Vorticity in superfluid helium nanodroplets

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Content 1 VORTICITY IN SUPERFLUID HELIUM NANODROPLETS by Charles Bernando 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 (PHYSICS) August 2016 2 Acknowledgements I would like to gratefully acknowledge my advisor, Prof. Andrey Vilesov for his guidance, support and motivation during my graduate study at University of Southern California (USC). He is a caring mentor, a brilliant scientist and a light-hearted person, who is always resourceful and patient in guiding me through challenging research projects. I could not have done my research projects extremely well without him. My gratitude extends to my fellow colleagues Dr. Luis F. Gomez, Rico Mayro P. Tanyag, Curtis Jones, Justin Kwok, Martin Seifrid, Deepak Verma, and Sean O’Connell who have helped in keeping a lively working atmosphere in the lab and for the insightful and fascinating discussions. I am thankful to Dr. Oliver Gessner and Dr. Christoph Bostedt who were very helpful and collaborative during the beam-time at SLAC National Laboratory. I would also like to thank Prof. Vitaly Kresin, Prof. Aiichiro Nakano, Prof. Alexander Benderskii, and Prof. Susumu Takahashi for numerous valuable comments and suggestions on this work and thesis. I am indebted to my late grandparents P.M. Sinaga and Sarmaia Saragih, my parents Rasmuli Sembiring and Roly Julia Nansy Sinaga, and my sister Maria Fransisca Sembiring, who always encourage and support me throughout my graduate study at USC. I am grateful to my aunts, uncles, and cousins for the thoughts and prayers for me. I am also grateful to my fiancé Moria Oktaviane, whose love and comfort always motivate me in difficult times. This thesis would not have been possible without funding from NSF and DOE. Portions of this research were carried out at the Linac Coherent Light Source (LCLS), a national user facility operated by Stanford University on behalf of the U.S. DOE OBES under beam-time grant 3 L549: Imaging of quantum vortices in superfluid helium droplets. This work was supported by the NSF Grant Nos. CHE-1362535 and DMR-1501276 of Andrey F. Vilesov. 4 Table of Contents Acknowledgements ..........................................................................................................................2 Table of Contents .............................................................................................................................4 Abstract ............................................................................................................................................7 Introduction ......................................................................................................................................9 Chapter 1: Properties of Large Helium Droplets ...........................................................................13 Chapter 2: X-Ray Diffraction Experiments on He Droplets ..........................................................22 2.1. Schematic of the X-Ray Diffraction Experiments at SLAC ..............................................22 2.2. X-Ray Scattering from Spheroidal He Droplets ................................................................29 Chapter 3: Shapes of Swiftly Spinning Superfluid He Droplets....................................................32 3.1. Introduction ........................................................................................................................32 3.2. Diffraction Images of Strongly Deformed He Droplets ....................................................35 3.3. Power Dependence of Intensity along the Streak ..............................................................41 3.4. Using the Inverse Fourier Transform for the Shape Reconstruction .................................47 Chapter 4: Shapes of Classical Rotating Droplets .........................................................................58 4.1. Shape Family of Classical Rotating Droplets ....................................................................58 4.2. Axially Symmetric Shapes .................................................................................................59 4.3. Two-Lobed Shapes ............................................................................................................63 4.4. Imaging of Droplets at Different Tilt Angles ....................................................................67 Chapter 5: X-Ray Scattering from Strongly Deformed He Droplets .............................................70 5.1. Discussion ..........................................................................................................................70 5.2. Conclusion .........................................................................................................................79 5 Chapter 6: Using He Droplets as Probes for the XFEL Beam .......................................................81 6.1. Beam Profile of X-Ray Free Electron Laser ......................................................................81 6.2. Calculation of XFEL Flux and Ionization Probability .......................................................84 Chapter 7: Decay of Surface Waves in Molecular Regime ...........................................................88 7.1. Introduction ........................................................................................................................88 7.2. Experimental Study of Ripplon Decay in Molecular Regime ...........................................89 7.3. Theoretical Studies on Damping of Ripplons in Three-Ripplon Interaction .....................93 7.4. Damping of Ripplons in One-Ripplon-Two-Phonon Interaction in Bulk Liquid He ........94 7.5. Damping of Ripplons in One-Ripplon-Two-Phonon Interaction in He Droplets ..............96 7.6. Lifetime of the Shape Oscillation during the X-ray Experiments ...................................100 7.7. Conclusion .......................................................................................................................102 Chapter 8: Kinematics of the Doped Vortices .............................................................................104 8.1. Rectilinear Vortices in a Cylinder....................................................................................106 8.2. Solvation Potential in Superfluid Helium ........................................................................107 8.3. Interaction between a Quantum Vortex and Xe Clusters in a Cylinder ...........................108 8.3.1. A Bare Quantum Vortex in a Cylinder .....................................................................108 8.3.2. A Quantum Vortex with Xe Filament in a Cylinder ................................................110 8.4. Doped Quantum Vortex in a Spherical Helium Droplet ..................................................113 8.5. Conclusion .......................................................................................................................119 Conclusions and Future Work .....................................................................................................120 Bibliography ................................................................................................................................123 Appendix ......................................................................................................................................129 A1: Other Streak Diffraction Images .......................................................................................129 6 A2: XFEL Flux and Ionization Probability of He and Xe in 2014 XFEL Experiments ..........136 7 Abstract Starting from Newton, the equilibrium shapes of rotating classical bodies held together by gravitation has been extensively studied theoretically. It was also shown that shapes of rotating droplets held together by surface tension belong to the same class of solutions and can serve as a system for laboratory emulation of astronomical objects. Equilibrium shapes of the classical rotating droplets have been extensively studied by calculations and experiments. However, surprisingly little is known about the rotation in viscosity-free, superfluid droplets, and their shape families, which is the focus of the present thesis. In this work, rotating superfluid helium droplets of up to about 1 μm in diameter, were produced upon expansion of helium fluid in vacuum and were studied via scattering of X-ray radiation from a free-electron laser (XFEL). This collaborative work was performed at Atomic, Molecular and Optical Science (AMO) Instrument at SLAC Linac Coherent Light Source (LCLS). The results show the existence of strongly deformed droplets as indicated by large asymmetry and intensity anomalies (streaks) in the obtained diffraction images. The analysis of the images shows that some droplets must have non-axisymmetric elongated shape which can be well modeled by classical droplet shapes along the two lobe stability line. The stability regions and shapes of superfluid and ordinary viscous droplets are compared. This thesis contains eight chapters. The first chapter provides introduction to the thermodynamics of He droplets. The second chapter contains description of the experimental set up and basics of the X-ray scattering from a spheroidal object. The third chapter describes the results of the X-ray coherent diffractive imaging experiments of superfluid He droplets at SLAC, along with the droplet shape reconstruction. The fourth chapter provides details on the 8 calculations of the shapes of classical rotating droplets along the axially symmetric and two- lobed shapes branches. The fifth chapter discusses the main finding from the results of the XFEL experiments. The sixth chapter contains a discussion of the XFEL beam profile and the ionization probability of He droplets in XFEL experiments. The seventh chapter describes droplet shape oscillations and their lifetime, which enables excluding the shape oscillations as a possible source for the observed deformations. The last chapter contains the results of model calculations of the geometry of quantum vortex containing a solid Xe filament in a cylinder and in a sphere. 9 Introduction Superfluid helium droplets have long been studied and used as a powerful technique to isolate atoms, molecules and ions inside the droplets at an ultra-cold temperature of T ≈ 0.38 K [1, 2]. Molecules have been embedded in superfluid helium droplets containing several thousands of He atoms and studied spectroscopically with high resolution. The observation of vibrational and rotational spectra of molecules embedded in He droplets has provided a novel microscopic probe of superfluidity in He droplets and its development as a function of droplet size. [3-6] There are several properties of superfluid helium that are attractive as a matrix material, such as very small viscosity and high heat conductivity. Therefore, Helium droplets effectively thermalize the embedded species. Moreover, superfluid He droplets are transparent to radiation in a broad spectral range from infrared (IR) to vacuum ultraviolet (VUV). Therefore, the observed absorption is solely due to the species embedded inside the droplets. The most dramatic hallmark of superfluidity is the existence of quantum vortices, which were first observed in 4 He [7-10]. In contrast to a normal fluid which rotates as a rigid body, a superfluid remains at rest when its container has low angular velocity. However, above a certain critical angular velocity the thermodynamically stable state of a superfluid includes one or more quantum vortices. Such a vortex can be characterized by a macroscopic wave function and quantized velocity circulation in units of κ = h/M, where h is Planck’s constant and M is the mass of the 4 He atom [7, 9]. Thus the velocity field in a superfluid droplet deviates from that in rigid body rotation, which may have an impact on the droplet shape and stability. Negligible viscosity may cause some unique shapes in superfluid droplets which are of particular interest. 10 Recently, the study of vorticity was extended to finite systems such as Bose-Einstein condensates confined to traps [9, 11]. The transfer of energy and angular momentum in finite systems between quantized vortices and surface excitations is of particular interest, as it defines the nucleation dynamics, shape and stability of the involved vortices [9, 11]. In comparison to confined Bose-Einstein condensates, He droplets are self-contained and present a case for the strongly interacting superfluid. Moreover, the radius of a vortex core which is approximately 0.1 nm in superfluid 4 He [7] is small relative to the droplet size with a radius of about R = 100 nm – 1 μm, suggesting a three-dimensionality of the vortices in the droplets. Vorticity in 4 He droplets has therefore attracted considerable interest [12, 13]. Early attempts to observe quantum vortices in mm-sized levitating He droplets in magnetic field did not give any conclusive results [14]. On the other hand theoretical calculations predicted that arc-shaped single vortex can exist in spherical superfluid helium droplets as small few nm in diameter [13, 15]. Due to its hydrodynamic attractive potential, a superfluid vortex can trap foreign species such as electrons and ions as well as atomic particles, a property that has facilitated the visualization of the vortex lines in bulk experiments. Very recently, vortices in He droplets were traced by introducing Ag atoms that clustered along the vortex lines [16]. The Ag clusters were subsequently surface-deposited and imaged via transmission electron microscopy. Although this experiment gives evidence for the presence of single and multiple vortices in helium droplets, these ex-situ deposition experiments are not suited for the study of the droplet shapes and vortex arrangements. At present He droplet beams remains the only known experimental technique for production of nano-droplets. However, in a typical beam experiment the outcome is averaged over billions of the droplets, which may have washed out possible signatures of vortices in the previous experiments. Thus in order to study the droplet shapes and 11 vortex arrangements, single nano-droplets in the He beam should be interrogated. However, precisely that became possible with the recent advent of the technique of single particle diffraction imaging with single coherent X-ray pulses from a free electron laser (XFEL) [17, 18], which is the main focus of this thesis. This work was performed at Atomic, Molecular and Optical Science (AMO) Instrument at SLAC Linac Coherent Light Source (LCLS), with collaborations from researchers and scientists from University of Southern California (USC), Lawrence Berkeley National Laboratory (LBNL), SLAC National Accelerator Laboratory and Max Planck Institute (MPI). [19]. Preliminary results published in Ref. [19] indicated that the superfluid 4 He droplets have shapes ranging from a spherical, a spheroidal and a wheel shape. The shapes allowed an estimate of the angular velocity and the number of vortices in a droplet. Doping of vortices with Xe atoms also enabled the observation of the diffraction images with Bragg spots, which confirmed the existence of triangular arrangement of vortices in large He droplets. Furthermore, a novel phase retrieval algorithm was developed by R. Tanyag in our group. This technique enabled to recover the phase information in the diffraction image and to reconstruct the shapes and arrangements of vortices in He droplets [20]. Some of the results of the reconstruction using DCDI show the symmetric arrangement of quantum vortices doped with Xe clusters [21]. It was obtained that the addition of Xe atoms, which takes up a large portion of droplet initial angular momentum, results in the system of vortex and Xe clusters stabilized at a distance close to the droplet surface. The main focus of this work is the study of the shapes of superfluid droplets executing rotation. Until few years ago, it was presumed that He droplets obtained upon expansion of gas and fluid in vacuum possess spherical shapes which minimizes the energy in a quiescent droplet [1]. Microscopic He droplets are held together by van der Waals binding forces between the He 12 atoms, which on a larger scale manifest themselves as surface tension. Here we present the first experimental study of the shapes of superfluid droplets via x-ray coherent diffractive imaging. We demonstrate that in addition to spherical droplets, a spheroid, an ellipsoid, a wheel shaped or two lobed shapes droplets are formed in the beam [19]. These shapes were assigned to centrifugal deformation of the rotating superfluid droplets. In general, a droplet may acquire a non-spherical shape due to rotational or vibrational excitation. The vibrational excitation exists in the form of capillary waves on the surface of the He droplets, which may deform the droplet shape [22]. Thus, the study of the shape oscillations is critical in determining the shapes of the droplets. Although the lifetime of shape oscillations at higher temperature of T > 1 K is well established [23], the lifetime of droplet shape oscillations at low temperature T < 1 K remains elusive. Therefore, in this thesis the decay of shape oscillation in a so-called molecular regime is discussed. The implication of the shape oscillation on the observed shape of the droplets in XFEL experiment is also presented in this thesis. Until very recently it was generally believed that when a number of atoms or molecules are embedded into He droplets they eventually aggregate into a compact cluster residing close to droplet's center [1]. However, recent experiments revealed that in the presence of quantum vortices, foreign particles are pinned to the core of quantum vortices forming nm-thin filaments [16, 24]. Doping of quantum vortices with Xe atoms enabled the visualization of shapes and positions of the vortices inside the droplet [20]. On the other hand, it was also discovered that excessive doping leads to a change of the equilibrium vortex configurations [21], which is also in the focus of this thesis. 13 Chapter 1: Properties of Large Helium Droplets This work is part of Experiments in Large Superfluid Helium Nanodroplets chapter in Low Energy and Low Temperature Molecular Scattering book, which was submitted in April 2016 and will be published by The Royal Society of Chemistry Publishing. Superfluid He droplets in the beam may contain from 10 2 to 10 12 He atoms.[25] In this thesis, we review properties of large He droplets consisting of more than about 10 8 atoms and having radius of R > 100 nm. Experimentally, large droplets are produced in the expansion of a liquid or a supercritical fluid into vacuum, whereas the smaller droplets usually originate from the condensation of atoms in the gas expansion. The large droplets are also interesting as a boundary between the degenerate quantum fluid in small droplets and superfluid helium in the bulk. Embedding large clusters in the droplet allows multiple experimental opportunities to study the properties of the clusters with respect to their size and also the properties of the droplet itself. The size of the embedded cluster is only limited by the droplet's evaporation enthalpy of about 7 K per He atom. In addition to ripplons, large droplets possess thermodynamic excitations characteristic of bulk superfluid 4 He, such as phonons, rotons, and quantum vortices [26, 27]. The existence of quantum vortices dramatically influences the morphology of the formed clusters and leads to preferential formation of filament-shaped clusters. [28-30] Liquid 4 He undergoes a phase transition from its normal liquid state into a superfluid phase at 2.17 K [1]. In the superfluid state helium has a vanishing small viscosity and a thermal conductivity 30 times larger than copper, the next most thermally conductive element. [29] The 14 lack of viscosity in liquid helium allows the fluid to pass through microscopic capillaries and spread over the entire inner surface of whatever container it occupies [26]. When superfluid He is rotated in a bucket, triangular lattices of quantum vortices are formed parallel to the axis of rotation.[30, 31] The total density, ρ, of helium can be viewed as the sum of the normal density, ρ n , and superfluid density, ρ s . [26]. As the temperature T decreases from the superfluid transition temperature, T λ , to absolute zero, the ratio of ρ s /ρ increases from 0 to 1. However, the total density, ρ, remains approximately constant with a value of 145 kg/m 3 .[28, 29, 32] The corresponding number density is n He = 21.8 nm -3 . For a droplet, the density at the surface drops from 0.9 to 0.1 of the bulk value within 0.7 nm.[33] Therefore, the density of large helium droplets, with R > 10 nm, could be taken as uniform and equal to that of the bulk liquid. Droplets devoid of global excitations, such as shape oscillation or rotation, are spherical and has a radius given by:[25, 34]   nm N R He 3 1 222 . 0   (1.1) Elementary excitations in the droplet are represented by phonons and rotons in its volume and ripplons on the surface.[26] The energy and momentum of the bulk elementary excitations in superfluid helium are represented by the dispersion curve, proposed by Landau [35], and later confirmed by neutron scattering experiments [36], which is shown in Fig. 1.1. 15 Fig. 1.1 Dispersion curve of elementary excitations in superfluid Helium at T = 1.12 K [37]. The phonon excitations are represented by the nearly linear part of the curve at small momentum. The parabolic part of the curve around p 0 /ħ is assigned to roton excitations. In the bulk, the energy of phonons is given by E = ħ∙q∙s, where ħ = h/2π, h is the Planck constant, q is the phonon wavevector and s the first sound velocity. In spherical droplets, the energy and angular momentum of the phonons are quantized.[38] However as the droplets become larger the thermodynamic quantities should converge to the bulk values. In small droplets, the largest possible phonon wavelength (smallest possible energy) is determined by the droplet radius. The energy of the lowest excitation is given by E min ≈ 26∙N -1/3 [K].[38] The smallest droplets considered in this chapter of about N = 10 6 have the lowest excitation energy of E min = 0.26 K, which is comparable to the experimental droplet temperature of 0.38 K.[1] 16 Therefore, these droplets represent an intermediate case where the heat capacity is influenced by the phonon cut-off at low temperature. However, for droplets with N > 10 8 , E min << 0.38 K and the bulk approximation is well justified. The enthalpy of phonons is given by: [26]   He B B ph N T K V T k s h T k H                     4 3 3 3 5 10 5 . 2 15 4  (1.2) in which k B is the Boltzmann constant, s = 238 m/s is the first sound velocity,[29] and V is the volume of the droplet. The average number of phonons inside He droplets is   He B ph N T K V s h T k N                   3 3 4 3 10 24 . 9 6 . 9  (1.3) The average wavelength for phonons can be obtained from ħ∙q∙s = k B ∙T, i.e. λ = h∙s/(k B ∙T) = 11.44 [nm∙K] ∙ T -1 , which is around 30 nm at T = 0.4 K. Similarly, the heat capacity due to rotons is given by:[26, 27]     dT T k T k T k V T k p dT C H T B B B B eff T V rot                                               0 2 3 2 0 0 4 3 exp 2 2 2 3 2 1 2 1    (1.4) In this equation, at T = 0.4 K, p 0 = 2.02 x 10 -24 kg∙m/s, is the roton momentum, at its minimum energy, μ = 1.068 x 10 -27 kg is the roton effective mass, and Δ = 8.71 K is the roton energy gap.[28] The average number of rotons is:         He B B rot N T T K K V T k T k p N                                 2 / 1 2 / 1 3 2 / 3 2 / 1 2 0 71 . 8 exp 46 . 2 2 exp 2    (1.5) At low temperature as in this work the wavelength of rotons is approximately that in the roton minimum of λ = 0.33 nm, i.e. much smaller than the droplet radius. The energy of the elementary ripplon excitations is given by: [39]   ) 2 ( ) 1 ( 61 . 3 ) 2 ( ) 1 ( 2 / 1 0                l l l N K l l l E rip   (1.6) 17 in which l = 2,3,4….. is the angular momentum quantum number, and ω 0 is defined as   2 / 1 0 61 . 3 3 4         N K N m    (1.7) where σ = 3.536 x 10 -4 N/m is the droplet surface tension, [29, 40] and m is the He atomic mass. The heat capacity due to surface ripplon excitations can be obtained using the canonical partition function Z, where [38]   3 / 2 3 / 4 3 / 4 3 4 0 3 . 