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Ultracold molecular plasma Schulz-Weiling, Markus 2017

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Ultracold Molecular PlasmabyMarkus Schulz-WeilingA THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017© Markus Schulz-Weiling 2017AbstractThe conditions a↵orded by a skimmed free-jet expansion intersected bytwo laser pulses, driving resonant transitions in nitric oxide, determine thephase-space volume of a dense molecular Rydberg ensemble. Spontaneousavalanche to plasma within this system leads to the development of twomacroscopic domains. These domains are clearly distinguished by their po-larizability as well as their locality within the plasma. The first domainappears at the system core, is polarized by fields exceeding 500 mV/cmand displays an ambipolar expansion character suggestive of initial electrontemperatures exceeding ⇠150 Kelvin. The second domain travels with thevelocity of the supersonic beam and is robust to the application of severalhundred V/cm pulses. It is further distinguished through the apparent ar-rest of relaxation channels, annealing the domain over a millisecond or morein a state far from thermal equilibrium. Both domains are linked via thespontaneous breaking of ellipsoidal symmetry to form bifurcating arrestedvolumes.iiLay summaryPlasma is the fourth state of matter next to solid, fluid and gas. Physicalsystems can transition between states of matter when energy is added orremoved. Plasmas form when neutral matter obtains enough energy to breakinto charged subatomic components - into ions and electrons.Plasma particles carry charge and can sense each other over distance,giving raise to collective behaviour. Such dynamic systems, when preparedfar from thermal equilibrium, must redistribute their energy along all acces-sible channels on a short timescale.Our group studies the evolution of plasma in an exotic regime. We ob-serve unexpected system behaviour and the arrest of energy redistributionfor uncharacteristic durations, suggesting new physics. We seek to under-stand these dynamics on the microscopic scale.iiiPrefaceThis thesis is the result of a collaborative research e↵ort in the lab of Profes-sor Ed Grant at the University of British Columbia. My involvement in thislab dates back to the year 2009 and consists of an undergraduate researchproject, my thesis project for the degree ’Diplomphysiker’ at the Universityof Freiburg as well as my PhD program. My work in this research group,prior to enrolling into the PhD program in January 2012, led to the followingpublications:• H. Sadeghi, M. Schulz-Weiling, J.P. Morrison, J.C.H. Yui, N. Saquet,C.J. Rennick and E.R. Grant. Molecular ion-electron recombinationin an expanding ultracold neutral plasma of NO+Phys. Chem. Chem. Phys.: 13(42):18872-9 (2011)• N. Saquet, J.P. Morrison, M. Schulz-Weiling, H. Sadeghi, J.C.H. Yui,C.J. Rennick and E.R. Grant. On the formation and decay of a molec-ular ultracold plasmaJ. Phys. B: At. Mol. Opt. Phys.: 44 184015 (2011)• C.J. Rennick, N. Saquet, J.P. Morrison, J. Ortega-Arroyo, P. Godin,L. Fu, M. Schulz-Weiling and E.R. Grant. Dissociative recombinationand the decay of a molecular ultracold plasmaJ. Phys.: Conf. Ser.: 300 012005 (2011)• M. Schulz-Weiling. Expansion dynamics of a non-spherical ultracoldplasma in a supersonic molecular beamDiplomarbeit (diploma thesis) - Faculty of Mathematics and PhysicsAlbert-Ludwigs-Universita¨t Freiburg (2011)ivPrefaceIn the following, I list six published articles reporting results from researchwork over the course of my PhD program. I state my relative contributions:• (1) H. Sadeghi, A. Kruyen, J. Gurian, J. Morrison, M. Schulz-Weiling,N. Saquet, C.J. Rennick and E.R. Grant. Dissociation and the devel-opment of spatial correlation in a molecular ultracold plasmaPhys. Rev. Lett.: 21;112(7):075001 (2013)Contributed technical lab support and took part in discussions as wellas the writing process.• (2) J. Hung, H. Sadeghi, M. Schulz-Weiling and E.R. Grant. The evo-lution from Rydberg gas to Plasma in an Atomic Beam of XeJ. Phys. B: At. Mol. Opt. Phys.: 47 155301 (2014)Converted our experimental setup to prepare a xenon plasma. Datacollection and analysis was done by J. Hung and H. Sadeghi. Con-tributed to discussions and writeup.• (3) M. Schulz-Weiling and E.R. Grant. Three-dimensional imaging ofthe ultracold plasma formed in a supersonic molecular beamAIP Conf. Proc. 1668, 050002 (2015)Developed new detector and collected all data published in this ar-ticle. Contributed key points of the interpretation. Creation of themanuscript was a combined e↵ort of the authors.• (4) M. Schulz-Weiling and E.R. Grant. Quantum State Control ofUltracold Plasma FissionJ. Phys. B: At. Mol. Opt. Phys. 49 (2016) 064009 (9pp) (2016)Collected all data published in this article. Contributed key points ofthe interpretation. Creation of the manuscript was a combined e↵ortof the authors.