ assuming a random droplet orientation as described in the following. In classical droplets, the largest aspect ratio of stable, axially symmetric droplets is ar = 1.47 [24, 25]. About 99% of the measurements in Figure 4.2 have AR < 1.4, in agreement with previous measurements in 4 He droplets [2, 4, 5, 26]. Here, we assume that an overwhelming majority of droplets with AR < 1.4 have oblate, axially symmetric shapes. We also assume that the data set contains less than ~10 events from prolate He droplets that are oriented in such a way that 60 their projections yield AR < 1.4 and cannot be distinguished from oblate droplets. This estimate is based on the number of events producing AR > 1.4, which are entirely ascribed to prolate droplets. For more details, see Reference [3], specifically Figure 11 and the corresponding text. For shapes with AR < 1.4, the average values for the observed major half axis A and aspect ratio AR are <A3> = 160 ± 3 nm, <AR3> = 1.049 ± 0.003, <A4> = 348 ± 14 nm, <AR4> = 1.059 ± 0.005, where the subscripts 3 and 4 refer to 3 He and 4 He, respectively. To translate the measured <AR> into the actual <ar>, we assume a spheroid with a well- defined ar and calculate its projection on the detector plane when its symmetry axis c forms an angle α with the normal to the plane. The aspect ratios of the diffraction pattern (AR) and of the spheroid off which the X-rays diffract (ar) are related by: 𝐴𝑅 = D𝑐𝑜𝑠 ; (𝛼)+𝑎𝑟 ; 𝑠𝑖𝑛 ; (𝛼) (see Eq. (S2.8) in the SM to [2]). The average AR of an ensemble of randomly aligned droplets is then calculated as 〈𝐴𝑅〉 = ∫ 𝐴𝑅(𝛼)∙𝑠𝑖𝑛(𝛼) , + * ∙𝑑𝛼, where sin(α) represents the probability of finding a spheroid at angle α. Integration and second-order expansion near 〈𝐴𝑅〉 = 1 yields 〈𝐴𝑅〉−1 = ; A (𝑎𝑟−1)+ ! !E (𝑎𝑟−1) ; . The expression is approximately linear within less than 5% error in the range of 1 ≤ ar ≤ 1.4. Due to the linear relationship between <AR> and ar, the same formula also applies when considering not just an orientation-averaged ensemble with one specific ar, but also averages over all orientations, thus 〈𝑎𝑟〉−1 ≈ A ; (〈𝐴𝑅〉−1). From this relationship, the average true aspect ratios for 3 He and 4 He droplets are derived as áarñ3 = 1.074 ± 0.005 and áarñ4 = 1.088 ± 0.008, respectively. 4.4.3 Average angular momenta and angular velocities of 3 He and 4 He droplets As previously described for 4 He droplets [2-6], we ascribe the shape deformation in 3 He droplets to centrifugal distortion. It has been reported that the shapes of rotating 4 He droplets 61 closely follow the equilibrium shapes of classical droplets having the same values of angular momentum [3, 5, 6, 9, 10]. This pattern is also expected to be the case of 3 He droplets, which at the temperature of these experiments (~0.15 K) [20] should behave classically due to the high viscosity of about 200 µP and small mean free path (a few nm) of elementary excitations at this temperature [22]. In recent density functional calculations, the shapes of rotating 3 He droplets were found to be very close to those predicted for classical droplets [23]. The blue curves in Figure 4.4 show the stability diagram of the classical droplets in terms of the reduced angular momentum (Λ) and reduced angular velocity (Ω), which were given in Chapter 1.2 as [24, 25]: Λ = F "#%&' ! (4.1) Ω = ' &' " #% ∙𝜔 (4.2) For liquid 4 He and 3 He at low temperature, the surface tensions are σ4 = 3.54×10 -4 N/m [39] and σ3 = 1.55×10 -4 N/m [40], respectively, while their densities are ρ4 = 145 kg/m 3 [38] and ρ3 = 82 kg/m 3 [22]. With increasing Λ, the droplet’s equilibrium shape transitions from spherical to oblate axially symmetric, which is shown by the solid blue curve. At Ω ≈ 0.56, Λ ≈ 1.2, ar ≈ 1.47, 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 droplets with 1.2 < Λ < 1.6, and dumbbell-shaped droplets at Λ > 1.6 [25, 30, 32, 41]. For Λ > 2, droplets become unstable and break up. Also shown in Figure 4 is the ar of droplets along the axisymmetric branch as a function of Λ, which is represented by the red curve [30]. Using an exponential distribution of the ar values: 𝑃(𝑎𝑟−1) = ! C?-G!D expP− ?-G! C?-G!D Q, and the functions of Λ(ar) and Ω(ar) in Figure 4, integration over ar gives the average Λ and Ω for 3 He and 4 He droplets to be <Λ3> = 0.47, <Ω3> = 0.27 and <Λ4> = 0.51, <Ω4>= 0.29. Those values are indicated 62 in Figure 4 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) = ! "#$(&)( !) exp.− #$( ! "#$(&)( !) / ! "&) exp.− & "&) /, where the values of ar(A) were obtained from the blue line in Figure 2(c) multiplied by 1.5. Although the values of <AR> vs A in Figure 2(c) lay somewhat lower for 4 He than for 3 He, the corresponding average values for 4 He are larger due to greater prevalence of large 4 He droplets. From the values of <Λ3,4> and <Ω3,4> and using Eqs. (1.1, 1.2), the angular momentum (L) is obtained as L3 = 1.5×10 9 ħ and L4 = 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. Last, ω was calculated as 1.6×10 7 rad/s and 5.9×10 6 rad/s for 3 He and 4 He, respectively. Although the 4 He droplets and 3 He droplets have similar <Λ>, 4 He droplets have about a factor of three times larger L per atom. Mathematically, this effect stems from the different factors of D𝜎𝜌𝑅 H in Eq. 1.1 in 3 He and 4 He droplets. 63 Figure 4.3. Red curve: Calculated aspect ratio as a function of reduced angular momentum (Λ) for axially symmetric oblate droplet shapes. Blue curve: stability diagram of rotating droplets in terms of reduced angular velocity (Ω) and reduced angular momentum (Λ). 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 (Ω, Λ) values corresponding to 3 He and 4 He droplets. 4.4.4 Formation of rotating droplets in the fluid jet expansion It is remarkable that despite their widely different physical properties, 3 He and 4 He droplets have very similar values of Ω and Λ on average. Previous XFEL experiments with 4 He droplets yielded average aspect ratios, <AR>, in the range of 1.06 – 1.08 at P0 = 20 bar and T0 = 4 - 7 K, which spans average droplet sizes from 200 nm to 1000 nm in diameter (see Figure 4.11 in Reference [35]). Thus, it is noteworthy that very similar average aspect ratios, and therefore Ω and 64 Λ, were obtained at different T0. Comparable <AR> were obtained in experiments involving different nozzle plates, including measurements with partially obstructed and intact nozzles [36]. Hence, it seems that the acquired <AR> is largely independent of particular nozzles used in the experiments. Similar results for non-superfluid 3 He and superfluid 4 He droplets indicate that the state of the droplets has a small effect on the resulting average reduced angular momentum. In previous works [3, 42], we conjectured 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 [26]. Accordingly, the estimated average angular velocity of 4 He and 3 He droplets is 3.4×10 7 rad/s and 4.5×10 7 rad/s, respectively. Such high angular velocities can only be sustained by rather small droplets. It is challenging to explain the similarities in reduced angular velocity and angular momentum in 3 He and 4 He droplets based on the stability diagram in Figure 4 and the estimated vorticities. Moreover, the half axis and shape distributions in Figure 2, as observed at high vacuum far downstream (~70 cm) from the nozzle, originate from processes in the high-density region inside or close to the nozzle, where collisions between droplets with the dense He gas must play an important role. For example, for a droplet with a radius of 300 nm, rotating at 10 7 rad/s, the peripheral velocity will be ~3 m/s, assuming rigid body rotation. In the regime of extensive jet atomization as in this work, a large spread of droplet velocities up to Δv/v ~ 5% has previously been observed [43]. Thus, with a characteristic droplet velocity on the order of 200 m/s, the droplets may have significant relative collision velocities of ~10 m/s, which are sufficient to produce rapidly spinning products. Further downstream, presumably a few mm away from the nozzle, the number density of the gas and droplets decrease, the collision rates decrease, and the angular momenta of individual droplets remain constant further downstream. 65 Although we are currently unable to provide a quantitative model of the processes close to the nozzle, it is instructive to consider the evolution of a droplet driven at some angular velocity as opposed to free droplets with a constant angular momentum. The corresponding driving force may originate from the aforementioned collisions. The prolate branch on the stability curve of driven droplets is unstable at constant ω [24]. Driven droplets will climb along the axially symmetric branch until they reach the bifurcation point at Ω = 0.56 (Figure 4) at which point they will enter the unstable prolate branch. Here, further elongation of the droplets occurs, culminating in their fission. The stable configurations beyond the fission point correspond to two spherical droplets, each having one half the volume of the parent droplet [24]. On the other hand, scission of dumbbell-shaped droplets will result in strongly deformed fragments. Related theoretical studies of nuclear fission indicate that such fragments contain sizable angular momentum [44, 45]. Similar to the parent droplets, daughter droplets will acquire angular momentum via collisions. The fission cycle continues until sufficiently small, stable droplets are formed or the droplets are far away from the nozzle, where the driving force diminishes. Because the occurrence of such a cycle is largely independent of the He isotope, the process should yield very similar values of <AR>, <Λ> and <Ω>, independent of the droplet size and composition. This model is also consistent with the trend apparent in Figure 2(c) that larger droplets exhibit larger values of <AR>. Conclusions In this work, bosonic 4 He and, for the first time, fermionic 3 He droplets are studied by single-pulse X-ray coherent diffractive imaging. Statistics of the droplets’ sizes, aspect ratios, reduced angular momenta and reduced angular velocities are compared for superfluid 4 He droplets and normal fluid 3 He droplets. Since the experiments only give access to projections of droplets onto the detector plane, estimates are made to determine the true average axes and aspect ratios. It 66 is found that, although the superfluid droplets have a much higher average angular momentum, the two kinds of droplets have very similar average aspect ratios and, thus, similar average reduced angular momenta and reduced angular velocities. This observation may result from the formation of the droplets through turbulent nozzle flow and the atomization regime in the immediate vicinity of the nozzle. We conjecture that the droplets’ rotation is driven by a combination of the liquid flow velocity gradient inside the nozzle and collisions close to it, leading to elongation and, ultimately, fragmentation into daughter droplets, which may undergo repeated collision- elongation-fragmentation cycles. Future studies will shed more light on the origin of angular momentum in droplets produced via fluid fragmentation. A large number of studies discuss the fragmentation of classical liquids upon jet expansion [46, 47]. However, to the best of our knowledge, the amount of angular momentum contained in the resulting droplets remains unknown. It is therefore interesting to see that the jet atomization of classical liquids produces highly rotating droplets similar to quantum He droplets. The availability of the large 3 He droplets suitable for single-pulse diffraction experiments also opens additional research directions. Vortex-induced cluster aggregation has so far been unique to superfluid 4 He. It is of high interest to expand diffraction experiments to non-superfluid 3 He and study the aggregation patterns in rotating fermionic droplets. Dopant aggregation mechanisms and the morphology of the phase separation in rotating mixed 3 He/ 4 He droplets presents another frontier [48]. 67 References [1] D. Verma et al., Shapes of Rotating Normal Fluid 3 He Versus Superfluid 4 He Droplets in Molecular Beams. Physical Review B 102, 014504 (2020). [2] L. F. Gomez et al., Shapes and Vorticities of Superfluid Helium Nanodroplets. Science 345, 906 (2014). [3] C. Bernando et al., Shapes of Rotating Superfluid Helium Nanodroplets. Physical Review B 95, 064510 (2017). [4] D. Rupp et al., Coherent Diffractive Imaging of Single Helium Nanodroplets with a High Harmonic Generation Source. Nature Communications 8, 493 (2017). [5] B. Langbehn et al., Three-Dimensional Shapes of Spinning Helium Nanodroplets. 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[1] Introduction At low temperatures, the angular momentum of a superfluid droplet may be stored in quantized vortices or, as much less frequently discussed, in capillary waves. Both phenomena are fundamentally different from rigid body rotation (RBR) that described rotation of classical droplets, which poses intriguing questions regarding their relative contributions when comparing classical and quantum mechanical kinematics. Previous experiments in our group and by others have established the existence of oblate axisymmetric and triaxial prolate superfluid droplets with shapes that are found in classical droplets [2-5]. However, previous experiments could not independently probe the angular momentum and angular velocity of rotating droplets, which is imperative for understanding their laws of motion. We address this challenge by simultaneously determining the angular momentum L, the angular velocity ω, and the corresponding shapes of the rotating droplets via imaging of both the shapes of 4 He droplets and the vortex configurations they contain. The shapes of classical droplets undergoing RBR follow a universal stability diagram in terms of reduced angular momentum ΛRBR and reduced angular velocity ΩRBR, as shown in Figure 5.1 [2-5]. The reduced quantities are linked to the angular momentum LRBR and angular velocity ωRBR according to Eqs. 5.1 and 5.2, respectively [4, 5], and are shown below: 73 𝛬 'I' = ! "#%&' ! 𝐿 'I' (5.1) 𝛺 'I' = ' &' " #% 𝜔 'I' (5.2) Here, σ is the surface tension of the liquid; ρ its mass density; and R the radius of a spherical droplet with the same volume as the deformed droplet. With increasing ΛRBR, the droplets become more oblate. At ΛRBR > 1.2, however, they assume prolate shapes such as pseudo-triaxial ellipsoids, capsules, and dumbbells. Figure 5.1. Stability diagram for classical droplets executing RBR. Red solid traces indicate stable shapes for specific combinations of reduced angular velocity Ω RBR and reduced angular momentum Λ RBR [4, 5]. The left branch corresponds to oblate axisymmetric shapes, the right branch to prolate shapes. The unstable portion of the branch is indicated by a dashed curve. The blue curve corresponds to droplets rotating solely through capillary wave motion [6]. Black circles mark the locations of classical droplets with the same aspect ratios as the superfluid droplets studied in this work. 74 Rotation of prolate superfluid droplets can be accomplished by capillary waves, which decay rapidly in classical droplets due to viscosity-induced energy dissipation but may be sustained indefinitely in superfluids. Capillary waves in levitating helium droplets have been investigated experimentally and theoretically [7, 8]. Traveling capillary waves give rise to a new branch in the stability diagram as shown by the blue line in Figure 5.1 [6]. Prolate superfluid droplets likely contain both capillary waves and vortices, whose combined action determines the droplet shape. The arrangement of vortices in rotating, nonaxisymmetric systems represents a general problem that has been discussed theoretically in connection with liquid helium [6, 9] and dilute Bose- Einstein condensates (BECs) [10-13]. The observation, however, of vortices in rotating systems with large asymmetries remains challenging [14]. We posit that in axisymmetric, oblate superfluid droplets, angular momentum originates solely from quantized vortices that arrange in a triangular lattice [15, 16] similar to those previously observed in BECs [17-19]. In contrast, the angular momentum in prolate superfluid droplets has significant contributions from vortices and capillary waves. We find that in capsule- shaped droplets, vortices form a triangular lattice, whereas they are arranged along elliptical contours in ellipsoidal droplets. The combined action of the vortex lattice and the capillary waves yields droplet shapes close to those in isochoric classical droplets rotating at the same angular velocity. Experimental Experiments were performed using the LAMP end station at the Atomic, Molecular, and Optical (AMO) beam line of the Linac Coherent Light Source (LCLS) XFEL [20, 21]. Sub- micrometer sized helium droplets containing NHe = 10 9 –10 11 atoms are produced via fragmentation of liquid 4 He expanded into vacuum through a 5 μm nozzle at a temperature of 4.5 K and a backing 75 pressure of 20 bar [15, 22, 23]. The droplets evaporatively cool in vacuum to T = 0.37 K [23, 24], becoming superfluid at T ≈ 2.17 K. To visualize vortices, the droplets are doped with Xe atoms, which cluster along the vortex cores [15, 25-28]. The XFEL is operated at 120 Hz with a photon energy of 1.5 keV (λ = 0.826 nm) and a pulse width of~100 fs.. Small angle diffraction images are recorded on a p-n charge-coupled device detector centered along the XFEL beam axis. The patterns predominantly contain column density information of the scattering object in the direction perpendicular to the detector plane. Previously, it was shown that densities of clusters encapsulated in helium droplets can be reconstructed from X-ray diffraction patterns using the so-called droplet coherent diffractive imaging algorithm (DCDI) [29]. The algorithm exploits the fact that the scattering amplitude from a doped He droplet can be calculated as the sum of the known amplitude from the He droplet itself and the unknown amplitude from the dopants. Since the experiments were performed at small scattering angles [15, 27, 29, 30], the diffraction amplitude can be well approximated by the two- dimensional Fourier Transform of the column density of the scattering object in the direction perpendicular to the detector plane. Accordingly, the DCDI algorithm requires the column density of the droplet as an input. For a spheroidal droplet, the column density can be calculated analytically from the half axes of the droplet’s projection on the detector plane. The half axes are obtained from the elliptical droplet diffraction contours as described previously [15, 29, 30]. The density is then normalized to fit the intensity of the diffraction signal from the droplet [29]. However, the shapes of prolate droplets cannot be obtained from the half axes because their shapes are not given by any analytical expression. Therefore, we apply a numerical determination of the column densities for prolate droplets. First, the contour of the droplet’s projection onto the detector plane is determined. It is obtained 76 via 2D inverse Fourier Transform (IFT) [30]. In Reference [30], before applying IFT, the phase was assigned to each of the diffraction contours from the droplet, which is a tedious procedure. Therefore, in this work we employ the IFT of the square root of the diffraction intensity itself, which yields similar results in a less cumbersome manner. Because the input is positive throughout, it can be seen as a smooth positive offset function decreasing towards larger scattering angles with an oscillating component. It is easy to show that the period of oscillation is a factor of two smaller than that in the diffraction amplitude. Therefore, the obtained droplet contour is scaled down by a factor of two. Figure 5.2 (a) shows the result of the application of the IFT to the square root of the diffraction intensity shown in Figure 5.2 (b). The diffraction pattern contains a bright streak along the diagonal of the image. This streak originates from the nearly parallel sides of the droplet contour in (a). The diffraction also shows closely spaced, elongated contours, indicating a highly deformed droplet. 77 Figure 5.2. (a) Droplet shape reconstruction from inverse Fourier Transform of the square root of the diffraction image in Panel (b). The result of the IFT is shown in a linear color scale. There are two contours of similar intensity with a superimposed red line in between, which marks the droplet contour. The signals inside the droplet are caused by a positive offset, Bragg peaks, and other features of the original diffraction as discussed in the text. Figure 5.2 (a) shows the contour of the droplet in light blue and yellow colors, the long axis of which is aligned diagonally. Intensities inside of the contour are due to the IFT of the offset function, Bragg spots and other low frequency features in the diffraction. Similar to previous work 78 [30], the procedure yields multiple contours because some low-frequency signals from the droplet contour are not recorded due to the central hole in the detector. The true droplet contour is expected to coincide with the center of gravity of the modulus of these contours. Figure 5.2 (a) shows a set of two contours of comparable amplitude. Therefore, the droplet’s contour lies in between, as indicated by the red line. Once the droplet contour is obtained, the density of the droplet is filled in axisymmetrically with respect to the long axis of the contour. This density is an approximation, because the droplet b-axis perpendicular to the rotation axis is slightly larger than the c-axis parallel to the rotation axis as noted in the main text [30]. Therefore, the actual density distribution of the prolate droplet is not strictly axisymmetric with respect to the long axis. Results Diffraction patterns from doped droplets with slightly oblate axisymmetric, triaxial pseudo-ellipsoidal, and capsule shapes are presented in a logarithmic color scale in Panels (a1), (b1) and (c1) of Figure 5.3, respectively. Panels (a2)–(c2) of Figure 5.3 show the corresponding column densities of He (blue) and Xe (red and yellow) obtained from the diffraction patterns using the droplet coherent diffractive imaging (DCDI) algorithm [28, 29]. The diffraction pattern in Figure 5.3(a1) exhibits a circular ring structure close to the center and a speckled pattern in the outer region, which is due to scattering off the He droplet and embedded Xe clusters, respectively. The projection of the droplet contour onto the detector plane is circular with a radius of 308 ± 6 nm [15, 27, 29]. The droplet in Figure 5.3(a2) contains 12 vortex cores that are aligned along the droplet rotational axis and extend between opposite surface points. The filaments appear shorter than the droplet diameter, indicating that the rotational axis is tilted with respect to the X-ray beam by ∼0.3 rad. 79 Panels (b1) and (c1) of Figure 5.3 exhibit elongated diffraction contours. Most noticeably, two parallel lines of four Bragg spots each are arranged symmetrically with respect to the streak. We assign the Bragg spots to X-ray scattering off a lattice of vortex-bound Xe clusters [15]. Figure 5.3 (c2) shows that the droplet contains ∼50 vortices arranged in four rows parallel to the droplet’s long axis. The Xe clusters in Figure 5.3 (c2) appear as compact spots, although they are likely filaments as in Figure 5.3 (a2). Therefore, the filaments must be aligned perpendicular to the detector plane and the droplet’s angular momentum along the X-ray beam. Figure 5.3. Diffraction patterns from Xe-doped droplets with various shapes: (a1) axisymmetric, nearly spherical, (b1) triaxial pseudo-ellipsoidal, and (c1) capsule shaped. The horizontal stripe in (a1)–(c1) results from the gap between the upper and lower detector panels. Panels (a2)–(c2) show column densities retrieved via the DCDI algorithm. The basis vectors of the vortex lattice in (c2) are shown in the upper right corner of the panel. The droplet in Figure 5.3(b2) has a pseudo-ellipsoidal shape and the pattern of vortices within resemble the droplet’s outer contour. The Xe clusters show some elongation along the long axis, indicating a tilt of ≤ 0.15 rad around the droplet’s intermediate axis. Since the angular momentum vectors of the droplets in Panels (b) and (c) of Figure 5.3 are aligned approximately along the X-ray beam, the droplet contours yield the actual values of the major and intermediate 80 half axes, a and b, as well as the droplet aspect ratio: AR = a/b. The minor half-axis c, parallel to the rotational axis, cannot be directly determined. Discussion 5.4.1 Droplet shape and Vortex Configurations Panels (a2), (b2) and (c2) of Figure 5.3 showcase the evolution of vortex arrangements with changing droplet shapes. Because of the small number of fast rotating droplets crossing the X-ray beam focus with proper orientation, only the two diffraction patterns shown in Panels (b1) and (c1) were suitable for reliable image phase retrieval. These two images are considered representative of droplets with similar aspect ratios since any nonequilibrium configurations decay before the droplets reach the interaction point with the imaging X-ray beam [15]. In the spheroidal droplet in Figure 5.3(a2), the vortices are distributed across the entire volume and at regular distances from each other. In Figs. 5.3(b2) and 5.3(c2), the vortices only occupy the central part of their respective droplets, leaving ∼100 and ∼150 nm wide boundary regions devoid of vortices, respectively. The vortices in the prolate droplet are arranged in four rows along the droplet’s long axis. The vortex lattice is reciprocal to that of the Bragg spots and is defined as: 𝑅 U⃑ = 𝑚 ! 𝑟 ! 𝚤 ⃑ ! +𝑚 ; 𝑟 ; 𝚤 ⃑ ; (5.3) where m1 and m2 are integers, r1 = 111 nm, r2 = 100 nm, and 𝚤 ⃑ ! and 𝚤 ⃑ ; are unit vectors with 𝚤 ⃑ ! perpendicular to the streak and 𝚤 ⃑ ; oriented at a 67° angle relative to 𝚤 ⃑ ! as illustrated in Figure 5.3(c2). The vortex lattice is triangular, but not equilateral as in the idealized 2D case [5] and as previously observed in axisymmetric rotating He droplets [15] and BECs [17-19]. These observations suggest that the droplet shapes and vortex patterns within are closely interrelated. Upon detailed inspection, the results in Figure 5.3(c) of the main text indicate some deviation between the calculated RBR and measured superfluid shapes. At ΛRBR = 1.5 as in Figure 81 2(c2), the classical shape has a weak depression along the b - axis, which gives rise to additional nodal structures in the diffraction signal that appear as multiple, radially emanating streaks (see SM of Bernando et al. [30]). The diffraction pattern in Figure 5.3(c1) does not exhibit any such nodes, which may indicate that the presence of the vortex array and capillary waves leads to a stabilization of the straight outline of the droplet. Moreover, the diffraction maxima along the streak are not oriented perpendicular to the streak axis, as is the case for RBR shapes (see SM from Bernando et al. [30]). Instead, the pattern is tilted counterclockwise. This tilt likely indicates that the two sides of the droplet are not exactly parallel but include a small angle on the order of a few degrees. Thus, the droplet in Fig 5.3(c2) is likely non-centrosymmetric, as is the case in classical droplets, which may be another indication of the interaction of the vortex lattice with the droplet shape. This wedging effect cannot be seen in the droplet contour in Figure 5.2(a) obtained from the Fourier Transform of the diffraction intensity, [30] which exclusively leads to axisymmetric shapes. 