0 685 . 1 ) ln( N T K T k Z B                    (1.8) The ripplons energy is defined as the first derivative of the log of partition function with respect to (k B .T) -1           3 / 2 3 / 7 3 / 4 3 / 4 0 3 / 7 1 41 . 0 25 . 2 ) ln( N T K T k T k Z H B B rip                 (1.9) Consequently, the number of ripplons is   3 / 2 3 / 4 3 / 4 3 / 4 0 41 . 0 25 . 2 N T K T k N B rip                   (1.10) Finally, besides the discussed elementary excitations, quantum vortices represent global excitations in the superfluid. In contrast with ordinary fluids that can rotate as a rigid body, similar rotation is not feasible in superfluid, where the rotation involves creation of a collection of quantum vortices. [8] In large He droplets, hundreds of quantum vortices can be formed.[19] The velocity circulation around each vortex core is quantized in units of κ = h/m = 9.97 x 10 -8 m 2 /s, where h is the Planck’s constant and m is the mass of a 4 He atom. A rectilinear quantum vortex passing through the center of the droplet has energy of: [13] 18       1 44 . 4 ln 7 . 3 1 2 ln 2 3 / 1 3 / 1 2 0                         N N K a R R E    (1.11) where ρ is the droplets density, R is the droplets radius, and a ≈ 0.1 nm is the effective core radius of a vortex, where the density is taken to be zero. The angular momentum of a rectilinear quantum vortex in the droplet is     2 0 h N L (1.12) In general, however, the vortices located at some distance from the rotational axis of the droplet and are curved and have smaller amount of energy and angular momentum compared to a straight rectilinear vortex in the middle of the droplet.[15] The curved vortices revolve around the center of the droplet. Fig. 1.2 (a) shows a plot of the enthalpy of various excitations at the experimentally relevant T = 0.4 K and T = 0.8 K as a function of the number of He atoms in the droplet. Fig. 1.2 (b) shows similar plot of the number of the phonon, roton and ripplon excitations. 19 Figure 1.2. Log-log scale plots of (a). Enthalpy of evaporation, energy of a rectilinear vortex, phonons, rotons, and ripplons as a function of number of He atoms in a droplet at a temperature of T = 0.4 K (shown as solid lines) and T = 0.8 K (shown as dashed lines). (b). Number of elementary excitations N exc at a temperature of 0.4 K (shown as solid lines) and 0.8 K (shown as dashed lines) as a function of number of He atoms in a droplet. In the droplets with N < 10 6 the enthalpy and the number of phonons is smaller than calculated by eq. (1.2) and (1.3) due to the finite size effects. In Fig. 1.2 (a), enthalpy due to rotons in small He droplets with N < 10 6 having T = 0.4 K is 6 orders of magnitude less than the enthalpy due to phonons, ripplons or energy of a single vortex. At a temperature of T = 0.8 K, however, the contribution of rotons to the enthalpy becomes comparable to that from phonons and ripplons. From Fig. 1.2 (b) it is also seen that the number of phonons and ripplons at T = 0.4 K are substantially higher compared to the number of rotons in both small and large droplets. Thus, rotons do not contribute significantly to the droplet’s heat capacity. At T = 0.8 K, however, the number of rotons increases significantly due to the exponential term in eq. (1.5), and it becomes comparable to the number of phonons and 20 ripplons. The number of rotons exceeds the number of phonons at T > 1.1 K, and at higher temperatures the rotons become the dominant excitations.[28] Small and large He droplets also differ by the relative importance of phonons and ripplons, which has a profound effect on the rate of evaporative cooling of the droplets. Figure 1.3 shows the droplet temperature versus time calculated for different droplet sizes of 100 nm, 1 μm and 10 μm in an absolute vacuum at an initial temperature of T = 4.5 K. This calculation has been done by Curtis Jones in our group and involve solution of the coupled differential equations for the temperature and the number of He atoms in the droplet [41, 42]. The characteristic kinks at T ≈ T λ ≈ 2.15 K and T ≈ 0.6 K signify the transition from classical to superfluid heat capacity and the freezing of the volume enthalpy of the droplet, respectively. Here we assumed that thermal equilibrium is established instantaneously within the droplet, but in reality heat transfer via phonons and rotons occurs at the speed of sound. Taking speed of sound s = 238 m/s and the diameter of a droplet, excitations travel across the length of a droplet with R = 10 nm is ~80 ps, but the same process takes ~8 ns in droplets with R = 1 m. Thus, cooling times faster than this are unphysical. It is seen that the larger droplets cool slower in agreement with larger contribution of phonons and rotons to their heat capacity. 21 Figure 1.3 Helium droplet temperature as a function of time for droplet diameters of 100 nm, 1 µm, and 10 µm in vacuum. 22 Chapter 2: X-Ray Diffraction Experiments on He Droplets The work in this chapter was published in L.F. Gomez et al., Science 345, 906 (2014) 2.1. Schematic of the X-Ray Diffraction Experiments at SLAC Fig. 2.1. Schematic Diagram of X-ray coherent diffractive imaging experiments at SLAC National Accelerator Laboratory. He droplets (R = 300 nm to 1000 nm) are produced from the free expansion of liquid He into vacuum at a nozzle temperature of 5 K. The droplets pass through a skimmer, before they pick up Xe atoms in the pick-up cell. The doped droplets are exposed to LCLS free electron laser at the interaction point. The diffraction images are recorded on the pn-CCD camera located at a distance of 565 mm from the interaction point. [19] 23 Fig. 2.2. Schematic of the He droplet apparatus. 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 the system 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 (RGA) The schematic of the XFEL experiment with He droplets and of the experimental apparatus are shown in Fig. 2.1 and 2.2. During our 2012 campaign the experiments were performed using the CFEL-ASG Multi-Purpose (CAMP) instrument at the Atomic, Molecular and Optical (AMO) Science beamline of the Linac Coherent Light Source (LCLS).[17, 18] Liquid He droplets with radii of R = 100 nm – 2000 nm were produced by the use of free jet expansions of research grade helium-4 gas or fluid (99.9999%) [16, 24, 25, 43, 44] at cryogenic temperatures into vacuum. The nozzle assembly was cooled by a Sumitomo SRDK 408 cryocooler. Helium gas was expanded continuously as it passes through a nozzle having an 24 orifice of 5 μm at a nozzle temperature of T 0 ≈ 5-10 K and a source pressure P 0 = 20 bar into a vacuum chamber. The average sizes of the droplets were determined by titrating the droplets with Xe atoms using a procedure described in Ref. [25]. At T 0 = 5.5, 6, 7, 8, 9, 9.5 K, the average size of the droplets were measured to be 1.7 x 10 10 , 3.1 x 10 8 , 1.0 x 10 7 , 5.3 x 10 6 , 1.8 x 10 6 , and 3.3 x 10 5 , respectively [25]. In vacuum the droplets cool down via evaporation and reach superfluid transition temperature (T λ = 2.17 K) [45]. Calculations in Fig. 1.3 show that within about 0.1 ms the temperature of the droplets reaches ≈0.4 K. The resulting droplets' beam was collimated when it passes through a skimmer having a diameter of 0.5 mm, and the beam was doped with Xe atoms in a 10 cm long pick-up cell placed 15 cm from the droplet source. The degree of Xe doping was controlled by varying the Xe pressure in the pick-up cell, which was measured in absolute units by a membrane manometer. After a time of flight of ≈3.5 ms across a distance of 640 mm from the nozzle, the droplets traversed the focus (≈25 μm 2 ) of the FEL beam (hν = 1.5 keV, λ= 0.826 nm, repetition rate 120 Hz). Diffraction images were recorded using a cooled pn-CCD detectors placed at ≈ 565 mm distance from the interaction volume. The detector contains about 10 6 pixels, 75x75 µm 2 each.[46] The detector has semicircular cuts to accommodate the primary X-ray beam, which crossed the detector approximately in its geometric center. One detected X-ray photon results in about 33 units of intensity. Experiments in 2014 involved a new version of the vacuum apparatus - LAMP. Accordingly, the droplet traversed a slightly larger distance from the nozzle to the interaction point (≈ 700 mm), and the tighter focus of the FEL beam (≈ 5 μm 2 ), which is comparable to the size of the droplets.[47] The 2014 experiments employed photons with smaller energy (h ν = 850 eV, λ= 1.46 nm); the distance from the interaction point to the rear PnCCD detector was ≈ 735 mm and to the front pnCCD detector was ≈ 370 mm. Longer wavelength employed in the recent 25 experiments, enabled detecting scattered photons at about a factor of two larger scattering angle, which implies that the scattered photons fill a larger area of the detector. The probability for a droplet to reside in the FEL focal volume during a laser pulse was estimated to be less than 10 -3 [25]. Therefore, the probability to find two droplets in the detection volume was negligible and each diffraction image originates from a single droplet irradiated by a single XFEL shot. In addition to scattering from He and Xe atoms, many X-ray photons within a single FEL shot are absorbed, leading to ionization of the He droplets and solid Xe [48], which was measured using a time-of-flight (TOF) mass spectrometer. The ionization of He droplets and solid Xe may lead to Coulomb explosion [49]. After passing through the scattering chamber, the droplet beam intensity was monitored via the partial He pressure, P He , in the most downstream chamber, where the beam terminates. The average number of Xe atoms captured per He droplet, <N Xe > and the average droplet size before doping, <N He >, were estimated by measuring the attenuation of the droplet beam intensity due to dopant-induced evaporation of He atoms [25]. Upon repeated capture of Xe atoms, <N He > decreases, and was monitored by a reduction in the average partial pressure of He ΔP He in the detection chamber using a residual gas analyzer (RGA). Thus, <N Xe > can be estimated from: Xe He He He He Xe E E P N P N     (2.1) in which E He and E Xe are the binding energy of He atoms to the droplet and the average energy released upon capture of one Xe atom, respectively. These values are E He = 0.6 meV and E Xe = 0.15 eV. [38] The XFEL diffractive imaging experiments at SLAC National Laboratory involved collaborations between scientists from University of Southern California (USC), Lawrence Berkeley National Laboratory (LBNL), SLAC National Laboratory and Max Planck Institute 26 (MPI) and Technical University Berlin in Germany. The researchers and scientists involved in XFEL experiments are Luis F. Gomez, Charles Bernando, Rico Mayro P. Tanyag, Curtis Jones, Martin Seifrid, Justin Kwok, and Andrey F. Vilesov (USC), Ken R. Ferguson, Maximilian Bucher, Sebastian Schorb, John Bozek, Sebastian Carron, Michele Swiggers, and Christoph Bostedt (SLAC), James P. Cryan, Camila Bacellar, Michael Ziemkiewicz, Adam Chatterley, Ali Belkacem, Gang Chen, Alexander Hexemer, Stephen R. Leone, Jonathan H. S. Ma, Filipe R. N. C. Maia, Erik Malmerberg, Stefano Marchesini, Daniel M. Neumark, Billy Poon, James Prell, Katrin R. Siefermann, Felix P. Sturm, Fabian Weise, Petrus Zwart and Oliver Gessner (LBNL), Denis Anielski, Rebecca Boll, Tjark Delmas, Lars Englert, Sascha W. Epp, Benjamin Erk, Lutz Foucar, Robert Hartmann, Martin Huth, Daniel Rolles, Benedikt Rudek, Artem Rudenko, and Joachim Ullrich (MPI), Maria Mueller, Daniela Rupp, Tais Gorkhover, Thomas Moeller (TU Berlin). In experiments, the USC team has provided the He source, consisting of a cryogenic refrigerator cold head and a He compressor, along with a temperature controller. Starting from two weeks before the beamtime, we have performed calibration measurements at USC in order to assure that the droplets of the desired size could be produced. We have measured the He drople size and flux by using a titration method [25]. Then we have disassembled our He source and cryogenic refrigeration system at USC. One week before the beamtime, we packed the equipment and brought it to the SLAC National Laboratory. At SLAC, in collaboration with our colleagues, we have assembled and tested the vacuum apparatus described in Fig. 2.2 which contains 5 vacuum chambers, 8 turbomolecular pumps, and 6 ion gauges. In particular, we were responsible in installing our He source into the vacuum system. After the installation we have aligned and characterized the He droplet beam. The alignment was done by observing the pressure rise in the detection chamber via the RGA. 27 The next step was to measure the He size at different nozzle temperatures. During experiments, we were responsible in maintaining the performance of our He source and in adjusting source parameters, such as the backing pressure and nozzle temperature. After the experiment, we have disassembled the vacuum apparatus and our He source from the system and packed it back into containers and brought it back to USC. A few days after, we installed and aligned our He source back into our vacuum system at USC. The beamtime follows by an extensive data analysis. I was particularly involved in conducting data analysis to collecting useful diffraction image data from raw data. A diffraction image of a He droplet is produced when the droplet interacts with XFEL beam at the focal point. This event is called a ‘hit’. Most of the time, the droplets do not scatter the XFEL since they reside beyond the focal volume of the XFEL. Therefore, no diffraction pattern is recorded on the detector. This event is called a ‘no hit’. The raw data (≈ 60 TB of data) at SLAC contains all information regarding hit and no hit events. The main purpose of this analysis is to extract hit events from the data. The first step is to extract all hit events from big raw data XTC files to obtain HDF5 files. To extract all hit events, The Max Planck Advanced Study Group developed software called CFEL-ASG Software Suite (CASS) to view, process and analyze multi-parameter experimental data acquired at SLAC. It can be used ‘on-line’ using a live data stream from the FEL data acquisition system and ‘off-line’ for the post-experiment data processing [50]. To perform the data analysis using CASS, one has to SSH (Secure Socket Shell) to SLAC server and modify the CASS code (.ini file) in UNIX/Linux environment. In Windows, one can use PuTTY, and in Mac OS X, one can use iTerm for this purpose. The CASS code contains some post processors to perform hit finding in a particular region of an image matrix originated from both front and rear 28 pn-CCD detectors by setting an image threshold value. Since we have small angle scattering, the diffraction pattern are mostly obtained from the rear detector. Thus, an image threshold is set only for the rear detector. For instance, an image threshold value of 10 5 ADU (which corresponds to 5.9 x 10 3 detected photons) was set in a particular region of the rear detector defined by row index from 550 to 800 and column index from 550 to 800. If the total intensity within this region is larger than 10 6 ADU, then the image belongs to a ‘hit’ event. Otherwise, it is a ‘no hit’ event. In the same code, one has to obtain dark calibration files (.cal files) before submitting a job to the SLAC cluster. A dark calibration file is the file that contains XFEL background intensity detected on the pn-CCD. It was obtained in the absence of He droplets at the interaction point, before the actual run was conducted. The dark calibration files are used to subtract the background signal to assure that the actual run produces diffraction images with almost no XFEL background noise. After the dark calibration files are created, the next step is to list all raw data XTC files in a text file (.txt file). Then one has to create an .SH file (shell executable file) to perform offline data analysis, with an input of a .txt file and an .ini file/CASS script. These three files have to be created along with the dark calibration file before submitting a job to the computer cluster at SLAC. After the job submission to SLAC cluster is finished, the HDF5 files are created in our home folder. The HDF5 files are then converted further into .DAT files and image files (.PNG files). This process can be done in the same UNIX/Linux environment, or by using an FTP (file transfer protocol), such as FileZilla. By using FileZilla, the HDF5 files can be downloaded to our local PC for further processing. The HDF5 files are converted to DAT files by using Matlab. In Matlab, the information in HDF5 files can be read and extracted. The information include the time of the hit event (in second and nanosecond), the image matrix from the rear and front pn- 29 CCD, and the time of flight (TOF) mass spectra. In total, I have processed ~ 30 runs in 2014 XFEL experiments and obtained ~ 2500 hits. 2.2. X-ray scattering from spheroidal He droplets He droplets in this work are optically thin, an approximation which holds if 1 1 2      n R ( λ: wavelength of light, R: typical radius of scattering object). The complex refractive index of liquid He at hν = 850 eV (λ= 1.46 nm) of   1 10 6 . 1 10 2 . 4 1 7 5         i n was obtained from He atomic scattering factors and the number density of liquid helium.[51] For the largest He droplets in this work, with R ≈ 1000 nm 06 . 0 1 2      n R . The total scattering amplitude can then be calculated as the superposition of scattering waves from all infinitesimal volume elements of the droplet as: [52]                   V d q i d e n n k i S      2 1 4 3 2 2 3 (2.2) Here,   d q i e    accounts for the phase difference between wavelets originating from different volume elements with coordinates d  ; k k   is the wave number of the incident wave; and q  is the change of the wave vector upon elastic scattering,   2 sin 2  k q   , in which θ is the scattering angle. Consequently, the intensity of the scattered light is: [52]       0 2 2 2 2 4 0 2 2 2 1 4 I P n r V k I r k S I                       (2.3) in which r is the distance between the scattering center and the detector, V is the volume of the droplet,    P is the form factor and I 0 is the incident photon flux. 30 For a spheroid with symmetry half axis, a, and two equal half axes, b, the form factor is:   3 2 2 / 3 ) ( )) ( ( 2 9 u u J P spheroid    (2.4) in which 2 3 J is the Bessel function of the order of 3/2 and ) , , , , ( 0      b a R q u eff .[52] In case of a sphere R eff is a constant equal to the droplet's radius. For a spheroid at small angle scattering, the effective radius, R eff , is given by: [53] ) ( cos ) ( sin ) ( sin ) ( cos ) , , , , ( 0 2 0 2 2 2 2 2 0                       b a b b a R eff (2.5) in which  is the angle between the symmetry axis a and the X-ray beam, while Φ and Φ 0 define the azimuthal angles of q and of the projection of a in the scattering plane, respectively, see Fig. S2 in Ref. [19]. Note that 1/R eff defines a parametric equation for ellipse. Therefore small angle diffraction from a spheroid consists of elliptic rings with constant intensity along some specific ring. From the eq. (2.15) it is also seen that the aspect ratio in the diffraction pattern is given by:         2 2 2 2 sin cos b a A B (2.6) in which A and B are the major and the minor image axes, respectively. The image aspect ratio turns into A/B for a prolate spheroid. Equations (2.15, 2.16) show that only the angle Φ 0 or (Φ 0 + π/2) and the half axes b (which do not contain the symmetry axes) can be determined from the elliptical diffraction pattern. In contrast, the symmetry half axis, a, and the angle  remain interrelated. As a result only the upper (lower) boundary on the actual value of the aspect ratio in an oblate (prolate) droplet could be determined. In addition, if a spheroidal droplet is oblate or prolate could not be determined from the small angle scattering experiments. 31 In experiments during our 2012 and 2014 beam times about 98% of the obtained diffraction images from bare He droplets show ring patterns or elliptic patterns with aspect ratio of less than 1.3.[19] Those patterns were assigned to spherical or oblate spheroidal droplets. This assignment is in agreement with the range of stability of the axisymmetric shapes, as it will be discussed in Chapter 4. On the other hand, droplets having angular momentum beyond the bifurcation point into two lobed shapes have aspect ratio exceeding 1.48. In addition estimates have shown that shape oscillations decay during the time of flight of about 3 ms before they reach the x-ray beam, are therefore unlikely candidates for droplet's deviation from the spherical shape, as discussed in Chapter 7. 32 Chapter 3: Shapes of Swiftly Spinning Superfluid He Droplets The work in this chapter, along with the work in chapter 4 and 5 are written in manuscript Shapes of Rotating Superfluid Helium Nanodroplets, which is going to be submitted to Physical Review B 3.1. Introduction The study of the shape of the rotating Globe, which is held together by gravitational forces, preoccupied several generations of great scientists including Newton, MacLaurin, Jacoby, Riemann, and Poincare. It was shown that following an increase in the angular velocity of a globe its shape can progressively change from a spheroid, to a wheel-like, to shapes resembling peanut or multi-lobed shapes.[54-58] It was also shown that the mathematics describing the rigid body rotation (RBR) of classical droplets held together by surface tension forces belongs to the same class of solutions as the rotating globe. Starting from seminal experiments by Plateau the shapes of rotating droplets (e.g. water) have been extensively studied. However, surprisingly little is known about the rotation in viscosity-free, superfluid droplets, which are the focus of the present thesis. In a rotating ordinary droplet, the liquid is at rest in the rotating body frame as defined by the equilibrium between the centrifugal and surface tension forces.[55, 56] In a superfluid droplet, the RBR is not feasible, and any rotational motion of the droplet should manifest itself as a collection of quantum vortices.[8] The shape of the rotating superfluid droplet and possible differences from the classical counterparts is of particular interest in this thesis. Each vortex is characterized by a quantized velocity circulation s m M h / 10 0 . 1 2 7      , which is the ratio of the Planck’s constant, h, and the mass of the 4 He atom, M. In a cylinder 33 rotating with angular velocity ω and filled with superfluid helium, an array of parallel vortices is formed with an average vorticity equal to that of RBR,         V n v curl 2 ) ( , where n V is the number of vortices per unit area in a plane perpendicular to the axis of rotation.[8] Multiple vortices form an array which is stationary in the frame rotating with ω. The rotational energy of the superfluid is larger than that for the RBR rotating at same ω, due to additional energy required to create vortices, E VORT . It can be shown that the ratio of the E VORT to the kinetic energy of the liquid in a cylinder of radius R and executing RBR, E RBR , is ) / ln( 2 2 a b R E E RBR VORT        (3.1) where b- is the distance between vortex lines (   2 /  b ) and a ≈ 0.14 nm is the vortex radius. For a typical rotating bucket experiment, R≈1 cm, and ω≈ 10 rad/s this additional energy is small: E VORT / E RBR ≈ 10 -3 and is negligible. E VORT could however make up to about 10% of E RBR in sub-micron sized rotating droplets of superfluid helium. R = 400 nm, ω = 3 x 10 7 rad/s, b = 60 nm, E VORT /E RBR = 0.08. This additional energy of vortex-vortex repulsion may have an impact on the droplet shape which remains to be explored. Thus the progression of shapes found in classical droplets with increasing angular momentum will not necessarily be replicated in the quantum droplets and requires further investigation. Whereas in cylinder the vortices array naturally takes parallel shape in a droplet vortices must terminate perpendicular to the curved surface, which adds complexity.[15] Therefore the droplet shape as well as the configuration of the vortices in the droplet having certain value of angular momentum is unknown. Early attempts to observe vortices in mm-sized He droplets levitating in magnetic field were inconclusive. Calculations, however, predict that single vortices could exist in spherical superfluid helium droplets where they form ark shaped filaments.