• (5) M. Schulz-Weiling, H. Sadeghi, J. Hung and E.R. Grant. On theevolution of the phase-space distributions of a non-spherical molecularultracold plasma in a supersonic beamJ. Phys. B: At. Mol. Opt. Phys. 49 193001 (2016)vPrefaceThis article is the expression of my multi-year e↵ort to provide a moresolid foundation to our supersonic beam apparatus, which led to mydiscussion of our system in lower geometry. I incorporated results frommy diploma thesis. The 3D hydrodynamic shell model computationis based on a 1D model by H. Sadeghi for a spherical plasma. Myco-authors provided editing.• (6) R. Haenel, M. Schulz-Weiling, J. Sous, H. Sadeghi, M. Agigh, L.Melo, J. S. Keller and E. R. Grant. Arrested relaxation in an isolatedmolecular ultracold plasmaAccepted for publication in Physics Review A (July 2017)Contributed central experimental results. Creation of manuscript byEd Grant. Contributed to data interpretation and provided editing.Chapter 4 in Part I of my thesis represents a literature review of plasmaphysics and has appeared in a similar form in my diploma thesis. Chapters5, 6 and 8 have appeared in comparable format in publication reference(5). The coupled-rate-equation model discussed in Chapter 7 was previouslydeveloped by J. Morrison and N. Saquet, J. Hung and H. Sadeghi. Mycolleagues were kind enough to share their code and allow me to developtheir method. Chapter 10 takes results from reference (3) and draws on ourdiscussion in reference (6). Chapter 11, likewise, borrows from the discussionin reference (6).viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . xxvList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . 1I Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Spectroscopy of nitric oxide . . . . . . . . . . . . . . . . . . . 83 Rydberg physics . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1 Rydberg molecules . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Dynamic processes in dense Rydberg gases . . . . . . . . . . 16viiTable of Contents3.3 Field ionization of Rydberg molecules . . . . . . . . . . . . . 204 Plasma theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.1 Introduction to plasma theory . . . . . . . . . . . . . . . . . 234.2 Dynamics in two-species neutral plasmas . . . . . . . . . . . 364.2.1 Moments of the Boltzmann equation . . . . . . . . . 374.2.2 Quasineutrality . . . . . . . . . . . . . . . . . . . . . 394.2.3 Equilibration . . . . . . . . . . . . . . . . . . . . . . . 404.2.4 Global energy . . . . . . . . . . . . . . . . . . . . . . 424.3 Plasma theory applied to UNPs . . . . . . . . . . . . . . . . 444.3.1 Kinetic theory in molecular ultracold plasma . . . . . 444.3.2 Correlation e↵ects . . . . . . . . . . . . . . . . . . . . 474.3.3 Disorder induced heating . . . . . . . . . . . . . . . . 48II Formation of a molecular Rydberg ensemble . . . . . 505 Molecular Beam Gas Dynamics . . . . . . . . . . . . . . . . . 535.1 Experimental setup overview . . . . . . . . . . . . . . . . . . 535.2 The supersonic beam . . . . . . . . . . . . . . . . . . . . . . 545.3 Gas properties . . . . . . . . . . . . . . . . . . . . . . . . . . 565.4 Continuum free-jet expansion . . . . . . . . . . . . . . . . . . 575.5 The method of characteristics . . . . . . . . . . . . . . . . . 605.6 Non-equilibrium e↵ects . . . . . . . . . . . . . . . . . . . . . 615.7 Sudden freeze and point-source approximation . . . . . . . . 675.8 Phase-space distribution of nitric oxide . . . . . . . . . . . . 736 The Rydberg ensemble . . . . . . . . . . . . . . . . . . . . . . 766.1 Excitation process . . . . . . . . . . . . . . . . . . . . . . . . 766.2 Phase-space distribution of the Rydberg ensemble . . . . . . 81III Theory work . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 Early-time dynamics in an UNP . . . . . . . . . . . . . . . . 88viiiTable of Contents7.1 Plasma evolution from a Rydberg gas . . . . . . . . . . . . . 897.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 928 Long-time dynamics in an UNP . . . . . . . . . . . . . . . . 1008.1 Analytic solution to plasma expansion . . . . . . . . . . . . . 1008.2 Expansion under the influence of dissociative recombination 1028.3 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . 106IV Experimental work . . . . . . . . . . . . . . . . . . . . . . 1129 Plasma detection in 1D . . . . . . . . . . . . . . . . . . . . . . 1159.1 The ZEKE and moving-grid detectors . . . . . . . . . . . . . 1159.2 Observations and discussion . . . . . . . . . . . . . . . . . . 1179.2.1 Rydberg spectrum . . . . . . . . . . . . . . . . . . . . 1189.2.2 Plasma formation . . . . . . . . . . . . . . . . . . . . 1239.2.3 Plasma expansion and decay in 1D . . . . . . . . . . 1299.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 14110 Plasma detection in 3D . . . . . . . . . . . . . . . . . . . . . . 14410.1 The plasma-tv detector . . . . . . . . . . . . . . . . . . . . . 