5.4.2 Kinematic Parameters of the Droplets We now compare the angular momentum and angular velocity in superfluid droplets in Figs. 5.3(b2) and 5.3(c2) to those of their classical counterparts. From the measured aspect ratios in Figs. 5.3(b2) and 5.3(c2), we first obtain the values of the reduced quantities ΛRBR and ΩRBR [4, 5, 30] for the corresponding classical rotating droplets and then calculate LRBR and ωRBR according to Eqs. (5.1) and (5.2). The angular momentum due to vortices LVORT is obtained from the number of vortices NV and their location within the droplet [31] as detailed later. The angular velocity of the vortex array is obtained from the average vorticity as [32]: 𝜔 JKLM = > - N ; (5.4) 82 in which nv is the areal number density of vortices and κ = h/m4 is the quantum of circulation, derived from Planck’s constant h and the mass of the 4 He atom m4. The angular velocity of the superfluid droplet is given by ωSF = ωVORT because the vortex array must be stationary with respect to the droplet contour at equilibrium. The angular momentum due to capillary waves LCAP is obtained as the product of the effective irrotational moment of inertia of the droplets multiplied by ωSF [33]. Contributions of Xe tracers to the angular momentum are neglected since the total mass of the embedded Xe atoms in Figure 5.3 is ∼100 times smaller than that of the He droplet and the vortices doped with Xe atoms mostly occupy the central region of the droplet. The impact of doping on vortex kinematics in spherical droplets has been studied previously [27, 34]. It was concluded that bare and Xe- decorated vortices have similar angular momenta when the vortices are located near the droplet’s center as is the case here. Table 5.1 lists the kinematic parameters of the rotating superfluid droplets in Figure 5.3. The total angular momentum in superfluid prolate droplets [(b2), (c2)] is defined as LSF = LVORT + LCAP. The LCAP/LVORT ratio decreases with decreasing aspect ratio, A, ultimately meeting the condition LCAP = 0 in axisymmetric droplets (a2). The analysis also reveals that, within the accuracy of our estimates, LSF ≈ LRBR and ωSF ≈ ωRBR for isochoric classical and vortex-hosting superfluid droplets of similar shape. More generally, the results in Table 5.1 suggest that rotation of large, vortex-hosting superfluid droplets can be described in terms of a reduced angular velocity and a reduced angular momentum in a similar fashion as the rotational motion of their classical counterparts. Quantity a2 b2 c2 a, nm 308 466 883 b, nm 308 290 389 83 AR 1.00 (exp) 1.007 (from ΩRBR) 1.61 2.27 NHe 2.7×10 9 3.1×10 9 1.45×10 10 ΛRBR 0.13 1.31 1.5 LRBR/(NHe×ℏ) 4.8 50 73 ΩRBR 0.08 0.52 0.46 ωRBR, rad/s 2.0×10 6 1.24×10 7 4.9×10 6 NV 12 50 50 SV, nm 170 77 109 nV, m -2 4.0×10 13 2.0×10 14 9.8×10 13 ω SF=ω VORT, rad/s 2.0×10 6 9.7×10 6 4.9×10 6 LVORT/(NHe×ℏ) 4.8 (RB = R) 31 (RB = .75∙R) 30 (RB = .78∙R) LCAP/(NHe×ℏ) 0 18 48 LSF/ (NHe×ℏ) 4.8 49 78 Table 5.1. Kinematic parameters for the droplets in Figure 5.3. The following describes all entries in Table 5.1 and how they were obtained. The first three rows of Table 5.1 show the long and short half-axis of the droplet and the aspect ratios, a, b and AR, which were obtained from the diffraction contour patterns [15, 29, 30]. The number of atoms in the droplet, NHe, is determined from the volume of a classical droplet having same a and b and the number density of liquid helium (nHe = 2.18×10 28 atoms/m 3 ), see Figure 8 in Bernando et al. [30]. The values of reduced angular momentum, ΛRBR, and reduced angular velocity, ΩRBR, are derived from AR, assuming the shapes of RBR, see Figure 8 in Bernando et al. [30]. The angular momentum, LRBR, and angular velocity, ωRBR, are deduced using Eqs. 5.1 and 5.2, respectively. LRBR is expressed in units of ℏ per He atom, and ωRBR is expressed in radians per second. For the droplet in column a2, ΛRBR and ΩRBR cannot be obtained in the same way, because its AR is unity within the experimental accuracy. Therefore, in this case, the results are based on ωSF, which is set to ωRBR from which ΛRBR and ΩRBR are deduced using Eqs. S1 and S2. The angular velocity of the 84 lattice is given by Eq. 5.4. Here, h is Planck’s constant and m4 is the mass of the 4 He atom. For droplet (a2), it is assumed that vortices are uniformly distributed throughout the spherical droplet. Thus, 𝑛 O = 6 - =' + . For droplets (b2) and (c2), a triangular lattice of vortices is assumed, so 𝑛 O = ; √A 𝑆 O G; , where SV is the average nearest neighbor distance between vortices. For (a2), nV is used to calculate SV, assuming a uniform distribution of vortex cores inside the spherical droplet. For b2, SV is estimated from the spacing between vortices along the left part of the vortex lattice. In c2, SV is derived from the vortex positions in a lattice as described by Eq. (5.3). In droplets (a2) and (c2), the number of vortices NV is found by direct count, and in (b2), NV is approximated as the product of nV and the area filled with vortices, which is approximated by an ellipse with half axes of 360 nm and 218 nm. The expression for the total angular momentum of NV linear vortices in spherical droplets was described by Nam et al. [31]. Here, we use their expression: 𝐿 BQ = ;&N A ∑ (𝑅−𝑟 0 ) " + 6 - 0R! (5.5) in which R is the radius of the droplet, ri is the displacement of the i th vortex from the center of the droplet, ρ is the mass density of superfluid helium, and 𝜅 is the quantum of circulation as described above. For this analysis, it is assumed that vortices with constant areal density fill a circular region with radius RB around the droplet center so that there is a boundary region of the droplet devoid of vortices between RB and R. In the limit of large numbers of vortices, the summation in Eq. (5.5) can be replaced by integration. This yields the angular momentum due to vortices in units of ℏ per He atom as: F ./ 6 '( ℏ = ;6 0 E ' 1 GT' + G' 2 + U 1 + ' " ' 2 + (5.6) 85 Here, we use Eq. (5.6) for a sphere to approximate non-spherical solutions for droplets (b2) and (c2). The value of RB is estimated by measuring the area filled with vortices, AV, dividing this area by the area of the droplet cross section, AD, and taking the square root of the result as: 𝑅 I = 𝑅' V 0 V 3 . The angular momentum of capillary waves in a triaxial ellipsoid rotating around the z-axis is given by [33]: F 456 6 '( ℏ = 𝜔 BQ 𝑚 < ] T〈W + 〉G〈X + 〉U + 〈W + 〉Y〈X + 〉 ^. (5.7) The estimated values of LSF contain some error due to uncertainties in determining LVORT and LCAP. One is associated with the use of Eq. (5.6), which is strictly only valid for spherical droplets. No corresponding analytical expression is available for droplets of general shape. In addition, LCAP is calculated using an irrotational moment of inertia for a liquid constrained to an ellipsoid [33], whereas the actual droplet shape is not strictly ellipsoidal. Furthermore, the values for ωSF are obtained from Feynman’s equation, 𝜔 BQ = > - N ; , which is valid for large 2D arrays, whereas this work considers finite 3D vortex arrays within droplets. We note, however, that the perfect match of ωSF and ωRBR as obtained for droplet in Figure 5.3 (c2), where nV can be determined with high accuracy, as well as the results of the DFT calculations described in the following text, indicate that the application of Feynman’s equation is likely accurate. To rationalize the similarities between the shapes of classical and superfluid droplets, we may recall that, for a given L, RBR results in the smallest kinetic energy. A vortex array in the central part of a droplet creates a region of liquid that effectively moves as in RBR and minimizes the required capillary engagement. Essentially, the prolate droplet attempts to approach RBR by combining the effect of vortices and capillary waves. It appears that for each L, there is an optimal partitioning into LVORT and LCAP that minimizes the total energy of the rotating droplet. 86 5.4.3 Density Functional Calculations To ascertain how the shape of a superfluid droplet changes with the number of vortices at constant angular momentum, Manuel Barranco, Marti Pi, and Frnacesco Ancilotto (BPA) have carried out density functional theory (DFT) calculations [11,36,37]. Computationally costly 3D calculations can only be performed for small droplets, which cannot support a large number of vortices [6]. Therefore, we study a freestanding, deformable 4 He cylinder, rotating around its symmetry axis. Calculations are performed in the corotating frame [35]. The total angular momentum is fixed at LSF/NHe = 7.83ℏ. Barranco, Pi, and Ancilotto (BPA) consider a self-bound superfluid 4 He cylinder rotating around its symmetry z-axis with a constant angular velocity w. The details of the calculations are described in Reference [35], where vortices in axisymmetric rotating cylinders were considered. Here, BPA extended that work to select configurations with the vortex-hosting cylinder allowed to deform into shapes with non-circular cross sections. Barranco, Pi, and Ancilotto assumed a uniform density along the z-direction in the calculation, which implies that the resulting vortices always remain linear. A complex order parameter, Y, is found to represent the superfluid helium state at zero temperature [35, 36], whose square modulus is the atomic density. The DFT equation is formulated in a rotating frame-of-reference with constant angular velocity w (corotating frame), which follows from the variational minimization of the energy, and is solved by looking for stationary solutions. Vortex structures and positions are optimized during the functional minimization, after having been phase-imprinted, as described in Refs. [35, 36]. Simultaneously, BPA allow for deviations from the circular cross section (“prolate” rotating cylinders) using the same method as previously applied to prolate spinning droplets [6], i.e. solve the variational equation by imposing a given value for the angular momentum per atom Lz and iteratively find the 87 associated value of the rotational frequency w. Classically, such fixed Lz calculations correspond to torque-free drops with an initially prescribed rotation (“isolated drops”) and result in stable prolate configurations like the ones shown in Figure 5.4. Figure 5.4. Calculated equilibrium density profiles of a deformable 4 He cylinder rotating around its symmetry axis at fixed L SF/N He = 7.83ℏ for different numbers of vortices. Streamlines are shown in black. The positions of the vortex cores are marked by red dots for visualization. The color bar shows the density in units of Å −3 . Figure 5.4 shows the results of the calculations with different numbers of vortices, ranging from NV = 2 to NV = 11. The deformation of the cylinder increases for decreasing numbers of vortices, as capillary waves contribute more angular momentum. Note that the energies of the configurations in Figure 5.4 are similar; they differ less than 0.01 K per atom, with the NV = 6 configuration [(Figure 5.4 (c)] representing the global minimum. Each Panel in Figure 5.4 shows the computed LCAP=LSF ratio and the cylinder’s aspect ratio and ω. In cylinders with relatively 88 small AR = 1.1 and 1.5, the vortices are arranged into elliptical patterns, whereas they form elongated arrays for larger aspect ratios, in agreement with the experimental observations in Figure 5.3. Figure 5.4 illustrates how the effects of vortices and capillary waves confer the appearance of RBR to the motion of the superfluid. Each panel shows the superfluid streamlines, illustrating the different irrotational velocity fields: the streamlines dominated by vortices wrap around the vortex cores, whereas those associated with capillary waves terminate at the surface. Figure 5.4(d) shows an elongated droplet containing eight vortices arranged in a pattern reminiscent of a distorted square lattice with an average nearest-neighbor distance of d = 4.7 nm and an areal vortex density of 1/d 2 = 4.5 × 10 16 m −2 . According to Eq. (5.4), the angular velocity is ω = 2.26 × 10 9 rad/s, in excellent agreement with the DFT result of ω = 2.23 × 10 9 rad/s. Thus, the results of the DFT calculations lend further support for using Eq. (5.4) to determine ω of rotating He droplets from the finite vortex arrays they contain. Conclusions This combined experimental and theoretical study provides the first direct comparison between the shapes, angular velocities, and angular momenta of rotating classical and superfluid droplets. The results indicate that the equilibrium figures of superfluid droplets hosting vortices may be described in a similar fashion as those of classical, viscous droplets by a series of oblate and prolate shapes that evolve along curves of stability as a function of reduced angular momentum Λ and reduced angular velocity Ω. In axisymmetric oblate superfluid droplets, all angular momentum is stored in triangular lattices of quantum vortices that extend throughout the entire droplet volume. In prolate droplets, however, capillary waves can contribute a substantial and, for large aspect ratios, even dominant amount of angular momentum. It appears that prolate superfluid 89 droplets approach the classical equilibrium figures through a combination of vortices and capillary motion, whereby the vortex arrangements are restricted to the central region of the droplet. Future experiments at high-repetition rate ultrafast X-ray light sources will provide larger sample sizes and explore more expansive ranges of Λ, Ω, and R. More advanced theoretical work is required to study angular velocities and angular momenta with higher accuracy and to derive a more detailed stability diagram of superfluid droplets from the measurements. Furthermore, probing smaller droplet sizes will reveal possible finite size effects. Corresponding work on self- bound quantum droplets in mixed BECs may provide complementary information by studying more dilute systems, potentially with greater control over their rotational states [39,40]. References [1] S. M. O. O'Connell et al., Angular Momentum in Rotating Superfluid Droplets. Physical Review Letters 124, 215301 (2020). [2] S. Chandrasekhar, The Stability of a Rotating Liquid Drop. Proc. Roy. Soc. London A 186, 1 (1965). [3] S. Cohen, F. Plasil, and W. J. Swiatecki, Equilibrium Configurations of Rotating Charged or Gravitating Liquid Masses with Surface-Tension .2. Annals of Physics 82, 557 (1974). [4] R. A. Brown and L. E. Scriven, The Shape and Stability of Rotating Liquid-Drops. Proceedings of the Royal Society of London Series A 371, 331 (1980). [5] S. L. Butler, M. R. Stauffer, G. Sinha, A. Lilly, and R. J. Spiteri, The Shape Distribution of Splash-Form Tektites Predicted by Numerical Simulations of Rotating Fluid Drops. Journal of Fluid Mechanics 667, 358 (2011). [6] F. Ancilotto, M. Barranco, and M. Pi, Spinning Superfluid He-4 Nanodroplets. Physical Review B 97, 184515 (2018). 90 [7] D. L. Whitaker, M. A. Weilert, C. L. Vicente, H. J. Maris, and G. M. Seidel, Oscillations of Charged Helium Ii Drops. Journal of Low Temperature Physics 110, 173 (1998). [8] L. Childress, M. P. Schmidt, A. D. Kashkanova, C. D. Brown, G. I. Harris, A. Aiello, F. Marquardt, and J. G. E. Harris, Cavity Optomechanics in a Levitated Helium Drop. Physical Review A 96, 063842 (2017). [9] A. L. Fetter, Vortex Nucleation in Deformed Rotating Cylinders. Journal of Low Temperature Physics 16, 533 (1974). [10] M. O. Oktel, Vortex Lattice of a Bose-Einstein Condensate in a Rotating Anisotropic Trap. Physical Review A 69, 023618 (2004). [11] P. Sanchez-Lotero and J. J. Palacios, Vortices in a Rotating Bose-Einstein Condensate under Extreme Elongation. Physical Review A 72, 043613 (2005). [12] S. Sinha and G. V. Shlyapnikov, Two-Dimensional Bose-Einstein Condensate under Extreme Rotation. Physical Review Letters 94, 150401 (2005). [13] N. Lo Gullo, T. Busch, and M. Paternostro, Structural Change of Vortex Patterns in Anisotropic Bose-Einstein Condensates. Physical Review A 83, 063632 (2011). [14] K. Deconde and R. E. Packard, Measurement of Equilibrium Critical Velocities for Vortex Formation in Superfluid-Helium. Physical Review Letters 35, 732 (1975). [15] L. F. Gomez et al., Shapes and Vorticities of Superfluid Helium Nanodroplets. Science 345, 906 (2014). [16] F. Ancilotto, M. Pi, and M. Barranco, Vortex Arrays in Nanoscopic Superfluid Helium Droplets. Physical Review B 91, 100503(R) (2015). [17] J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates. Science 292, 476 (2001). [18] A. L. Fetter, Rotating Trapped Bose-Einstein Condensates. Reviews of Modern Physics 81, 647 (2009). 91 [19] L. Pitaevskii and S. Stringari, Bose-Einstein Condensation and Superfluidity. Bose- Einstein Condensation and Superfluidity 164, Oxford University Press (2016). [20] L. Struder et al., Large-Format, High-Speed, X-Ray Pnccds Combined with Electron and Ion Imaging Spectrometers in a Multipurpose Chamber for Experiments at 4th Generation Light Sources. Nuclear Instruments & Methods in Physics Research Section a-Accelerators Spectrometers Detectors and Associated Equipment 614, 483 (2010). [21] K. R. Ferguson et al., The Atomic, Molecular and Optical Science Instrument at the Linac Coherent Light Source. Journal of Synchrotron Radiation 22, 492 (2015). [22] L. F. Gomez, E. Loginov, R. Sliter, and A. F. Vilesov, Sizes of Large He Droplets. Journal of Chemical Physics 135, 154201 (2011). [23] R. M. P. Tanyag, C. F. Jones, C. Bernando, D. Verma, S. M. O. O'Connell, and A. F. Vilesov, in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, edited by A. Osterwalder, and O. Dulieu (Royal Society of Chemistry, Cambridge, 2018), p. 389. [24] M. Hartmann, R. E. Miller, J. P. Toennies, and A. Vilesov, Rotationally Resolved Spectroscopy of Sf6 in Liquid-Helium Clusters - a Molecular Probe of Cluster Temperature. Physical Review Letters 75, 1566 (1995). [25] F. Dalfovo, R. Mayol, M. Pi, and M. Barranco, Pinning of Quantized Vortices in Helium Drops by Dopant Atoms and Molecules. Physical Review Letters 85, 1028 (2000). [26] G. P. Bewley, D. P. Lathrop, and K. R. Sreenivasan, Superfluid Helium - Visualization of Quantized Vortices. Nature 441, 588 (2006). [27] C. F. Jones et al., Coupled Motion of Xe Clusters and Quantum Vortices in He Nanodroplets. Physical Review B 93, 180510 (2016). [28] O. Gessner and A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Droplets. Annual Review of Physical Chemistry 70, 173 (2019). [29] R. M. P. Tanyag et al., Communication: X-Ray Coherent Diffractive Imaging by Immersion in Nanodroplets. Structural Dynamics 2, 051102 (2015). 92 [30] C. Bernando et al., Shapes of Rotating Superfluid Helium Nanodroplets. Physical Review B 95, 064510 (2017). [31] S. T. Nam, G. H. Bauer, and R. J. Donnelly, Vortex Patterns in a Freely Rotating Superfluid. Journal of the Korean Physical Society 29, 755 (1996). [32] R. P. Feynman, Chapter Ii Application of Quantum Mechanics to Liquid Helium, C. J. Gorter Ed., North-Holland Publishing Company: Amsterdam, 1955: Vol. 2, p. 17. [33] F. Coppens, F. Ancilotto, M. Barranco, N. Halberstadt, and M. Pi, Capture of Xe and Ar Atoms by Quantized Vortices in He-4 Nanodroplets. Physical Chemistry Chemical Physics 19, 24805 (2017). [34] C. Bernando and A. F. Vilesov, Kinematics of the Doped Quantum Vortices in Superfluid Helium Droplets. Journal of Low Temperature Physics 191, 242 (2018). [35] F. Ancilotto, M. Pi, and M. Barranco, Vortex Arrays in a Rotating Superfluid He-4 Nanocylinder. Physical Review B 90, 174512 (2014). [36] F. Ancilotto, M. Barranco, F. Coppens, J. Eloranta, N. Halberstadt, A. Hernando, D. Mateo, and M. Pi, Density Functional Theory of Doped Superfluid Liquid Helium and Nanodroplets. International Reviews in Physical Chemistry 36, 174512 (2017). 93 Mixed 3 He/ 4 He Droplets Introduction Mixtures of bulk 3 He and 4 He liquids provide a vivid example of phase separation due to quantum mechanical effects [1, 2]. The two isotopic fluids are miscible at higher temperatures, T, but they separate into stratified 3 He-rich and 4 He-rich phases below 0.87 K. Close to T = 0 K, the 4 He-rich phase contains ~6.6% 3 He, whereas the 4 He content in 3 He-rich phase is negligible [1]. Figure 1, plotted using data from Refs. [3, 4], illustrates the temperature dependence of the phase separation. It was proposed that phase separation starts from the creation of a 3 He-rich nucleus which could be produced via thermal or quantal nucleation [2] or inside the cores of quantum vortices within the mixed supersaturated superfluid state [2, 5-8]. The container walls may also act as nucleation centers. However, no attempts were made to determine more complex non- equilibrium morphologies of the bulk phases. Here, we expand the study of the phase separation to freely-rotating He droplets a few hundred nanometers in diameter. Previously, mixed 3 He/ 4 He droplets a few nanometers in diameter were studied via infrared spectroscopy of the embedded atoms and molecules. It was found that molecules reside in a 4 He-rich phase immersed in the 3 He-rich phase, which has a smaller surface tension coefficient [9, 10]. On the other hand, Ca atoms were found to sit at the interface [11]. The temperature of mixed droplets, T ≈ 0.15 K, was obtained from rotationally resolved spectra of the embedded molecules and complexes and is assumed to be the same in this work [9, 10]. Density functional theory (DFT) calculations yielded a spherical shell structure for such mixed droplets [10-13]. The equilibrium shapes of classical rotating droplets consisting of two immiscible fluids have recently been investigated by numerical solution of the Navier-Stokes 94 equations by S. Butler [14], yielding similar shapes in the inner and outer phases. Quantum mechanical DFT calculations on nanometer sized rotating mixed 3 He/ 4 He droplets by Pi et al. [15] yielded a variety of the shapes of the inner 4 He-rich phase at different angular velocities such as a sphere in the droplet's center, a torus and multi-lobed entities. Here, 3 He/ 4 He phase separation is studied via small-angle X-ray scattering. We found that mixed droplets with X ≈ 0.5 do not show any signs of the phase separation. However, results for X ≈ 0.75 indicate phase separation in which the 4 He-rich phase forms one or two inner droplets displaced from the center of the outer 3 He droplet, configurations which are at variance with the previous theoretical studies. Figure 6.1. The phase diagram of 3 He/ 4 He mixtures using data from Chaudhry et al. [4] and Qin et al. [3]. The concentration of 3 He is given as X = N 3/(N 3+N 4). Arrows indicate possible trajectories droplets may take upon evaporative cooling in vacuum. The arrows are tilted to indicate predominant evaporation of 3 He atoms. 95 Experimental The experiments were performed at the Atomic, Molecular and Optical (AMO) end station using the LAMP instrument of the Linac Coherent Light Source (LCLS) X-ray free electron laser (XFEL) [16, 17]. Helium droplets were produced by fragmentation of a mixture of liquid 3 He and 4 He expanded into vacuum through a 5 µm orifice at temperature T0 = 2.1 K and backing pressure P0 = 20 bar [9, 18], see elsewhere for more information [19-21]. In vacuum the droplets cool to T ≈ 0.15 K [9, 18] via evaporation within few microseconds. The focused XFEL beam (~2 μm in diameter) crosses the He droplet beam ~70 cm downstream from the nozzle after ~ 3.5 ms flight time. The XFEL was tuned to 1.5 keV (λ = 0.826 nm) with a pulse duration of ~100 fs FWHM and energy of ~1.5 mJ. Due to the short pulse duration, the diffraction pattern captures the instantaneous configuration of the He droplet. The pattern was detected by a pn-charged-coupled device (pnCCD) located 735 mm downstream from the interaction point along the XFEL axis. Small angle scattering diffraction images predominantly contain column density information of droplets in the direction perpendicular to the detector plane. In addition to diffraction images, time of flight (TOF) mass spectra of He + and small Hen + splitter ions were recorded from each droplet exposed to X-rays. A He flowing cryostat, Advanced Research Systems model LT-3-110, was used to obtain the low nozzle temperatures required to produce 3 He/ 4 He droplets. A gas recycling system was employed during these experiments to account for the considerable cost of 3 He, as described in Appendices 2 and 3. The He gas was continuously collected from the backing pumps, purified, pressurized by a two-stage metal membrane compressor, the output of which was resupplied to the nozzle. Due to the limited amount of 3 He gas available, mixtures having different ratios of 3 He and 4 He could not be prepared ahead of the experiments. Therefore, the isotopic mixture was made 96 during the beamtime by adding 4 He gas in small amounts to the 3 He gas inside the recycling apparatus, which was then circulated for 15-20 minutes for mixing. However, that time was not sufficient for the complete equilibration of the mixture and the fraction of 3 He, in the expanding fluid, X0, varied during the time of the experiments. Therefore, similar to Reference [11], the X values for each of the imaged droplets were estimated from the mass spectra, as described later. The measurements with mixed droplets were obtained at two nominal values of X0 ≈ 0.8 and 0.5 in the entire circulated mixture; those values are deemed accurate within one significant figure. Results 6.3.1 Density Reconstructions from Diffraction Patterns Panels (a1) and (a2) of Figure 6.2 show the central portion of diffraction patterns of mixed He droplets (X = 0.69±0.04 and 0.74±0.04, respectively, see Section 6.3.2 for details). The images exhibit a pattern of elliptical rings characteristic of spheroidal droplets [22, 23] with some pronounced intensity modulation along the rings. About half of the ~30 intense images obtained at X0 = 0.8 show similar azimuthal intensity modulation. In contrast, the diffraction of neat 3 He and 4 He droplets have rings of constant intensity [20, 22-26]. Diffraction rings devoid of azimuthal modulations were also obtained in this work at X0 = 0.5. The modulation indicates an inhomogeneous number density distribution inside the mixed droplets due to phase separation in which the morphology of the inner droplet does not follow the spheroidal shape of the outer droplet. The sensitivity of the diffraction to the phase separation comes from the different refractive index in the 3 He- and 4 He-rich phases. The scattering intensity from a droplet approximately scales as a number density of He atoms squared. The number densities neat 4 He and 3 He at low temperature are known to be 2.18×10 28 m -3 [27] and 1.62×10 28 m -3 [28], 97 respectively. The number densities in the two phases at equilibrium are tabulated in Reference [4]. Figure 6.2. Representative diffraction patterns of mixed 3 He/ 4 He droplets and corresponding traces along the major and minor axes of the diffraction. Panels (a1) and (a2) show the central (400×400 pixels) portion of the diffraction patterns. Continuous traces in Panels (b1)-b2) and (c1)-c2) show the radial intensity along the corresponding wedges in the upper Panels, which were scaled as described in the text. Dotted traces are calculations based on the reconstructed density, see in the text. The abscissas are in pixel units. One pixel corresponds to the change of X-ray wavevector by 7.76×10 -4 nm -1 . 