[13, 15, 59] Subsequently, 34 vortices in He droplets were traced by introducing Ag atoms that clustered along the vortex lines forming wire shaped clusters.[16] The clusters were surface-deposited and imaged via electron microscopy giving evidence for the presence of single and multiple vortices in helium droplets.[16, 43] Very recently we have reported an investigation of rotation in single, isolated superfluid He nanodroplets via scattering of X-ray radiation from a free-electron laser (FEL).[19] This thesis is built upon these results and describes the analysis of the extensive data set on the strongly deformed droplets. We found that large fraction of the droplets (≈40%) give rise to elliptic diffraction contours with aspect ratio in the range of 1.05 <AR < 1.3. Elliptic diffraction patterns are ascribed pseudo-spheroidal shape of the rotating droplets with aspect ratio of up to AR = 1.5. At larger aspect ratios the classical axially symmetric shape became unstable and the droplets acquire two lobed shapes which will be discussed in the following. We have found that approximately 1% of the diffraction images show streaks, pronounced intensity anomaly radiating away from the diffraction center, see Fig. 3.1 (c-l). Corresponding diffraction images cannot be described by elliptical diffraction contours and exhibit very high aspect ratios of 1.6 < AR < 2.3. In Ref. [19] we have assigned such images to axially symmetric droplets beyond the classical stability range, in agreement with predicted extended range of stability in rotating inviscid droplets [60] and recent DFT calculations [61]. However, the question of the existence of the superfluid droplets along the two lobed branch remained open. This chapter presents the detailed analysis of the diffraction images, resulting in the discovery of such shapes and discusses the abundance of the different classes of strongly deformed droplets in the beam. 35 3.2. Diffraction Images of Strongly Deformed He Droplets Some of the diffraction images from bare He droplets are shown in figure 3.1 in the logarithmic scale. The images are characterized by the aspect ratio, AR, of the diffraction contours which are the inverses of the ratio of the droplet's extension along the corresponding directions. The aspect ratio is defined as AR = (long half-axis)/(short half-axis). The circular diffraction contours in Fig. 3.1 A are consistent with diffraction from a spherical droplet with AR ≈ 1, and has been discussed in Ref [19]. Fig. 3.1 B shows an image with noticeable ellipticity of the diffraction contours and aspect ratio of about AR = 1.105 which will be used to verify our shape reconstruction technique. Panels C-E show strongly distorted diffraction images having AR in the range of 1.7 – 2.4 and well recognizable diffraction contours. The images reveal some pronounced streaks, i.e., the region of high intensity along the direction of the long axis in the diffraction, which extends well beyond the diffraction contours. Large aspect ratio and streaks in the diffraction images indicate some pronounced deformations of the droplets which are in the focus of the present thesis. Many of the images, such as in Fig. 3.1 F, have low intensity and likely originate from droplets which resided beyond the focal volume of the FEL beam. Due to low intensity, such images could not be used for the full reconstruction of the shape, as it will be discussed in the following, but could still be used to obtain the half axes of the droplets and their orientation in the lab frame. The image in Fig. 3.1 G has been shown in our previous paper [19], and it will be a subject of more detailed analysis together with other images. Finally, the images H and I do not show any resolved diffraction contours, however the intensity alternation along the streaks could be resolved and originate from very large droplets. The resolution of the diffraction contours in this work is limited by the pixel size of the detector. For example, the diffraction pattern originating from the droplets with R = 1.55 µm at λ = 0.826 nm have the 36 distance between the consecutive diffraction maxima of about 150 µm, which equals to two pixels along the horizontal or vertical direction. Remarkably, Fig. 3.1 H shows doublet of streaks. Figs. 3.1 I, J and K represent curved streaks, which appear in some diffraction images. Images with streaks in Fig. 3.1 K and L show some vertical ranges of missing intensity. Similar ranges were observed in other intense images and are ascribed to inconsistencies of the detector operation during our 2014 measurements. The analysis of our 2012 and 2014 data indicated that about 1% of the images contained streaks. In this thesis, we will focus on the images in panels (C-G) and image (K), which show high intensity and clear resolution of the diffraction contours. 37 (a) Run 104 20333 (b) Run112 13745 (c) Run043 11900 (d) Run104 59657 (e) Run112 55685 (f) Run104 45083 (g) Run104 95285 (h) Run140 13925 (i) Run140 81692 38 (j) Run156 48275 (k) R158 5973684581118622464 (l) R162 5973700798894034432 Fig. 3.1. Diffraction images of rotating superfluid He droplets. Image size: 500x500 pixels (a-j), 800x800 pixels (k), and 1000x1000 pixels (l). The logarithmic intensity color scale is shown on the left. The horizontal stripe separates the upper and lower plates of the pn-CCD detector. The pn-CCD plates have a semicircular cut in the middle sections to accommodate the incoming primary x-ray beam. [46] As it was discussed in Chapter 2, the diffraction images in figure 3.1 mainly reflect the projection of a single droplet’s density profile onto the detector plane; the longer axis in the diffraction pattern corresponds to the shorter droplet axis and vice versa. Thus, the half minor, a, and half major axis, b, of the droplet’s projection onto the detector plane are represented by the half major and half minor axis in the reciprocal space of the diffraction image, respectively. Because the a axis subtends an arbitrary angle with the X-ray beam, only the b axis and an upper boundary a ≤ b/AR can be deduced from each image for an oblate axially symmetric shape. Accordingly: max 2      a (3.2) 39 min 2      b (3.3) where λ is the wavelength of the x-ray free electron laser, and Δθ max(min) is the average difference in the scattering angle between the adjacent diffraction contours along the major (minor) axis of the image. From the fits of the diffraction images along the principal axes, the projection of the droplet in figure 3.1 (C) has a major half axis of b = 439 ± 5 nm and a minor half axis of a = 254 ± 5 nm, which corresponds to the aspect ratio of the droplet AR = 1.73. The projection of the droplet in figure 3.1 (D) has a major half axis of b = 501 ± 5 nm and a minor half axis of a = 210 ± 5 nm. It corresponds to an aspect ratio of AR = 2.39, which is the largest AR found in these experiments. The projection of the droplet in figure 3.1 (E) has a major half axis of b = 470 ± 5 nm and a minor half axis of a = 214 ± 5 nm, which corresponds to an aspect ratio of AR = 2.19. Fig. 3.2 shows the histogram of the frequency count vs aspect ratio of the diffraction contours as obtained for all images with strong streaks (red bars). Fig. 3.2 also contains frequency count for all images with AR > 1.3 which do not show well defined streaks, such as in Fig. 3.1 (c) and (j). In the following these data will be referred to as "large AR events". The plots contain the results of all measurements from the 2012 and 2014 experimental runs for which the aspect ratio could be determined. It is seen that the streaked images concentrate at AR > 1.6 with only two images obtained with smaller AR and gradual decrease of the appearance frequency at larger AR. The largest AR obtained was AR = 2.4. On the other hand the probability of the large AR events increases with AR and diminishes abruptly at AR > 1.8. Fig. 3.3 shows the plot of the major halve axis vs minor halve axis for the streaked images with discernible diffraction rings. It is seen that streaks are due to droplets with the minor and major halve axis in the range of 200 - 400 nm and 400 - 700 nm, respectively. The average aspect ratio of 1.82 is indicated by the linear fit of the data. This rather tight concentration of the 40 points remains to be explained. In comparison an ample amount of droplets having average radius in the range of R = 100 - 300 nm and R = 500 - 1500 nm have been observed. 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Streaks Weak streak, large AR 1.00 + 0.1 binning Count Bin Center Fig. 3.2. Frequency count of the streaked and large AR events vs their aspect ratio shown by red and blue bars, respectively. The data includes measurements with bare droplets as well as droplets doped by Xe to less than 15% depletion. 0 100 200 300 400 500 0 100 200 300 400 500 600 700 800 900 1000 b, nm a, nm Fig. 3.3. Major vs minor axis for images with streaks. Solid line represents the linear fit of the data points by y = a∙x with a = 1.82± 0.05. 41 3.3. Power Dependence of Intensity along the Streak The appearance of a streak along some particular direction indicates different radial dependence of the diffraction intensity and relates to the scattering object shape. At some fixed azimuthal angle away from the center the intensity scales as a negative power, α, of the scattering angle, θ, i.e.,     I . Fig. 3.4 shows the values of α obtained by fits along the long axis (streaks) from ~80 images showing weak and intense streaks. Along the azimuthal angles away from the long axis of the images the fits consistently gave α = 4 as expected for a spheroid or an ellipsoid.[53, 62] However, along the long axis in the streak region, α in the range of 3 – 4 were found, signifying a considerable deviation of the droplets from ellipsoidal shape as it will be discussed in the following. The data points stemming from the images demonstrating visible streaks are marked by red circles, whereas the data points from the large AR events are shown by blue points. The data points contain the results of measurements from the bare He droplets, which are marked by cyan dots, and from the He droplets with small level of Xe doping. It is seen that the data points from bare and from Xe doped droplets do not show any systematic deviations from each other. 42 1.0 1.5 2.0 2.5 3.0 2.8 3.2 3.6 4.0 4.4 Streaks He and Xe<15% Large AR He and Xe<15% Large AR He only Axisymmetric  AR Fig. 3.4. Plot of α as a function of AR for streaked (red) and large AR images (blue). The continuous curve is the results of the fits to the calculated diffraction from the axisymmetric classical rotating droplets imaged edge on. The large AR data points measured without Xe are marked by cyan dots. The complete characteristics of the diffraction images in streaked and large AR events are given in Table 3.1, which shows values the averages of the results obtained independently from the upper and lower plate of the detector. The entries are labeled by the run and image numbers and show the amount of Xe doping. The entries correspond to the diffraction images in Fig. 3.1 are highlighted in yellow. The rest of the diffraction images are shown in Appendix A1. The major and minor halve axis, b and a, of the droplet projections on the diffraction plane are shown in the unit of μm. The aspect ratio AR = b/a is also shown. Angle of the long axis of the diffraction image is obtained with respect to the droplet's beam axis, from -π/2 to π/2 radian. α is the power of the diffraction intensity dependence vs scattering angle along the long axis of the diffraction image. Total number of the detected photons in each diffraction image is also shown. 43 Run Number % Xe Average Major Half Axis Average Minor Half Axis AR Total Number of Photons Average Angle of the Long Axis of Diffraction Image Average Alpha Along the Long Axis of Diffraction Image 43 11900 0 0.439 0.254 1.730 3.703E+5 0.95 3.863 51 70730 11 0.486 0.284 1.711 5.677E+4 -1.00 3.801 51 104042 11 0.512 0.259 1.982 1.527E+5 1.16 3.681 52 57920 11 0.391 0.197 1.990 1.173E+5 1.41 3.623 52 95603 11 0.384 0.238 1.617 1.290E+5 0.38 3.503 52 126101 11 0.472 0.292 1.616 1.931E+5 0.82 3.780 54 78524 27 0.516 0.289 1.784 5.227E+4 -0.40 3.482 62 115892 10 0.372 0.280 1.329 4.107E+4 -0.44 3.979 64 44582 10 0.352 0.191 1.843 4.573E+4 0.43 4.055 64 55751 10 0.404 0.232 1.743 4.823E+4 -1.22 3.923 64 82523 10 0.424 0.240 1.770 4.633E+4 -1.44 3.878 64 84620 10 0.503 0.304 1.653 7.257E+4 -1.08 3.859 64 110027 10 0.344 0.246 1.398 4.595E+4 -0.56 3.915 71 60794 0 0.161 0.108 1.491 3.637E+4 -1.04 3.955 99 81143 0 0.306 0.225 1.360 1.837E+5 0.87 3.967 99 93038 0 0.163 0.120 1.358 4.397E+4 -1.24 4.020 99 100715 0 0.271 0.179 1.514 4.084E+4 -0.66 3.798 104 20144 0 0.354 0.246 1.439 4.191E+4 0.05 N/A 104 45083 0 0.447 0.245 1.829 5.457E+4 -1.27 3.481 104 59657 0 0.501 0.210 2.386 8.880E+4 1.50 3.343 104 81302 0 0.282 0.174 1.626 4.160E+4 1.13 3.957 104 95285 0 0.424 0.219 1.934 2.106E+5 0.79 3.554 44 108 22766 5 0.340 0.289 1.175 9.118E+4 -0.20 3.874 109 6917 0 0.419 0.290 1.447 1.136E+5 1.25 4.022 109 82772 0 0.281 0.213 1.319 5.517E+5 1.25 3.951 109 125096 0 0.590 0.459 1.285 4.419E+4 -0.89 3.988 109 128543 0 0.420 0.205 2.049 6.287E+4 1.51 3.504 112 55685 0 0.470 0.214 2.194 6.903E+4 -0.90 3.422 112 56702 0 0.434 0.271 1.602 4.773E+4 -0.20 4.129 112 60137 0 0.369 0.292 1.264 9.419E+4 0.45 3.966 112 83090 0 0.529 0.381 1.388 5.820E+4 -1.04 3.870 112 90101 0 0.381 0.248 1.534 4.427E+4 1.40 4.090 112 121646 0 0.503 0.273 1.846 4.857E+4 1.28 3.849 119 83759 0 N/A 1.305 N/A 7.426E+5 -1.37 3.595 119 104804 0 0.858 0.687 1.249 1.535E+5 -0.10 3.719 125 10235 11 0.718 0.354 2.029 1.605E+5 -0.40 3.722 125 88535 11 0.575 0.445 1.291 2.266E+5 0.85 4.112 134 5036 0 0.347 0.279 1.243 7.622E+4 -0.22 4.062 134 53324 0 0.176 0.147 1.198 7.432E+4 -0.39 4.055 134 55268 0 0.644 0.541 1.192 8.694E+4 -0.26 4.004 134 59669 0 N/A 0.431 N/A 4.546E+4 0.25 3.635 134 60578 0 0.661 0.466 1.418 3.650E+4 0.65 3.846 134 62111 0 0.680 0.400 1.701 7.560E+4 0.20 3.868 134 120677 0 0.647 0.357 1.811 4.127E+4 -0.75 3.949 140 13925 0 N/A 1.126 N/A 2.328E+5 0.71 3.055 0 N/A 1.127 N/A 2.328E+5 0.78 3.325 140 15980 0 1.035 0.791 1.308 1.876E+5 1.15 4.042 45 140 81692 0 N/A 1.086 N/A 3.423E+5 0.72 2.981 144 40232 36 1.028 0.609 1.688 1.769E+5 -0.84 4.069 149 88769 18 N/A 0.759 N/A 2.075E+5 1.10 3.530 149 95843 18 N/A 1.086 N/A 1.634E+6 -1.33 3.661 155 130742 0 0.383 0.225 1.700 1.585E+5 -0.94 3.985 156 27629 15 0.636 0.355 1.792 2.034E+5 0.82 3.621 156 48275 15 0.464 0.332 1.400 4.400E+5 1.48 3.611 156 76583 15 0.306 0.226 1.354 1.349E+5 0.80 4.007 156 88868 15 0.478 0.379 1.263 1.652E+5 -1.09 4.088 46 Run Number % Xe Average Major Half Axis Average Minor Half Axis AR Total Number of Photons Average Angle of the Long Axis of Diffraction Image Average Alpha Along the Long Axis of Diffraction Image 155 52416 0 0.350 0.267 1.311 1.512E+5 -0.74 4.073 155 41536 0 0.472 0.352 1.340 4.389E+6 0.10 3.995 157 33312 0 0.366 0.229 1.597 8.453E+5 0.17 3.681 158 22464 0 0.452 0.236 1.920 2.070E+6 -0.86 3.169 158 44064 0 0.334 0.205 1.629 3.005E+5 -1.39 4.049 158 83552 0 0.235 0.147 1.594 3.598E+5 1.25 4.046 159 58720 0 0.444 0.283 1.567 3.574E+5 -1.55 4.038 160 43680 0 0.396 0.244 1.623 5.546E+5 -0.99 3.949 160 73152 0 0.504 0.373 1.352 3.161E+6 -0.64 3.894 160 76384 0 0.285 0.207 1.378 4.972E+5 0.75 3.866 162 34432 0 0.553 0.331 1.671 3.172E+5 0.71 3.489 162 92352 0 0.660 0.398 1.659 9.376E+5 0.76 3.703 174 44512 13 0.550 0.308 1.787 1.459E+6 0.44 3.485 174 88608 13 0.301 0.193 1.560 2.002E+5 0.45 3.943 185 14496 0 0.488 0.345 1.413 6.340E+5 0.60 3.949 47 Table 3.1. Parameters of streaked diffraction images from the 2012 (upper table) and 2014 experiments (lower table). See texts for explanation. 3.4. Using the Inverse Fourier Transform for the Shape Reconstruction We proceed with the reconstruction of the droplet's shapes from images in Fig. 3.1. He droplets studied in this work are optically thin objects for which Born approximation holds [63]. Therefore the scattering amplitude is given by the Fourier transform of the droplet's density. For small angle scattering as in our case (Θ < 0.02 rad) the scattering amplitude is well approximated by the 2D Fourier transform of the projection of the droplet density onto the detector plane. Therefore, in general the diffraction patterns allow a reconstruction of only of the projection of the density onto the detector plane. In order to reconstruct the phase information, we assumed that droplets have centro- symmetric shape, as per a classical rotating droplet, and therefore have real-valued scattering amplitude. Accordingly, we assigned alternating (+/-) signs to the scattering amplitudes of consecutive diffraction maxima in Figure 3.1. The scattering amplitude is expressed as:   E sign I A   (3.4) where I the diffraction intensity and sign(E) is the alternating positive-negative sign assigned to the consecutive diffraction contours. The signs of E were obtained by fitting of the diffraction intensity at different azimuthal angles using the expression for the scattering amplitude, E, for diffraction from a sphere:[52]     q R q E R q I 2 ) , ( ) , ( (3.5) ) ( ) , ( 2 / 3 2 / 2 / 1 R q J q q R q E       (3.6) 48 where q is the change of the wave vector upon elastic scattering, R is the radius of the droplet, and J 3/2 is the Bessel function in the order of 3/2. The intensity in eq. (3.5) is proportional to q -α which is introduced to account for deviation of the intensity dependence from q -4 as in the case of a sphere or a spheroid. Except for the region around the streak fits consistently gave α = 4 in agreement with scattering intensity from the sphere or spheroid which is inversely proportional to q 4 . However in the long diffraction axis region, α in the range of 3 – 4 were found, see table 3.1, signifying a considerable deviation of the droplets from spheroidal shape. In addition to eq. (3.6), we have also used a function described by eq. (3.7) ) cos( ) , ( 2 /         R q q R q E (3.7) to fit the experimental amplitude, which is an asymptotic expression for eq. (3.6) except it has an additional phase variable, φ. This phase φ = 0 for a sphere or a spheroid, but deviates for other shapes, as it will be discussed in the following. Fig. 3.5 (a), (b) and (c) shows the azimuthal dependences of the φ for the diffraction in Figs. 3.1 (c), (g) and (k), respectively. The fitted phase of a cosine function in figure 3.5 (a) has some rather large noise of about ± 0.2 rad (rms). Nevertheless, the phase show some pronounced anomaly in the region of the streak, such as phase increments at an angle between 0.85 and 1.05 rad, where the phase increases to ~0.5 rad. Outside the streak region, the phase in Fig. 3.5 (b) and (c) shows some fluctuations around 0 within the noise level. Fig. 3.5 (b), indicates a significant phase drop at an angle of around 1 rad. This effect is more pronounced in fig. 3.5 (c), since the phase drops occur at 2.0 and 2.6 rad, which is the region between ring diffraction pattern and streak diffraction pattern. In the streak region, however, the phase increases abruptly to φ ≈ 0.6 and forms an arc shape. 49 (a) (b) (c) Fig. 3.5. Phases (in radian units) of the cosine function, obtained from the fit of diffraction intensity of fig. 3.1 (a), (e) and (j) at each angle, from 0 to π. The streak region is marked by 2 vertical dashed lines. (a) (b) (c) (d) Fig. 3.6. (a) Phase of theoretical shapes at AR = 1.88, (b) AR = 2.00, (c) AR = 2.32, and (d) AR = 3.08. The dashed lines represent the region of the streak. In order to understand the origin of the phase discontinuities, the diffraction images from the calculated axially symmetric shapes were fitted using equation (3.7). The calculation of the shapes is discussed in Chapter 4. The resulting phase dependences are shown in Fig. 3.6 for the axially symmetric shapes at aspect ratio of AR = 1.88, 2.00, 2.32, and 3.08. Figure 3.6 shows that the phase increases about π/4 rad in the streak region for all diffraction images with streaks from the oblate droplets. The region outside the streak has a constant phase range of φ = 0 ± 0.2 50 rad. In addition to the phase jump, there is a clear distinction between phases in droplets with different AR. In droplets with AR = 1.88 and AR = 2.00, the phase around the streak (around 1.2 – 1.5 rad and 1.7-2.0 rad) is close to 0, whereas in droplets with AR = 2.32, the phase around the streak decreases gradually from 0 to around -0.2 rad before it jumps and reach π/4 rad at the center of the streak. In droplets with AR = 3.08, the phase around the streak decreases abruptly, from 0.2 to -π/4 rad, before it jumps and reach 1 rad in the streak region. Unlike the case in droplets with AR = 1.88, 2.00 and 2.32, droplets with AR = 3.08 have constant phase in the streak region, instead of forming a peak at the center of the streak. This characteristic is attributed to droplets with depression, which have different shapes compared to droplets with flat parallel surface, which will be explained later. The obtained amplitudes A of the diffraction pattern using both Bessel and cosine functions (eqs. 3.4, 3.6, 3.7) were then Inverse Fourier transformed to obtain droplets’ shapes. The phase information in the amplitude is restored by assigning phase to each diffraction contour alternating positive or negative sign, since diffraction amplitude of a center-symmetric object is real. The shapes after Inverse Fourier Transform contain oscillations due to truncation of the diffraction pattern at the central hole. It produces sidelobes or feet in the vicinity of the droplets shapes, which may conceal the true shapes of the droplets. Therefore, the amplitudes A are multiplied by a Gaussian apodization function in the form of               2 2 25 . 0 4 exp N c N q to minimize this effect before applying the Inverse Fourier Transform. In the equation above, N equals to the matrix size of the diffraction image, and c is the constant that determines the width of the apodization function, which was typically set to 0.07. The shapes obtained from diffraction in Fig. 3.1 (c) are shown in figure 3.7 (a) and (b) which were obtained using phase fits 51 by Bessel and cosine functions, respectively. Both fits yield very similar contours within scattering of the data points. (a) (b) Fig. 3.7. Inverse Fourier Transform of 1000x1000 pixels diffraction amplitude using (a) Bessel function and (b) cosine function, which show the cross sections of the droplet from diffraction image in figure 3.1 (c). (a) (b) Fig. 3.8. The cross section of the (a) Bessel and (b) cosine contour of the droplet in Fig. 3.7 at 0.95 rad 52 The density of the droplet could not be reconstructed because the corresponding low frequency amplitude information could not be recorded due to the central hole in the detector. [46] Therefore, the IFT shows droplet's boundary as a transient contour with positive and negative values shown by red and blue lines, respectively. The cross section of the contour at the angle of the streak (0.95 rad) with respect to the horizontal is shown in figure 3.8. From the transients, such as in Fig. 3.8 contours of the droplets were obtained from the weighted average modulus value of the transient curve at each azimuthal angle, from 0 to 2π. The contours are shown in figure 3.9 as small squares, from Bessel and cosine functions, for upper and lower half images. In model calculations we found that the weighted average modulus method applied to calculated diffraction pattern from classical axisymmetric rotating droplet with the same aspect ratio of AR = 1.73 (purple line), produces transient contours which have size of the 0.98 of the original droplet. Therefore to correct for this effect the obtained contours are enlarged by a factor 1.02. It is seen that different fit results coincide within the scattering of the data points and that the calculated shape (see details in Chapter 4) gives an excellent representation of the measured shape. The complete set of reconstructions is shown in Table 3.2 for images in Fig. 3.1 C – G. The theoretical contours were obtained from rotating drops equation developed by Chandrasekhar with parameters aspect ratio (AR), angle of the streaks, and half major axis taken from Table 3.1. 