98 In addition to the azimuthal modulation, the diffraction shows some pronounced modulation in the radial direction. The bottom Panels of Figure 6.2 show intensity plots along the major (red) and minor (blue) axes of their respective diffractions. The intensities were averaged over the highlighted filled angles (40 o full width) which are illustrated in the top Panels. In both Figure 6.2 (a1) and (a2), the maxima along the major diffraction axis have noticeably smaller intensities as compared with those along the short axis. Diffraction intensity of a sharp-edged homogeneous spheroid scales as q -4 , where q is the modulus of the change of the wavevector of X-rays upon scattering, which is in turn approximately proportional to the distance from the diffraction center. Therefore, to visualize the behavior at larger q, the averaged radial intensities were multiplied by q 4 [29, 30]. Similarly, to account for the effect of the aspect ratio, 𝒶, the blue traces along the diffraction short axis were multiplied by 𝒶 < . The traces tend to become jagged at large q due to photon shot noise. An error reduction (ER) algorithm [30, 31] was used to reconstruct the density of the mixed droplets, as described in Section 6.4.2. The application of the algorithm to the diffractions of pure 3 He droplets with half axes less than ~250 nm gave centrosymmetric density patterns with radial density profiles as expected. Figure 6.3 i) and ii) show the reconstructed density profiles from the diffraction in Figure 6.2 (a1) and (b1), respectively, as well as those obtained from other diffraction images in Panels iii) - xii). In general, axially symmetric diffraction images yield density profiles with maximum density in the middle of the droplet, such as in Figure 6.3 Panels vi), ix) and xi). On the other hand, diffraction images showing some noticeable azimuthal modulation yield density profiles with maxima shifted from the center. Some images with articulated radial modulation along the diffraction short axis, such as in Figure 6.2 (a1), give rise to two approximately 99 symmetric density maxima along the long axis of the droplet, seen in Panels i), iii) and x). It is seen that in most cases, the density maxima lay on the major axis of the droplet. Figure 6.3. Density reconstruction of mixed droplets shown in a linear color scale. Panels i), ii) are reconstructed from the diffraction images in Figure 6.2 (a1) and (a2), respectively. The distance between each tick mark is 100 nm. We assign the off-center density maxima to an inner droplet of 4 He-rich phase immersed in an outer 3 He droplet. In some images, such as Figure 3 iv), vi), ix), xi), and xii), the maxima were found close to the center of the corresponding droplets. It is feasible that those droplets still contain a 4 He-rich pocket, but the droplet was imaged parallel to the line that connects the inner droplet and the center of the outer droplet. Due to this uncertainty of the imaging angle, the physical distance of the pocket center from the droplet center could not, in general, be obtained. It is, however, likely that some images, such as Figure 3 i), ii) iii), v), vi), vii) and x), where the inner droplets are far from the center and lie on the long axis of the droplet, stem from the fortuitous events when the droplet was imaged approximately perpendicular to the line connecting the centers of the droplet and of the pocket. Accordingly, most of the droplets have a single off-center 4 He- rich pocket, whereas about 10% of droplets studied in this work have two symmetrical pockets, such as schematized in Figure 6.4 (a) and b), respectively. Model density calculations in support of this conclusion are detailed in Section 6.4.3. All images except in Figure 6.3 x) have a small 100 aspect ratio of a < 1.15, with the average over the 20 most intense images of = 1.054. A similar was found during the same beamtime measurements for neat 3 He droplets [23] which was assigned to rotating spheroidal droplets. Recent DFT calculations [32] confirmed that the sequence of rotating 3 He droplet shapes as a function of angular momentum agrees with that of rotating classical droplets. Therefore, most of the droplets observed in this work have shapes close to a spheroid. One may argue that the actual shape of the mixed droplets is triaxial due to the asymmetry caused by the 4 He pocket, however, this effect cannot be quantified based on the small angle scattering results in this work. The only exception is the droplet's image in Figure 6.3 x) which has an aspect ratio of 1.39 and likely corresponds to a triaxial prolate shape [22, 23, 26, 33]. Figure 6.4. A pictogram of inner droplet shapes seen in this work. The inner droplets are shown as spheres with a linear color scaling. The outline of the outer droplet is given in black and is based on the droplet in Figure 3 i). 6.3.2 Determining 3 He Content of Droplets via Time-of-Flight Spectra For each diffraction image, an ion time-of-flight (TOF) mass spectrum was also recorded. The ions are generated by the ionization of the He droplets by X-rays. Unfortunately, the TOF spectra corresponding to the brightest diffraction images, such as in Figure 6.2 (a1), (a2) of the main text, were partially saturated such that the isotope ratio could not be determined. Therefore, 101 to determine the ratio of the numbers of 3 He and 4 He atoms in the mixed droplets, F, we used a weaker hit within less than 10 seconds to those in Figure 6.2. The fraction of the 3 He atoms is then calculated as X=F/(1+F). Figure 6.5 (a) shows the TOF spectrum for these “weak” hits a few seconds after recording the images in Figure 6.2. Herein, Droplet 1 refers to the droplet imaged in Panel (a1) of Figure 6.2, and Droplet 2 refers to the droplet imaged in Panel (a2) of Figure 6.2. The peaks are marked by inverted blue triangles and labeled by the possible ion contributors. The peak around ~0.9 µs is due to protons, likely from residual H2 or H2O in the interaction region. This peak was present in each TOF trace that was studied. The graph in Figure 6.5 b) shows a plot of time versus the square root of mass per charge, m/z. The black squares and red circles each denote a peak in Figure 6.5 (a), while the blue line shows the linear fit for the peaks. The heights of the atomic peaks can be used to estimate the 3 He/ 4 He ratio. Because the peaks due to 3 He + and 4 He + are not completely resolved the true ratio of the numbers of the 3 He and 4 He atoms can be within the range of X = 0.69±0.04 and X= 0.76±0.04 for Droplets 1 and 2, respectively. The low limit is obtained via the ratio of the peak heights from the baseline, and the high limit is the ratio of the peak heights from the valley between the 3 He and 4 He peaks. Similar analyses were performed for other droplets in Figure 6.3 in the main text that did not have saturated TOF traces. Those results, along with size and spatial orientation information for the droplets is presented in Table 6.1. 102 Figure 6.5. Panel (a) shows the TOF traces for weak events: one recorded ~4 seconds after the event for Droplet 1 (black trace) and the other recorded ~10 s before the event for Droplet 2 (red trace) in the main text. The peaks in the TOF are indicated with an inverted blue triangle, and their assignments are given with black text. Panel b) shows a graph of time of flight versus the square root of mass per charge used to verify peak assignments in the TOF spectrum shown in (a). The blue line shows the linear fit to the data points. 103 Table 6.1. The size of the major and minor axes, orientation of the major axis with respect to the x-axis of the image, and the X values of the droplets displayed in Figure 6.3. Discussion 6.4.1 Comparison Between Theory and Experiment Recently, the morphologies of core-shell rotating droplets were calculated [14, 15]. Classical Navier-Stokes calculations show that the shape of the outer phase follows very similar stability diagram in terms of scaled reduced angular momentum, Λ* , and angular velocity, Ω*, calculated using same equations as for single-phase droplets with effective values of the density and surface tension [14]. Pursuant to the 3 He/ 4 He droplets, the values of Λ* and Ω* are very similar to the corresponding reduced Λ and Ω for single component droplets. The droplets with an aspect ratio of 1.05 (1.11), and major axes of 200 nm (182 nm), such as Droplets 1 and 2, have Panel Major axis (nm) Minor Axis (nm) Orientation of Major Axis (°) X i) Droplet 1 200 190 18.67 0.69±0.04 ii) Droplet 2 182 164 -68.81 0.76±0.04 iii) 193 189 -45.78 - iv) 154 148 13.50 - v) 108 102 71.77 - vi) 114 111 34.37 - vii) 94 89 -50.92 0.80±0.01 viii) 178 167 -38.06 - ix) 133 131 42.88 - x) 241 173 55.13 0.85±0.01 xi) 83 80 9.04 0.71±0.01 xii) 119 116 58.29 0.75±0.01 104 corresponding values of Λ ≈ 0.35 (0.55) and Ω ≈ 0.25 (0.30). The angular velocity can be estimated to be 1.1×10 7 rad/s (1.5×10 7 rad/s). The actual values of the aspect ratio, and therefore Λ and Ω, are likely larger, but could not be determined due to the unknown angle between the figure axis and the X-ray beam [23]. Classical calculations were done on rotating mixed droplets with Λ*= 1.34 and densities and surface tensions mimicking a 4 He (core) - 3 He (shell) droplet [14]. The results show that the core and shell drops are oblate pseudo-spheroids with similar aspect ratios of ~1.46. The denser-phase core resides at the center of the less dense droplet [14]. It is reasonable to assume that at smaller Λ*, classical calculations will also give core-shell droplets of very similar shape, but smaller aspect ratio. This finding is somewhat surprising because moving the inner droplet off-center will increase the total moment of inertia of the system and thus decrease the rotational energy at constant angular momentum. The results of this experimental work agree with symmetry breaking due to centrifugal effects. DFT calculations on mixed droplets containing N3 = 6000 and N4 = 1500 at rest form a spherical 4 He-rich core with radius of ~2.5 nm immersed into an outer 3 He droplet with radius of ~5 nm [15]. At Λ* < 1, the outer droplets are oblate pseudo-spheroids with aspect ratios close to those in classical droplets. In the absence of vortices, the inner superfluid droplet is spherical and bears no angular momentum. At Λ* = 0.33, the solution with a quiescent 4 He phase has a total energy smaller than that containing a central vortex by ~50 K. However, at Λ* = 1.33, the solution with a singly quantized, ~1 nm diameter vortex becomes energetically favorable, whereas those with a quiescent core or containing a multiply quantized vortex have higher energy. At Λ* = 1 – 2, DFT calculations predict a rich variety of oblate and prolate shapes of the inner phase [15]. In particular, calculations give the two symmetric pocket configurations in rotating prolate droplets having Λ>1.6 and aspect ratio of larger than about 2. This finding is broadly in agreement with the 105 results in Figure 6.3 x) for the prolate droplet with the true aspect ratio of ≥1.39. However, here we found two pocket configurations in droplets having a much smaller aspect ratio of ≈1.05, such as in Figure 6.3 i). Some discrepancies may be related to the manifestation of the finite size effects in small droplets of about 10 nm in diameter that were invoked in the DFT calculations, whereas the experiments in this work involve much larger droplets ~400 nm in diameter. The fact that only the centrosymmetric configurations were found in the classical and DFT calculations relates to the explicit symmetry constraint in the calculation [14, 15, 34]. Another striking difference to the calculations is the observation of the two symmetric inner droplets within the outer droplet with small aspect ratio of ~1.05. In particular, the diffraction in Figure 6.2 (a1) could be quantitatively modeled by two spherical inner droplets 75 nm in diameter each and symmetrically shifted with respect to the center by 110 nm, see Section 6.4.3. This configuration may correspond to the largest possible shift since the distance between the phase separation interface and the outer surface of the 3 He droplet is less than ~10 nm. The formation of the two pockets of 4 He-rich phase allows the system to further increase its moment of inertia, leading to further decrease of the kinetic energy at constant angular momentum. However, the fission of the 4 He core is associated with an increase of the surface energy. We performed model calculations to compare the total energy of the system containing a single spherical 4 He-rich inner droplet displaced to the boundary of the outer spherical 3 He-rich droplet with that when the 4 He-rich phase is equally divided between the two spherical inner droplets, such as in Figure 6.7 (a1). The phase compositions and the corresponding surface tension coefficients used were the same as in the bulk at 0.15 K (σ3= 1.55 10 -4 N/m [35], σ34=0.22 10 -4 N/m [36]), and assumed that the entire system rotated as a rigid body. The results show that the fission of one spherical droplet into two droplets becomes energetically favorable at a factor of 106 ~4.5 times higher angular momentum than the estimates for the droplet in Figure 6.3 i). If the inner droplets are devoid of vortices and bear no angular momentum via internal rotation, the threshold angular momentum decreases, but only to a factor of 2.5 times larger than the experimental estimate. A smaller σ34 may facilitate the fission. The value of σ34 decreases with temperature, and is ~0.11 10 -4 N/m at 0.5 K [36]. However, it is unlikely that the temperature of the droplets in the beam deviates that much from our previous measurements [9, 10, 18] and calculations based on the evaporation rate [21]. Even with phase densities and surface tensions characteristic for 0.5 K, the threshold angular momentum is a factor of 1.7 times larger than the experimental estimate. The width of the 3 He- 4 He interphase region where the number density of 4 He drops from 90% to 10% of its value, was found in DFT calculations to be about 0.7 nm [11, 13], which is much less than the extension of one pixel in Figure 6.3 of ~7 nm and resolution of the density reconstruction of ~20 nm [30]. The experimental diffraction in Figure 6.2 (c) shows periodic oscillations up to q = 450 pixels, indicating that the phase boundary has a width of less than ~6 nm, in agreement with the results of DFT calculations. This conclusion is based on the study of a mixed droplet model as discussed in the SM, with different width of the interface. Analysis of the diffraction pattern could give the ratio of the number density of He atoms in the two phases. Quantitative agreement of the experiment and calculations in Figure 6.7 (a1)- (d1) is consistent with the number density ratio of 1.30 as in the bulk at 0.15 K [4]. The smaller ratio of 1.22, such as in equilibrium at T = 0.5 K [4] will give rise to the noticeable deviation of the measured and calculated diffraction pattern, as discussed in the SM. Therefore, we concluded that the diffraction pattern in Figure 6.2 (a) is consistent with the number density ratio of 1.30±0.05. 107 Our estimates show that the lowest energy configuration at angular momentum consistent with the aspect ratio in Figure 6.3 i) should be a single 4 He-rich inner droplet displaced from the center, while the two-lobed configuration is likely metastable. The formation mechanism of the metastable configuration remains to be elucidated. A possible mechanism involves the formation of two nucleation centers which, upon evaporative cooling, evolve into the two separate 4 He-rich droplets such that the system becomes kinetically trapped in this metastable state. The details of the phase separation may also depend on the cooling path of the mixed droplets. For instance, the system may cross into the forbidden region via the superfluid phase or the normal fluid phase as it is schematized in Figure 6.1 by blue arrows. At higher X, the path crosses into the phase separation (“forbidden”) region directly from the normal fluid phase, whereas at lower X the path first crosses into the superfluid phase before entering the phase separation region. It is interesting that no azimuthal modulations of the diffraction were observed upon measurement of the X0 = 0.5 mixture, although the TOF mass spectra recorded simultaneously with some strong diffraction images indicated equal content of 3 He and 4 He atoms in the droplet (X ≈ 0.5). This observation may indicate the lacking or less-than-complete degree of phase separation. While the exact trajectories of mixed droplet cooling remain unknown, more 3 He atoms are evaporated than 4 He atoms due to the lower binding energy of 3 He compared to 4 He [1]. Therefore, the initial value of X upon expansion into vacuum is greater than upon the cooling as obtained from the TOF spectra. Figure 6.1 shows that the cooling of the droplets with X >~0.66 proceeds through the non-superfluid phase before entering the phase separation region at 0.7 - 0.8 K[1]. On the other hand, droplets with X ≈ 0.5 first pass through the mixed superfluid phase in the range of temperatures from ~1.2 to 0.7 K[1]. In previous bulk experiments, only small levels of 108 supersaturation of the 4 He-rich phase of ~1% or less have been reported [37-39]. Cooling in a jet expansion is about 8 orders of magnitude faster and may provide an opportunity for observation of some non-equilibrium phases. It is conceivable that 4 He atoms in the non-superfluid phase are more likely to form one or few critical nuclei that continue growing at lower temperature, whereas the formation of a critical nuclei in the superfluid phase may be inefficient during ~3.5 ms time of flight from the nozzle to the interaction point. This scenario may result in the formation of a metastable homogeneous 3 He/ 4 He mixture in the elusive forbidden range, see Figure 6.1, which has been searched after in previous works in the bulk [37-42]. The rotation at angular velocities as found in this work is a small perturbation to the phase separation energy balance. The amount of 3 He atoms within the 4 He rich phase at T = 0 K is determined by their Fermi energy, which reaches ~0.3 K in equilibrium [1]. An additional energy per 3 He atom due to rotation in a droplet with radius of R = 200 nm could be estimated based on the angular velocity of ~10 7 rad/s to be ~4×10 -3 K, about a factor of 100 less than the Fermi energy of the 3 He atoms in the 4 He-rich phase. Therefore, we conclude that the equilibrium content of the phases is not substantially influenced by the rotation, and the concentration of the 3 He atoms is approximately constant throughout the 4 He-rich phase. 6.4.2 Density Reconstruction via the Error Reduction Algorithm. Upon analysis of many diffraction images, it appears that the results from the upper detector plate were unreliable. The top left quadrant contains a rectangular region of low intensity such as in Figure 6.2, the shape of which changes from image to image. In addition, the diffraction rings show a noticeable jump in intensity on the boundary between the top left and right quadrants. Therefore, to facilitate density reconstructions, we replaced the data in the upper part of the 109 detector with the lower plate upon rotation by 180 degrees with respect to the diffraction center. This procedure is justified because the imaginary part of the He refractive index is negligible, and the diffraction images must be centrosymmetric at the small scattering angles in this work. The density is reconstructed from the diffraction image upon application of the error reduction algorithm [30, 31]. It consists of multiple consecutive 2D inverse Fourier transform (IFT) of the diffraction amplitude and Fourier transform (FT) of the density, each represented by 1025×1025 matrices. In reciprocal space, the modulus amplitude of the diffraction was kept for the matrix elements belonging to the gap and the central cut of the detector and set to the observed values otherwise. In both cases, the phase was kept as calculated from the FT. The droplet contour was determined from the fits of the diffraction rings to the analytic expression for the homogeneous reference droplet [22]. In the density domain, the real positive part from the output was kept inside the droplet contour and set to zero outside of the contour or for any negative values. Additionally, the reconstruction of the small density on the droplet's contour is unreliable. So, that density was set to the values calculated for a homogeneous reference droplet, resulting in the same scattering intensity as in the observed diffraction. The algorithm starts with some small trial density within the contour, which usually was taken to be a fraction of a reference droplet, similar to that used in Tanyag et al. [30], or some small random density points within the contour. We found that the process typically converges within ~1,000 iterations to a density profile that is consistent across different initial conditions. The algorithm was tested using neat 3 He droplets whose diffraction consists of elliptical rings of constant azimuthal intensity [23]. Their column densities must be centrosymmetric and are given by that of constant-density spheroids [30]. Figure 6.6 shows the results of the 2D reconstruction together with the density profiles along the main axis averaged over 60-degree 110 wedges. The plots also show the density profiles obtained from the spheroid model and scaled to give the observed diffraction intensity. It is seen that for the droplets with a half axis of less than about 250 nm, the reconstructions give centrosymmetric density patterns with radial density profiles in close agreement with those obtained from the spheroid model. The deviation typically affects the central region where the reconstruction often gives flatter profiles with density of up to about 10% less than expected. The density reconstruction for droplets larger than ~250 nm (~30 pixels in Figure 6.6) is less reliable due the effect of the missed central part of the diffraction. 111 Figure 6.6. Left column: Reconstructed 2D density for some representative 3 He droplets of different size. Right column: Corresponding density profiles along the minor (blue) and major axis (red) integrated within 60 o wedges. In the left column the abscissa is given in pixel units with a single pixel corresponding to 7.8 nm. Continuous and dotted curves correspond to the spheroid of constant density scaled to give the observed diffraction intensity and result from the reconstruction, respectively. 6.4.3 Modeling Experimental Results Here, we present modeling of the diffraction in Panel (a1) of Figure 6.2 in the main text, the reconstruction of which yields the two-lobed density profile in Figure 6.3 i). To verify the 112 reconstruction, 3 different 3 He droplet- 4 He pocket configurations were modeled: a single offset inner lobe, 2 symmetric inner lobes, and a torus shape. A torus-shaped configuration was observed, for example, in a rapidly rotating quantum gas in a trap [43]. The models consist of 3 He density from the host droplet plus the excess density from the 4 He-rich phase. The model assumes the ratio of the He number densities in the 4 He-rich and 3 He-rich phases of 1.30 as at equilibrium at 0.15 K. Each droplet has the same size and aspect ratio as in Figure 6.3 i), see Table S1. The two-lobe excess density models 4 He lobes as spheres of 75 nm (10 pixels) in diameter offset along the long axis of the hosting 3 He droplet by 110 nm (14 pixels). The one-lobe model was created in a similar way by only including one of the spheres. The modeled torus has a major radius of 112 nm (14.5 pixels) and a minor (tube) radius of 90 nm (11.5 pixels). The torus is tilted π/16 radians with respect to the x-z plane since the flat sides of a torus that is perpendicular to the XFEL beam would produce streaks that were not observed in this work [22]. Panels (a1) - (a3) of Figure 6.7 show the column density of the models of the two-lobe, one-lobe, and torus models, respectively, in a linear color scale. In (b1)-(b3), the diffraction intensities were calculated as a modulus squared of the 2D Fourier transforms of the corresponding densities using 1025×1025 matrices. It is seen that the diffraction from all three models have similar patterns. However, the relative intensities of particular features differ, which enables the selection of the most appropriate model. Similar to Figure 6.2 of the main text, the highlighted regions represent the areas that were summed to produce the radial intensity dependences shown at the bottom of the figure. Note that the model lacks the 45-degree tilt present in the experimentally obtained image in Figure 6.2 (a1). Panels (c1) and (d1) show the comparison of the intensities along the minor (red) and major (blue) axes of the diffraction from the model (continuous curves) with the experimental one (dashed curves). The diffraction intensity was 113 scaled to give the same intensity of the third maximum in Panel (d1). The same scaling was used in all panels. It is seen that the model amplitude and radial modulation patterns are in quantitative agreement with the experimental one for both along the minor and major axis, indicating that the density for the two-lobe model is very similar to the experimental one. This good agreement lends further credence to the two-lobe morphology, given complete phase separation. In Panels (c2), (d2) and (c3), (d3) the outcomes of the one-lobe and torus models were compared to the two-lobe model. Panels (c2) and (d2) compare the long and short axes, respectively, of the diffractions of the two-lobe and one-lobe density models. The two-lobe model is illustrated by a solid trace, while the one-lobe model is given by dashes. Similarly, Panels (c3) and (d3) compare the two-lobe and torus models, with the two-lobe model being represented by a solid trace and the torus being represented by dots. It is seen that the torus model does not match the intensity modulations present in the two-lobe droplet. The one-lobe model has some of the same features as the two-lobe model. However, the intensity modulation is less articulated as compared with the two lobed model, which is in good agreement with the experiment, as shown in Panels (b1)-(d1). In addition, the X values for the model densities Panels (a1) – (c1) were obtained to be 0.75, 0.88 and 0.35, respectively. It is seen that the X = 0.75 for the two-lobe model is in reasonable agreement with the ratio obtained from the TOF spectra of 0.69±0.04, see Table 6.1. On the other hand, the one lobe and torus models give values that are too large and too small, respectively. 114 Figure 6.7. (a1)-3): Densities representing 3 models for the morphology of the 4 He rich phase: two lobes, one lobe, and a torus in Panels (a1), (a2) and (a3), respectively. b1-3): the diffractions from the models in the top row displayed in a logarithmic color scale. The red and blue wedges represent the areas summed to produce the red and blue traces shown in the bottom of the figure. The traces shown compare the two-lobe model and experiment in (c1) and (d1), the two-lobe and one-lobe models (c2) and (d2), and the two-lobe and torus models in (c3) and (d3). The solid trace marks the two-lobe model, and the dashed traces mark the experimental trace in (c1) and (d1), the one-lobe model in (c2) and (d2), and the torus model in (c3) and (d3). One pixel corresponds to the change of wavevector by 7.76×10 -4 nm -1 . 115 The appearance of the diffraction pattern is sensitive to the ratio of the number density of He atoms in the two phases. For a quantitative assessment, we studied the outcomes of the two- lobe model using representative number density ratios of the 4 He-rich and 3 He rich phases in equilibrium, which are known to be 1.30 and 1.22 at T = 0.15 and 0.5 K, respectively [4]. Figure 6.7 shows the comparison of the radial modulation along the short and long diffraction axes in Panels (a) and b), respectively. The T = 0.15 K traces are shown in black, and the T = 0.5 K results are shown in red and blue for the traces along the short and long diffraction axes, respectively. The red trace was normalized to equal the first peak on the T = 0.15 trace for the short axis. This same normalization factor was used on the blue curve for the long axis trace. The comparison of the traces along the short diffraction axes show that the intensities of many peaks are higher with larger density ratio, characteristic for T = 0.15 K. In comparison, the change of the intensities along the long axis trace is smaller and mostly affect the first two peaks which experimental observation is obscured by the central hole of the detector. It is seen that while these results show that X-ray diffractive imaging is sensitive to the number density ratio of the phases, the rather large noise level in the present experimental diffraction does not allow very accurate determination. The noise level could be evaluated from the deviation of the measured and calculated peak heights in Figure 6.7 (d1). Accordingly, we estimated that the density ratio in the phases in Figure 6.3 i) of the main text is 1.30±0.05. Brighter diffraction images which may be obtained by using larger droplets, longer wavelength of the X- rays and higher brilliance of the next generation XFELs will enable more accurate determination of the phase content. 116 Figure 6.8. Intensity traces from 2-lobe models using representative density ratios of the 4 He-rich and 3 He- rich phases at T = 0.15 K (black traces) and T = 0.5 K (colored traces). The graphs with the red and blue traces are along the short and long axis of the diffraction, respectively. Conclusions This work reports the first experimental study of phase separation in rotating 3 He/ 4 He droplets via X-ray scattering. Diffractions from droplets consisting of about 50% 3 He and 50% 4 He do not show azimuthal modulations, which suggests homogeneity. On the other hand, 117 diffractions from droplets containing 75% 3 He and 25% 4 He show characteristic modulation patterns in the diffraction, which indicate phase separation. This discrepancy hints that homogenous phases may remain metastable in droplets with lower content of 3 He for as long as ~3.5 ms. In many droplets with larger 3 He content, the 4 He-rich phase forms a single inner droplet shifted from the center of the hosting 3 He droplet. This observation is at variance with the results of both classical and quantum DFT calculations, which place the inner droplet center- symmetrically. In addition, in some droplets the 4 He phase forms two symmetric, off-center inner droplets, which we tentatively assigned to a kinetically trapped metastable state originated from two independent nucleation centers. This observation possibly opens a way to study the single events of quantum nucleation and the kinetics of the ensuring phase separation. Indeed, the droplet beam- X-ray experiments described in this work enable imaging of the droplets during the large interval of time from ~1 μs to ~10 ms upon their formation, which could be varied by changing the speed of the droplets and the flight distance to the observation point. These unexpected experimental results hint at the rich physics that underly these phase separation experiments. This work may open two new fields of study: the configurations of self- contained multiphasic liquids and the effects of rotation on such systems. Future work should explore the phase separation in droplets of different sizes, originating at different temperatures and having different mixing ratios. Employing smaller X-ray energies of ~500 eV will enable attaining a factor of 100 brighter X-ray diffraction images and more accurate determination of the shapes and densities of the phases. Employing XUV lasers may also provide access to 3D configurations of mixed droplets. 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Reppy, and Richards.Rc, Low-Temperature Density and Solubility of He-3 in Liquid He-4 under Pressure. Physical Review 188, 384 (1969). [42] P. Seligmann, D. O. Edwards, R. E. Sarwinski, and J. T. Tough, Heat of Mixing and Ground-State Energy of Liquid 3 He- 4 He Mixtures. Physical Review 181, 415 (1969). [43] Y. Guo, R. Dubessy, M. d. G. de Herve, A. Kumar, T. Badr, A. Perrin, L. Longchambon, and H. Perrin, Supersonic Rotation of a Superfluid: A Long-Lived Dynamical Ring. Physical Review Letters 124, 025301 (2020). 122 Laser-induced Reconstruction of Ag Clusters in Helium Droplets This chapter uses text from our paper cited below. The experiment was performed by Luis Gomez, and my contribution to this work was in the data analysis. Luis F. Gomez, Sean M. O. O'Connell, Curtis F. Jones, Justin Kwok, & Andrey F. Vilesov. (2016). “Laser-induced Reconstruction of Ag Clusters in Helium Droplets”. The Journal of Chemical Physics, 145 (11), 114304. [1] Introduction Helium nanodroplets are a versatile spectroscopic cryo-matrix, and have been employed in many spectroscopic studies [2-6]. Some findings suggest that He may facilitate the formation of unusual, metastable clusters such as linear chains of HCN molecules [7] that small Mgn (n = 5-53) clusters form metastable weakly bound complexes in which individual metal atoms are separated by a He layer resembling a “bubble foam [8-10].” Given the unique properties of superfluid 4 He as a quantum solvent, it is of great interest to study the kinetics and mechanisms of particle combination in He nanodroplets and the structure of the resulting clusters. In the past several years, the focus of our studies of aggregation has shifted to very large He droplets consisting of 10 6 -10 10 atoms with corresponding diameters of about 0.1-1 µm [5, 11, 12]. These droplets are large enough to display bulk superfluid properties such as quantum vorticity [13-17], but small enough that finite size and surface effects can appreciably alter their physical nature. Droplets of this size can facilitate the formation of correspondingly large Ag aggregates containing up to 10 6 atoms [13, 18, 19]. The photoabsorption spectra of Ag clusters indicate a different structure of the clusters obtained in droplets of different size [18]. In smaller droplets of less than 30 nm in diameter and containing NAg < 300, the isolated absorption peak at 344 nm was assigned to the surface plasmon resonance of spherical silver nanoparticles. For larger droplets having diameters in the range of 30-150 nm and containing 1000–6000 Ag atoms, there was an 123 increase in the infrared absorption with a concomitant decrease in the UV absorption, an effect that was initially ascribed to a shift in the growth mechanism from a single-centered to multiple- centered aggregation [18]. Ultrafast X-ray free electron laser (XFEL) diffractive imaging of individual He nanodroplets doped with Xe atoms [14-17, 20] revealed that quantum vortices exist in droplets larger than about 200 nm in diameter. Vortices serve as dominant aggregation centers, changing the growth mechanism; the existence of multiple vortices results in the formation of widely spaced, filament-shaped clusters [13-17, 21-23]. It is feasible that small droplets of significantly less than 200 nm in diameter are devoid of vortices [24], but at present we are unable to give an accurate number for the threshold. During spectroscopic measurements of Ag clusters in He droplets [18], previous students noticed a pronounced saturation of the absorption signal above a certain threshold of incident laser power. This phenomenon may indicate reconstruction of embedded clusters upon sufficiently intense laser irradiation and may provide additional details of heat exchange between excited nano- clusters and the surrounding superfluid environment. In this work, we study the laser power dependence of the absorption by Ag clusters following the pulsed irradiation of the Ag clusters in He droplets at two different wavelengths. Photons at 532 nm excite the broad absorption peak characteristic of large Ag clusters, whereas 355 nm is close to the maximum of the surface plasmon resonance. Measurements were also conducted using a continuous wave (CW) laser operating at 532 nm. It was found that the absorption saturated at much lower laser pulse energy at 532 nm than at 355 nm, whereas the CW measurements did not show any saturation behavior. Such results suggest that the large clusters are ramified upon formation in the droplets then melt and reconstruct into more compact entities upon laser irradiation. 124 Experimental Figure 7.1. (a) Schematic of the He droplet beam vacuum apparatus. NZ—5 µm diameter nozzle, SK—1 mm diameter skimmer, PC1 and PC2—upstream and downstream pickup cells, respectively, SH—beam shutter, A1 and A2—6 mm diameter apertures, GV1 and GV2—gate valves, EI—electron impact ionizer, IB—ion bender, QMS—quadruple mass spectrometer. A pyroelectric detector, PD, was attached to SH. (b) Cross section of the oven inside of the pickup cell: CW—cold water jacket, CR—alumina ceramic crucible, RS—radiation shield, TF—tungsten filament, Ag—metallic silver. (c) Typical depletion dip upon laser excitation at t = 0, as measured at mass M = 8 at T 0 = 8 K upon 532 nm excitation. Weak secondary pulse at about t = 7 ms shows the effect of laser pulse heating of the nozzle. Figure 7.1(a) shows a schematic of the molecular beam apparatus, which is the earlier version of the setup described in Chapter 2 of this thesis. Helium nanodroplets of average radii 125 between 15 nm and 600 nm (⟨NHe⟩ = 3 × 10 5 – 2 × 10 10 ) are formed by expanding high purity (99.9999%) He gas at a pressure of 20 bar into vacuum through a 5 µm diameter nozzle at temperatures in the range T0 = 9.5 – 5.5 K [12]. A skimmer 0.5 mm in diameter collimates the droplet beam, which then captures Ag atoms during traversal of a heated alumina oven filled with Ag, see Figure 7.1(b). The average number of Ag atoms captured per He droplet, ⟨NAg⟩, has been estimated using measurements of the droplet beam attenuation [19]. The flux of He atoms transported by the droplets is monitored as a rise in the partial pressure of He, PHe, in the terminal UHV analysis chamber, with values typically in the range of 10 −8 -10 −7 mbar. Repeated capture of Ag atoms causes the average size of the droplets to decrease by evaporation of He atoms, resulting in a reduced flux and concomitant pressure drop, ∆PHe, in the terminal chamber. Equation 7.1 gives the average number of Ag atoms captured by droplets of initial average size,⟨NHe⟩, as a function of He beam attenuation, A = ∆PHe/PHe, upon pickup of Ag atoms, 〈𝑁 V1 〉 = 𝐴〈𝑁 3( 〉 8 '( 8 57 (7.1) in which EHe = 0.76 meV is the evaporation enthalpy of He atoms at T = 0.65 K [25] (the average temperature of droplets in the pickup cell [12]), EAg is the energy associated with the pickup of a single Ag atom, which combines the thermal energy of about 0.15 eV and the Ag–AgN cohesion energy released during the cluster formation. The inverse of the ratio of energies in Eq. 7.1 gives the number of evaporated He atoms per one added particle. Because of the possible formation of non-compact clusters, such as via multi-center or vortex-assisted aggregation, the cohesion energy cannot be accurately quantified without knowing the microscopic structure of the clusters. Therefore, similarly to Refs. [19] and [26], in this work we assumed that the addition of each Ag atom causes evaporation of 3500 He atoms from the droplet. All measurements were performed at the same attenuation, A = 0.7, to ensure a constant ratio of He to Ag atoms in the doped droplets 126 of about 1500. The five He droplet source temperatures employed, along with corresponding droplet sizes and number of doped silver clusters, are given in Table 7.1. T0, K 〈𝑁 3( 〉 〈𝑁 V1 〉 V, m/s S(355) cm 2 /J σ(355) Å 2 , pulsed S(532) cm 2 /J pulsed (cw) σ(532) Å 2 , pulsed (cw) 5.5 1.7 × 10 10 1.13 × 10 7 173 170 0.4844 -- (29.88) -- (0.08366) 6 3.1 × 10 8 2.1 × 10 5 175 170 0.4900 -- (49.31) -- (0.1381) 7 1× 10 7 6.7 × 10 3 200 120 0.3360 37 (67.27) 0.1036 (0.1884) 9 1.8 × 10 6 1.2 × 10 3 223 250 0.7000 12 (22.30) 0.0336 (0.06244) 9.5 3.3 × 10 5 220 236 --- --- --- (11.16) --- (0.03125) Table 7.1. Initial He droplet size, size of the obtained Ag clusters, and overall absorption cross section per Ag atom at 355 nm and 532 nm excitations. The values obtained with CW excitation at 532 nm are given in parenthesis. To evaluate the amount of energy absorbed by Ag clusters from the laser beam, the He droplet flux was measured 110 cm downstream from the pickup cell with a quadrupole mass spectrometer equipped with a crossbeam electron ionizer. The signal at m/z = 8 amu, I8, due primarily to He2 + splitter ions, was used to monitor the intensity of the droplet beam. The laser beam propagates anti-collinearly to the droplet beam, and its energy was measured with a calibrated pyroelectric detector, which could be placed on the beam axis inside the vacuum apparatus about halfway between the pickup cell and the mass spectrometer. The average fluence, F, is the quotient of the energy and the area of the 6 mm orifice in front of the detector and has 127 units of mJ/cm 2 for pulsed experiments. Irradiance, Φ, which has units of mW/cm 2 , was used in the case of the CW measurements. Following absorption of multiple photons by the Ag clusters and their concomitant heating of the droplet, a large number of He atoms evaporate from the droplet, resulting in a transient decrease in I8. The typical shape of this depletion dip following pulsed excitation is shown in Figure 7.1(c). The depletion lasts ∼5.5 ms, which is the time of flight of the droplets from the pickup cell to the ionizer of the mass spectrometer. The shape of the signal reflects changing of the effective laser fluence along the flight length. The tail of the dip corresponds to clusters excited close to the pickup cell at the center of the droplet beam. Such clusters interact with the central part of the laser beam, which has higher fluence, thus larger depletion at the tail of the dip. On the other hand, the initial part of the pulse originates from the even overlap of the droplet and laser beams, each about 6 mm in diameter close to the ionizer of the mass spectrometer; therefore, the averaged fluence is smaller, giving rise to the smaller level of the depletion signal. The dip was integrated to obtain the fractional depletion, 𝐷 = Z # 8 GZ # 9 Z # 8 , in which the superscripts i and f refer to the average signal before the depletion dip and during the dip, respectively. In the case of the CW laser measurements, the laser output was chopped electronically at a frequency of 50 Hz, and the depletion signal was measured using a lock-in amplifier, the output of which was multiplied by a factor of two to compare with the pulsed measurements, relating to the duty cycle of the signal. Estimates based on the evaporation rate from liquid helium and the heat capacity of the helium droplets of about 10 7 atoms show that upon heating, the temperature of the droplet returns to its stationary value of about 0.4 K within about 10 −5 s; i.e., much faster than the time of flight in this work. 128 Results Figure 7.2(a) shows the depletion of the signal, D, versus laser fluence, F, resulting from pulsed excitation of embedded Ag clusters of different sizes at 532 nm. In the smallest clusters with ⟨NAg⟩ = 340, D rises approximately linearly up to F ≈ 25 mJ/cm2. At higher F the signal reaches a constant level at D ≈ 0.20. For ⟨NAg⟩ = 2000, D again rises linearly up to a few mJ/cm 2 and becomes independent of F above D ≈ 0.14. The initial linear part of the dependence can be used to calculate the absorption cross section of the Ag clusters as discussed in the following. The slope, S, at small F is larger in the Ag2000 clusters, which indicates a larger absorption cross section per Ag atom than for the smaller Ag340 clusters. For ⟨NAg⟩ = 6.2 × 10 4 and 3.4 × 10 6 atoms, the signal rises sharply at F < 2 mJ/cm 2 and then continues growing slower, reaching D ≈ 0.12 at F = 40 mJ/cm 2 . In contrast to the smaller clusters, in the largest clusters the signal does not reach a clear saturation even at the highest F values employed. Figure 7.2(b) shows similar measurements of D at 355 nm. At F < 2 mJ/cm 2 , all four cluster sizes show similar linear dependences. At higher F, the dependence becomes sub-linear, reaching a value of D ≈ 0.8 for the largest clusters, 3.4 × 10 6 and 6.2 × 10 4 Ag atoms, at F ≈ 15 mJ/cm 2 . For smaller Ag2000 clusters, D ≈ 0.55. For the smallest Ag340 Ag1200 clusters, D shows a sharp change in slope at F ≈ 2 mJ/cm 2 , with a slower rise at higher values of F. Figure 7.3 shows the D vs. Φ dependences using CW excitation at 532 nm for five different cluster sizes, indicated in the inset. In comparison to the results obtained with pulsed excitation, the dependence of D is approximately linear in all cases through the range of laser irradiances measured and shows only small signs of sub-linear behavior at the highest irradiances. However, the slope, S, of the initial part of the curves decreases by about a factor of 3 in going from ⟨NAg⟩ = 2000 to 66. 129 Figure 7.2. Depletion of the M=8 signal with laser fluence obtained for droplets with initial average size of <N Ag> = 1.13 × 10 7 , 2.1 × 10 5 , 6700, 1200 produced at T 0 = 5.5, 6, 7, and 9 K used to grow Ag clusters of <N Ag> = 1.13 × 10 7 , 2.1 × 10 5 , 6700, and 1200 atoms, respectively. Pulsed laser excitation is at 532 nm and 355 nm in (a) and b), respectively. 0 5 10 15 20 25 0.0 0.2 0.4 0.6 0.8 <N Ag >= 1.13 x 10 7 <N Ag >= 2.1 x 10 5 <N Ag >= 6700 <N Ag >= 1200 Depletion Fluence, mJ/cm 2 b) 0 10 20 30 40 50 0.00 0.05 0.10 0.15 0.20 <N Ag >= 1.13 x 10 7 <N Ag >= 2.1 x 10 5 <N Ag >= 6700 <N Ag >= 1200 Depletion Fluence, mJ/cm 2 a) 130 Figure 7.3. Depletion in M=8 signal with laser irradiance for different Ag cluster sizes as measured upon continuous laser excitation at 532 nm. Discussion Pulsed excitation of Ag clusters occurs at a rate that is ~6 orders of magnitude larger than CW laser excitation. Whereas the pulsed excitation proceeds during the 7 ns laser pulse, the CW excitation occurs during the entire ~5.5 ms flight time between the pick-up cell and QMS ionizer. The plasmon excitations in Ag clusters dephase within ~10 fs [27], while the electron-electron and electron-phonon relaxation times are on the order of 1 ps [28, 29], both much shorter than either of the two excitation times. Thus, for the purpose of this discussion we assume that upon photon absorption its energy instantly releases as heat within Ag clusters. At the highest CW laser irradiance, 1600 mW/cm 2 , shown in Figure 7.3, the clusters accumulate a dose of radiation of ~9 mJ/cm 2 during their time-of-flight. This dose corresponds to about a quarter of the maximum 0 400 800 1200 1600 0.0 0.1 0.2 0.3 0.4 0.5 0.6 <N Ag > = 1.13 x 10 7 <N Ag > = 2.1 x 10 5 <N Ag > = 6700 <N Ag > = 1200 <N Ag > = 220 Depletion Irradiance, mW/cm 2 131 fluence used in the pulsed excitation shown in Figure 7.2(a) for which the depletion reaches a saturation level at about D = 0.14 for T0 = 7 K. In comparison, a much larger level of D = 0.40 is reached in the case of CW irradiation, for which the signal shows only small signs of deviations from linearity. The large difference in the depletion magnitudes between pulsed and CW excitations for equal amounts of absorption may indicate the bleaching of the 532 nm absorption in Ag clusters during the laser pulse as discussed in the following. 7.4.1 Absorption Cross Sections Figures 7.2(a) and 7.2(b) show that at small fluences, D is proportional to F. The corresponding values of the slope, S [cm 2 /mJ], can be used to obtain the per-atom absorption cross sections of the Ag clusters, σ[Å 2 ]. As described later, the clusters absorb the first photon at 0.4 K and become hot for the duration of laser pulses. Thus, the cross sections calculated for pulsed experiments are overall cross sections, which ignore any change of the cross section upon the increase of the cluster’s temperature. In the limit of small values, the magnitude of D can be expressed as follows: 𝐷 = Q%〈6 57 〉 [〈6 57 〉 5 8 '( (7.2) where ⟨NHe⟩A is the average number of He atoms in a doped droplet and β = 1.5 is a constant that is related to the ionization cross section [26]. From Eq. (7.2), noting that ⟨NHe⟩A = (1 − (a)⟨NHe⟩ and S = DF , the following expression is obtained: 𝜎 = 𝛽𝑆 !GV V 𝐸 V1 (7.3) Taking EAg = 2.7 eV and A = 0.7 the following numeric expression is obtained: 𝜎dÅ ; f = 2.8×10 GA 𝑆i @\ + ] j (7.4) 132 The factor of β = 1.5 comes from the assumed NHe 2/3 dependence of the ionization cross section upon electron impact in large droplets; the factor should be unity in the case of NHe dependence. Our recent measurements of attenuation coefficients for He indicate somewhat faster scaling, which can be approximated with NHe 1/β , with the average value of β = 1.25 for the He droplets obtained in the range of T0 = 7–9 K [12]. Therefore, the obtained values of σ in smaller clusters could be overestimated by up to 20%. In the case of the CW excitation the slopes [cm 2 /W] obtained from Figure 7.3 are divided by the irradiation time of t = L/vD, where vD is the velocity of the droplet beam [12] listed in Table 7.1, and L = 1.10 m is the distance from the pickup cell to the ionizer of the mass spectrometer. Table 7.1 shows the calculated slopes and cross sections for the five droplet sizes studied and the two wavelengths of light used in these experiments. The values obtained with CW excitation at 532 nm are shown in parentheses. Our previous measurements [18] show that the plasmon resonance at ~3.5 eV dominates the spectrum of Ag clusters formed in smaller droplets obtained at T0 = 9 K, indicating the formation of predominantly spherical, compact clusters. For comparison, σ = 0.45 Å 2 has been measured for (Ag70±5) + ionic clusters in the beam at a maximum absorption of about 3.7 eV [30]. Furthermore, σ for Ag300 clusters at the maximum of the plasmon band, 3.5 eV, was calculated to be ~0.4 Å 2 [18]. Here, we note that the values of σ obtained in this work should have a rather large uncertainty of up to a factor of 2, due to the unknown effective value of F along the droplet beam and the estimate made for EAg. The accuracy could be improved by measuring the laser beam profile and obtaining overlap integrals with the droplet beam. Additionally, a more accurate value for EAg could be found if the size and shape of the Ag clusters were known. In this work, however, 133 no attempt was made to quantify either of these values. On the other hand, the ratios of the σ obtained at different ⟨NHe⟩ are accurate to within about 20%. Table 7.1 shows that σ(355) is about a factor of 2 smaller for clusters with ⟨NAg⟩ = 2000 than for ⟨NAg⟩ = 340. This result is consistent with our previous observation of an emerging infrared band for larger clusters, which leads to a decrease in the intensity at the plasmon frequency [18]. It is seen that the σ(355) increases again in the largest ⟨NAg⟩ = 3.4 × 10 4 and 6.2 × 10 6 clusters. The cluster size dependence of the σ(532) is opposite and has a maximum at ⟨NAg⟩ = 2000 embedded and is less for both larger and smaller clusters. This behavior is expected if the infrared band borrows intensity from the plasmon resonance band. In addition, the values of σ(532) measured for CW excitation are approximately a factor of 1.5 larger than those for pulsed excitation at T0 = 7 K and 9 K, for which meaningful σ values could be extracted. Some deviation of the results is not unexpected in view of the different shapes of the CW and pulsed laser beams, which could not be quantified in this work. The ratio of the cross sections, R = σ(532)/σ(355), was found to be 0.3 and 0.05 for measurements at T0 = 7 K and 9 K, respectively. Calculations for spherical dense Ag clusters containing 6000 and 60 atoms give R = 0.02 and 0.03, respectively [18]. Therefore, in smaller droplets, the ratio approaches the prediction for spherical clusters, whereas in larger droplets σ(532) is a factor of 10 larger than expected. Even though values of R obtained from this experiment have large uncertainties, these uncertainties cannot account for the order of magnitude difference. This trend is consistent with a transition of the droplet size from a regime in which single, compact-cluster formation is favored to one in which multicenter cluster growth dominates, leading to cluster-cluster aggregates. The cluster-cluster aggregates show extensive broadband visible and near IR absorption due to the interaction between the plasmon modes of the clusters. 134 In larger droplets obtained at T0 = 6 and 5.5 K the R further decreases to ~0.3 and ~0.2, respectively, suggesting a further evolution of the growth mechanism to the paradigm of many compact clusters attached to quantum vortices, which eliminates the cluster-cluster aggregation found at intermediate droplet sizes. The structure of the Ag clusters attached to quantum vortices remains to be determined. Quantum vortices drive the formation of clusters into multiple elongated, wire-like structures [13, 14, 16, 17, 21, 22, 31]. The spectra of the elongated silver clusters show visible and IR absorptions [32] consistent with our previous results. Thus, it is possible that the broadband absorption in the large droplets is due to an ensemble of wire-like Ag clusters in droplets with vortices rather than cluster-cluster aggregates. 7.4.2 Heat Transfer in Superfluid Ag Droplets Although superfluid He is characterized by an exceptionally large heat conductivity, heat transfer from a solid to superfluid He is also known to be inefficient due to a large mismatch in the velocity of sound, an effect known as Kapitza resistance [33]. Above certain values of the heat flux on the order of ~1 W/cm 2 [34], a bubble of He gas will be formed which effectively insulates the hot body from the liquid He environment. This effect was dramatically manifested in early experiments in which a tungsten filament was heated while inside a vessel filled with superfluid He [35]. Previous experiments have shown the formation of bubbles in liquid He formed by laser ablation of silver alloy clusters [36]. These bubbles grow to be several millimeters in diameter over a few milliseconds, which is comparable to the flight time of He droplets studied in the current work. 135 We begin with estimates of the energy deposited during the laser pulse. For the sake of an estimate we take spherical clusters containing ~1000 Ag atoms with a cluster surface area of CA ≈ 32 nm 2 . Upon excitation at 532 nm with F = 0.03 mJ/cm 2 as in this work at the threshold of saturation of signal in Figure 7.2(a), such clusters will absorb 340 eV during the 7 ns laser pulse. The energy absorbed per atom can be obtained using the following equation: 𝐸 = [^〈6 '( 〉 5 8 '( 〈6 57 〉 (7.5) Typical depletion of D = 0.2 leads to absorption of about 0.34 eV energy per atom and concomitant increase of the cluster temperature to T ≈ 1200 K, based on the specific heat of metallic Ag [37]. Release of the entire energy to the He environment during the laser pulse would require a heat flux of ~2 × 10 4 W/cm2. This flux is several orders of magnitude above the threshold for the bubble formation of ~1 W/cm2 [34]. Even if the bubble is not formed the heat flux is limited by the Kapitza resistance. Taking a typical Kapitza resistance of silver to superfluid He, R ≈ 10 −4 m 2 K/W [33], the cooling time can be estimated as: 𝜏 = _' (BV)∆b (7.6) where ∆T is the temperature difference between the silver clusters and He and SA is the surface area of the clusters. The initial value of ∆T is taken to be 740 K, and the clusters will absorb Q = 340 eV during the laser pulse. Accordingly, the value for τ is found to be ~230 ns, much longer than the excitation length of the laser. These estimates show that under any circumstances the silver clusters are not able to dissipate the entire heat during the laser pulse. Because the estimated temperature of the Ag cluster of several hundreds of K is much larger than the superfluid transition and critical temperatures for He of 2.17 K and 5.2 K, respectively, a bubble will likely be formed around the hot Ag cluster, which will further decrease the cooling rate. A prolonged residence of the clusters at high temperature will likely result in the melting of the clusters as discussed below. 136 On the other hand, the estimated heat flux during the CW excitation is 2 × 10 −2 W/cm 2 , which is well below the bubble formation threshold. Therefore, CW irradiation of the Ag clusters allows heat to be effectively transferred to the He bath and the clusters will remain at low temperature. Here we note that the maximum fluence achieved in this experiment is F = 0.008 mJ/cm 2 ; this amount is less than a third of the pulsed fluence, which is the threshold for saturation in this work. The CW fluence was limited by the maximum output of the laser. A CW laser output of 1.5 W is required to achieve F = 0.03 J/cm 2 with our experimental setup. The duration of the initial fast drop of the I8 signal immediately after the time of the laser pulse in Figure 7.1(c) of about 100 µs gives an estimate for the cooling time of the Ag clusters. This time is comparable with the 20 µs dwell time of the AD converter and presents an upper boundary for the measurable cooling time. Measurements of the initial fall with better time resolution at different droplet sizes and levels of the depletion may be helpful for understanding the physics of the heat exchange between nano- clusters and superfluid He droplets. Moreover, such measurements could reveal the magnitude of the initial temperature rise in droplets following laser excitation, allowing determination of the normal fraction and the superfluid fraction of nanodroplets. The above estimates indicate the possibility for the formation of a gaseous bubble around the hot Ag clusters. The existence of a bubble may change the mechanism for the droplet cooling. In particular a large bubble close to the droplet surface may open at the surface and eject its hot content in vacuum, giving rise to non-evaporative cooling. Finally, the formation of multiple large bubbles may tear the large droplet apart leading to the fission into a number of smaller droplets. 