53 Fig. 3.9. Experimental shapes from Bessel and cosine functions, as obtained independently from the reconstruction of upper and lower half diffraction images with 1000x1000 pixels in Fig. 3.1 (c), and the calculated shape (purple) of an axisymmetric He droplet rotating at ω = 1.23 x 10 7 rad/s and imaged perpendicular to its symmetry axis. 54 Run No. 043 11900 104 45083 104 59657 104 95285 112 55685 112 13745 Diffraction Image Shape Reconstruction (Upper) Shape Reconstruction (Lower) Contours 55 Run No. 043 11900 104 45083 104 59657 104 95285 112 55685 158 22464 Diffraction Image Shape Reconstruction (Upper) Shape Reconstruction (Lower) Contours 56 Table 3.2. Shape reconstructions and contours of diffraction images in Fig. 3.1 (c) – (g) from Bessel (upper table) and Cosine phase (lower table) fits. The calculated shapes are also shown as discussed in the text. The horizontal axis is parallel to the droplet beam. Table 3.2 shows the obtained droplet contours from images in Fig. 3.1 C-G. It is seen that in weaker images such as in (D), (E), and (G) reconstruction of the contours parts which are parallel to X axis has larger scattering. This is because the corresponding parts of the diffraction image are lost in the gap between the pn-CCD plates. For comparison, Fig. 3.1 (B) and the right most entry in Table 3.2 shows the results of similar reconstruction of the droplet’s contour from the diffraction image with a small aspect ratio of AR = 1.11 which has no streak. Again the image could be nicely represented with the calculated contour for the rotating droplet (red), which shows a pronounced deviation of the droplet shape from a spheroid. A notable finding from these shape reconstructions is on the appearance of small depression in the middle of the droplets in run158 5973684581118622464. The same phenomena can also be observed in run104 59657, although the depression is less pronounced. The appearance of the depression could also be implied from the diffraction images. Both diffraction images have streak region which is separated from the diffraction ring region by region of very low intensity (nodes) which extends radially. The diffraction ring patterns have a discontinuity over the nodes, since the maximum of a diffraction ring is followed by a minimum of a diffraction pattern in the streak region. These appearances do not exist in the other diffraction images, which only produce droplets shapes with no depression after Inverse Fourier Transform. 57 Fig. 3.10. The combined diffraction image in Fig. 3.1 (j) (within red rectangle) as obtained with the rear detector combined with the appropriately scaled image from the front detector (outside the red rectangle). The logarithmic intensity color scale is shown on the left. The diffraction image in Fig. 3.1 (k) has high intensity and extends to large scattering angle detectable with the front CCD detector. Fig. 3.10 shows the image which was obtained by combining the data from the front and rear detectors upon appropriate scaling. The data from the rear detector, same as that in Fig. 3.1 (k), are within the red rectangle. The streak shows some noticeable curvature. In comparison most of the other recorded streaks appear as straight stripes within the accuracy of the experiment. The average curvature, 𝜒 = 𝛥𝜃 𝛥𝑠 , could be obtained from Fig. 3.10 to be χ = 0.90 m -1 , where as usual Δθ is the degree of deviation of the streak along its length Δs. Moreover in distinction to most of the observed streaks which merge smoothly into the ringing pattern at small scattering angle, the image in Fig. 3.10 shows nodes (weak intensity region) between the streak and diffraction rings over which the diffraction pattern has a discontinuity. At large angle the streak has two satellites. 58 Chapter 4: Shapes of Classical Rotating Droplets 4.1. Shape Family of Classical Rotating Droplets For the purpose of the assigning of the observed diffraction images to some particular droplet shapes it is instructive to review the shapes of rotating classical droplets. Equilibrium shape of a droplet rotating as rigid body is defined by the balance between the capillary force from surface tension of the curved drop surface, and centrifugal force.[55, 56] The equilibrium shapes of rotating droplets have been extensively studied by calculations and experiments.[55, 56, 58, 64-66] Similar model was used to the study of shapes of rotating atomic nuclei.[54, 57, 67-72] The stability diagram and corresponding representative shapes for the axially symmetric and two lobed (C 2h ) branches are shown in Fig. 4.1. Angular velocity, ω, is given in reduced units:[55]            32 3 V (4.1) where, ρ is the density of liquid helium, V is the volume of the droplet, σ is the surface tension of liquid helium and ω is the absolute unit of angular velocity. A related quantity, rotational number,[56]         8 3 2 b (4.2) is used for axially symmetric droplets. Another important parameter is the reduced angular momentum of the droplet Λ:[73] l R R        3 2 8 1    (4.3) where l is the angular momentum of the droplet in the absolute unit. 59 Fig. 4.1. Angular velocity Ω vs angular momentum Λ, stability diagram for rotating droplets in equilibrium. The upper branch corresponds to oblate axisymmetric shapes, whereas the lower branch to prolate two lobed shapes. Bifurcation point is at Λ = 1.2, Ω = 0.56 with AR = 1.47. Fig. 4.1 shows that with increasing Λ the equilibrium shape of the droplet evolves from spherical to oblate axially symmetric. At large Λ the shapes show considerable flattening in the polar regions and even depression at Λ > 2.03. However, beyond Λ ≈ 1.2 the axially symmetric shapes become unstable with respect to the two lobed deformations. At larger Λ the stable shapes correspond to the lower branch representing prolate droplets, which resemble pills at 1.2 < Λ < 1.6, has a dumb-bell shape at Λ > 1.6 and finally becomes unstable against fission at Λ > 2. 4.2. Axially Symmetric Shapes The calculation of axially symmetric shapes of classical rotating droplets was developed by Chandrasekhar [56]. The calculation along the axially symmetric line is derived from the force balance between centrifugal force and force due to surface tension. The parameter Σ as 60 shown in eq. (4.2) is introduced to account for the degree of shape deformation of the droplet from a spherical shape. The equation of the boundary of the drop, for a given Σ, is:                                                                      cos 1 sin 1 sin , 2 , 1 2 1 , 2 2 k k E A k F A A x y (4.4) where y represents the height of the drop boundary at a particular horizontal position x, ) ), ( (   k F and ) ), ( (   k E are the incomplete elliptic integrals of the first kind and second kind, respectively [74]. The variables A(Σ) and k(Σ) are defined as:         2 1 A (4.5)                         1 2 1 1 1 2 1 A k (4.6) and ) cos(  is defined as:       2 2 1 1 cos x A x A         (4.7) The calculated axially symmetric shapes at certain values of Λ are shown in Fig. 4.4 (a). The values of Λ are related to Σ via eqs. (4.1 – 4.3). The diffraction images of axially symmetric shapes at AR = 1.5, 1.8, 2.0, 2.32, 3.08, and 3.87 or Λ = 1.20, 1.56, 1.74, 1.96, 2.23, and 2.43 are shown in Fig. 4.2. The 2D density projection matrix has a size of 200 x 200 pixels. The shapes can be rotated with respect to three different axes: x and y axes are the long axis, and z is the short axes (the angular momentum axis). The parameters in this calculation are: the wavelength of XFEL is λ = 1.4748 nm (841.35 eV), the distance between the interaction point and the front pn-CCD detector is 0.371 m, the size of 1 pixel in the diffraction images is 75 μm, and the size of the diffraction image matrix is 61 1000 x 1000 pixels. These parameters produce the size of 1 pixel in the density projection matrix of 7.3 nm. To model the diffraction image in Fig. 3.1 (k) having a long axis of the density projection of 450 nm, the half major axis of the axially symmetric density projection was kept constant at 450 nm. The density projection and the diffraction images of axially symmetric shapes are shown in Fig. 4.2 with XFEL directed perpendicular to the direction of the short and long axis. The diffraction amplitude is obtained from the 2D Fourier Transform of density projection matrix. The diffraction amplitude is then squared to obtain the diffraction intensity. 62 (a) Λ = 1.20 (b) Λ = 1.56 (c) Λ = 1.74 (d) Λ = 1.96 (e) Λ = 2.23 (f) Λ = 2.43 Fig. 4.2. Density projections on the detector plane and the diffraction images of axially symmetric droplets with (a) Λ = 1.20, (b) Λ = 1.56, (c) Λ = 1.74, (d) Λ = 1.96, (e) Λ = 2.23, and (f) Λ = 2.43, obtained from [56]. 63 4.3. Two-Lobed Shapes The calculations of two-lobed shapes of classical rotating droplets have been conducted by several groups.[73, 75] Most recently, Butler’s group at University of Saskatchewan calculated the shape families of tektites, which resemble the shapes of classical rotating droplets.[73, 76] From the shapes provided by S. Butler along the two-lobed branch at Λ = 1.23, 1.26, 1.3, 1.35, 1.4, 1.5, 1.6, 1.7 and 2.0, we have calculated the diffraction images at different droplets’ orientation with respect to the x-ray beam. The data points of the shapes are obtained from numerical solution of the Navier-Stokes equation for rotating fluid drops [73]. The scattered data points of the shapes were then interpolated to ensure the shapes continuity in 2D density projection matrix with a size of 200 x 200 pixels. The shapes can be rotated with respect to three different axes: x axis is the long axis, y axis is the median axis and z axis is the short axis of the shape (the direction of the angular momentum). The parameters in this calculation are: the wavelength of XFEL is λ = 1.4748 nm (841.35 eV), the distance between the interaction point and the front pn-CCD detector is 0.371 m, the size of 1 pixel in the diffraction images is 75 μm, and the size of the diffraction image matrix is 1000 x 1000 pixels. These parameters produce the size of 1 pixel in the density projection matrix of 7.3 nm. To model the diffraction image in Fig. 3.1 (k) having a long axis of the density projection of 450 nm, the half major axis of the two- lobed density projection was kept constant at 450 nm. The density projection images and the corresponding diffraction images at different Λ are shown in Fig. 4.3. The density projection matrix shows the projection of the shape density with XFEL directed perpendicular to the direction of the short and long axis. The diffraction images are obtained from the three dimensional Fourier Transform of the density by using a Slice Fourier Theorem. In this method, each slice of the density projection matrix along the XFEL 64 direction is individually Fourier Transformed before being multiplied by a phase difference along z.     2     z Z (4.8) where z is the distance of each density slice from the center, and θ is the scattering angle. The resulting matrix is the diffraction amplitude matrix from each slice. The total diffraction amplitude is obtained by adding the diffraction amplitude matrix from each slice. Consequently, the diffraction intensity is obtained from the square of the diffraction amplitude. The cross sections of axially symmetric droplets at various values of Λ are shown in Fig. 4.4 (a). The droplets along the two lobed branch are triaxial bodies having C 2h symmetry. The cross sections of the two lobed droplets at 1.3 < Λ < 2 obtained via numeric calculations are shown in Fig. 4.4 b) and c) for projections in the equatorial plane and planes containing the rotation axis and long axis, respectively. Corresponding aspect ratios are shown in Fig. 4.5 (a). Fig. 4.5 (b) represents the ratios of the droplet volume to the cube of the long half axis in the unit of 4π/3 for axially symmetric and two lobed shapes. 65 (a) (b) (c) (d) (e) (f) (g) (h) 66 (i) Fig. 4.3. Density projections on the detector plane and the diffraction images of two-lobed droplets with (a) Λ = 1.23, (b) Λ = 1.26, (c) Λ = 1.3, (d) Λ = 1.35, (e) Λ = 1.4, (f) Λ = 1.5, (g) Λ = 1.6, (h) Λ = 1.7, (i) Λ = 2, obtained from [73] a) b) c) Fig. 4.4 a). Cross sections of rotating droplets at indicated values of the reduced angular momentum Λ. All droplets have the same volume of 4π/3. a) for axisymmetric shapes, b), and c) for two lobed shapes in the equatorial plane and plane containing the rotational axis and long axis, respectively. Note different scale in a) and b, c) 67 a) b) Fig. 4.5 a) Aspect ratios vs Λ: red curve - equatorial to polar extension for axially symmetric droplets; green - long to polar axis, blue - long to axis perpendicular to polar and long axis for two lobed shapes and brown – axis perpendicular to polar and long axis to polar axis. b) The ratios of the droplet volume to the cube of the long halve axis in units of 4π/3 for the axially symmetric and two lobbed shapes are shown by red and blue curves, respectively. 4.4. Imaging of Droplets at Different Tilt Angles The angle from which the droplets are observed is important since random imaging angles of swiftly rotating droplets with respect to the droplets’ beam axis may produce ellipsoidal, spheroidal, or even spherical droplets. The range of angles with respect to the droplet’s beam axis in which streaked images could still be produced was examined theoretically. Fig. 4.6 shows the diffraction images of the axially symmetric droplets as a function of rotation angle with respect to the droplet’s beam axis. The parameters of the XFEL and the droplets are the same as in Section 4.2. It is seen that the streak is the most pronounce upon the edge on imaging, but disappear at θ > 15 O . 68 (a) (b) (c) (d) (e) (f) Fig. 4.6 (a) The rotation angle with respect to the droplet’s beam axis for an axisymmetric shape with Λ = 1.96 (AR = 2.32), and (b) the diffraction images as the rotation angle θ is 0 0 (side on observation), (c) 5 0 , (d) 10 0 , (e) 15 0 , (f) 20 0 . As the rotation angle increases, the streak disappears, starting from an angle of around 10 0 - 15 0 . The diffraction images of the tilted two-lobed shapes have also been calculated. The images of a two-lobed shape at Λ = 1.5 which is tilted at an angle of 0 0 , 22.5 0 , 45 0 , 67.5 0 and 90 0 are shown in Fig. 4.7. The parameters of the XFEL and the droplets are the same as in Section 4.3. It is seen that at any tilt angle (except very close to 90 O ) the diffraction always contains a streak. Upon the tilt the streak also becomes curved. The streak disappears at an angle of 85 0 -90 0 , 69 which means that a two-lobed shape produces streaks in a larger range of tilt angles as compared with the axisymmetric shapes. (a) (b) (c) (d) (e) Fig. 4.7. Diffraction images of a two-lobed shape with Λ = 1.5 when it is rotated at an angle of (a) 0 0 (b) 22.5 0 (c) 45 0 , (d) 67.5 0 , and (e) 90 0 . As the rotation angle increases, the streak is still observable, but it vanishes at an angle of 85 0 - 90 0 . 70 Chapter 5: X-ray Scattering from Strongly Deformed He Droplets 5.1. Discussion Droplets which are in the focus of this work produce non-elliptic diffraction contour and have some pronounced intensity anomalies, which most dramatically manifest in streaks, such as in Fig. 3.1. The diffraction intensity along the streak,     I , has slower decrease vs. the scattering angle as compared with the rest of the diffraction image. Eq. (2.4) shows that in case of elliptic diffraction contours, away from the center, the diffraction intensity scales with the power of α = 4 in agreement with the experimental observations. Same dependence was observed in the strongly deformed images away from the long axis. On the other hand, Fig. 3.4 shows that the values of α are significantly smaller along the long axis of the diffraction. In case of streaks the values were found in the range α = 3.1 - 3.5. For strongly deformed images the values were found to be at α = 3.5 - 4.1, signifying a considerable deviation of the droplets from ellipsoidal shape. In general a streak indicates singularity in diffraction where the rays having same path length difference originate from a group of points on the surface of the droplet, such as from two opposite flat surfaces of a cylinder or from two lines of points on the opposite sides a cylinder, where the limiting values of α are known to be 2 and 3, respectively. The shapes of the droplets are related to the diffraction amplitude via the inverse Fourier transform, see eqs. (2.2, 2.3). For small scattering angle, θ, the phase difference acquired due to the extension of the object along the x-ray beam, z, can be obtained as in eq. (4.8). For some characteristic length of z = 500 nm and θ < 0.02 rad (as in most of our images) Δφ Z < 0.14 π which is small and the form factor in eq. (2.3) is well approximated by the two dimensional Fourier transform of the projection of the density onto the detector plane. Therefore we have 71 used an inverse Fourier transform of the diffraction amplitudes, as described in Section 3.4, in order to obtain the droplet shape. The shapes obtained from the diffraction images in Fig. 3.1 k), j) and e) are shown in Fig. 5.1 a), b) and c), respectively, by red squares. Unfortunately, only the contours and not the entire density of the droplets could be obtained from the diffraction data, due to the missing scattering amplitude at very small scattering angle falling into the detector central hole. The contours in Fig. 5.1 (a, c) are not elliptic and show pronounced regions of low curvature, where the opposite sides of the contours run nearly parallel. This behavior is consistent with the observed streaks as discussed earlier. Finally, the contour in Fig. 5.1 b) shows some noticeable depression in the flat region. a) b) c) Fig. 5.1. Experimental contours are from the diffraction images in Fig. 3.1 k), j) and e) in panels a), b) and c), respectively. Red continuous curves are calculated contours for the axially symmetric shapes with the same aspect ratio as found experimentally. The green contour in b) is the calculation with the two lobed shape with Λ = 1.5 and 45 O tilt as described in the text. Fig. 5.1 also shows the calculated contours for axially symmetric rotating droplets, such as shown in Fig. 4.4 a) with experimental aspect ratios of AR = 1.67, 1.92 and 1.93, (Λ= 1.41, 1.68, and 1.69) respectively. The contours represent the edge-on view of the droplets, i.e., with 72 axis of rotation in the picture plane. The distorted experimental contours in Fig. 5.1 (a, c) are also in good agreement with the calculations. In comparison, experimental and calculated contours in Fig. 5.1 (b) show considerable differences in that the calculated contour does not show any depression. The appearance of the depression in the droplet manifest itself in diffraction by multiple streaks as discussed in relation to Fig. 3.10. Besides the image in Fig. 3.10, only one additional image in 2012 with AR = 2.4 had similar nodes, indicating very small abundance of the droplets with depression in the beam. Previously, we have assigned the shapes such as in Fig. 5.1 (c) to axially symmetric shapes imaged edge on based on the close resemblance to the expected shapes.[19] However the axially symmetric shapes develop the depression only at large Λ > 2.1, and the aspect ratio for such droplets exceeds ≈2.5, see Fig. 4.5 a). Accordingly, the axially symmetric droplets are unlikely candidates for the shape observed in Fig. 5.1 (b) which has AR = 1.92. Therefore we must consider closely the shapes along the two lobed family. The presence of such shapes in the beam is consistent with the observation of the curved streak in Fig. 3.10. In the two dimensional picture the diffraction by a homogenous body of any shape give rise to the center symmetric diffraction picture. The curvature signifies the departure from the two dimensional approximation and indicates that the phase acquired along z-direction, see eq. (4.8), cannot be neglected. Moreover even under the 3D treatment of the diffraction, a curved streak is not forthcoming at any tilt of the axially symmetric shape. In general if an object has a plane of symmetry along the x-ray beam, such as an axisymmetric shape at any tilt angle, the corresponding diffraction image should be center-symmetric. On the other hand curved streaks in diffraction may naturally originate from the prolate shapes tilted with respect to the detector plane. 73 In order to gain more insights into the diffraction which can arise from the two lobed shapes large number of diffraction images have been calculated for each of the representative two lobed shapes. For this purpose the three dimensional shapes of constant density which cross sections are shown in Fig. 4.4 b) and c) were tilted around the corresponding short axis and three dimensional Fourier transforms were calculated and presented. We have obtained that the triple streak as observed in Fig. 3.10 is consistent with shapes having small depression in droplets with angular momentum of Λ = 1.50 ± 0.05. Shapes with Λ < 1.5 have no depression and produce a single streak. The shapes with Λ >1.5 have a large depression and produce multiple streaks or even X-shaped streaks in the diffraction which were not observed in this work. The most close match with the experimental diffraction is produced by the two lobed shape with Λ = 1.50 with long halve axis of 550 nm and short (angular momentum) halve axis of 230 nm tilted out of plane around the short axis by π/4. Fig. 5.2 shows the comparison of the measured and simulated diffraction images which show very good agreement. Fig. 5.1 b) also contains the contour of the projection of the tilted density onto the detector plane, which is in a very good agreement with the experimental contour. a) b) Fig. 5.2. Panel a) has reproduced the image from Fig. 3.10. Panel (b) shows a simulated diffraction image obtained from a two lobe shape at Λ = 1.5 with long halve axis of 550 nm and short halve axis of 230 nm tilted out of plane by π/4. See the text for more details. 74 These results also show that diffraction images with single streaks could also originate from the two lobed shapes with 1.3 < Λ < 1.5 whereas the aspect ratio of the diffraction could be tuned (reduced) by the appropriate tilt of the shapes, to account for the contours such as in Fig. 5.1 (a, c). Therefore we need to reevaluate our previous assignment of the streaked diffraction images to axially symmetric shapes. Here we analyze the appearance statistics for the streaks which may contain information on the abundance of the oblate and prolate shapes in the beam. This analysis is based on the fact that axially symmetric oblate and prolate shapes differ in the appearance of the streaks at different tilt angle which translates at different probability for observation of the streaks vs the aspect ratio. In the case of the oblate axially symmetric shapes the streak is observed if the droplet is imaged edge on and fades away upon tilt by more than ≈10 o . As a result no streaks are expected to be observed in diffraction images with aspect ratio of < 1.6. On the other hand, images with large aspect ratio devoid of any streaks will result from the tilted axially symmetric shapes. In comparison, the streaks will still be observed upon the tilt of the prolate shapes down to very small aspect ratio of about 1.1. 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 0.0 0.1 0.2 Probability Aspect Ratio L=2 random orientation AR Hystogram 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Streaks Weak streak, large AR 1.00 + 0.1 binning Count Bin Center (a) (b) Fig. 5.3. (a) Probability to find certain aspect ratio for a random orientation of the prolate droplet with Λ = 1.3 Bin size is 0.1. (b) The occurrence of strong streaks (red) and weak streaks with large AR (blue) in experiments. 75 Upon the tilt the aspect ratio of the prolate droplet changes from its maximum value to nearly 1 when the droplet is viewed along its long axis. Fig. 5.3 shows the probability to find an aspect ratio from 1 to 2.0, binned with intervals of 0.1 for a random orientation of two lobed droplet with Λ = 1.3, see Fig. 4.4. It is seen that there is a considerable probability of about 40% to find diffraction with streak, but with a small aspect ratio of AR < 1.6. This prediction could be compared with the results of experimental observations in Fig. 3.1. From the total of 16 streaks at AR > 1.6 statistically about 10 streaks should be observed with AR < 1.6, whereas only 2 were observed experimentally. Although the number of the counts is not excessive, this discrepancy indicates that the prolate shapes alone are not able to account for all of the experimental deficiency of the small AR events. This shows that in addition to the prolate shapes the distribution may contain axisymmetric oblate shapes as we have postulated previously. 