137 7.4.3 Absorption Bleaching ad Reconstruction of the Ag Clusters Upon Laser Irradiation The different laser fluency dependences of the signal upon 532 and 355 nm irradiations may indicate reconstruction of the Ag clusters upon pulsed laser irradiation. According to the estimate above, absorption of high intensity light imparts energy to clusters faster than it can be dissipated to the droplet. Thus, during the laser pulse, the clusters can reconstruct, leading to a compact configuration. Compact clusters have very weak absorption at 532 nm, which will result in bleaching, consistent with the observed saturation of the signal. On the other hand, no bleaching is expected at 355 nm because compact clusters have a strong plasmon resonance absorption in this spectral range. Estimates in Section 7.4.2 show that cluster temperatures during the laser pulse can rise up to about 1200 K. The melting point of bulk silver is 1233 K, but silver nanoclusters with radii ∼3 nm, such as those found in droplets formed at ⟨NAg⟩ = 2000, have their melting point depressed to 60% of the bulk value by confinement effects, pushing it down to 740 K [38]. Melting of the Ag cluster will require another 0.12 eV/atom, based on the enthalpy of fusion for metallic Ag [37]. Thus, reconstruction of the clusters during the laser pulse into spherical form seems to provide a satisfying explanation of the absorption bleaching upon 532 nm excitation. The absence of the bleaching following the CW excitation at 532 nm indicates that the absorbed energy is dissipated into the He bath faster than noticeable before reconstruction of the clusters occurs, which is in agreement with the factor of 106 times slower rate of CW laser excitation. Finally, the D vs F dependence upon excitation at 355 nm shows some pronounced sublinear behavior. In ⟨NAg⟩ = 3.4 × 10 6 and ⟨NAg⟩ = 6.2 × 10 4 experiments the maximum values of D ≈ 0.8 were achieved and the deviation from linearity occurs at much larger values of D as compared with the 532 nm excitation, underscoring different saturation mechanisms. This 138 behavior could be explained by a benign saturation effect, as the value of D cannot exceed 1, upon evaporation of the entire droplet. Indeed, the dependence of the D vs F in Figure 7.2(a) for the two largest clusters can be modelled assuming that the droplets have an exponential size distribution [39, 40] and are exposed to different laser fluency due to the Gaussian intensity distribution in the laser beam. On the other hand, the results in Figure 7.2(b) at ⟨NAg⟩ = 340 and to lesser degree at ⟨NAg⟩ = 2000 show fairly sharp kinks at F ≈ 2.5 mJ/cm 2 . In particular, the largest values for D at ⟨NAg⟩ = 340 were obtained to be about 0.4, which is a factor of two times smaller than in larger clusters. This difference is surprising in view of similar absorption cross sections in Table 7.1 and similar numbers of He atoms per Ag atom in the droplets studied in this work. This effect may indicate the existence of some other bleaching mechanism, such as ejection of the entire clusters. For example, an ejection of molecular ions upon multiphoton infrared excitations has been well documented [41-43]. Finally, another bleaching mechanism may include the laser-induced evaporation of the entire Ag cluster; however, understanding this phenomenon will require much larger energies of the order of 3 eV/atom, which will require much stronger laser fluency than available in this work. Conclusions In this work, large silver clusters were grown in superfluid helium droplets upon doping with silver atoms. The absorption of the clusters was then studied upon irradiation with laser light of varying fluence and irradiance. In agreement with our earlier results [18] we found that in addition to the surface plasmon absorption around 355 nm, Ag clusters formed in large He droplets have some additional low frequency absorption band that was probed by 532 nm radiation. 139 In the medium sized droplets of about 10 6 atoms this additional band was assigned to absorption of the cluster-cluster aggregates resulting from the multi-centered growth mechanism. Similar but somewhat weaker absorption at 532 nm was observed in this work in even larger droplets of up to 10 10 atoms and may stem from the absorption of the elongated clusters formed on quantum vortices [13-17]. We found that in contrast to the excitation at 355 nm, the absorption at 532 nm shows a saturation of the signal with increasing fluence. This saturation was not present in 532 nm CW laser experiments at similar laser fluence. It was shown that, upon pulsed excitation, the clusters may reach high temperatures of the order of 1200 K due to ineffective heat transfer to the helium environment. Hot clusters likely reconstruct during the laser pulse from a ramified form into a compact spherical form, which has only weak absorption at 532 nm. Therefore, the reconstruction provides a likely explanation for the bleaching of the absorption at 532 nm. We also conjectured the formation of the gaseous nanobubbles of helium around the hot Ag clusters. 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Vilesov, Traces of Vortices in Superfluid Helium Droplets. Physical Review Letters 108, 155302 (2012). [14] L. F. Gomez et al., Shapes and Vorticities of Superfluid Helium Nanodroplets. Science 345, 906 (2014). [15] R. M. P. Tanyag et al., Communication: X-Ray Coherent Diffractive Imaging by Immersion in Nanodroplets. Structural Dynamics 2, 051102 (2015). [16] C. F. Jones et al., Coupled Motion of Xe Clusters and Quantum Vortices in He Nanodroplets. Physical Review B 93, 180510 (2016). [17] S. M. O. O'Connell et al., Angular Momentum in Rotating Superfluid Droplets. Physical Review Letters 124, 215301 (2020). [18] E. Loginov, L. F. Gomez, N. Chiang, A. Halder, N. Guggemos, V. V. Kresin, and A. F. Vilesov, Photoabsorption of Ag-N(N Similar to 6-6000) Nanoclusters Formed in Helium Droplets: Transition from Compact to Multicenter Aggregation. Physical Review Letters 106, (2011). [19] E. Loginov, L. F. Gomez, and A. F. Vilesov, Surface Deposition and Imaging of Large Ag Clusters Formed in He Droplets. Journal of Physical Chemistry A 115, 7199 (2011). [20] O. Gessner and A. F. Vilesov, Imaging Quantum Vortices in Superfluid Helium Droplets. Annual Review of Physical Chemistry 70, 173 (2019). [21] D. Spence, E. Latimer, C. Feng, A. Boatwright, A. M. Ellis, and S. F. Yang, Vortex- Induced Aggregation in Superfluid Helium Droplets. Physical Chemistry Chemical Physics 16, 6903 (2014). 142 [22] P. Thaler, A. Volk, F. Lackner, J. Steurer, D. Knez, W. Grogger, F. Hofer, and W. E. Ernst, Formation of Bimetallic Core-Shell Nanowires Along Vortices in Superfluid He Nanodroplets. Physical Review B 90, (2014). [23] A. Volk, P. Thaler, D. Knez, A. W. Hauser, J. Steurer, W. Grogger, F. Hofer, and W. E. Ernst, The Impact of Doping Rates on the Morphologies of Silver and Gold Nanowires Grown in Helium Nanodroplets. Physical Chemistry Chemical Physics 18, 1451 (2016). [24] M. N. Slipchenko, H. Hoshina, D. Stolyarov, B. G. Sartakov, and A. F. Vilesov, Internal Rotation of Methane Molecules in Large Clusters. Journal of Physical Chemistry Letters 7, 47 (2016). [25] R. J. Donnelly and C. F. Barenghi, The Observed Properties of Liquid Helium at the Saturated Vapor Pressure. Journal of Physical and Chemical Reference Data 27, 1217 (1998). [26] E. Loginov, L. F. Gomez, and A. F. Vilesov, Formation of Core-Shell Silver-Ethane Clusters in He Droplets. Journal of Physical Chemistry A 117, 11774 (2013). [27] J. Bosbach, C. Hendrich, F. Stietz, T. Vartanyan, and F. Trager, Ultrafast Dephasing of Surface Plasmon Excitation in Silver Nanoparticles: Influence of Particle Size, Shape, and Chemical Surrounding. Physical Review Letters 89, (2002). [28] C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallee, Ultrafast Electron Dynamics and Optical Nonlinearities in Metal Nanoparticles. Journal of Physical Chemistry B 105, 2264 (2001). [29] M. Maillard, M. P. Pileni, S. Link, and M. A. El-Sayed, Picosecond Self-Induced Thermal Lensing, from Colloidal Silver Nanodisks (Vol 108b, Pg 5233, 2004). Journal of Physical Chemistry B 108, 11876 (2004). [30] J. Tiggesbaumker, L. Koller, K. H. Meiwesbroer, and A. Liebsch, Blue-Shift of the Mie Plasma Frequency in Ag Clusters and Particles. Physical Review A 48, R1749 (1993). [31] V. Lebedev, P. Moroshkin, B. Grobety, E. Gordon, and A. Weis, Formation of Metallic Nanowires by Laser Ablation in Liquid Helium. Journal of Low Temperature Physics 165, 166 (2011). [32] S. Link and M. A. El-Sayed, Size and Temperature Dependence of the Plasmon Absorption of Colloidal Gold Nanoparticles. Journal of Physical Chemistry B 103, 4212 (1999). 143 [33] G. L. Pollack, Kapitza Resistance. Reviews of Modern Physics 41, 48 (1969). [34] J. S. Vinson, F. J. Agee, R. J. Manning, and F. L. Hereford, Phenomena Resulting from Heating of Small Wires in He 2. Physical Review 168, 180 (1968). [35] G. E. Spangler and F. L. Hereford, Injection of Electrons into He 2 from an Immersed Tungsten Filament. Physical Review Letters 20, 1229 (1968). [36] E. Popov, M. Mammetkuliyev, and J. Eloranta, Dynamics of Vortex Assisted Metal Condensation in Superfluid Helium. Journal of Chemical Physics 138, (2013). [37] R. S. a. F. R. Fickett, Nist Standard Reference Database 69: Nist Chemistry Webbook (National Institute of Standards and Technology, 2009). [38] T. Castro, R. Reifenberger, E. Choi, and R. P. Andres, Size-Dependent Melting Temperature of Individual Nanometer-Sized Metallic Clusters. Physical Review B 42, 8548 (1990). [39] U. Henne and J. P. Toennies, Electron Capture by Large Helium Droplets. Journal of Chemical Physics 108, 9327 (1998). [40] R. Sliter, L. F. Gomez, J. Kwok, and A. Vilesov, Sizes Distributions of Large He Droplets. Chemical Physics Letters 600, 29 (2014). [41] N. B. Brauer, S. Smolarek, X. H. Zhang, W. J. Buma, and M. Drabbels, Electronic Spectroscopy of Aniline Ions Embedded in Helium Nanodroplets. Journal of Physical Chemistry Letters 2, 1563 (2011). [42] X. H. Zhang, N. B. Brauer, G. Berden, A. M. Rijs, and M. Drabbels, Mid-Infrared Spectroscopy of Molecular Ions in Helium Nanodroplets. Journal of Chemical Physics 136, (2012). [43] A. I. G. Florez, D. S. Ahn, S. Gewinner, W. Schollkopf, and G. von Helden, Ir Spectroscopy of Protonated Leu-Enkephalin and Its 18-Crown-6 Complex Embedded in Helium Droplets. Physical Chemistry Chemical Physics 17, 21902 (2015). 144 Magnetic Circular Dichroism Spectroscopy in He Nanodroplets The following section describes an experiment that was interrupted by the COVID-19 breakout in 2020. While the experiment was unable to be completed, the following provides motivation of the experiment, description the physical setup, notes on what I was able to accomplish as well as future directions. Motivation Magnetic circular dichroism spectroscopy studies the difference in absorption between left and right circularly polarized light (LCP and RCP, respectively) of a substrate in a magnetic field. The two different polarizations have different selection rules, with Δml = +1 for LCP and Δml = - 1 for RCP. Mathematically, the spectroscopic signal of MCD proportional to an A, B, and C term as follows [1]: 𝐼 c+^ ≈ l𝐴P ,)(8) ,8 Q+P𝐵+𝑒 4 :; Q𝑓(𝐸)p𝜇 I q𝐻 UU⃗ q (8.1) in which IMCD is the difference in signal between LCP and RCP light, A, B, and C, are substrate- specific values, describing the MCD effect, f(E) is the shape of the absorption curve vs energy of the absorbed photons, E, in the absence of a magnetic field, k is the Boltzmann constant, T is the temperature of the substrate, µB is the Bohr magneton, and q𝐻 UU⃗ q is the strength of the magnetic field. The MCD spectra are usually given as the difference of LCP and RCP spectra. Figure 8.1 illustrates four kinds of absorption: absorption with no field, A-Term absorption, B-Term absorption, and C-Term absorption in Panels i, ii, iii, and iv, respectively. At q𝐻 UU⃗ q = 0, the RCP and LCP absorptions are identical, so their difference is zero. For A-Term absorption, the excited state undergoes Zeeman splitting, resulting in a shift of the RCP and LCP absorption energies. The B-Term absorption arises due to a so-called |J⟩ state that undergoes Zeeman mixing with either the 145 ground or excited state in the presence of a magnetic field. The case in which the |K⟩ state mixes with the excited state is illustrated in Figure 8.1. Neither the A-Term nor B-term absorptions have temperature dependencies because the Zeeman splitting does not affect the ground state. However, C-Term absorptions stem from the splitting of the ground state. If there is a larger population in one ml sublevel than another, the LCP signal will be much greater than the RCP signal. Therefore, the MCD signal due to the C-Term will be significant, as shown in Figure 8.1 (iv). The difference in the population can be induced by larger ΔE at higher q𝐻 UU⃗ q or due to smaller population of the higher level at lower T. In He droplets the temperature of the trapped dopants is ~0.4 K [2]. Therefore, they are expected to give rise to a sizable MCD signal due to the C-Term. Indeed, He droplets have been used to obtain MCD spectra of K atoms [3]. Magnetic circular dichroism spectroscopy gives information on the magnetic moments of clusters. Current magnetic moment data in free clusters remain incomplete and rely on the measurements of the deflection of clusters in an inhomogeneous magnetic field [4-7]. Measurements with Fe, Co and Ni clusters showed that ferromagnetism is already recognizable in the smallest clusters and rather rapidly evolves to its bulk-like manifestation (n ~100-500). It was shown that in free clusters, the effect of spin coupling with clusters’ rotation (T = 25-100 K) leads to a Langevin-like dependence of the magnetization on the applied magnetic field and cluster temperature. Recently, in addition to spectroscopy in optical range, study of small cationic clusters (n < 20) by X-ray magnetic circular dichroism (MCD) have become available [8-11] using synchrotron radiation. He droplets can not only provide lower temperatures, but rotational relaxation of the cluster via coupling to the He bath may lead to spin relaxation and concomitantly larger magnetization in weaker fields. 146 Figure 8.1. Energy level diagrams (left) and sample illustrations of MCD spectra (right). For the energy level diagrams, the left side shows the energy states in the absence of a magnetic field, and the right side of the energy diagrams shows how the degenerate levels split in the presence of a magnetic field. The |A⟩ state denotes the ground state while the |J⟩ and |K⟩ states denote excited states. The RCP and LCP absorptions are illustrated as red and blue arrows, respectively. On the right side of the figure, the RCP and LCP signals are shown as red and blue dotted lines, respectively. Their difference gives rise to the black MCD spectra. In i), there is no magnetic field applied, so there is no MCD signal. Panel ii) shows A-term MCD signal in which the excited |J⟩ state splits. B-term MCD signal is shown in iii), in which the |J⟩ and |K⟩ states mix due to being close in energy. Lastly, Panel iv) shows the C-term absorption in which the ground level splits and there is a greater population in the lower level, giving rise to more LCP than RCP signal. In the following, I describe my attempts to obtain MCD spectra of zinc phthalocyanine (ZnPc) in order to characterize the experimental setup described in Chapter 2.3. ZnPc has a 147 degenerate ground state, which makes it ideal for C-type MCD spectroscopy [12]. Previous studies at T ≈ 15 K have shown MCD signals on the order of a few percent of the optical signal [13-15], which is a manageable range for He droplet experiments. An experimental study of Fe clusters has been planned for some time, but it has yet to be realized. As described herein, and in Appendix 1, several alterations needed to be made to the setup, including changes to the heated pickup cell and heat shield designs. Experimental The experiments were attempted using the apparatus described in Chapter 2.3, with some modifications. The main changes were to the first pickup chamber. Powdered ZnPc was held in a cylindrical crucible with holes on either side to accommodate the He beam, see Figure 8.2 (a). This type of crucible has been employed in several previous experiments [16, 17]. The crucible contained a wire mesh to prevent the ZnPc powder from blowing away during pumping and held in place by a restively heated filament, which was wrapped with foil to mitigate radiative heat loss. The ends of the tungsten element were then attached to copper rods that were attached to feedthroughs at the top of the chamber, which were connected to the outputs of an MDC Re-Vap 900 power supply. The copper rods are visible in Figure 8.2 b). Thermocouple wires, seen in Figure 8.2 b), were sandwiched between the top and bottom halves of the crucible to read the temperature of the oven. The temperature was selected using the vapor pressure data of ZnPc from Basova et al. [18]. To prevent chamber contamination from accidental over-evaporation of ZnPc, two stainless steel shields were installed on either end of the crucible. One of these plates is seen on the left side of Figure 8.2 b). The shields each contained 5 mm holes to accommodate the He beam. The shields were mounted to a stainless-steel bar that was laid across the cross along the direction of the He beam. 148 Figure 8.2. Panel (a) left: a sketch of the crucible halves in their tungsten heater. Right: a sketch of the two crucible halves showing the opening in the top half to accommodate the He beam. Panel b): The evaporative source inside the vacuum chamber. The He beam traverses from left to right. The red and yellow wire are leads to a type-K thermocouple. The insulation on the leads was only rated to ~900° C, so the insulation was stripped close to the cell. The copper rods connect to a feedthrough mounted to the top of the chamber. There is a stainless-steel plate with a central hole to accommodate the beam shown on the left side of the image. Another such plate is mounted at the right side of the chamber but is not in view of this picture. 149 The tunable output of the Ekspla NT242 laser was aligned to propagate anti-colinearly to the He beam. Alignment was achieved using an iris external to the vacuum chamber and aligned along the beam axis using the external telescope. Once well-aligned, the laser would impinge upon the nozzle plate, creating a bright reflection. The laser could be coupled out of the second pickup chamber and its power measured via a pyrometer. Successful doping of the droplets with ZnPc was indicated by the decrease of the intensity of the He droplet beam as monitored by the intensity of the I8 signal due to He2+ ions. We were, however, unable to detect the depletion spectra due to absorption of ZnPc in the droplets. We used several references of previous ZnPc studies, including light-induced fluorescence [19] and MCD/standard absorption [13, 14, 20] works to find the optical absorptions of ZnPc. Unfortunately, the absorption lines seemed to be very sharp (~1 cm -1 ), whereas the output of the Ekspla system is broad (~10 cm -1 ) in the range of 600 nm. It is likely that this broadness led to less absorption, and the signal to noise ratio of our experiment was too high. However, the COVID-19 pandemic forced us to abandon the experiment at this stage, so no further work could be completed. While the experiments never proceeded to this point, the magnetic tube that is intended to be used for these experiments is described below. The stainless-steel tube, pictured in Figure 8.3, houses 6 permanent NdFeB magnets having 5 cm length, 5 mm diameter axial bores and axial magnetization. The resulting magnetic field along the beam axis is shown in the lower panel of Figure 8.3. When doped droplets traverse the magnetic system, they are exposed to periodic ± 0.4 T variation of the axial magnetic field. The circularly polarized pulsed laser beam will enter anti- collinearly to the He droplet beam. 150 Figure 8.3. Top: manufactured holder with 6 magnets inside. Bottom: magnetic field along the axis [21]. Future Directions Before running MCD experiments, it is imperative to obtain standard depletion signal to ensure that the apparatus is working as intended. More time needs to be spent calibrating and tuning the Ekspla laser or exploring the other laser options in the lab, but they may not operate at high enough energies or repetition rates. Another future issue is obtaining a sufficient vapor pressure of Fe atoms for the prospected experiments on iron clusters. I designed and tested a few different oven designs for the study of metal clusters in helium droplets. The best design I was able to produce was only able to reach a temperature of ~1100 °C, see Appendix 1 for details. The temperatures needed for evaporation of Fe are closer to 1500 °C. The group of Paul Scheier in 151 Austria has designed an oven capable of melting gold [22], so it would be worthwhile to reproduce their design in our lab. References [1] J. Mack, M. J. Stillman, and N. Kobayashi, Application of Mcd Spectroscopy to Porphyrinoids. Coordination Chemistry Reviews 251, 429 (2007). [2] R. M. P. Tanyag, C. F. Jones, C. Bernando, D. Verma, S. M. O. O'Connell, and A. F. Vilesov, in Cold Chemistry: Molecular Scattering and Reactivity near Absolute Zero, edited by A. Osterwalder, and O. Dulieu (Royal Society of Chemistry, Cambridge, 2018), p. 389. [3] J. Nagl, G. Auböck, C. Callegari, and W. E. Ernst, Magnetic Dichroism of Potassium Atoms on the Surface of Helium Nanodroplets. Physical Review Letters 98, 075301 (2007). [4] I. M. L. Billas, A. Chatelain, and W. A. de Heer, Magnetism from the Atom to the Bulk in Iron, Cobalt, and Nickel Clusters. Science 265, 1682 (1994). [5] X. S. Xu, S. Y. Yin, R. Moro, and W. A. de Heer, Magnetic Moments and Adiabatic Magnetization of Free Cobalt Clusters. Physical Review Letters 95, 237209 (2005). [6] X. S. Xu, S. Y. Yin, R. Moro, and W. A. de Heer, Distribution of Magnetization of a Cold Ferromagnetic Cluster Beam. Physical Review B 78, 054430 (2008). [7] X. S. Xu, S. Y. Yin, R. Moro, A. Liang, J. Bowlan, and W. A. de Heer, Metastability of Free Cobalt and Iron Clusters: A Possible Precursor to Bulk Ferromagnetism. Physical Review Letters 107, 057203 (2011). [8] M. Niemeyer et al., Spin Coupling and Orbital Angular Momentum Quenching in Free Iron Clusters. Physical Review Letters 108, 057201 (2012). [9] A. Langenberg et al., Spin and Orbital Magnetic Moments of Size-Selected Iron, Cobalt, and Nickel Clusters. Physical Review B 90, 184420 (2014). 152 [10] J. Meyer et al., The Spin and Orbital Contributions to the Total Magnetic Moments of Free Fe, Co, and Ni Clusters. Journal of Chemical Physics 143, 104302 (2015). [11] S. Peredkov, M. Neeb, W. Eberhardt, J. Meyer, M. Tombers, H. Kampschulte, and G. Niedner-Schatteburg, Spin and Orbital Magnetic Moments of Free Nanoparticles. Physical Review Letters 107, (2011). [12] G. Ricciardi, A. Rosa, and E. J. Baerends, Ground and Excited States of Zinc Phthalocyanine Studied by Density Functional Methods. The Journal of Physical Chemistry A 105, 5242 (2001). [13] T. C. VanCott, University of Virginia, 1989. [14] T. C. VanCott, M. Koralewski, D. H. Metcalf, P. N. Schatz, and B. E. Williamson, Magnetooptical Spectroscopy of Zinc Tetrabenzoporphyrin in an Argon Matrix. The Journal of Physical Chemistry 97, 7417 (1993). [15] M. T. Tiedemann and M. J. Stillman, Application of Magnetic Circular Dichroism Spectroscopy to Porphyrins, Phthalocyanines and Hemes. Journal of Porphyrins and Phthalocyanines 15, 1134 (2011). [16] R. M. P. Tanyag, Imaging Supefluid Nanodroplets, University of Southern California, 2018. [17] C. F. Jones, C. Bernando, S. Erukala, and A. F. Vilesov, Evaporation Dynamics from Ag- Doped He Droplets Upon Laser Excitation. The Journal of Physical Chemistry A 123, 5859 (2019). [18] T. Basova, P. Semyannikov, V. Plyashkevich, A. Hassan, and I. Igumenov, Volatile Phthalocyanines: Vapor Pressure and Thermodynamics. Critical Reviews in Solid State and Materials Sciences 34, 180 (2009). [19] S. Kuma, H. Goto, M. N. Slipchenko, A. F. Vilesov, A. Khramov, and T. Momose, Laser Induced Fluorescence of Mg-Phthalocyanine in He Droplets: Evidence for Fluxionality of Large H2 Clusters at 0.38k. The Journal of Chemical Physics 127, 214301 (2007). [20] L. Edwards and M. Gouterman, Porphyrins: Xv. Vapor Absorption Spectra and Stability: Phthalocyanines. Journal of Molecular Spectroscopy 33, 292 (1970). 153 [21] C. Bernando, Vorticity in Superfluid Helium Nanodroplets, University of Southern California, 2016. [22] L. Lundberg, P. Martini, M. Goulart, M. Gatchell, D. K. Bohme, and P. Scheier, Hydrogenated Gold Clusters from Helium Nanodroplets: Cluster Ionization and Affinities for Protons and Hydrogen Molecules. Journal of the American Society for Mass Spectrometry 30, 1906 (2019). 154 Conclusions and Outlooks Summary Helium nanodroplets continue to be a unique vessel for spectroscopic and superfluid studies. As He droplet spectroscopy continues to evolve into more complex techniques, it is crucial to understand how these quantum systems behave. In this thesis, large 3 He, 4 He, and mixed 3 He/ 4 He droplets were investigated by a variety of experimental techniques. Optical imaging was used to identify the nature of jet breakup upon expansion of fluid He into vacuum. In addition, X- ray coherent diffractive imaging was used to quantify angular momentum of 4 He droplets, compare size and shape characteristics of 3 He and 4 He droplets, and provide benchmark results on phase separation in large, rotating, mixed 3 He/ 4 He droplets. Last, spectroscopy was used to investigate the effects of heat transfer from dopants to He droplets upon increasing laser fluence, and the groundwork was laid for magnetic circular dichroism experiments in He droplets. The optical imaging studies of jet breakup helped shed light on the operation of He beams. Several different operational modes of the beam were observed (spraying, jet branching, and capillary breakup) at various initial source conditions. Under normal operational conditions in our lab (T0 = 3.5 K, P0 = 20 bar) the jet split into multiple branches. While this effect is intriguing in its own right, the practical outcome of these findings is to be more careful in the alignment of the beam. These preliminary results are expanded upon by the group of Daniela Rupp at MBI, Berlin, although there is nothing published at the time of writing. X-ray diffraction results show that most of the large droplets are non-spherical, which implies that they are rotating. Angular momentum can only be imparted to the droplets upon their formation in the nozzle expansion. It remains unclear whether angular momentum is imparted in the channel leading up to the nozzle, or by the 155 breakup of the liquid He jet in vacuum. It is also possible that turbulent flow within the nozzle channel imparts angular momentum to the He fluid. X-ray diffraction experiments also show that 3 He and 4 He droplets produced under the same conditions both have the same average reduced angular momentum and average reduced angular velocity. So, the angular momentum is imparted to both normal fluids and superfluids to the same extent. This result leads credence to the idea that angular momentum is imparted to the jet via flow through the nozzle. As droplets fragment, they may be driven by the chaotic environment outside of the nozzle to fragment and form daughter droplets. While the results cannot confirm the exact cause of the angular momentum, it is remarkable that superfluid and normal fluids both have similar amounts of angular momentum despite such disparate mechanisms of rotation. Of course, both fluids extrude from the nozzle as normal fluids. The shapes of classical rotating droplets executing so-called rigid body rotation can be mapped onto a universal stability diagram. A classical droplet’s shape is determined by balancing the kinetic rotational energy and the surface tension of the droplet. It was found that superfluid droplets follow this same shape relation, even though superfluids cannot execute rigid body rotation. Instead, superfluid droplets manifest angular momentum through quantized vortices and capillary waves. A droplet at rest is spherical and becomes oblate when first set into rotation. Oblate superfluid droplets host only vortices, as capillary waves are not forthcoming in axisymmetric systems. As a droplet acquires more angular momentum, it becomes tri-axial elliptically prolate, breaking the axial symmetry of oblate droplets. As the droplet acquires yet more angular momentum, it becomes capsule-shaped, peanut-shaped, and finally fissions into two daughter droplets. 156 In this work, three kinds of droplets were studied in great detail: an oblate spheroid, a triaxial prolate, and capsule-shaped droplet. The prolate droplets were imaged with the XFEL aligned with the droplets’ axes of rotation, which revealed the vortex pattern within the droplets. From this pattern, the angular momentum of the droplet could be estimated. It was found that the angular momentum in prolate droplets was partitioned between vortices and capillary waves, with capillary waves contributing an increasing percentage as angular momentum increases. These insights indicate that superfluid droplets seem to follow the same shape curve as classical droplets while having different sources of angular momentum. Another interesting question is how rotation affects phase separation of 3 He and 4 He liquids. The phase separation of these two quantum fluids is well-documented in the bulk, but no experiments have been done to study the morphology of the phase-separated fluids in droplets. More generally, to the best of my knowledge, no experiments have investigated the morphology of rotating classical or quantum multiphasic droplets. X-ray diffraction imaging experiments revealed the morphology of phases in mixed 3 He and 4 He droplets created from the expansion of homogeneous 3 He/ 4 He fluid. For droplets containing 50% 3 He and 50% 4 He, the droplets seem to be homogeneous, indicating that no phase separation occurred within the ~5 ms time-of-flight from the droplet source to the interaction point with the laser. In contrast, phase separation was seen in droplets containing ~75% 3 He and ~25% 4 He. Three main types of density configurations were observed: a centrally located 4 He inner droplet, an off-center 4 He inner droplet, and two off-center, symmetric, spherical 4 He inner droplets. The outer droplet consists of pure 3 He, while the inner 4 He droplet contains ~8% 3 He. These results are at variance with the results of classical and quantum DFT calculations for mixed 3 He/ 4 He droplets, which predicted only centrally located 4 He inner droplets, except at high values of angular momenta not observed in this experiment. The 157 discrepancy with the DFT results comes from an axisymmetric constraint in the calculations [1]. The classical Navier-Stokes calculation did not have this explicit constraint, but the inner droplets were initially placed in the center of the outer droplets. Since there is no force to displace a symmetrically placed object from the axis of rotation, in the calculations, the inner droplet may be trapped at the center of the outer droplet. The results of this work show that mixed droplets can be observed in experiment, and that they can take on a variety of different configurations. There is still much to be learned using He droplets as a spectroscopic tool. Depletion spectroscopy of He droplets doped with Ag was taken at several different laser fluences. It was found that Ag clusters absorbed in the plasmon band at 355 nm, and lower-frequency absorption was also observed at 532 nm. This lower-frequency absorption is ascribed to interactions between cluster-cluster aggregates in droplets containing ~10 6 atoms and longitudinal absorptions of elongated clusters in droplets containing 10 10 atoms. It was shown that the pulsed 532 nm signal saturated with laser fluence. This effect was not observed for CW 532 nm or for 355 nm signals. It was shown that clusters may reach temperatures as high as 1200 K upon pulsed excitation due to inefficient heat transfer to the He matrix. At these high temperatures, the ramified cluster aggregates may form a more compact, spherical shape that no longer absorbs at 532 nm. It is also possible that the hot clusters form a bubble of He around themselves, which may then lead to the clusters being ejected from the droplets. Last, this work shows the initial ideas and setup of magnetic circular dichroism studies of zinc phthalocyanine, though the work could also be expanded to other clusters, such as Fe, with modifications to the pickup cell. Unfortunately, the experiment was interrupted before any data was obtained. This area of research remains promising once the machine is available for experiments. 158 Future Outlooks Helium droplets have become a well-established, productive area of research for spectroscopy and fundamental studies of superfluids. To keep growing, researchers need to continue to use innovative insights to address previously difficult or inaccessible problems. Understanding the properties of 3 He, 4 He, and mixed 3 He/ 4 He droplets will therefore open avenues to future discoveries. There are also immediate extensions of current experiments that can be done as well. The experiments presented employed X-ray free electron lasers (XFEL). The first generation of these experiments were done at LCLS at SLAC, but the AMO instrument that featured the first four of these experiments was demolished and replaced with a different instrument that is unable to support imaging experiments until at least 2023. However, this prospected instrument is designed for a new, brighter XFEL with a higher repetition rate. These improvements will greatly boost the efficacy of future experiments. For droplet imaging experiments, very bright hits are needed for trustworthy analysis, and those images are fairly rare with the presented experiments at 120 Hz. By upgrading the repetition rate of the laser to 1 MHz, many more useful images can be taken. However, the repetition rate of the camera will still limit the number of images taken and is still likely to be ~1-10 kHz for the time being. Still, a 10-fold increase of processible data will greatly improve the data analysis for these kinds of experiments. I have also participated at some experiments that were been done at the European XFEL (which has a repetition rate of 1 MHz), but my role in those experiments was smaller than those in previous LCLS experiments. The data from these projects is still being analyzed. Stray light was a large issue in these experiments that needs to be mitigated in the future. 159 The recycling system built during this work will enable further XFEL studies of pure and doped 3 He droplets, as well as mixed 3 He/ 4 He droplets. The system can also be applied at USC to enable spectroscopy experiments in 3 He droplets. One of the immediate studies to be done should be size characterizations of large 3 He droplets. While this work presents XFEL sizing, that sizing was only sensitive to droplets larger than 50 nm, and only investigated source temperatures around 2 K. The next spectroscopic works to be done include finishing the zinc phthalocyanine project. Ordinary depletion spectra should be taken, then MCD studies can be undertaken. If the zinc phthalocyanine validates our ideas, then the system can be modified to examine species that have not been investigated by other methods, such as Fe clusters. However, the current pickup cells likely need to be modified to reach the high temperatures needed to melt Fe. Last, more fundamental work on rotation of superfluid droplets and rotating mixed droplets may find applications outside of the helium droplet community. Groups studying Bose-Einstein condensates may also benefit from the studies we have done on superfluid rotation, for example. The multiphasic 3 He/ 4 He droplet works are interesting in that they may provide a timescale of phase separation at different mixing ratios. Further studies should use pre-mixed gases with known ratios for more repeatable studies. Additionally, the time-of-flight from the source to the laser interaction point can be varied by moving the source closer to or further from the beamline or varying source conditions. Mapping out the timescale of phase separation will be very useful to both theorists and experimentalists. References [1] M. Barranco, Personal Communication, 2021. 160 Appendix 1: Pickup Cell Design This appendix includes descriptions and operating instructions of the He recycling system, tables of pressures during normal beam operation, and notes on the design of the metal evaporation oven. During this work, I attempted to modify our crucible-based metal evaporation oven for reaching higher temperatures required for evaporation of Al atoms for the planned spectroscopic measurements on Al clusters in He droplets. To make large Al clusters, vapor pressure of up to ~10 -3 mbar is required, which corresponds to an oven temperature, Toven ≈ 1200 C. [1] With the previous oven design, temperatures of up to 1000 C could be achieved sufficient for experiments with Ag clusters [2]. Further increase of the temperature proved to be difficult. Several problems must be overcome for the oven to work: the Al supply must last for at least the length of one experiment (~12 hours); the Al atoms must be added to the droplets isotropically, otherwise the beam may get deflected if a large number of hot Al atoms are added to droplets; and heat loss due to radiative emission needs to be mitigated. Previous iterations of metal ovens in the lab included whetting tungsten electrodes with metal, then evaporating the metal into the beam. This approach required frequent changing of the metal load, which required venting of the apparatus, an impractical outcome. So, we opted to use a crucible to heat up the metal (see Figs. 10.2 and 10.3). However, this option leads to complications with beam deflection and heat loss, as described below. 161 Figure 0.1. A plot of vapor pressures of metals versus temperature. Al has10 -8 mm Hg at ~900 K, and Fe is 3 lines to the right of Al. Beam Deflection via Anisotropic Capture of Atoms The magnitude of the deflection may be estimated by considering the inelastic collision between a droplet of, say 10 10 He atoms upon the picking up of 10 6 Al having T = 1500 K that are all moving upwards from a crucible placed below the droplet beam: 𝜃 ,()d(@:0e> = O <= O >?@8%?$A* = - <= C@?= D C@?= EF 5* - <= 5* D 5* D C@?= EF 5* D 5* O C@?= >?@8%?$GA* (0.1) In which vhorizontal is given by Gomez et al. to be ~175 m/s at T0 = 5 K, vup drop is assumed to be zero, mdrop is the mass of a droplet containing 10 10 atoms, NAl = 10 6 is the number of Al atoms 162 colliding with the droplet, mAl is the mass of an aluminum atom, and 𝑣 fg Vd = ' Ai H b 5* \ 5* = 1690 m/s. Plugging in these values results in θdeflection = 0.003 radians. While this amount may seem insignificant, the beam traverses a distance of 1 m and is collimated by a hole 6 mm in diameter. Under these conditions, a droplet would experience an upward displacement of 3 mm, which would mean that much of the beam would be clipped by the collimating orifice and not detected via QMS. It was suggested that this effect could be mitigated by placing a cap or dome of Mo foil over the hot crucible to deflect hot atoms back downwards and get closer to isotropic addition of Al atoms. Radiative Heat Loss Estimates The heating assembly consisted of a power supply (MDC Re-Vap 900), large-gauge copper cables, vacuum feedthroughs, copper mounts, a ~25 Ω tungsten resistive heating element and a ceramic crucible. The heater is designed for a 120 VAC single-phase input, but we modified the cable to take 220 V single-phase input to boot the voltage out of the supply. Indeed, this modification was able to dramatically increase the heating output of the supply. A few different types of heaters and baskets were used for different experiments. The Al experiments described herein used a heater and basket from RD Mathis: the heater was a customized Tungsten Evaporation Basket Heater – B8A and the crucible was Crucible C1-BNC. A BNC crucible was used, as melted Al would react with an alumina crucible. To reach a crucible temperature of 1200 C, several heat shields were installed. There were 3 iterations of the heat shield design. The first iteration was built for the custom heater and crucible described in Chapter 8 and featured two stainless-steel interlocking tubes that were centered by ceramic screws. This design failed because we were not able to achieve sufficiently high 163 temperatures. However, similar designs were previously successful, so it may be useful return to a similar scheme. The second design was based around the RD Mathis crucible, and consisted of a cylinder with holes to accommodate the He beam. The cylinder was suspended on a lid that rested on ceramic spacers placed on the leads of the heater. The cylinder was made of two windings of Mo foil that were spot-welded to maintain the shape. The cylinder had foldable tabs at the top that were used to hang off the lid. While this design was able to get to ~1200 C, the Mo foil deformed at higher temperatures, and this design was abandoned because the integrity of the beam would always be a question. 164 Figure 0.2. A sketch of the final heater design. The heater and crucible are suspended in the cup by the feedthroughs attached to the top of the vacuum chamber (omitted in the drawing). The height and tilt of the heat shield can be adjusted via nuts that rest on a stainless-steel plate that spans the chamber. Originally, we planned to use hex nuts, but we opted for knurled knobs (McMaster-Carr part number 91833A112). The heat shield was custom-built by the machine shop. The final design, shown in Figure 10.2, used the idea of its predecessor, but was mounted via a plate at the bottom of the chamber. The heat shield consisted of a stainless-steel cylinder with 4 legs that mounted to a stainless-steel base mounted across the chamber. This same base was then modified to fit the shields shown in Figure 8.2. The shield had holes to accommodate the He beam, and a lid with slots and holes to accommodate the heater leads without shorting. Indeed, this design 165 was able to achieve 1170 K with a voltage of 6V and a current of 65 A (hot shield shown in Figure 10.3), and it is what we used to attempt Al cluster experiments. However, we found that the Al seemed to give some initial signal that decayed quickly. During a discussion with Gary Douberly at a conference in January 2020, he revealed that molten Al tends to creep up the walls of the crucible, which leads to quick depletion of the Al pellets, which explains the effects we saw in our experiments. At this point, we decided to abandon the experiment. 166 Figure 0.3. The heat shield assembly with crucible enclosed. The temperature of the crucible is ~1500 K. 167 Appendix 2: Overview of 3 He Recycling System Due to the considerable cost of 3 He, a recycling system was designed and manufactured as a part of this thesis work. The described design of the recycling system was inspired by a similar system used for experiments with 3 He droplets in Goettingen [3-5] and 3 He gas circulation system in Blue Fors model LD dilution refrigerator model. Figure 11.1 shows the schematics of the recycling system. The blue valves connect the manifold itself, while the green valves connect the manifold to auxiliary parts of the recycling system (i.e. the compressor or liquid nitrogen traps, see Figure 11.2). The purple valves denote connections to storage cylinders that are mounted to the manifold. The orange valves are used for evacuating the system, and the red valves are currently unused. Blue arrows in Figure S1 indicate the direction of the helium flow during operation. Starting with valve G5 on the far left of the board, near the bottom, 3 He gas enters the gas manifold as it is collected from the output of the scroll pumps. To clean the recycled gas, 3 He then travels to the liquid nitrogen (LN2) traps through valve G6. Upon cleaning, the gas goes to the compressor through G7, B8, G8. The compressor output travels via G2 and G1 to pressure reducer 1 on the upper left part of the manifold, which is connected to the He droplet source. Figure 11.1 shows the photograph of the recycling system consisting of a gas manifold and a compressor. 3 He recycling system fulfills the following functions: 1) collection of recycled gas, 2) cleaning of recycled gas, 3) pressurization of clean gas, and 4) storage of clean gas as described in more detail in the following. 168 Figure 0.1. A color-coded schematic of the manifold pictured in Figure 11.2. Blue arrows indicate the direction of helium flow during operation. Blue valves denote connections between parts of the manifold, while green valves denote connections between the gas manifold and other parts of the recycling system. The black lines indicate tubing connections on the board. The dotted green lines illustrate recycling system connections that are off the board. Purple valves and lines denote connections to storage cylinders that are mounted to the manifold. Orange valves connect the gas manifold to the vacuum pump used to evacuate the recycling system. The orange lines denote tubing that is under vacuum while the evacuation pump is turned on. Red valves are currently unused. Pressure gauges, shown as PG, are installed on the lecture bottles and after the LN 2 traps. The pressure reducers similarly have pressure gauges, the input being indicated as H and the output indicated as L. 169 Figure 0.2. A picture of the recycling system compressor (left), LN 2 trap (middle) and gas manifold (see Figure S1 for schematic) at the AMO hutch at LCLS in 2017. Collection of Gas During the experiments, helium is introduced into the vacuum system through a nozzle orifice at a flow rate of about 3 bar∙atm/s and pumped out from the source chamber and downstream in differentially pumped chambers (see Figure 2.5) by turbo-molecular pumps that are back by several scroll pumps such as Leybold SC 30D and Anest Iawata ISP 250C, 500C. The output from the scroll pumps is passed through Alcatel DFT-25 dust filters and enters the gas handling system through an in-line gas filter, Adixen DFT-25, to prohibit any scroll pump debris from entering the gas handling system. Purification of Gas The collected helium gas is contaminated by gases that may have been used as dopants as well as other trace gases that may stem from rest gas in the vacuum apparatus such as water or 170 minute leaks in the vacuum system. The bulk of the freezable dopant gases, such as xenon, are removed by freezing on a u-shaped ½” tube passing through a container filled with liquid nitrogen. More volatile gases, such as nitrogen, oxygen and hydrogen are removed in a subsequent liquid nitrogen (LN2) trap filled with a zeolite, shown in Figure 11.3. Zeolite is regenerated before the experiments upon extended heated pumping by Agilent IDP3. When the traps are heated, the for- vacuum pressure rises as the volatile gases are desorbed from the zeolite. Once the pressure begins to fall, heating is stopped, and the traps are allowed to cool under vacuum. Once immersed in LN2 dewars, these traps can operate for at least 12 hours under high doping conditions with minimal losses to efficacies. Additionally, the traps can be isolated from the rest of the system by closing valves G6 and G7 during the regeneration. Figure 0.3. Cross-sectional view of the cyrogenic trap. It is filled with zeolite pellets (MDC part number 500003) and immersed in LN 2. The filter (Swagelok SS-4F-VCR-2) is used to avoid dust contamination from zeolite in other parts of the recycling system. 171 Pressurization of Gas Compressing the purified He gas is performed by a triple metal membrane two-stage compressor, a Fluitron S1-20/150, which has a maximum output pressure of 110 bar. To prevent damage to the membranes by residual particles in the system, an in-line gas filter (Swagelok SS- 4F-VCR-2) is used before the first stage of the compressor. Filling the System Before the experiment, the evacuated system is filled with 3 He gas until the pressure in the low-pressure line as measured with an electronic capacitance pressure gauge 2, Wika S-10, 0-2 bar absolute (bara), is above atmospheric pressure (1-1.3 bara) with the compressor operating and the nozzle stagnation pressure set to the desired value, 20 bar in the current experiments. Our estimates show that the system takes ~7 L⸱bar of gas to fill when the nozzle is at room temperature, and an additional 2 L⸱bar is needed as it reaches operation temperatures of T0 < 5 K. If a mixture of isotopes is needed, the system is first filled with pure 3 He gas, then 4 He gas is added using valve G4, depicted in Figure 11.1. Small amounts of 4 He gas can be added to the system by opening valve G4, closing the connection to the cylinder, then adjusting Pressure Reducer 2 so that the needle moves slightly off 0 bar. Valve B6 is then used to regulate the introduction of gas into the system so that the reading of Electronic Pressure Gauge 2 stays below 1.3 bara. The gas mixture is then stirred for about 30 minutes by slightly opening valve B1, effectively bypassing the nozzle to assure fast equilibration of the mixture 4 He is often used for leak and operational testing. Since the system does not differentiate between different isotopes of helium, it needs to be purged when switching from 4 He to 3 He operation. Purging is done when the source traps are at room temperature. During the purging 172 procedure the system was filled with Ar then pumped out. The procedure is repeated several times. The same procedure was used by Grebenev et al. [6] to assure a fraction of 4 He gas in 3 He on the order of 10 -5 . In those previous experiments, high purity 3 He gas (6.0) was used. Storage of Gas It is practical to store helium while experiments are not being run. Storage containers, shown in the upper right portion of Figs. 11.1 and 11.2 are added to the output of the compressor line. The compressor can be used effectively if the inlet pressure is higher than approximately 0.5 bar. To collect the gas at lower pressure, an additional pump (Agilent IDP-3) is installed, as shown in the bottom right of Figure 11.1. Operation Parameters and System Components During the experiments, the recycling system ran continuously for 24-hour periods without noticeable deterioration of performance. The limiting factors to extended use are likely the capacity of the LN2 traps when droplets are doped with Xe or other atoms and molecules. One of the main indicators of stable operation is the pressure of the inlet gas after the traps (PG 4), which is an indicator of the amount of gas in the system. During normal operation, the pressure stayed constant to within 2 mbar, sometimes jumping up ~ 50 mbar, then falling back down to its normal value. This pressure should be in the range of 800 - 1200 mbar (absolute). Above this value, the KF o-ring-based fitting at the output of the scroll pumps may start to leak, and below this value, the compressor begins to complain, as indicated by a change in the pitch of the compressor and loud mechanical clicks. In 2017, we found that during operation the reading of P4 decreased slowly by about 20 mbar/hour, which may indicate some minute leaks in the high-pressure part of the system. The more likely cause is that during the experiment the 3 He gas was not collected from some downstream differentially pumped chambers, which is another loss source. In contrast, gas 173 was collected from all chambers in 2018 and there were minimal losses as long as pressure PG 4 remained below 1.2 bara. Importantly, the biggest cause of the loss (1-2 bar/L depending on the usage of the helping pump) is associated with the storage of the gas, because some sizable fraction of the gas inevitably stays in the low- and high-pressure parts of the system and does not enter into the storage cylinder. The recycling system in Figure 11.1 employs 2 Swagelok pressure reducers (model KPR1JRF411A20020), 11 Swagelok SS-DSS4 valves, 13 Swagelok SS-4BK-V51 valves, 3 Swagelok SS-DSM4F4A valves used attached to the cylinders, 2 electronic pressure gauges (Wika Tronic Line 891.23.510 and S-10), a 500 mL gas cylinder (behind board, near top right of picture), a 1000 mL gas cylinder (double-ended, middle-right of picture), a compound pressure gauge (model unknown), 2 Swagelok pressure gauges, model number PGI-63B-PG100-LAQX-J, 2 Swagelok Swagelok SS-4F-VCR-2 inline filters, an Agilent IDP-3 scroll pump (underneath manifold), 2 custom LN2 traps diagrammed in Figure 10.6 (inside of silver LN2 dewars), and a Fluitron S1-20/150 compressor, shown in blue and yellow. Leak Testing the Recycling System at SLAC Figure 11.4 shows me, along with the recycling system, after the conclusion of the October 2018 beamtime. The LCLS beamline has a He leak tester that is capable of positive and negative pressure leak testing. A similar tester was used at USC in preparation experiment. The manifold is tested in sections, starting with all valves closed and slowly opening valves, starting with the “to external vacuum,” valves. In addition to the manifold and compressor, ~10 m of 1/4” tubing for return gas and 5 m of 1/8” tubing for high pressure gas was added to the recycling system and tested at SLAC. Such long tubing was required because space close to the cryogenic source was limited. It took roughly a week to build and leak-test all components of the recycling system. The 174 Adixen DFT-25 filters were the biggest source of leaks with a leak rate of ~10 -7 mbar L s -1 . The next biggest leak was somewhere in the storage section of the manifold, but I was never able to pin it down fully, and that leak was ~5 × 10 -8 mbar L s -1 . The rest of the system leaked at a very low ~10 -9 mbar L s -1 . The leak in the storage section is not critical since the system is only used briefly to introduce He to the rest of the system. Figure 0.4. A boy and his 3 He recycling system. Taken at the AMO hutch at LCLS, SLAC, October 2018. References [1] R. E. Honig, Radio Corporation of America RCA Labratories Division, Vapor Pressure Data for the More Common Elements (David Sarnoff Research Center, Princeton, NJ, 1957). [2] L. F. Gomez, S. M. O. O’Connell, C. F. Jones, J. Kwok, and A. F. Vilesov, Laser- Induced Reconstruction of Ag Clusters in Helium Droplets. The Journal of Chemical Physics 145, 114304 (2016). 175 [3] M. Hartmann, University of Göttingen, 1997. [4] J. Harms, M. Hartmann, B. Sartakov, J. P. Toennies, and A. F. Vilesov, High Resolution Infrared Spectroscopy of Single Sf6 Molecules in Helium Droplets. Ii. The Effect of Small Amounts of He-4 in Large He-3 Droplets. Journal of Chemical Physics 110, 5124 (1999). [5] J. Harms, J. P. Toennies, M. Barranco, and M. Pi, Experimental and Theoretical Study of the Radial Density Distributions of Large He-3 Droplets. Physical Review B 63, (2001). [6] S. Grebenev, J. P. Toennies, and A. F. Vilesov, Superfluidity within a Small Helium-4 Cluster: The Microscopic Andronikashvili Experiment. Science 279, 2083 (1998). 176 Appendix 3: 3 He Recycling System Operation Manual 177 Originally written for the LP05 Beamtime at LCLS, SLAC, 8/25/17-8/29/17 System Introduction The helium recycling system is designed to intake the output of a vacuum pump, trap any stray gases, such as nitrogen and oxygen, and compress the remaining helium so that it may be reused in further experiments. The system is designed to recycle 3 He and 3 He/ 4 He mixtures, and once the gases are mixed, the system cannot unmix them. Note also that this procedure was written for use with the Fluitron S1-20/150 compressor on loan from Wolfgang Jaeger at the University of Alberta. This compressor is a 2-stage compressor capable of achieving an output greater than 120 bar. When using the Sundyne PPI 1560 compressor, other methods may be needed since the Sundyne unit is a single-stage compressor capable of pumping to 22 bar from atmosphere. The system is sectioned into three separate parts: the so-called low-pressure system, the high- pressure system and the cylinder/storage system. The low-pressure system intakes and cleans the vacuum exhaust in a cryogenic trap before feeding into the compressor. The high-pressure system is used in the recycling mode of operation and regulates the pressure of the helium coming out of the compressor and going into the cold head. The cylinder system is used to introduce gas into the system at start-up and store gas between experiments. Gas mixtures may also be made in the cylinder system. There are several components of the recycling system that exist off the board: a cryogenic trap and corresponding nitrogen dewars, gas compressor, dry vacuum pump, and vacuum chamber assembly, including cold head. The low-pressure system is built with VCR flex lines, which can only sustain 6 bar over-pressure. Therefore, much care must be taken not to put much pressure in the system. Under typical operation, the pressure in the low-pressure system (which may be monitored by an electronic gauge – NOTE: The electronic gauge reads absolute pressure in absolute mbar!) should not exceed 2 bar absolute (14 psig) of pressure. The high pressure and cylinder systems connect directly to the low- pressure system, so be VERY careful when using valves. Always open valves slowly and be sure to closely monitor pressures. The following manual instructs the user to operate the system in 4 basic modes: (I) system evacuation, (II) filling the system with a gas, (III) normal recycling operation and (IV) stopping operation to store helium. A diagram of the manifold is shown in Figure 12.1. Note the color coding: RED valves lead directly to atmosphere and should always remain closed. ORANGE valves lead to vacuum pumping, and ORANGE lines are pumped while the external vacuum pump is operational. GREEN valves connect from a system on the board to a system external to the board, such as the compressor, cold head, or vacuum exhaust. BLUE valves connect one system on the board to another, like the line that leads from the high-pressure system to the low-pressure system, or the high-pressure system to the cylinder system. BLUE lines highlight the parts of the board that are connections. PURPLE lines highlight helium storage cylinders and their connections to the lines. These connections include the 3 He lecture bottle, and the two storage cylinders: a ~500 mL cylinder mounted to the top of the system and a ~1000 mL cylinder mounted to the front of the board. 