1.0 1.5 2.0 2.5 3.0 2.8 3.2 3.6 4.0 4.4 Streaks He and Xe<15% Large AR He and Xe<15% Large AR He only Axisymmetric Two lobed, Xray  Rot Two lobed, Xray // Rot  AR Fig. 5.4. Similar to Fig. 3.4, but with the alpha data for two-lobed shapes. See text for explanations. 76 Another important piece of information on the abundance of the different shapes comes from the analysis of the power dependence of the intensity along the streak as quantified by the exponents, α. Fig. 5.4 shows the plot of α vs AR for the relevant diffraction images. The red circles show the results obtained from the clear streaked patterns and the blue circles from the large aspect ratio images which do not reveal obvious streaks. The squares show the (AR, α) values calculated from the computed shapes with Λ= 1.23, 1.26. 1.3, 1.35, 1.4, 1.5, 1.6, 1.7 and 2.0 as described earlier. Three points correspond to three different orientations of the shapes: orange - with long axis parallel to the x-ray beam; green - with long and short (angular momentum) axis perpendicular to the x-ray beam; brown - with the short (angular momentum axis) parallel to the x-ray beam. The lines in Fig. 5.4 connect the data points belonging to the shapes having the same Λ = 1.23, 1.26, 1.3, 1.35, 1.4, 1.5. The resulting triangle approximately delimits the locus of (AR, α) points which can be obtained from a given shape with some particular Λ at an arbitrary orientation with respect to the X-ray beam. Continuous pink curve in Fig. 5.4 shows the (AR, α) computed from the classical axisymmetric shapes having the same aspect ratio as in experiments and placed with their short axis perpendicular to the X-ray beam (edge on). Upon 10-15 o tilt of the axially symmetric shapes the streak in the diffraction disappears, the values of α approach 4 and the AR decreases. Therefore the (AR, α) points for axially symmetric shapes at arbitrary orientation fill the space between the pink curve and the pink line at AR = 4. On the other hand Fig. 5.4 shows that most of the (AR, α) points for the prolate shapes should fill the space below the pink curve. The inspection of Fig. 5.4 shows that a number of streaked events have their (AR, α) values well below the axisymmetric curve and consistent with the values expected for the two 77 lobed shapes having Λ in the range of 1.26 - 1.5. At larger Λ = 1.6, 1.7 and 2.0 (unconnected squares in Fig. 5.4) the droplets have some pronounced depression leading to multiple streaks in the diffraction, which have not been observed in this work. This analysis confirms the existence of the non-axially symmetric two lobbed shapes in the helium droplet beam. On the other hand Fig. 5.4 indicates that a large number of events with AR in the range of 1.5 - 1.85 and α ≈ 4 within the error of about ±0.1. These points lay significantly higher than the range expected for the two lobed shapes. The AR of these events are beyond that for the last stable axially symmetric shape with Λ = 1.2, AR = 1.48. These points are consistent with the existence of the axially symmetric droplets beyond the stability point as we have postulated previously. However, as we have stated before the full three dimensional shape could not be reconstructed from the small angle scattering as in this work. Therefore the degree of deviation from the axial symmetry cannot be quantified. More accurate information on the droplet shapes could be obtained based on the measurements at large scattering angle, where the deviation of the diffraction from centro-symmetricity may be used to quantify the droplet's shape. This however would require using much longer wavelength of radiation of about 12 nm (100 eV) to assure strong enough scattering at large angle. The long wavelength range is unavailable at LCLS, but is common at FLASH and ELETRA free electron lasers. Recent wide angle scattering XFEL study of Ag nano-particles enabled reconstruction of their three dimensional shapes [77]. At the wavelengths currently available (λ = 1.46 and 0.826 nm) most of the images, such as in Fig. 3.1 appear centro-symmetric. In particular, most of the streaks do not show curvature within the accuracy of the measurements, which may however reflect our inability to measure the curvature at small scattering angle. 78 Good agreement between the experimental and calculated droplet's contours in Fig. 5.1 and of the diffraction in Fig. 5.2 indicates that the classical calculations give a fair representation of the shapes of the superfluid helium droplets. From the shapes the angular velocity, ω, and the angular momenta, L, of the droplets can be estimated. For example, assuming that the diffraction in Fig. 5.2 stems from the two lobbed shape with Λ = 1.5 and major halve axis a = 550 nm as it was discussed earlier. The resulting values of ω, L, as well as other parameters of the droplet are presented in the second column of the Table 5.1. For comparison same parameters have been calculated for the axially symmetric droplets of the same volume and having Λ=1.5 (unstable) and Λ=1.2 (last unstable). Table 5.1 also lists the values of L in units of ħ per droplet as well as per He atom in the droplet. These estimates show that the rotating droplet have large angular momentum of about 61 ħ per single He atom and thus must contain large number of quantum vortices. Obtaining the precise number of vortices, N V , in a droplet of a given shape and angular momentum requires some calculations which are beyond the current state of the art [78, 79]. Therefore we estimate the N V based on the two dimensional model. According to Feynman, in the limit when parallel vortices are present in a superfluid He slab, the velocity field vector v far away from the vortex cores is approximately the same as that for classical rigid body rotation, [8] where vorticity is defined as M h n v V        2  (5.1) In this equation, n v is the density of vortices, h is the Planck constant, and M is the mass of the 4 He atom. The number of vortices can be estimated as N V = n V ∙A, where A is the area of the equatorial cross section of the droplet, see Table 5.1. This expression likely underestimates the N V because the vortices close to the equatorial contour have smaller length as compared with those in the interior, and thus carry smaller angular momentum. 79 Table 5.1. Parameters of a two-lobed shape with Λ = 1.5 and axisymmetric shapes with Λ = 1.2 and Λ = 1.5 having the same volume. Table 5.1. Λ = 1.5 (two lobed) Λ = 1.2 (axisymmetric) Λ = 1.5 (axisymmetric) Volume (nm 3 ) 1.63 x 10 8 1.63 x 10 8 1.63 x 10 8 Half major axis (nm) 550 372 387 N He 3.558 x 10 9 3.558 x 10 9 3.558 x 10 9 R spherical (nm) 338.9 338.9 338.9 Moment of inertia (kg.m 2 ) 4.061 x 10 -30 1.846 x 10 -30 2.050 x 10 -30 L (kg.m 2 /s) 2.178 x 10 -23 1.742 x 10 -23 2.178 x 10 -23 L (ħ) 2.178 x 10 11 1.742 x 10 11 2.178 x 10 11 L (ħ)/N He 61 49 61 Omega (rad/s) 1.028 x 10 7 1.245 x 10 7 1.409 x 10 7 Number of vortices 122 108 133 Equatorial Area (m 2 ) 5.939 x 10 -13 4.347 x 10 -13 4.705 x 10 -13 5.2. Conclusion In general, in the presence of the large number of vortices the shapes of rotating superfluid droplets in equilibrium are expected to resemble classical droplets rotating at the same ω due to similar velocity fields far from the vortex cores. For example, the shape of a rotating superfluid in a cylindrical container adopts a parabolic shape similar to that of a classically rotating liquid.[7, 26] However the question on how accurate the classical shapes and the obtained classical ω approximate superfluid droplets calls for future studies. In distinction to the classical droplets the rotating superfluid droplets contain additional energy contribution due to vortex cores which is approximately proportional to the length of the vortex filament. One may speculate that in order to minimize the total length of the filaments the superfluid droplet may take somewhat flatter shape in the polar regions. In a free droplet the total angular momentum, L, is an independent variable. Therefore in the future it would be desirable to obtain the shapes of the superfluid droplet and the configuration of vortices as a function of L and N He in order to 80 constitute the stability diagram for superfluid droplets, as it is established for their classical counterparts, see Fig. 4.1. Our measurements indicate that in axially symmetric pseudo- spheroidal droplets the vortices form an equilateral triangular lattice similar as observed in Bose- Einstein condensates.[9, 80, 81] In the future similar measurements may help evaluating the arrangement of the vortices in the strongly deformed helium droplets. 81 Chapter 6: Using He Droplets as Probes for the XFEL Beam The diffraction from the neat He droplets shows some simple diffraction pattern which could be represented analytically via equation (2.3). Therefore the deviation of the measured pattern from calculated could be used to evaluate the profile of the X-ray beam. In addition as the refractive index for liquid He is known a-priori, the radius of the droplets is obtained from the diffraction and the quantum efficiency of the CCD detection system is known from calibrations, the number of the scattered photons gives the absolute x-ray flux in the droplet’s position for each single shot event. 6.1. Beam profile of X-ray Free Electron Laser A considerable fraction of the diffraction images obtained during the 2014 campaign contained some pronounced low image contrasts which were not observed during our 2012 campaign. These may be related to the tighter focusing of the beam during 2014 or to the inferior x-ray beam quality during 2014. Actually the 2014 experiments were focused on the x-ray pump - x-ray probe measurements which required some special tuning of the XFEL. As discussed earlier most of the droplets should have a spherical or a spheroidal shape and should produce a series of circular or elliptical diffraction contours of high contrast. The beam profile of XFEL can be evaluated by analyzing the low contrast in the diffraction images from the expected. To characterize the beam, diffraction images of pure He droplets in single-shot X-ray diffraction experiment were obtained. Some representative images are shown in Fig. 6.1, which show some blurry pattern along one particular angle. 82 (a) Run157 5973672653973428224 (b) Run159 5973688214638325760 Fig. 6.1. Some representative images showing blurry diffraction pattern along some angle. For instance, in Fig. 6.1 the image in (a) shows blurry pattern or low image contrast along an angle around 50 0 , whereas the image in (b) shows the low image contrast along an angle around 45 0 . This pattern is similar to the smoothing of high contrast diffraction images with a Gaussian kernel. This Gaussian is the representation of the XFEL beam profile. Therefore, these diffraction images can be modelled by taking the product of the matrix of density projection of a spherical droplet in the real space with a 2D Gaussian function which is shifted from the center of the droplet density along one particular angle, and then Fourier Transformed. Mathematically, the density matrix of the droplet DG, taking into account its position with respect to XFEL beam is obtained by:                        2 2 0 2 0 2 2 2 , , , 2 exp  y y x x y x R G D DG y x y x y x (6.1) where D is the density projection matrix of a spherical droplet on the detector, G is the matrix of the XFEL Gaussian beam profile, R is the radius of the droplet, x and y are the indices of a 83 matrix, x 0 and y 0 are the positions of the Gaussian peak, and σ is the standard deviation of the Gaussian. The low contrast diffraction image is obtained by taking the Fourier Transform of droplet density DG. If the 2D Gaussian is shifted along the horizontal direction, the blurry pattern will exist along the horizontal direction of the image. Fig. 6.2 shows the images before and after applying the Gaussian function for a droplet with radius of 360 nm. (a) (b) Fig. 6.2. (a) Original diffraction image of a spherical droplet with a radius of 360 nm (b) The image showing blurry diffraction pattern is due to the tighter XFEL beam focus. The original diffraction image of a He droplet with a radius of 360 nm is shown in Fig. 6.2 (a). Assuming 1 pixel in density projection matrix is equal to 14.4 nm, the image in Fig. 6.2 (b) is obtained from eq. (6.1) with R = 360 nm (25 pixels), σ = 455 nm (31.6 pixels), and x 0 = y 0 = 360 nm (25 pixels). In other words, the Gaussian was shifted by 510 nm from the center of the droplet density along an angle of 45 0 . As the result, the diffraction image in (b) shows similar image with low contrast as in Fig. 6.1 along an angle of 45 0 . Thus, the images shown in Fig. 6.1 84 are likely caused by the very tight beam profile of the XFEL, which is comparable to the size of the droplet, or a non-Gaussian beam with hot spots. As the XFEL focal area becomes tighter (the XFEL Gaussian beam profile has smaller σ), there is a larger probability for the diffraction image to have a low image contrast. This is due to the comparable size between the XFEL focal area and the cross section of the droplets. Thus, droplets may interact more with the tail of the XFEL Gaussian, which is less intense and produce low contrast images. This modeling could be used to estimate parts of the XFEL beam which interact with the droplets. 6.2. Calculation of XFEL Flux and Ionization Probability The diffraction images can also be used to obtain accurate information about the flux of XFEL from the known shape and refractive index of the droplet. The refractive index of liquid He at λ = 1.46 nm are: n He = 1 – 4.203 x 10 -5 + 1.642 x 10 -7 i.[51, 82] The flux of XFEL, Φ [photons/m 2 ] is given by:[52] 4 2 3 2 1 8 R n I He total          (6.2) where I total is the total number of photons scattered by the droplet, and R the radius of the droplet. The total number of photons is obtained by dividing the number of detected photons by the fraction of photons detected on the pn-CCD detector. The fraction of photons detected on the detector is related to the geometry of the pn-CCD detectors. Since the detectors have central hole in the middle and a horizontal stripe that separates the upper and lower plate, there is a fraction of scattered photons that moves through the central hole and horizontal space (undetected), and a fraction that falls on the detector (detected). The flux of XFEL Φ can in turn be used to estimate the ionization probability of He and embedded Xe atoms. The ionization probability also depends on the He and Xe absorption cross sections, which are obtained from:[51] 85 0 2 2 f r e abs       (6.3) where r e = 2.814 x 10 -15 m is the classical electron radius, and f 2 0 is the imaginary part of complex atomic scattering factor. For He, f 2 0 (λ = 1.46 nm) = 0.0079, and for Xe the corresponding f 2 0 = 28.6. Thus, the He and Xe absorption cross sections are 6.48 x 10 -26 m 2 and 2.35 x 10 -22 m 2 , respectively. The ionization probability is defined by:        abs ion P  exp 1 (6.4) The complete list of the XFEL flux, droplet size, aspect ratio, and He and Xe ionization probabilities from 108 diffraction images obtained during the 2014 XFEL experiment for bare He droplets are shown in table A2 in the Appendix. Note that Xe was not present in these image data, and the calculated ionization probability of Xe is given to estimate the ionization probability for some typical experimental flux. Fig. 6.3. Plots of frequency counts of (a) XFEL flux from all diffraction images and (b) XFEL flux from high-contrast images showing clear resolution of the diffraction contour with droplets AR < 1.2. 86 From table A2, frequency count histograms of XFEL flux in 2014 experiments are shown in Fig. 6.3. Panel (a) shows that the XFEL flux is less than 3 x 10 22 photons/m 2 for majority of hits and only about 7% of the hits contains a flux of larger than 5 x 10 22 photons/m 2 . This calculated flux is about two magnitude smaller than it was predicted by tighter focus of 1 x 10 24 photons/m 2 [47]. To estimate the flux more accurately, only the XFEL flux from nearly spherical droplets (AR < 1.2) and from images showing clear resolution of the diffraction contour are taken into account. This is because eq. (6.2) is strictly valid only for spherical shapes. Moreover, images showing clear diffraction contour suggest that the droplets scatter the maximum XFEL intensity, as discussed in Section 6.1. The counts of XFEL flux based on these two parameters (AR < 1.2 and for images showing clear diffraction contour) are shown in panel (b). It is shown that the flux distribution taking only the images with only AR < 1.2 is approximately the same as the flux distribution in panel (a). However, the flux distribution with the high contrast image data resembles a distribution which peaks at ~3x10 22 photons/m 2 . It is also shown that the distribution of flux larger than 5x10 22 photons/m 2 stay the same in panel (a) and (b), which implies that the maximum flux is obtained due to positions of the droplets at the center of the X-ray beam. On the other hand, the small flux of < 5x10 22 photons/m 2 could originate due to positions of the droplets away from the center of the X-ray beam or periphery of the beam. 87 Fig. 6.4. Histogram of He and Xe ionization probability. He atoms are barely ionized during the 60 fs duration of the XFEL pulse, whereas Xe atoms are completely ionized. Another main finding from the table is the ionization probability of He and Xe atoms. The frequency counts of these probabilities are shown in Fig. 6.4. In this figure, the ionization probability of Xe is more than two orders of magnitude higher than the ionization probability of He. Thus, during one XFEL pulse with a flux of larger than 2 x 10 22 photons/m 2 , the Xe atoms are completely ionized (ionization probability is ≈ 1), whereas the He atoms are barely ionized, with maximum ionization probability of 0.006. It is also shown that the ionization probability for He in most cases is less than 0.002, whereas 72% of the ionization probability for Xe is between 0.9 to 1.0, which implies that Xe atoms are completely ionized during the XFEL pulse. 88 Chapter 7: Decay of Surface Waves in Molecular Regime 7.1. Introduction In 2012 XFEL experiment, approximately 40% of the diffraction images from bare He droplets have non-axially symmetric shapes and are better represented by diffraction from spheroidal, wheel shapes and two-lobed shapes. However similar droplets shapes may originate from rotational or vibrational excitation. The distortions from rotational and vibrational excitations can not be distinguished in small angle X-ray scattering experiments. Therefore, an estimate on the vibrational excitation and its lifetime is important. This vibrational excitation exists on the droplet in a form of collective (multiple) quadrupolar excitations, which causes oscillations on the droplets shape. For this purpose, the study of shape oscillations and its lifetime at low temperature is presented in this chapter. There are several studies on the lifetime of the capillary waves on the surface of the superfluid Helium (ripplons) at high and low temperatures.[83-92] At low temperature between 0.1 and 1 K, the mean free path of the elementary excitations, which are mainly represented by phonons, are larger than the wavelength of the ripplons. Therefore, the decay is described by so- called molecular regime, rather than viscous regime at higher temperatures. The notable experimental studies were done by Roche et. al.[84], and the notable theoretical studies were done by Saam [86, 87], and were corrected later by Roche et. al.[85] Early theoretical studies of the damping of ripplons by several authors revealed large discrepancy between experimental and theoretical values of the lifetime of about 2 orders of magnitude: the experimental is larger than the theoretical lifetime.[84-87] The discrepancy was later explained in [85] by some mistakes in imposing the boundary condition constraint at the 89 unperturbed [86, 87], rather than perturbed surface position, which would change the Hamiltonian terms in the scattering process. Decay of ripplons is affected by two important mechanisms. At very low temperature (T < 1.1 K), excitations inside superfluid He droplets are governed by phonons. The scattering of phonons at the free surface of liquid He could influence the decay of ripplons. In addition to scattering by phonons, the decay of one ripplon into two ripplons may also occur. Thus, two different processes could influence the lifetime of ripplons: the three-ripplon interaction and the one-ripplon-two-phonon interaction. Although some authors have suggested that the first interaction is the dominant one [83, 86, 87], recent measurement supported the opposite case. [84, 85, 93, 94] 7.2. Experimental Study of Ripplon Decay in Molecular Regime To obtain the experimental decay of surface waves, Roche et. al. [84] used a horizontal array of coplanar interdigital capacitors (IDC’s), below which there is a superfluid layer of Helium. Interdigital capacitor is an element for producing a capacitor-like, high pass characteristic using microstrip lines, as shown in figure 7.1. 90 Fig. 7.1. Interdigital capacitor with the gaps between finger G, gaps at the end of the finger GE, width and length of the finger W and L, respectively. The capacitance increases as the gaps are decreased. Reducing the width of the fingers reduces the required area, but lowers the effective capacitance. Increasing the length of the fingers increases the capacitance, but increases the required board area. Fig. 7.2. Schematic of the ripplon detection experiment. There are three interdigital capacitors (IDC A, B and C), which are located at a distance of h = 8 mm above the free 4 He surface. The top view of the capacitors is shown in the inset. The repeat distance λ 0 = 20 μm or 40 μm. The total number of fingers in a capacitor of length L is N = 2L/λ 0 = 50, 100 or 150. 91 The ripplons [22] were excited by AC voltage applied across the capacitors and the waves are detected by AC capacitance measurement. The schematic of the experiment is shown in figure 7.2. The thickness d in figure 7.2 is the equilibrium thickness of liquid He above the capacitors, which occurs when a constant voltage V DC is applied. The ripplons were generated here with the use of an AC voltage V AC << V DC . The attenuation time of the ripplons was deduced from the linewidth of a ripplon resonance or from the reduction in wave amplitude in time. The idea is to produce waves with an AC signal V AC e -iωt applied to IDC A, which then propagate onto IDC B and onto IDC C. The capacitance induces a charge variation proportional to V DC A V DC C V AC e i(kL- ωt) , where L = (L A +2L B +L C )/2 is the effective propagation length of the waves from IDC A to IDC C. Introducing a complex wavevector k = k’ + i k”, the amplitude of the ripplons can be written as: ) ' ' exp( ) ' cos( ) ' ' , ' ( 0 L k L k A k k A       (7.1) Thus, the damping as a function of temperature T is defined in terms of attenuation factor, k’’/k’:  T c k k   ' ' ' . (7.2) In this equation, c is a constant. It was found that the results follow the power-law behavior in the range of 300<T<700 mK, with ν = (4.5±1). The plot of the attenuation factor is shown in figure 7.3. 92 Fig. 7.3. Plot of experimental attenuation factor k’’(T)/k’ as a function of temperature T for ripplon wavelength of λ = 20, 10, 5, and 3.3 μm. In the temperature range of 300 mK<T<700 mK, the attenuation factor follows a power-law behavior of k’’(T) = c.T ν with ν = 4.5±1 The range of temperature 300 mK<T<700 mK was studied based on the detected peak ripplon signal S as a function temperature T, which is shown in figure 7.4. The lowest detected signal is at a temperature of around 700 mK, whereas the signal at temperature T < 300 mK saturates at around 400 nV, which implies that the signal is not thought to be intrinsic to the ripplons. The signal at T<300 mK is coupled to a low frequency mode caused by vibration of the apparatus, and is independent of temperature. 