178 Figure 0.1. A color-coded schematic of the manifold, copied from Figure 11.1. Blue arrows indicate the direction of helium flow during operation. Blue valves denote connections between parts of the manifold, while green valves denote connections between the gas manifold and other parts of the recycling system. The black lines indicate tubing connections on the board. The dotted green lines illustrate recycling system connections that are off the board. Purple valves and lines denote connections to storage cylinders that are mounted to the manifold. Orange valves connect the gas manifold to the vacuum pump used to evacuate the recycling system. The orange lines denote tubing that is under vacuum while the evacuation pump is turned on. Red valves are currently unused. Pressure gauges, shown as PG, are installed on the lecture bottles and after the LN 2 traps. The pressure reducers similarly have pressure gauges, the input being indicated as H and the output indicated as L. 179 Evacuating the System 12.2.1 Purpose The system must be evacuated after being moved, or after standing unused for an extended time. Evacuating the system provides a clean system that can easily be filled and purified with helium. 12.2.2 Introduction The system needs to be pumped by an external oil-free pump that is attached to the cross below valve O3. A pressure gauge should be attached to the pump to monitor the pressure of the system. The Agilent IDP3 pump that was used for testing could achieve an ultimate pressure of ~0.03 mbar, but the system has gone down to 10 -3 mbar when larger pumps are used. Evacuation of the system should take place before the cold head is in operation and while the trap is at room temperature. The trap should be pumped separately from the rest of the system, so always keep valves G6 and G7 closed during system evacuation. 12.2.3 Procedure 1. To start evacuation, ensure that the cold head and compressor are not running and the trap is at room temperature. 2. Make sure that all valves are closed, except for B7 and B8, which remain open. 3. Turn on the external vacuum pump and wait until a pressure less than 0.05 mbar is reached in the vacuum line. 4. The system should be pumped in sections. First, pump the trap. Open valve O2. Let the trap pump for 5-10 minutes to purge the zeolite of any residual gases. Once the trap has reached a good pressure, close valve O2. a. In case of extended operation, or operation at room temperature, the trap should be baked before further use. To bake the trap, open valve O2. b. Heat the trap evenly with a heat gun, heating straps, or another method that evenly and safely heats the trap. Always use care when heating the trap. Be aware of your surroundings, including other personnel in your vicinity. Never use open flames to heat the trap. c. Once the trap is heated, allow it to cool to room temperature, then close valve O2. 5. Once the trap is pumped and the valves are closed, immerse the trap in LN2. 6. Next, pump the low-pressure section. Slowly open valve O3. 7. Once a low pressure is achieved in the vacuum line and on Pressure Gauge 4 and Electronic Pressure Gauge 2, the high-pressure line may be pumped. Carefully open valve B1, making sure to keep the pressure on Pressure Gauge 4 and Electronic Pressure Gauge 2 below 2 bar absolute (14 psig). Pressure Reducer 1 and valve G1 may be fully opened to ensure pumping through the line. 8. Once a low pressure is reached in the low-pressure system and the vacuum line, the cylinder line may be pumped. If the cylinder line does not need to be pumped, skip to step 11. 180 9. To start pumping ensure valve B6 is closed. Open valve B4 and Pressure Regulator 2. Use valve B6 to regulate flow into the low-pressure part, as the regulator is rather imprecise. Watch Pressure Gauge 4 and Electronic Pressure Gauge 2 for pressure readings. 10. Once valve B6 is fully opened, valves B5 and O1 may also be opened. 11. Next, pump the compressor. Slowly open O4, G2, and G8. The compressor usually takes 5 to 10 minutes to pump, but it can be longer if water is present or the pumping line is long. 12. Close the vacuum exhaust valve and open G5. The vacuum exhaust section takes a while to pump, possibly up to 30 minutes. 181 Filling the System 12.3.1 Purpose Filling the system with helium allows for the operation of experiments with clean helium of limited quantity. The system should be filled quickly with this procedure so that experiments may proceed. 12.3.2 Introduction In principle, one can fill the system with any gas through the lines extending from valves B2 and B3. However, it is recommended that the system only be used with 3 He, 4 He, and Ar. The system is not designed to handle any corrosive or reactive gases, so the use of these gases with the current system is prohibited. By filling the system through the backfill, one avoids the time-consuming task of letting the gas run through the cold head (at ~3 cm 3 /s) and filling the low-pressure system. The following directions guide the user through filling the system. It is important that the user be mindful to not let the low-pressure system exceed 2 bar absolute (14 psig) at any time. 12.3.3 Procedure 1) Once the compressor is under vacuum, ensure that the cylinder system is isolated by closing valves B5, B4, and O1. Check that valves B2, B3, B9, G3 and G4 are also closed. Then, open the valve corresponding to the gas you want to fill. i. To fill with 3 He, ensure tank is attached, open the lecture bottle valve, note the pressure, then open valve B2. ii. To fill from Mixed 3 He/ 4 He storage: i. If stored in lecture bottle, open the lecture bottle valve, note the pressure, then open valve B3. ii. If stored in 1000 mL cylinder, open the large cylinder valve, note the pressure, then open valve B9. iii. To fill from external cylinder: i. External cylinder must use a regulator. ii. Connect output of regulator to either valve G3 or G4. iii. Set pressure to desired value on regulator. iv. Open the valve to which the regulated cylinder is attached. 2) Once the cylinder line is filled, ensure that Pressure Reducer 2 is entirely closed and that valves B1, B6, O2, O3, and O4 are closed. Once valve B6 is closed, open valve B4. 3) Carefully open Pressure Regulator 2 until the needle on the output pressure gauge just registers a pressure. NOTE- This pressure is imprecise. 4) Carefully open valve B6 until the pressure in the low-pressure system begins to rise. DO NOT LET THIS PRESSURE RISE ABOVE 2 BAR ABSOLUTE (14 PSIG). DO 182 NOT LEAVE VALVE B6 OPEN. Pay attention to Pressure Gauge 4 and Electronic Pressure Gauge 2. 5) At this point, the compressor should be filling with helium as well. Ensure that valves G1 and G2 are open, then turn the compressor on. Normal compressor operation is indicated audibly by consistent mechanical clicks, and visually by 2 streams of oil flowing into the reservoir. Note that the second stage takes 30-45 seconds to prime after turning on the compressor. 6) Once the compressor is running 1 , fill the system so that the pressure after the first stage of the compressor (not pictured in diagram, but labeled on compressor) reads 390 psig and the low-pressure system is at ~1 bar absolute (0 psig). The maximum pressure after the first stage of the compressor is set to 390 psig. 7) Slowly increase the output of Pressure Regulator 1 to desired nozzle pressure, P0. 8) Again, fill the system until the first stage compressor pressure is 390 psig and the low- pressure system is at ~1 bar absolute (0 psig). Monitor Pressure Gauge 4 and Electronic Pressure Gauge 2. 9) To fill the trap, open valves G6 and G7. If the system is not immersed in LN2, then immerse it now and let it cool. 10) Again, fill the system until the first stage compressor pressure is 390 psig and the low- pressure system is at ~1 bar absolute (0 psig). Monitor Pressure Gauge 4 and Electronic Pressure Gauge 2. 11) To circulate gas through the trap, close valve B7. 12) At this point, the system should be in normal recycling operation. Turn on the cold head to begin experiments. The system will need to be refilled once the cold head has reached the experimental temperature. Note that even slight changes in temperature when the cold head is cold (3.5~30 K) will affect the pressure in the low-pressure system. 1 If compressor does not operate but makes a loud buzzing sound and quickly shuts off, the flywheel must be rotated by hand. Always disconnect the power cable before working on the pump. Refer to the compressor notes for details and troubleshooting. 183 Normal Recycling Operation Purpose The system should be able to recycle helium for at least 12 consecutive hours, the length of a shift at LCLS. By recycling helium, losses of 3 He are minimized, and many experiments can be performed. Introduction Normal system operation entails a few different aspects: tracking losses of helium, making helium mixtures, and circulating helium through the system itself. Each task will be described below, but it should be noted that the instructions for making 3 He/ 4 He mixtures may be altered as more experience is gained in this area. Normal Run Procedure 1. Normal operation is defined as helium cycling through the system, including through the trap, while the compressor is running. The pressure after the first compressor head should be 390 psig, and the pressure after the second compressor head should be 110 bar. 2. To maintain this operation, the pressure in the low-pressure system should be between 750 mbar and 1200 mbar, as read on Pressure Gauge 4 or Electronic Pressure Gauge 2. Add more gas as necessary to maintain proper pressure. 3. At USC, the system lost ~15 cm 3 ·bar·hr -1 . The loss rate at SLAC has yet to be determined. Electronic Pressure Gauge 2 should be used to monitor helium losses. Gas Mixing Gas mixing takes place in the cylinder line. Mixtures are formed by filling the cylinder line with a gas and leaving the gas reservoir connected to the line. If needed, pump the line by opening valve O1. Ensure that valves O1 and B4 are closed before beginning operation. In the following, two methods of gas mixing are proposed. It was found that the first method was more prone to overshoot the desired mixture level, and the second method was a more careful procedure. Neither method produced a stable mixture, even after an hour or so of operation. The results tend to suggest that the recycling system is not a good way to mix gases, and one should mix gases before adding them to the recycling system. IMPORTANT DISCLAIMER Neither of these methods produced satisfying results. Method 1 was not feasible due to equipment malfunctions, and Method 2 resulted in unstable mixing ratios that varied over ~15-minute periods. Future operators should invest time in finding better mixing methods. 184 Method 1 Only open the valves corresponding to the type of gases you wish to mix: i. If pure 3 He is being used, ensure that B2 and Lecture Bottle Valve 1 are both open. ii. If a mixture of 3 He/ 4 He is being used, open valve B3 and Lecture Bottle Valve 1 to use the mixture in the 500 mL lecture bottle. Or, to use the mixture in the 1000 mL cylinder, open valve B9 and Cylinder Valve 1. iii. Different gases may also be mixed by adding gas through valve G3 if desired. Using this valve is not applicable to Method 1 but proved useful in Method 2. Once you have filled the line with gas, note the pressure of Electronic Pressure Gauge 1. This pressure will be called Pi. Next, the amount of gas to be added needs to be determined. Use the following formula to determine the amount of pressure to be added to the system. 𝑀𝑖𝑥% = z𝑃 ) −𝑃 0 { 𝑃 ) Rearranging this formula for the final pressure, one arrives at the following equation: 𝑃 ) = 𝑃 0 (1−𝑀𝑖𝑥%) Note that Electronic Pressure Gauge 1 only has one decimal of precision (in bar), so mixtures less than 5% likely cannot be made reliably. A more precise measure of the mixture percent may be obtained using a mass spectrometer. Note that if the volumes of the gas reservoirs are of different sizes, which will almost always be the case, the mixture percent will change with time. The mixture ratio can be reset by closing all valves, evacuating the line, then repeating the procedure. To use a gas mixture in the system, follow the filling instructions. Method 2 i. While running, ensure that PG 4 or Electronic Pressure Gauge 2 both read a value that is less than your desired final operation pressure. For example, if one wants to make a ¾ mixture of the current gas and some other gas, the initial pressure should be ¾ of the desired final operational pressure of the 2 gases. ii. If pure 3 He is being added to the recycled gas, ensure that B2 and Lecture Bottle Valve 1 are both closed. iii. If a mixture of 3 He/ 4 He is being used, open Lecture Bottle Valve 1 to use the mixture in the 500 mL lecture bottle. Or, to use the mixture in the 1000 mL cylinder, open Cylinder Valve 1. 185 iv. Different gases may also be mixed by adding gas through valve G3 if desired. Using this valve is not applicable to Method 1 but proved useful in Method 2. v. Using the gas source of choice, open the valve closest to the container. 1. For 3 He, open Lecture Bottle Valve 1. 2. For 4 He or other gases, open the valve on the cylinder and regulate to the desired pressure (cylinder valve and regulator must be external to the system and are not shown). 3. For mixtures in 500 mL and 1000 mL storage cylinders, open Lecture Bottle Valve 2 or Cylinder Valve 1, respectively. vi. Close the valve you just opened. Now there is a small section of the gas line filled with the desired doping gas. This “shot” of gas is what will be added to the recycling gas. vii. Open the valve connecting your “shot” of gas to the storage line: 1. For 3 He, open B2. 2. For 4 He or other gases, open valve G3. 3. For mixtures in 500 mL and 1000 mL storage cylinders, open valve B3 or B9, respectively. vii. Open B4, and carefully open Pressure Reducer 2 until a rise is seen in Electronic Pressure Gauge 2. viii. Close the valve opened in step vii. Then close the pressure reducer and valve B4. ix. Wait for a few minutes for the system to equilibrate. Then repeat steps i-viii as necessary. Repeating the steps ensures that there is not too much gas added to the system accidentally. One can also simply regulate the gas using the pressure regulator between valves B4 and B6. 186 Stopping Recycling and Filling Cylinders 12.11.1 Purpose While not running experiments, it is important to try to collect as much gas as possible into the cylinders. The collected gas can be reintroduced to the system by following item 1) b. of the filling procedure. 12.11.2 Introduction Filling of the cylinders should only be done while the cold head is above ~80 K. At low temperatures, a large amount of helium is sucked in the nozzle reservoir as a fluid, so heating up the cold head will allow a lot of this helium to be recaptured. If there is an auxiliary, oil-free pre-compression pump available, connect its input to G9 and its output to G10. Using an auxiliary pump is a great way to accumulate all the helium that the compressor cannot collect. 12.11.3 Procedure 1) Ensure that the helium recycling compressor is running and turn off the cold head compressor if it is running. Then wait for the cold head to warm up. Incremental heating to 80 K may be used to speed up the warming time. 2) To start filling the cylinders, ensure that valves G3 and G4 are closed. If there is any unwanted gas in the cylinder fill line, carefully open valve O1 to pump this gas away. Close valve O1 after the line has been evacuated. 3) Ensure that valve O1 and B4 are closed. Then, open valve B5 and close valve G1. At this point, the compressor will now be filling the cylinder line. 4) Choose which cylinder you wish to fill. For the lecture bottle, open the lecture bottle valve, then open valve B3. To fill the 1000 mL cylinder, open the large cylinder valve, then open valve B9. Only fill one cylinder at a time. NOTE – THE 1000 mL CYLINDER CANNOT HOLD MORE THAN 27 ATM OF PRESSURE. CAREFULLY WATCH PRESSURE GAUGE 3 WHEN FILLING THE 1000 mL CYLINDER. 5) Completely actuate Pressure Reducer 2 and open valve B7, then slowly open valve B6 to release the pressure in the line. Once the line is depressurized, leave valve B6 open. 6) To release the pressure from the high-pressure line, completely actuate Pressure Reducer 1, then slowly open valve B1. When the line has been depressurized, leave valve B1 open. 7) Allow the compressor to pump until the low-pressure system is operating at ~250 mbar absolute, or if the compressor begins to complain too much. Note that the trap should remain immersed in LN2, even though a large chunk of gas will remain in the trap. Removing the trap from LN2 could contaminate the system with nitrogen and other gases. 187 8) If an oil-free auxiliary pump is available and connected as described in the Introduction to this section, then close valve B8 and turn on the compressor to collect the helium gas that the compressor cannot gather. 9) Once the low-pressure system has reached ~250 mbar, (or lower, with the use of a scroll pump) close valves G8, G2, and G5, then shut off the compressor. Then close the valves corresponding to the bottle that was filled and open the scroll exhaust valve. 10) Before removing the trap from LN2, close valve G6 and G7, then open valve O2. Once the trap has been pumped to a low pressure, remove it from the LN2. Observe the trap to ensure the pressure of the vacuum line rises and falls as gases heat up and desorb from the zeolite then get pumped away. 188 Index 12.12.1 List of Valves Note that the description for the red valves details the intended purpose of the valve. Some of the systems were not implemented, so the valves were blanked. Valve Name Description of Valve R1 Valve for a reservoir that may regulate pressure fluctuations R2 To the inlet for a helping compressor (Agilent scroll pump or similar) O1 Cylinder line pumping valve O2 Trap pumping valve O3 Low-pressure system pumping valve O4 Compressor pumping valve (attached between first and second compressor stages) G1 Valve to cold head (open for normal recycling operation) G2 Separates outlet of compressor from main recycling line and cylinder line G3 Gas inlet (He) G4 Gas inlet (Ar) G5 Vacuum exhaust inlet (open for normal recycling operation) G6 To inlet of trap G7 From outlet of trap G8 To the inlet of the first stage of the compressor G9 Connection to inlet of Pre-Compression Pump G10 Connection from outlet of Pre-Compression Pump B1 Used to pump the high-pressure line B2 Connects 3 He lecture bottle to cylinder line B3 Connects 3 He/ 4 He storage bottle to cylinder line B4 Used to backfill system B5 Open when storing helium 189 B6 Used to regulate the backfill of the system. Operate very carefully. B7 Trap Shortcut (open when trap is closed) B8 Helping compressor shortcut (This valve may be left open) B9 Connects 1000 mL storage cylinder to cylinder line LBV 1 Valve attached to 3 He lecture bottle LBV 2 Valve attached to small 3 He/ 4 He storage bottle CV 1 Inlet valve to 1000 mL cylinder CV 2 Outlet valve of 1000 mL cylinder (outlet currently blanked) 190 12.12.2 Compressor Notes The manual for the compressor is very general, so it is important to read it and understand the basic principles of the compressor. However, it does not contain many integral points for the compressor operation. If anything in this section is unclear, Fluitron has been extremely helpful in answering questions. Bryan Waltz (bwaltz@fluitron.com) has been a very good contact. The company can be reached by phone at 215-355-9970. Note that they are based on the East Coast. These notes are written under the assumption that the reader has some familiarity with the compressor. 12.12.3 Introduction to Priming the Compressor Heads Priming the heads can be done while the compressor is on or off. It is sometimes helpful to prime when the compressor is on so that the compressor may take over and achieve overpump. Overpump is the term used by Fluitron to describe the normal operation of the compressor heads. When overpump is achieved, one can see oil coming from the compressor heads and into the main reservoir. Usually, the oil from the first head comes in as a stream, and the oil from the second head can come in as a stream, or it may stick to the side of the reservoir. Priming of the heads is only required if overpump is not achieved after a few minutes of operation. Before priming the system, one must pay attention to the orientation of the yellow handles in the lower plumbing section. The handle is in normal operation mode when it is perpendicular to the back of the compressor, see Figure 6.1, and the handle is in priming mode when it is parallel to the back of the compressor mount, see Figure 6.2. Note that these valves are not on/off valves; they always output to the compressor head. Changing their orientation selects the line that feeds to the compressor head, as indicated by the arrows printed on the valve handles. Figure 6.1. A simplified top-down view of the compressor with both valves in the normal run position. Note that the sizes and distances of components are not representative of the real components. Flywheel Cover Motor Back of Compressor Mount First Stage Valve Second Stage Valve Manual Priming Pump 191 Figure 6.2. A simple top-down view of the compressor with both valves in the priming position. Note that the sizes and distances of components are not representative of the real components. Also note that one valve must always be in the run position. Both heads cannot be primed at the same time. Flywheel Cover Motor Back of Compressor Mount First Stage Valve Second Stage Valve Manual Priming Pump 192 12.12.4 Procedure: 1. Choose which head to prime. The order in which the heads are primed does not matter, but ONLY ONE HEAD CAN BE PRIMED AT A TIME. 2. To prime the system, first ensure that both valves are in normal run mode (Figure 6.1). Once the proper valve configuration is reached, pull up on the manual priming pump (black/silver unit with black spherical handle). Pulling up on the manual pump in the priming position pulls a vacuum on the compressor head, which could damage the diaphragms and un-prime the system. Take care when pulling up on the manual pump. 3. Once a head is chosen, move the corresponding valve to the prime position, then push down on the manual pump. Some resistance should be felt, especially when the pump is close to being primed. 4. Once the stroke is complete, return the valve to the normal run position and go back to step 2. a. If the compressor is running, the priming may only take a few cycles to achieve overpump, and the compressor will take over easily. b. If the compressor is not running, some overpump may be seen. Note that some oil returns to the reservoir from a port in the bottom, and this may sometimes appear as overpump, but it is just a return from the line. See troubleshooting section for details. 5. Once overpump is achieved in one head, prime the other head by following steps 2-4. Figure 6.3 provides the positions of the valves for the upstroke, priming of the first head, and priming of the second head of the compressor. 193 Flywheel Cover Motor Back of Compressor Mount First Stage Valve Second Stage Valve Manual Priming Pump UPSTROKE Flywheel Cover Motor Back of Compressor Mount First Stage Valve Second Stage Valve Manual Priming Pump PRIMING FIRST HEAD Flywheel Cover Motor Back of Compressor Mount First Stage Valve Second Stage Valve Manual Priming Pump PRIMING SECOND HEAD 194 Figure 6.3: The Priming cycle of the compressor head. Upstroke ONLY when both valves are in the normal run position. Then, downstroke when the valve for the desired head is in the prime position. Only prime one head at a time. 12.12.5 Troubleshooting the Compressor i. Compressor will not run If the compressor is not running, ensure that the unit is plugged in and that the breaker box is set correctly by checking the breaker rating in the lower right corner of the fuse unit. ONLY INSPECT FUSE BOX WHEN THE CORD IS UNPLUGGED. Wear all necessary PPE and follow all appropriate procedures when working with electrical components. If this inspection reveals no flaws, unplug the compressor and remove the yellow flywheel cover. While the cover is off, ensure that everyone stays back of the flywheel. Do not approach the flywheel assembly while the compressor is plugged in. Do not work on the compressor while it is plugged in. Plug in the compressor, turn it on, and observe the rotation of the motor. I. If the motor does not rotate, and a loud buzzing is heard before the breaker trips and the motor shuts off, the flywheel must be rotated by hand. Ensure that the compressor is unplugged, then turn the large flywheel counterclockwise (when looking at the front of the compressor) by hand. If great resistance is encountered, push through it. After the resistance is alleviated, the flywheel should turn smoothly before encountering a lesser resistance. This is the most common problem when starting the compressor, especially after the compressor has been under vacuum. II. If the motor rotates clockwise (when looking at the front of the compressor), then unplug the compressor, interchange any two of the leads in the plug, plug the compressor back in, and test the compressor again. III. If the motor rotates counterclockwise (when looking at the front of the compressor), then the system should be operating normally. If problems persist, contact Fluitron with pictures and videos of the issue. ii. A Stage of the Compressor will not Prime This issue was encountered in the second stage when starting the unit for the first time, and after changing the filters on the unit. To solve this problem, turn the compressor off and unplug it. Ensure that the lockrings on the oil relief valve (refer to compressor manual) are tight, then back off (loosen) the nut in the oil relief valve. DO NOT LOOSEN THE NUT ON THE OIL RELIEF VALVE WHILE THE COMPRESSOR IS RUNNING. Be careful with how the system is braced so that no leaks are caused or develop. Once the nut is loosened, perform the priming procedure. Plug the compressor back in, then continue to prime until overpump is achieved. Once overpump is achieved, and while the compressor is running, tighten the nut back to its original position indicated by the lock ring. 195 If the head will not overpump, shut down and unplug the compressor again. Remove the nut from the oil relief valve, and remove the inside components of the oil relief valve, but be sure to note the way they are oriented. Wearing appropriate PPE, especially goggles, carefully manually prime the head. If you hear a pop when actuating the head, then the system is priming. At a certain point, oil should start to fill the oil check valve. At this time, the check valve assembly may be reinstalled, and the nut may be tightened slightly onto the valve. Plug in and turn on the compressor, then repeat the priming procedure if need be. If the issue persists, contact Fluitron. iii. Second Stage of Compressor Stopped Running During Operation This issue was first noticed after running the compressor continuously for more than 12 hours. At this point, the cause of this issue is unknown, but it may be a leak somewhere in the oil plumbing line. The fix is simple: prime the second stage while the compressor is running. The compressor should take over after one stroke of the manual pump.

Abstract (if available)

Abstract Helium nanodroplets have been used as spectroscopic matrices for the last few decades. These unique systems can also shed light on more fundamental questions about quantum mechanics and superfluidity. This thesis features the results from several experiments that explored helium droplets as a vehicle to study molecules and clusters, and as a unit of study itself. The following contains excerpts from previously published abstracts [1-4]. ❧ The formation of large droplets upon expansion of supercritical fluid remains poorly understood. The phenomenon of liquid jets disintegrating into droplets has attracted the attention of researchers for more than 200 years. Most of these studies considered classical viscous liquid jets issuing into ambient atmospheric gases, such as air. Here, optical shadowgraphy was applied to study the disintegration of a cryogenic liquid ⁴He jet produced with a 5 µm diameter nozzle into vacuum. The physical properties of liquid helium, such as its density, surface tension, and viscosity, change dramatically as the fluid moves through the nozzle and evaporatively cools in vacuum, eventually reaching the superfluid state. This study demonstrates that, at different stagnation pressures and temperatures, droplet formation may involve spraying, capillary breakup, jet branching, and/or flashing and cavitation. The average droplet sizes reported in this optical imaging experiment range from 3.4 × 10¹² to 6.5 × 10¹² helium atoms, equating to 6.7–8.3 µm in diameter. ❧ Large ⁴He droplets have been employed by different groups for at least a decade, but similar ³He droplets remain to be studied, due in part to the significant cost of ³He gas. Previous single-pulse extreme ultraviolet and X-ray diffraction studies revealed that superfluid ⁴He droplets obtained in a free jet expansion acquire sizable angular momentum, resulting in significant centrifugal distortion. Similar experiments with normal fluid ³He droplets may help elucidate the origin of the large degree of rotational excitation and highlight similarities and differences of dynamics in normal and superfluid droplets. ❧ Another experiment presented in this thesis compares the shapes of isolated ³He and ⁴He droplets following expansion of the corresponding fluids in vacuum at temperatures as low as ∼2 K. Large ³He and ⁴He droplets with average radii of ∼160 and ∼350 nm, respectively, were produced. It is found that most of the shapes of ³He droplets in the beam correspond to rotating oblate spheroids, in agreement with previous observations for ⁴He droplets. The aspect ratio of the droplets is related to the degree of their rotational excitation, which is discussed in terms of reduced angular momenta (Λ) and reduced angular velocities (Ω), the average values of which are found to be similar in both isotopes. This similarity suggests that comparable mechanisms induce rotation regardless of the isotope. It is hypothesized that the observed distribution of droplet sizes and angular momenta originate from processes in the dense region close to the nozzle, where a significant velocity spread and frequent collisions between droplets induce excessive rotation followed by droplet fission. ❧ The study of rotation in bulk superfluids is often complicated by interaction with container walls, which could not be well characterized. Helium nanodroplets offer unique opportunities to study quantum systems as they freely rotate in vacuum. The angular momentum of rotating superfluid droplets originates from quantized vortices and capillary waves, the interplay among which remains to be uncovered. Here, the rotation of isolated sub-micrometer superfluid ⁴He droplets is studied by ultrafast X-ray diffraction using a free electron laser. The diffraction patterns provide simultaneous access to the morphology of the droplets and the vortex arrays they host. In capsule-shaped droplets, vortices form a distorted triangular lattice, whereas they arrange along elliptical contours in ellipsoidal droplets. The combined action of vortices and capillary waves results in droplet shapes close to those of classical droplets rotating with the same angular velocity. The findings are corroborated by density functional theory calculations describing the velocity fields and shape deformations of a rotating superfluid cylinder. ❧ Mixed ³He/⁴He droplets present another interesting rotational system. It is well known that a mixture of ³He and ⁴He separates into discrete phases below ~0.9 K. While this phase separation has been characterized in the bulk, so far, the phase separation studies were limited to small quiescent droplets of few nm in diameter. In fact, no experimental studies of rotation in classical or quantum multiphasic droplets could be found. Herein, large mixed droplets of a few hundred nm in diameter were imaged via X-rays from a free electron laser, similar to the previous ³He and ⁴He studies. It was found that droplets containing equal amounts of ³He and ⁴He did not show any signs of phase separation during the ~4 ms time of flight from the nozzle to the point of interaction with the X-rays. However, droplets containing ~75% ³He and 25% ⁴He did undergo phase separation. In these phase-separated droplets, a variety of configurations were seen: i) a centrally located inner droplet of ⁴He, ii) an off-center inner ⁴He droplet, and iii) two off-center inner ⁴He droplets. In each case, these inner ⁴He-rich droplets were encased in an outer ³He droplet. These results are compared to classical and density functional theory calculations of general multiphasic and ³He/⁴He droplets, respectively. ❧ Heat transfer from embedded species to He droplets is still not fully understood. In another experiment in this thesis, silver clusters were assembled in helium droplets of different sizes ranging from 10⁵ to 10¹⁰ atoms. The absorption of the clusters was studied upon laser irradiation at 355 nm and 532 nm, which is close to the plasmon resonance maximum in spherical Ag clusters and in the range of the absorption of the ramified Ag clusters, respectively. The absorption of the pulsed (7 ns) radiation at 532 nm shows some pronounced saturation effects that are absent upon continuous irradiation. This phenomenon has been discussed in terms of the melting of the complex Ag clusters at high laser fluence, resulting in a loss of the 532 nm absorption. Estimates of the heat transfer also indicate that a bubble may be formed around the hot cluster at high fluences, which may result in ejection of the cluster from the droplet, or disintegration of the droplet entirely. ❧ This thesis also contains plans for a magnetic circular dichroism (MCD) experiment that was interrupted due to the COVID-19 pandemic. Signals derived from MCD spectroscopy generally investigate the energy difference in degenerate states that split in the presence of a magnetic field. While there are several ways for these states to split, the one most advantageous for study using He droplets involves a ground state splitting that is directly proportional to the magnetic field strength and inversely proportional to temperature. While no data was taken, the pickup cell containing zinc phthalocyanine, a common MCD analyte, was aligned and the system was set up for standard depletion absorption spectroscopy measurements. If these standard measurements would have been completed, a magnet tube with a field that rapidly switches between ±0.4 T would have been inserted to perform MCD experiments using zinc phthalocyanine. Upon successful proof-of-concept experiments, the MCD setup may then be adapted to studies of Fe clusters. ❧ References: ❧ [1] L. F. Gomez, S. M. O. O’Connell, C. F. Jones, J. Kwok, and A. F. Vilesov, Laser-Induced Reconstruction of Ag Clusters in Helium Droplets. The Journal of Chemical Physics 145, 114304 (2016). ❧ [2] S. M. O. O'Connell et al., Angular Momentum in Rotating Superfluid Droplets. Physical Review Letters 124, 215301 (2020). ❧ [3] R. M. P. Tanyag, A. J. Feinberg, S. M. O. O’Connell, and A. F. Vilesov, Disintegration of Diminutive Liquid Helium Jets in Vacuum. The Journal of Chemical Physics 152, 234306 (2020). ❧ [4] D. Verma et al., Shapes of Rotating Normal Fluid ³He Versus Superfluid ⁴He Droplets in Molecular Beams. Physical Review B 102, 014504 (2020).

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Asset Metadata

Creator O'Connell, Sean Marcus Olson (author)

Core Title Study of rotation and phase separation in ³He, ⁴He, and mixed ³He/⁴He droplets by X-ray diffraction

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Degree Conferral Date 2021-08

Publication Date 07/08/2023

Defense Date 05/19/2021

Tag ³He/⁴He,droplets,helium,helium nanodroplets,imaging,nanodroplets,OAI-PMH Harvest,quantum vortices,rotation,spectroscopy,superfluid,superfluidity,XFEL,X-ray diffraction

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Vilesov, Andrey (committee chair), Dawlaty, Jahan (committee member), Tanguay, Armand Jr. (committee member)

Creator Email sean.mo.oconnell@gmail.com,smoconne@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC15293610

Unique identifier UC15293610

Legacy Identifier etd-OConnellSe-9711

Document Type Dissertation

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Rights O'Connell\, Sean Marcus Olson

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Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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