93 Fig. 7.4. Peak ripplon signal S as a function of temperature T for λ = 10 μm. The signal at T < 300 mK saturates to around 400 nV, which implies that the signal below T = 300 mK is mainly caused by vibration of the apparatus, instead of ripplons 7.3. Theoretical Studies on Damping of Ripplons in Three-Ripplon Interaction The first calculation of damping of ripplons in three-ripplon interaction was done in 1974 by Saam [86, 87] using the hydrodynamic equation for a compressible fluid to deduce the Lagrangian and Hamiltonian corresponding to the system. Five years later, the calculation of ripplon damping in a three-ripplon interaction was improved by Gould and Wong.[83] However, the most recent work of the ripplon damping in three-ripplon mechanism [85] suggests that there was a mistake in Saam’s paper in imposing the boundary condition constraint at the unperturbed surface. The correction was applied and the result of the lifetime of ripplons due to the decay of one ripplon into two ripplons has been revised. The lifetime of subthermal ripplons for 10 -5 < ħ∙ω Q /k B ∙T < 10 -3 was obtained to be: [85]     2 0 2 1 18 2 . 1 a T k Q a B Q Q             (7.3) where a is defined as the Wigner-Seitz radius (≈ 2.2 Å), Q is the wave vector of ripplons, and σ 0 is the surface tension of the superfluid He at temperature T = 0. 94 7.4. Damping of Ripplons in One-Ripplon-Two-Phonon Interaction in Bulk Liquid He Another important mechanism that influences the decay of ripplons is the collision of a phonon with a free surface of liquid He. The first calculation was done by Saam.[87] He formulated the Lagrangian and Hamiltonian of the system and calculated the transition rate of the mechanism to occur. The Hamiltonian is given by: 2 1 0 H H H H    (7.4)                                  2 0 2 0 2 2 0 0 0 ) , ( 2 1 ) ( 2 2 y x dxdy r s r v dz dxdy H          (7.5)                                0 3 2 1 6 1 2 ) ( ) ( ) ( r s r v r r v dz dxdy H           (7.6)   2 0 0 0 2 ) , ( ) , ( 2 1 ) ( ) , ( ) ( 2 y x y x C C r v y x r v dxdy H z                                (7.7) where H 0 is the quadratic term of the Hamiltonian (harmonic Hamiltonian), H 1 is the third-order perturbation used for uniform system, and H 2 is the perturbations localized at the surface, which contains the curvature dependence C of the surface tension. The deviation δC(x,y) in the curvature C is related to the surface displacement by                2 2 2 2 ) , ( ) , ( ) , ( y y x x y x y x C    . In these Hamiltonians, ρ 0 is the bulk He density at zero temperature, v(r) represents the velocity field, which is the gradient of the velocity potential; s is the zero-temperature sound velocity, δρ(r) is the deviation of the liquid density from its equilibrium value ρ 0 , and ζ(x,y) is the position of the surface on the x,y plane relative to the unperturbed position (ζ(x,y) = 0). These Hamiltonians are almost the same as the ones obtained by Roche two decades later [85], except for the first term in H 2 . 95 According to Roche, the Hamiltonian for the one-ripplon-two-phonon interaction is given by:            k q k q k q q q k q k k H , ' , ' ' , ' ' , 0 ' 2 '     (7.8) where k and k’ are the wave vectors of the incoming and outgoing phonon, ψ is the conjugate momentum and η is the Fourier transform coefficient that defines the position of the liquid surface in x-y plane. They obtained the transition rate of the one-ripplon-two-phonon interaction, which is given by:   0 ' 2 0 2 0 ' " " ' ' 2 2 " ' 2 , ' " , " ; ' 2 q qk q k q qk q k q k s k q H k q q R                            (7.9) In this equation, we assume that the incoming phonon is characterized by wave vectors q and k, the outgoing phonon by wave vectors q” and k”, and the ripplon by the single wave vector q’. The wave vector k” of an outgoing phonon is written as:   2 2 2 / 3 2 / 1 2 0 0 2 ' ' ' ' q q q s s k qk                            (7.10) Differential power dP being transmitted to the surface per unit solid angle of the ripplon is:        R dq A d dP q q ' 2 0 ' '    (7.11) where A is the surface area of the liquid. From this result, fraction of the incoming energy deposited on the surface of the liquid can be calculated:   4 0 2 1 90 3 2                   s T k Q E E B Q Q Q phonon        (7.12) where Q is the wavevector of the ripplon. In other words, the lifetime of the ripplons is inversely proportional to the fourth power of temperature: 96 4 2 0 60                T k s Q B Q      (7.13) This deposited energy result is in accordance with the phonon energy, which is proportional to T, multiplied by phonon density in bulk liquid He: [28] 3 6 . 9            s h T k n B  (7.14) It was concluded that a one-ripplon-two-phonon interaction is a more efficient process compared to the three-ripplon mechanism, with a lifetime of 4 orders of magnitude larger than its counterpart. Thus, at very low temperature, the decay of ripplons’ excitations is mainly caused by the scattering of single phonons from the surface of superfluid He. 7.5. Damping of Ripplons in One-Ripplon-Two-Phonon Interaction in He Droplets Eq. 7.13 is valid for collisions between phonons with the flat free surface of liquid He, which is characterized by the following dispersion relation for plane surface waves with a wave vector of Q, [87] 0 3 0    Q Q   (7.15) To utilize this result in case of collisions between phonons and spherical surface of He droplets, we use the eigen frequency of ripplons on a sphere [23], which depends on the oscillation mode multipole number l:     0 3 0 2 1           R l l l l (7.16) where R is the radius of the droplets. By equating equation (7.15) and (7.16), we obtained the effective wave vector of ripplons on the sphere Q in terms of l: 97     R l l l Q 3 2 1      (7.17) By plugging in Q into the lifetime equation (7.13), we obtain:     4 3 2 0 2 1 60                     T k s l l l R B l      (7.18) Fig. 7.5 shows the temperature dependence of τ 2 (for the lowest and most long lived quadrupole modes) in He droplets with radius R = 100, 300, 1000, and 3000 nm. At low temperature T = 0.1 K to the temperature at which phonon mean free path equals to droplets’ radius, the calculations follow eq. (7.18) for molecular regime. The data are mainly obtained from Refs. [28, 29] Fig. 7.5. Lifetime of ripplons in viscous droplets and molecular regime as a function of temperature for droplets with radius of R = 3 μm, 1 μm, 0.3 μm, 0.1 μm. The horizontal dashed lines represent the time of flight between the nozzle and the skimmer (C), time to traverse the pick up cell (B), and time of flight between the nozzle and the interaction point (A). The vertical dashed lines represent their corresponding temperatures. Note that the temperature of the droplets decreases from 5 K around the nozzle to 0.8 K at the skimmer. 98 The plot also shows the lifetime of ripplons in a viscous droplet with radius R = 100, 300, 1000, and 3000 nm and range of temperature T from the temperature at which phonon mean free path equals to droplets’ radius to T = 1.5 K, according to [23]:         ) 1 ( ) 1 2 ( 2 l l R (7.19) where ν is the kinematic viscosity of the droplets, which depends on temperature T. This equation is valid when the mean free path of the droplet’s elementary excitation is smaller than the radius of the droplet, whereas the equation of lifetime of ripplons in molecular regime is valid when the mean free path of the droplet’s elementary excitation is larger than the radius of the droplet. Thus, we expect the shape oscillations in molecular regime to decay much longer. Here we did not attempt calculating the lifetime in the intermediate regime, where the molecular regime should in principle smoothly converge into the viscous regime. The mean free path of elementary excitations (phonons and rotons) in superfluid He droplets at low temperatures was discussed by Barenghi et. al.[95] Although they focused on mean free path of rotons, it can be applied to phonons as well. The phonon mean free path L is defined as: s L n      3 (7.20) where η is the normal fluid viscosity, ρ n the normal fluid density and s the speed of sound in He droplets. At high T>1.1 K, when rotons effect dominate, one should use the average group velocity of thermal rotons in place of the speed of sound s. The group velocity of rotons is given by: 99       T k v B G 2 (7.21) where μ is the roton effective mass. The value of v G is about 100 m/s, which will increase L by about a factor of two. Since these parameters are a function of temperature, L depends on temperature. The plot of mean free path of phonons and rotons in a temperature range of 0.65 to 1.5 K is shown in figure 7.6. The data at T < 0.7 K are obtained from Ref. [96]. Fig. 7.6. Plot of phonon (red line) and roton (blue line) mean free path L as a function of temperature T, from 0.65 K to 1.5 K. The main elementary excitation is phonons at temperatures T < 1.1 K, and rotons at temperatures T > 1.1 K. The horizontal dashed lines represent mean free path of phonons equals to droplets with radius of 3 μm (A), 1 μm (B), 0.3 μm (C), and 0.1 μm (D), whereas the vertical dashed lines represent their corresponding temperatures. 100 Figure 7.6 shows that the mean free path of elementary excitations increases as temperature decreases. At 1.5 K, the mean free path is about 2 nm, whereas at 0.65 K, the mean free path is around 0.05 mm. Thus, for a droplet with a radius of R = 3 μm, the molecular regime is valid at temperatures below 0.77 K, whereas the viscous regime is valid at temperatures above 0.77 K, which is in accordance with the plot of lifetime in figure 7.5. For droplets with radius of R = 1000, 300, and 100 nm, the molecular regime is valid at temperatures below 0.82 K, below 0.88 K, and below 0.95 K, respectively. 7.6. Lifetime of the shape oscillation during the X-ray experiments. During our SLAC experiments starting from the nozzle, He droplets move to the interaction point with a velocity v of around 170 m/s, and crosses three different regions: Region 1: The distance between the nozzle and the skimmer (x ≈ 20 mm), which the droplet traverses within t ≈ 120 μs. Region 2: The region inside pick up cell, x = 100 mm, t = 0.6 ms. Region 3: The whole region from the nozzle to the interaction point, x = 640 mm, t = 3.8 ms. In region 1, temperature of the droplets drops from 5 K close to the nozzle to around 0.9 K at the skimmer, assuming the time of flight of x/v ≈ 10 -4 s, due to the evaporative cooling of the droplet, see Fig. 1.3. After they traverse the skimmer, the temperature drops until it reaches 0.4 K. In the pick up cell, the temperature increases to about 0.8 K during the doping with Xe clusters. Around the interaction point, however, the temperature decreases again to around 0.4 K. In region 1, the decay of ripplons mainly occurs in viscous regime, since the temperature drops from 5 K at the nozzle to 0.9 K at the skimmer. The only droplets that experience the 101 molecular regime at a temperature of 0.9 K are the ones with radius smaller than 200 nm (see fig. 7.6). For the largest droplets in the experiment (radius of 1 μm), the lifetime of ripplons is ~10 -5 s at 0.9 K. This lifetime is still below the time of flight between the nozzle and skimmer (120 μs). Even for the largest possible droplets with a radius of 3 μm, the lifetime of ripplons is ~ 6 x 10 -5 s, which is one half the time of flight between the nozzle and skimmer. Therefore, the ripplons should decay before droplets pass through the skimmer. For droplets with a radius of 100 nm, however, the lifetime of ripplon is ~ 10 -5 s, whereas droplets with a radius of 200 nm has a ripplon lifetime of ~ 2 x 10 -5 s, since they have already experienced molecular regime at 0.9 K. These lifetimes are still below the time of flight between the nozzle and skimmer. Therefore, ripplons should decay long before droplets reach the skimmer for any droplet size in region 1. In region 2, droplets traverse the pick up cell for 0.6 ms. If Xe gas is present the temperature therein can increase up to about 0.8 K. Contrary to region 1, the decay of ripplons in region 2 mainly occurs in molecular regime. The only droplets that experience the viscous regime at a temperature of 0.8 K are the ones with radius larger than 2 μm. For the largest droplets in the experiment (radius of 1 μm), the lifetime of ripplons is 10 -4 s at 0.8 K. This lifetime is about one sixth the time it takes for droplets to traverse the pick up cell. Even for droplets with radius of 2 μm, the lifetime is still one half the time of flight to traverse the pick up cell. For droplets larger than 2 μm, the decay of ripplons occurs in viscous regime. For instance, the largest possible droplets with radius 3 μm has a lifetime of ripplons of ~2 x 10 -5 s, which is still way below the time it takes for droplets to pass through the pick up cell. Therefore, ripplons should decay before droplets reach the far end of the pick up cell for any droplet size in region 2. In the whole region from the nozzle to the interaction point (region 3), the temperature of the droplets varies around the nozzle and the pick up cell. To determine the average temperature 102 in this region, we consider the extreme case, in which the temperature of droplets drops significantly around the nozzle to 0.4 K and there is no Xe doping in the pick up cell. Therefore, the average temperature is 0.4 K. At 0.4 K, all droplets of any size experience the molecular regime of ripplon decay. Droplets with radius of 100 nm, 300 nm, and 1 μm have lifetimes of ripplons of 2 x 10 -4 s, 7 x 10 -4 s and 2 x 10 -3 s, respectively. These lifetimes are still less than the time of flight from the nozzle to the interaction point. However, the ripplon lifetime of a droplet with radius of 1 μm is comparable to the time of flight in this region. Our estimation reveals that the time of flight from the nozzle to the interaction point is about the same as ripplon lifetime of a droplet with radius of 2 μm. However, ripplons may not affect the droplet’s shape since most of the wheel shape droplets in the experiment have radius of around 100-300 nm. At 0.4 K, the maximum achievable lifetime is 7 x 10 -4 s, which is about one-fifth of the time of flight of the droplets in region 3. Another possible reason is the increase of temperature to 0.8 K during the doping process in the pick up cell (region 2), which means that the maximum achievable lifetime is 3.3 x 10 -5 s. Therefore, we conclude that the shape oscillations in the molecular regime could be ignored at the interaction point. 7.7. Conclusion The lifetime of shape oscillations of He droplets in molecular regime has been presented to estimate whether shape oscillations contribute to the droplet shapes obtained in XFEL diffraction imaging experiments. The lifetime estimate is derived from the calculation of a phonon collision with a free surface of He, as shown in eq. (7.13), which is modified to eq. (7.18) by considering the system of a spherical He droplet. The lifetime of shape oscillations depends on the size and temperature of the droplets. The results show that the larger the droplet 103 and the lower the temperature of the droplet, the longer the lifetime of shape oscillations. The lowest temperature that can be reached by the droplet in XFEL experiments is T = 0.4 K. At this temperature, the lifetime of shape oscillations for droplets with radius R < 1 μm is smaller compared to the time of flight between the nozzle and the interaction point. However, the lifetime of shape oscillations exceeds the nozzle-interaction point time of flight for droplets with radius R > 2 μm, which implies that the shape oscillations may contribute to the shapes obtained in XFEL experiments. This is not the case in our experiments, since the majority of droplets have radius of R < 1 μm, which signifies that the shape oscillations decay before the droplets interact with XFEL at the interaction point. Therefore, we conclude that the shape oscillation is not important at the interaction point, and the shape deformation of the droplets is solely due to rotation. 104 Chapter 8: Kinematics of the doped vortices Part of the work in this chapter was published in C. Jones et al., Phys. Rev. B 93, 180510 (2016). Our recent X-ray diffractive imaging experiments at SLAC enabled obtaining positions of vortices in the droplets by attachment of Xe clusters [20]. Xe atoms captured by the droplet are affected by the velocity field of the vortices. Because velocity of the fluid increases as 1/R towards the vortex core, the associated Bernoulli forces lead to attraction of impurities to the vortex core and formation of clusters. To simplify this case, any excitations that may affect the interaction between vortices and clusters, such as Kelvin waves [97, 98] or vortex density waves are neglected, along with the vortex-vortex interaction [99-101]. Calculations indicate that Xe atoms are bound to vortex cores by about 6 K [102, 103]. Therefore, the imaged system consists of vortices and pinned Xe clusters. In an equilibrium state, vortices are stationary in a frame rotating with some angular velocity with respect to the laboratory frame. Therefore, Xe clusters share the rotational kinetic energy and angular momentum of the droplets. This chapter contains some model calculations on the equilibrium state of the doped vortices in a free cylindrical and spherical droplet. In our 2012 XFEL experiments, we observed some diffraction images showing symmetric arrangements of quantum vortices and Xe clusters inside He droplets [21], as shown in Fig. 8.1. The images in Fig. 8.1 show two-fold, four-fold and six-fold symmetries, which represent diffraction from droplets containing systems of vortices and Xe clusters forming a line, a square and a hexagon, respectively. All clusters in Figs. 8.1 reside at symmetrically arranged 105 positions far away from each other and at large distances relative to the droplet center (d ≈ 0.7 R He – 0.8 R He ). (a) (b) (c) Fig. 8.1. The diffraction images of doped He droplets showing a high degree of a (a) two-fold, (b) four-fold and (c) six-fold symmetry, which originate from droplets containing 2, 4 and 6 vortices, respectively. The reconstruction was done by using DCDI algorithm [20]. The dopants are located at symmetrically arranged positions far away from the droplet center. In the absence of vortices, the van der Waals interaction between an embedded particle and an isotropic, spherical droplet leads to a potential energy minimum at the droplet’s center [104], which is expected to confine the dopant clusters to d < 0.3·R He for He droplets and Xe 106 clusters with sizes of R He ≈ 100 nm and R Xe ≈ 5 nm, respectively. However, the Xe clusters in Fig. 8.1 are located more than twice as far away from the droplet center as predicted by the isotropic droplet model. On the other hand, Packard found a much smaller equilibrium distance between the two vortices in a cylinder (d ≈ 0.12 R He ). Similarly, Ancilotto performed a DFT calculation and found a smaller equilibrium distance for bare vortices in a spherical He droplet. In order to study possible physical origin of this discrepancy in this chapter we have performed some model calculations on doped single vortices in a cylinder and a sphere. 8.1. Rectilinear Vortices in a Cylinder Vortices in a cylinder filled with a superfluid present the simplest possible model system which can be treated via analytic equations [105]. If the cylinder is rotated with a constant angular velocity Ω, such as in previous low temperature physics experiments [15], the equilibrium is described by minimization of its free energy F:     L E F (8.1) where E and L are the kinetic energy and the angular momentum of the system. The equilibrium position of a quantum vortex in a cylinder rotated with a constant angular velocity has been studied, both experimentally and theoretically [15, 30, 105, 106] However in the case of free droplets the total angular momentum, L, is an independent variable and the equilibrium corresponds to the minimum in the energy in the laboratory frame, E. Here our goal is to study how the addition of Xe clusters to the vortex cores affects the stability, energy and angular velocity of the system at a certain angular momentum. For simplicity we consider a system consisting of a single doped vortex in a droplet. Later this model could be expanded to the case of multiple vortices occupying symmetric configurations in the droplet. 107 8.2. Solvation Potential in Superfluid Helium In case of a doped vortex, in addition to kinetic energy of the fluid, E has a contribution from solvation potential energy of the embedded Xe cluster. In the close proximity (< 1 nm) of the Xe cluster the He density is influenced by the van der Waals attraction between He and Xe atoms. However, far away the He is treated as a continuum incompressible liquid and the interaction between Xe clusters and He atoms is long range attractive interaction.[104] This model allows a solvation potential inside the droplets to be calculated. It is approximately constant in the droplet’s interior and increasing as the Xe clusters approach the He droplets boundary. This solvation potential is dominated by the leading power of the pairwise Xe - He van der Waals attraction: ... ) ( 8 8 6 6     r C r C r V (8.2) where C 6 and C 8 are the long range dispersion interaction constants and r is the distance between the atoms. To simplify the calculation, we ignore the expansion members in Eq. (8.2) higher than C 6 . Let us consider a system consists of a cylindrical container with a radius of R He = 100 nm filled with liquid He, and a cylindrical Xe cluster with a radius of R Xe . To simplify the integration, consider the summation of all interaction between the cluster and the He atoms, where the smallest unit is defined as a 1 nm x 1 nm x 1 nm cube. The solvation potential per one nm length of the Xe filament is defined as:                              2 / 2 / 3 2 2 2 1 111 ) ( He He He He He He Xe Xe Xe Xe h h zd R R xd R R yd R x R x xc R R yc zd yd yc xd xc x V (K) (8.3) 108 where x is the horizontal position of the center of the cylindrical Xe cluster, xc and yc are the ranges of the cluster summation along x and y, respectively, xd, yd and zd are the range of summation of He along x, y and z, respectively, and h He is the height of the cylindrical container filled with liquid He. The solvation potential for R Xe = 3, 6, and 9 nm are shown in Figure 8.2. Here h He should be taken to be ∞, however for practical reasons h He = 20 nm was taken which is large enough for the missing summation range to be negligible. The potential was set to zero at x = 0. The flattening of the potential beyond R He -R Xe corresponds to partial solvation. This region is unimportant to the calculations described in the following. Fig. 8.2. Solvation potential per 1 nm length in K units for a system of a cylindrical liquid He with a cylindrical Xe cluster, having a radius of R Xe = 3, 6, and 9 nm. 8.3. Interaction between a Quantum Vortex and Xe Clusters in a Cylinder 8.3.1. A Bare Quantum Vortex in a Cylinder The kinetic energy and the angular momentum for a single rectilinear quantum vortex in a cylinder have been calculated in Ref. [105] by an image technique. If one vortex having a 109 quantum circulation κ is created at a distance of r from the center of the cylinder, an image vortex has to be created to satisfy the condition of zero flow through the cylinder surface.[13, 15] The image vortex has a quantum circulation of κ’ = -κ, and is located at a distance of R 2 /r from the center of the cylinder. The energy and angular momentum of the vortex per unit length are:               0 2 2 2 ln 4 a R r R E    (8.4)   2 2 2 1 r R L        (8.5) where ρ is the density of liquid helium, R is the radius of the cylinder, r is the distance of the vortex from the center of the cylinder, and a 0 is the radius of the vortex core. In the free cylinder the angular momentum, L is the independent variable. The equilibrium position of a vortex is given by:              He r v L L L r 0 0 2 (8.6) where 2 2 1 ) 0 ( R L He       In this equation, L r is the reduced angular momentum per unit length, which is defined as:   0 L L L r  (8.7) where L is the angular momentum per unit length in the absolute unit and L(0) is the angular momentum per unit length of the vortex at the center of the cylinder, ρ He is the density of liquid He = 145 kg/m 3 , and κ is the quantum circulation = 9.97 x 10 -8 J.s/kg. Energy of the vortex in equilibrium is: 110               0 2 2 2 ln 4 a R r R E v He v    (8.8) where a 0 is the vortex core diameter = 0.1 nm. The angular velocity of the vortex in equilibrium is:   2 2 2 v v r R        (8.9) Thus, r v , E v , and ω v depend only on reduced angular momentum L r . 8.3.2. Quantum Vortex with Xe Filament in a Cylinder. Here we assume that a filament of solid Xe with radius R Xe is placed into the vortex core. Then the energy and the angular momentum contain additional contribution due to Xe filaments. The total angular momentum of the system is:                             ) ( 2 ) ( ) ( 2 1 2 2 2 2 0 2 2 2 2 r R r a r r R r r L He Xe He Xe Xe           (8.10) where ρ Xe is the density of solid Xe = 3781 kg/m 3 , r Xe is the radius of Xe clusters = 3 nm, ω is the angular velocity of the cylinder, ρ He is the density of liquid He = 145 kg/m 3 , and a 0 is the vortex core diameter = 0.1 nm. The first term represents the angular momentum of the cluster, the second term is the angular momentum contribution from the vortex, and the last term describes the hydrodynamic backflow contribution because in general the speed of the relative motion between the superfluid and the filament is non zero. The total energy of the system is:   2 2 2 2 0 2 2 2 2 2 2 2 ln 4 2 1                                       r R r r r V a R r R r r E He Xe He Xe Xe            (8.11) where V is the solvation potential, which is obtained from equation 8.3. The equilibrium condition is obtained from the total force of the system per unit length which should be equal to 111 zero. The total force on the filament is the centrifugal force of Xe clusters, the force resulting from the solvation potential, and the Magnus force due to nonzero velocity of the superfluid with respect to the Xe filament:   0 2 2 2 2 2                        r r R r dr dV r r F He Xe Xe         (8.12) From these equations, the equilibrium position r at an angular momentum L is obtained. The equilibrium position r can be used to find ω since ω is a function of L and r. By substituting r and ω into the energy equation, E can also be obtained as a function of L. To simplify the calculation, the reduced values of r, L, E, and ω are introduced: R r r r  (8.13)   0 L L L r  (8.14)   0 E E E r  (8.15)   0     r (8.16) where L(0), E(0) and ω(0) are the angular momentum per unit length, energy per unit length and angular velocity of a straight rectilinear vortex at the center of the cylinder, respectively:               s m kg R L He 20 2 10 23 . 7 2 1 ) 0 (   (8.17)                     m K a R E He 10 0 2 10 74 . 5 ln 4 ) 0 (    (8.18)            s rad R 6 2 10 59 . 1 2 ) 0 (    (8.19) By taking the reduced angular momentum L r from 0 to 1, the plots in Fig. 8.3 are obtained. 112 (a) (b) (c) Fig. 8.3. Plots of (a) reduced equilibrium position r r , (b) reduced energy E r , and (c) reduced angular velocity ω r as a function of reduced angular momentum L r for a system of quantum vortex with and without Xe filament. The Xe filament has a radius of 3 nm. From the calculations, the system of vortex with Xe occupies the equilibrium position at a reduced distance of 0.92, from L r = 0 to 0.2, and follow the equilibrium position of the bare vortex system at higher L r . Similarly, at L r < 0.3, the energy and angular velocity of the system of vortex and Xe deviate from its counterpart, in which the energy is approximately constant at 0.8, whereas the angular velocity goes to zero at L r = 0.2, which means that Xe involves in taking up the angular momentum of the original system. At L r < 0.2 there is a case where the angular momentum of the system is smaller compared to the angular momentum of the vortex alone, which explains the negative values of the angular velocity. Finally, we note that the same results could be obtained without introducing an explicit Magnus force, by minimization of energy, as it is done in Section 8.4 for vortex in a sphere. 113 8.4. Doped Quantum Vortex in a Spherical Helium Droplet In contrast to a quantum vortex in a cylinder, a vortex in a spherical droplet is rectilinear at the center and is curved when it is displaced by a distance of r from the center, resembling a vortex ring [107, 108]. Here r is the distance of the vortex from the rotational axis in the equatorial plane. The curvature results from the boundary conditions of no flux through the droplet's surface. Therefore a vortex must terminate perpendicular to the droplet surface. The calculation on the shapes, energy, angular momentum and angular velocity of a vortex in a droplet has been conducted in Refs. [13, 15, 109-112] Shape of the vortex is defined by the Biot- Savart law and the condition that the shape is stationary in the rotating frame. The velocity of a superfluid flow of a vortex at a point r is given by:           image vortex s r s s d r s r v 3 4 ) (         (8.20) Since the velocity of the vortex core is the same as the local fluid velocity, the core velocity at a point s is given by putting r = s in eq. (8.20), which creates a singularity. To avoid singularity, the approximation that we can use is the Local Induction Approximation (LIA). The local induction approximation produces a vortex core velocity v(s) at a distance of s from the rotational axis:       s s a e l l s v                        0 4 / 1 2 / 1 2 ln 4 (8.21) where l + and l - are the length of the two adjacent line elements that hold the point s in between, a 0 is the radius of the vortex core, and χ is the vortex curvature. From this equation, the shape of the vortex core is obtained as follows: 114               2 / 1 2 2 0 2 2 0 2 2 1 1 2 1 0 s r s r dr s z s s                    (8.22) where z(s) is the height of a vortex core at a distance of s from the rotational axis, s 0 is the perpendicular distance of the vortex core from the droplet center, and Ω is a constant related to the angular velocity of the droplet ω, as defined by:            0 4 / 1 ln a e l total  (8.23) In this equation, l total is the total length of the curve of a vortex. The vortex curve located at a distance of s 0 = 0.5 R from the rotational axis of the droplet is shown in Fig. 8.4. Fig. 8.4. A vortex located at a distance of s 0 = 0.5 R from the rotational axis of the droplet. The height of a vortex element at a distance of s is given by z(s). The vortex can be made to terminate perpendicularly on the droplet surface by adjusting Ω. At this Ω, the angular velocity L v and energy E v of the vortex are obtained:      dS r L s v   (8.24) 115       dS v E s v   2 1 (8.25) where ρ s is the superfluid density, v = v(s) and dS is the droplet surface element. The integration limit is taken over a closely spaced surface S1, which is bounded by the core of the vortex line and the boundary of the fluid, see Fig. 8.5. [15] Fig. 8.5. The surface S1 in between the boundary of the fluid and the curve along the vortex core AB is taken as the limit of integration in eqs. (8.24) and (8.25). The plot of L v and E v as a function of distance s from 0 to R = 100 nm is shown in Fig. 8.6, along with the plot of ω as a function of x. The value of E(0) for a rectilinear vortex is 1.1x10 4 K. 116 (a) (b) Fig. 8.6. The calculation of (a) reduced energy, reduced angular momentum and (b) angular velocity as a function of the vortex position in the droplet having R = 100 nm. The local induction approximation breaks down as x approaches the droplet surface, since R-x is comparable to a 0 . In case of a vortex with Xe filament in the core is described by equations which are analogous to Section 8.3 as in the case of a cylinder. However the shape of the vortex with Xe is unknown. Therefore we need to apply some approximations. Here we consider a spherical droplet with a radius of R = 100 nm, containing a quantum vortex. We have assumed that of 1.37 x 10 5 Xe atoms as in the experiment, a filament has a radius of 5 nm which has the same shape as the bare vortex core at same distance s. The radius is estimated from the number density of solid Xe of 1.739 x 10 28 atoms/m 3 and assuming the length of the filament equals R. Strictly because the length of the filament depends on the distance s, the radius will change for the same mass of the filament. However this effect is disregarded and the calculations at different distances s were done with the fixed mass and radius of the filament. The total angular momentum of the system is: 117           r R r a r r L r R r r L v He Xe v Xe Xe                        2 2 0 2 2 , (8.26) where L v (r) is the vortex angular momentum in the absolute units and ω v (r) is the angular velocity in the absolute unit. The first term represents the angular momentum of the clusters and the third term is the angular momentum due to liquid backflow. Based on the conservation of angular momentum, this equation can be used to obtain the angular velocity of the system:                       R r r a r r r R r r r R r r a r r r L L l r l r He Xe r Xe Xe r v r He Xe r v v r r r                     2 2 0 2 2 2 2 0 0 ,         (8.27) where l r and r r are the reduced angular momentum and reduced distance, respectively. The reduced distance is defined as r r = r(r r )/R. The energy of the system is given by:                           2 2 2 2 2 , , 2 1 , r r v r r r He Xe r r v r r r Xe Xe r r r r r r r r r l R r r r V r r E r r r l R r r l E                         (8.28) where V(r(r r )) is the solvation potential. Now V(r) requires integration of the pair interactions along the curved filament in the spherical droplet which requires more complex calculations. In addition, it remains unclear how close the ends of the filament could actually approach the surface of the droplets. Therefore for the sake of an estimate we took V(r) = 100 V(d) as given by eq. (8.3) for the cylinder for the 100 nm long filament. For a system with no Xe clusters, the equilibrium position of the vortex r eq is obtained from the equation of:   r r v l r L  (8.29) and the reduced energy and reduced angular velocity of the vortex are E v (r eq )/E(0) and ω v (r eq )/ω(0), respectively. The plots of equilibrium position, energy and angular velocity of the system as a function of reduced angular momentum with and without the addition of Xe are shown in Fig. 8.7. The energy of a rectilinear vortex is 1.1 x 10 4 K and the angular velocity of a rectilinear vortex is 9.24 x 10 6 rad/s. 118 (a) (b) (c) Fig. 8.7. The plots of (a) reduced equilibrium position, (b) reduced energy and (c) reduced angular velocity at equilibrium position as a function of reduced angular momentum for a system of bare vortex and a system of vortex and Xe in Helium droplets. Fig. 8.7 in sphere shows similarities and differences with respect to similar plots for vortex in a cylinder, see Fig. 8.3. Similar to the in cylinder the plots for the doped and undoped vortices are close together at 0.65 < l r < 1. However Fig. 8.7 (a) shows that at smaller L r there is a discontinuity in the r r which jumps abruptly from 0.4 to 0.85. Fig. 8.7 (c) also shows that ω jumps by about a factor of two and then decreases approximately linearly. As the angular momentum l r increases from 0 to 0.64, the vortex and Xe systems stay at approximately the same equilibrium position at r = 0.86, while the vortex system with no Xe moves continuously closer to the center of the droplet. The vortex and cluster system immediately move closer to the center of the droplet at l r = 0.66, which implies that the system acquires velocity from the relative motion between the spinning liquid in the droplet and the motion of the vortex. This motion is caused by the Magnus force. At l r > 0.66, the vortex system approaches the center of the droplet, similar to the system with no Xe clusters. Similar trend is also observed in energy and angular velocity plots. At l r < 0.66, the energy of the vortex and Xe at equilibrium position is generally less than the energy of the vortex. This is because at a particular angular momentum, the Xe 119 clusters take up a large portion of angular momentum from the system. Consequently, the vortex has less angular momentum and energy, which makes the system less energetic compared to the bare vortex system. The angular velocity of the vortex and Xe system is quite different compared to the angular velocity of bare vortex system at l r < 0.66. The angular velocity of the bare vortex always decreases as angular momentum increases. On the other hand, the angular velocity of the vortex and cluster system increases as angular momentum increases up to l r = 0.65, similar to the rigid body rotation. From this plot, it is estimated that the system has a reduced moment of inertia of 0.216 or 2.32 x 10 -34 kg.m 2 . At l r > 0.65, the vortex and Xe system has an angular velocity that is approximately similar to the bare vortex system, within about 40%. 8.5. Conclusion Both experiments and model calculations show that loading of the vortices leads to substantial change of their equilibrium positions. The simplest model includes loaded rectilinear vortex in a cylindrical He container. Modeling shows that the equilibrium distance with Xe is larger than in the case of a bare vortex which could be associated with the centrifugal force. The vortex cannot approach the boundary of the cylinder beyond about 15 nm due to solvation potential of the containing Xe vortex. In He droplets, the system is in a spherical shape, and any motion, energy and angular momentum of the vortex system have to satisfy the condition that the vortex has to terminate perpendicularly on the droplet surface. This condition produces a curved vortex, instead of a straight vortex as in the case of vortex in a cylindrical container. From the results of the calculation, there is a discontinuity in the equilibrium position plot at L r = 0.66, at which the position of the vortex jumps from 44 nm to 87 nm. 120 Conclusions and Future Work Conclusions This thesis discusses the shapes and vorticity of superfluid helium droplets of up to about 1 μm in diameter, which were studied via scattering of X-ray radiation from a free-electron laser (XFEL). The measurements show that most of diffraction images have concentric circular pattern which correspond to droplets with an aspect ratio AR ≈ 1.0, and some of them show noticeable ellipticity of the diffraction contours and droplets aspect ratio of 1.1 < AR < 1.6. About 1% of the images have droplets AR in the range of 1.7 – 2.4 and show some pronounced streaks which extends well beyond the diffraction contours. Large aspect ratio and streaks in the diffraction images indicate some pronounced centrifugal deformations in the droplets. To obtain the shapes of the droplets, the inverse Fourier Transform technique has been used. In this technique, the required phase information is obtained assuming the centro- symmetric shape of the droplets, as per a classical rotating droplet. The shape reconstruction of images showing streaks reveals shapes with two nearly parallel surfaces. For the purpose of assigning the observed diffraction images to some particular droplet shapes, the axially symmetric and two-lobed shapes of rotating classical droplets have been investigated. By using the 3D Fourier Transform, the diffraction images of such shapes can be modelled and compared with the experimental images. The diffraction images from axially symmetric and two-lobed shapes are similar at the edge on x-ray impact; however, they differ considerably upon the tilt of the shapes. Furthermore, there is a difference in the power dependence of the intensity along the streak produced from axially symmetric and two-lobed 121 droplet shapes. The analysis confirms that in addition to strongly deformed axially symmetric droplets the beam also contains two-lobed shape droplets. The diffraction images of the droplets have been analyzed to obtain the beam profile of XFEL. Moreover, the XFEL flux and ionization probability of He and Xe atoms were estimated from the images. The analysis verifies the tighter XFEL focal volume in 2014 XFEL experiments, with a flux of less than 10 23 photons/m 2 . It is also estimated that He atoms are barely ionized whereas Xe atoms are completely ionized during the 60 fs duration of the XFEL pulse. The estimated lifetime of shape oscillations on He droplets was also reported. The lifetime depends on the size and temperature of the droplets. The calculation shows that the shape oscillations in the droplets relevant for this work decay long before they reach the interaction point with the XFEL beam. This result confirms that the observed shape deformations originate from the rotation of the droplets. Lastly, the kinematics of the doped vortices in a cylinder and a sphere has also been computed. In a cylindrical system, modeling shows that the equilibrium distance of a vortex with Xe is larger than in the case of a bare vortex which could be associated with the centrifugal force. In a sphere, however, there is a discontinuity in the plot of equilibrium position vs L, which likely results from the vortex curvature. Future Work In the presence of the large number of vortices the shapes of rotating superfluid droplets in equilibrium are expected to resemble classical droplets rotating at the same angular velocity ω due to similar velocity fields far from the vortex cores. 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Proceedings of the National Academy of Sciences of the United States of America, 2014. 111: p. 4667-4674. 129 Appendix A1: Other Streak Diffraction Images Run051 70730 Run051 104042 Run052 57920 Run052 95603 Run052 126101 Run054 78524 Run062 115892 Run064 44582 Run064 55751 130 Run064 82523 Run064 84620 Run064 110027 Run071 60794 Run099 81143 Run099 93038 Run099 100715 Run104 20144 Run104 81302 131 Run108 22766 Run109 6917 Run109 82772 Run109 125096 Run109 128543 Run112 56702 Run112 60137 Run112 83090 Run112 90101 132 Run112 121646 Run119 83759 Run119 104804 Run125 10235 Run125 88535 Run134 5036 Run134 53324 Run134 55268 Run134 59669 133 Run134 60578 Run134 62111 Run134 120677 Run140 15980 Run144 40232 Run149 88769 Run149 95843 Run155 130742 Run156 27629 134 Run156 76583 Run156 88868 R155 5973667109185852416 R155 5973667564428641536 R157 5973672220205933312 R158 5973682425032344064 R158 5973679199509783552 R159 5973691118031358720 R160 5973693398674343680 135 R160 5973698015762673152 R160 5973695374368176384 R162 5973702710161892352 R174 5973754202522144512 R174 5973755602677988608 R185 5973780964467114496 136 Appendix A2: XFEL Flux and Ionization Probability of He and Xe in 2014 XFEL experiments Run No Fraction of Detected Photons Half Major Axis (m) Half Minor Axis (m) Average Radius (m) Total Number of Photons Flux (photons/m 2 ) Low Contrast Image Aspect Ratio He Ionization Probability Xe Ionization Probability 155 5973666340370219008 0.12502 1.716E-07 1.672E-07 1.694E-07 7876360 4.65279E+22 N 1.0263158 0.003009073 0.999982076 155 5973666550828120064 0.1389 1.958E-07 1.518E-07 1.738E-07 1456970 7.7677E+21 N 1.2898551 0.000502988 0.838723273 155 5973666988924206080 0.17736 1.528E-07 1.357E-07 1.443E-07 7765660 8.72481E+22 N 1.1260133 0.005635125 0.999999999 155 5973667066235247104 0.02244 3.715E-07 3.715E-07 3.715E-07 6042270 1.54314E+21 Y 1 9.99443E-05 0.304054643 155 5973667564428641536 0.02057 4.518E-07 3.128E-07 3.823E-07 210276000 4.78866E+22 N 1.4443734 0.003096807 0.999986974 155 5973667650329827584 0.03182 3.388E-07 3.138E-07 3.263E-07 4163330 1.78654E+21 Y 1.0796686 0.000115708 0.34272918 155 5973667654624836352 0.04698 2.383E-07 2.219E-07 2.301E-07 11809300 2.04927E+22 Y 1.0739072 0.001326431 0.991882484 155 5973667933803713280 0.04556 2.413E-07 2.276E-07 2.345E-07 12692700 2.04359E+22 N 1.0601933 0.001322759 0.991773525 155 5973667946688930816 0.01373 4.827E-07 4.371E-07 4.599E-07 14519700 1.57887E+21 N 1.104324 0.000102258 0.309870145 155 5973667963869160960 0.05463 2.348E-07 1.971E-07 2.160E-07 13238500 2.9612E+22 N 1.1912735 0.001916128 0.999046955 155 5973668371899756288 0.04389 2.512E-07 2.340E-07 2.426E-07 15384400 2.16052E+22 Y 1.0735043 0.001398391 0.993749313 155 5973668930257502464 0.0254 3.759E-07 3.368E-07 3.564E-07 16206500 4.88904E+21 N 1.1160926 0.000316613 0.682867226 155 5973669398385445376 0.03513 3.138E-07 3.005E-07 3.072E-07 73440700 4.01399E+22 Y 1.0442596 0.002596482 0.999919629 155 5973669720514908416 0.03396 3.593E-07 2.688E-07 3.141E-07 42876300 2.14419E+22 Y 1.3366815 0.001387828 0.993504866 155 5973669776350649856 0.05323 2.302E-07 2.033E-07 2.168E-07 19934800 4.39356E+22 N 1.1323168 0.002841664 0.999967048 157 5973672026928281600 0.01391 4.680E-07 4.442E-07 4.561E-07 50447100 5.6707E+21 Y 1.0535795 0.000367224 0.736063984 157 5973672044108466432 0.04698 2.382E-07 2.225E-07 2.304E-07 31649900 5.4684E+22 N 1.0705618 0.00353562 0.999997361 157 5973672533711745024 0.03997 2.857E-07 2.567E-07 2.712E-07 16843300 1.51464E+22 Y 1.1129723 0.000980552 0.971501122 157 5973672555187017728 0.02284 4.424E-07 2.949E-07 3.687E-07 14078500 3.70801E+21 N 1.5001695 0.000240139 0.581471743 157 5973672585252403968 0.04089 2.960E-07 2.267E-07 2.614E-07 7188840 7.49566E+21 N 1.3056903 0.000485376 0.828080808 157 5973672653973428224 0.01956 4.286E-07 3.490E-07 3.888E-07 114737000 2.44253E+22 Y 1.2280802 0.001580775 0.996777181 157 5973672838660977920 0.04343 2.443E-07 2.443E-07 2.443E-07 3294630 4.49939E+21 Y 1 0.000291383 0.65247107 157 5973672881611574016 0.03882 2.860E-07 2.791E-07 2.826E-07 34927600 2.66581E+22 Y 1.0247223 0.001725153 0.998092534 157 5973673225216328192 0.03396 3.451E-07 2.860E-07 3.156E-07 50494700 2.4775E+22 N 1.2066434 0.001603392 0.997031369 157 5973673263871869440 0.0327 3.258E-07 3.151E-07 3.205E-07 36734400 1.69462E+22 Y 1.0339575 0.001097005 0.981326844 157 5973673289642223872 0.0355 3.251E-07 2.857E-07 3.054E-07 11018200 6.16135E+21 Y 1.1379069 0.000398991 0.764795596 137 158 5973679126493768192 0.02022 4.065E-07 3.632E-07 3.849E-07 18516500 4.10614E+21 Y 1.1192181 0.00026592 0.61883883 158 5973679242460428032 0.16963 1.639E-07 1.457E-07 1.548E-07 9390600 7.95519E+22 N 1.1249142 0.005139326 0.999999992 158 5973680333371943680 0.02166 3.957E-07 3.559E-07 3.758E-07 8607970 2.09949E+21 Y 1.1118292 0.000135975 0.389313344 158 5973680835893913088 0.02325 3.881E-07 3.482E-07 3.682E-07 68151000 1.80474E+22 N 1.1145893 0.001168246 0.985582742 158 5973680926090114048 0.03586 3.112E-07 2.960E-07 3.036E-07 20538700 1.176E+22 N 1.0513514 0.000761405 0.93686134 158 5973681325530681856 0.05323 2.259E-07 2.101E-07 2.180E-07 11739200 2.52845E+22 N 1.0752023 0.001636334 0.997366178 158 5973681488742964224 0.14819 1.629E-07 1.570E-07 1.600E-07 5788090 4.30169E+22 N 1.0375796 0.002782329 0.999959112 158 5973682175918915584 0.03312 3.438E-07 2.936E-07 3.187E-07 56984200 2.687E+22 Y 1.1709809 0.001738855 0.99818516 158 5973682300475658240 0.01493 4.518E-07 4.175E-07 4.347E-07 49626500 6.76388E+21 Y 1.0821557 0.000438001 0.795836868 158 5973682425032344064 0.04263 3.112E-07 1.844E-07 2.478E-07 5266540 6.79456E+21 Y 1.6876356 0.000439987 0.797302797 158 5973683464403146240 0.04389 2.650E-07 2.206E-07 2.428E-07 24517400 3.43179E+22 N 1.2012693 0.002220302 0.999684476 158 5973684443676692992 0.03046 3.508E-07 3.151E-07 3.330E-07 5976770 2.36587E+21 N 1.1132974 0.000153226 0.426354563 158 5973684757182521600 0.04624 2.487E-07 2.159E-07 2.323E-07 5776120 9.64896E+21 Y 1.1519222 0.000624768 0.896329808 158 5973684967640405760 0.0111 5.681E-07 4.962E-07 5.322E-07 34453200 2.08994E+21 Y 1.1449012 0.000135356 0.38794161 158 5973685495932744192 0.05194 2.467E-07 1.941E-07 2.204E-07 9558270 1.97049E+22 Y 1.2709943 0.001275472 0.990232348 159 5973686870318226944 0.0355 3.251E-07 2.869E-07 3.060E-07 10225600 5.67341E+21 N 1.1331474 0.000367399 0.736231688 159 5973686981989729792 0.11804 1.727E-07 1.684E-07 1.706E-07 14215400 8.17322E+22 N 1.0255344 0.005279809 0.999999995 159 5973687342774744832 0.26007 1.333E-07 1.289E-07 1.311E-07 3305700 5.4437E+22 N 1.034135 0.003519673 0.999997204 159 5973687458741385472 0.02722 3.559E-07 3.388E-07 3.474E-07 10154800 3.39346E+21 Y 1.0504723 0.000219771 0.549377253 159 5973687664904222464 0.03355 3.434E-07 2.907E-07 3.171E-07 22247300 1.07104E+22 Y 1.1812865 0.000693475 0.919208451 159 5973687896803859200 0.0254 3.617E-07 3.490E-07 3.554E-07 54185100 1.65309E+22 N 1.0363897 0.001070131 0.979413124 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Asset Metadata

Creator Bernando, Charles (author)

Core Title Vorticity in superfluid helium nanodroplets

Contributor Electronically uploaded by the author (provenance)

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program physics

Publication Date 07/25/2017

Defense Date 05/11/2016

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag axisymmetric shapes,Fourier transform,helium nanodroplets,OAI-PMH Harvest,quantum vortices,rotation,shape oscillation,superfluid,two-lobed shapes,vorticity,xenon clusters,X-ray diffraction imaging,X-ray free electron laser

Format application/pdf (imt)

Language English

Advisor Vilesov, Andrey (committee chair), Benderskii, Alexander (committee member), Kresin, Vitaly (committee member), Nakano, Aiichiro (committee member), Takahashi, Susumu (committee member)

Creator Email bernando@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-281623

Unique identifier UC11281375

Identifier etd-BernandoCh-4623.pdf (filename),usctheses-c40-281623 (legacy record id)

Legacy Identifier etd-BernandoCh-4623.pdf

Dmrecord 281623

Document Type Dissertation

Format application/pdf (imt)

Rights Bernando, Charles

Type texts

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

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA

Abstract (if available)

Abstract Starting from Newton, the equilibrium shapes of rotating classical bodies held together by gravitation has been extensively studied theoretically. It was also shown that shapes of rotating droplets held together by surface tension belong to the same class of solutions and can serve as a system for laboratory emulation of astronomical objects. Equilibrium shapes of the classical rotating droplets have been extensively studied by calculations and experiments. However, surprisingly little is known about the rotation in viscosity-free, superfluid droplets, and their shape families, which is the focus of the present thesis. In this work, rotating superfluid helium droplets of up to about 1 μm in diameter, were produced upon expansion of helium fluid in vacuum and were studied via scattering of X-ray radiation from a free-electron laser (XFEL). This collaborative work was performed at Atomic, Molecular and Optical Science (AMO) Instrument at SLAC Linac Coherent Light Source (LCLS). The results show the existence of strongly deformed droplets as indicated by large asymmetry and intensity anomalies (streaks) in the obtained diffraction images. The analysis of the images shows that some droplets must have non-axisymmetric elongated shape which can be well modeled by classical droplet shapes along the two lobe stability line. The stability regions and shapes of superfluid and ordinary viscous droplets are compared. This thesis contains eight chapters. The first chapter provides introduction to the thermodynamics of He droplets. The second chapter contains description of the experimental set up and basics of the X-ray scattering from a spheroidal object. The third chapter describes the results of the X-ray coherent diffractive imaging experiments of superfluid He droplets at SLAC, along with the droplet shape reconstruction. The fourth chapter provides details on the calculations of the shapes of classical rotating droplets along the axially symmetric and two-lobed shapes branches. The fifth chapter discusses the main finding from the results of the XFEL experiments. The sixth chapter contains a discussion of the XFEL beam profile and the ionization probability of He droplets in XFEL experiments. The seventh chapter describes droplet shape oscillations and their lifetime, which enables excluding the shape oscillations as a possible source for the observed deformations. The last chapter contains the results of model calculations of the geometry of quantum vortex containing a solid Xe filament in a cylinder and in a sphere.