14410.2 Observations and discussion . . . . . . . . . . . . . . . . . . 14910.2.1 Plasma bifurcation . . . . . . . . . . . . . . . . . . . 14910.2.2 Ultracold plasma hydrodynamics in three dimensions 16010.2.3 Recent work . . . . . . . . . . . . . . . . . . . . . . . 16510.3 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 17111 Arrested relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 17511.1 Comparison of model calculations and experiment . . . . . . 17511.2 The inadequacy of ’classic’ models . . . . . . . . . . . . . . 18111.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 183Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . 189Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192ixTable of ContentsAppendicesA Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202A.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202A.2 Laser and optics . . . . . . . . . . . . . . . . . . . . . . . . . 204A.2.1 Pump- and dye-lasers . . . . . . . . . . . . . . . . . . 204A.2.2 Frequency-doubling and colour filtering . . . . . . . . 205A.2.3 Additional optics . . . . . . . . . . . . . . . . . . . . 205A.3 The vacuum chamber . . . . . . . . . . . . . . . . . . . . . . 206A.3.1 Chamber dimensions . . . . . . . . . . . . . . . . . . 206A.3.2 Mu-metal shielding . . . . . . . . . . . . . . . . . . . 207A.3.3 Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . 207A.3.4 The supersonic beam . . . . . . . . . . . . . . . . . . 207A.4 The PlasmaTV detector . . . . . . . . . . . . . . . . . . . . . 208A.4.1 The chevron detector . . . . . . . . . . . . . . . . . . 208A.4.2 The lens element . . . . . . . . . . . . . . . . . . . . . 208A.5 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209A.5.1 PlasmaTV voltage supply and output coupler . . . . 209A.5.2 Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . 210A.5.3 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . 210A.5.4 Camera . . . . . . . . . . . . . . . . . . . . . . . . . . 211B Computation source-code . . . . . . . . . . . . . . . . . . . . . 212B.1 Early-time dynamics model . . . . . . . . . . . . . . . . . . . 212B.2 Long-time dynamics model . . . . . . . . . . . . . . . . . . . 219B.3 Simulate detector response to 3D Gaussian ellipsoid . . . . . 224xList of Tables3.1 Some general properties of Rydberg atoms . . . . . . . . . . . 153.2 Predissociation rates for NO (atomic units) . . . . . . . . . . 173.3 Predissociation lifetimes ⌧n for NO . . . . . . . . . . . . . . . 185.1 State properties of the seeded supersonic expansion stagna-tion mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.1 Fractional population in F1 rotational states of 2⇧ NO as afunction of rotational temperature in the molecular beam . . 776.2 Initial conditions imposed on Rydberg molecule position andmomentum based on underlying molecular beam and exci-tation scheme. Below values in combination with equation6.10 give a full account of the phase-space distribution of ourRydberg ensemble. . . . . . . . . . . . . . . . . . . . . . . . . 837.1 Initial conditions imposed on plasma particle position andmomentum based on underlying molecular beam and excita-tion scheme. Below values in combination with equation 7.11give a full account of the phase-space distribution of our plasma. 999.1 This table details typical voltages applied to the componentsof the ZEKE detector - see Figure 9.1 . . . . . . . . . . . . . 1159.2 Possible angular momentum coupling for Rydberg states pop-ulated from the ground rovibronic level of the NO A-state. . . 12210.1 This table details typical voltages applied to the componentsof the PlasmaTV detector - see Figures 10.1, c and 10.2 . . . 149xiList of Tables10.2 Experimental measures of FWHM in x, y and z dimensionsfor excited volumes striking the imaging detector followingshort (unprimed) and long (primed) flight times, as derivedfrom Gaussian fits to intensity distributions pictured in Fig-ure 10.6. All values are expressed in mm. Note that we candetermine unperturbed x0 = 10.25 and y0 = 3.3, from theshort flight path divergence of the beam, marked simply byits intersection with !1. We further predict z0 = 6.9 basedon the z-dimension thermal velocity of ’marked’ molecules aswell as !1-laser size. We convert the temporal widths in zand z0 to spatial widths using vbeam. . . . . . . . . . . . . . . 15411.1 Initial conditions corresponding to experimental data discussedin Figure 10.6 (left-hand side). . . . . . . . . . . . . . . . . . 17611.2 Plasma rise-time calculations for uniform density Rydbergsystems. Density in [1011 cm3]. . . . . . . . . . . . . . . . . 17611.3 I consider a spherical Rydberg volume with peak density, ⇢0 =2.7 · 1011 cm3, and FWHM of 3.2 mm. I model the systemthrough ten non-interacting shells with densities ⇢0, 0.9 · ⇢0,0.8 · rho0,...,0.1 · ⇢0. I calculate each shell centre position, x,and divide by the computed avalanche time for each shell, t(cf. Chapter 7) to find the velocity of the ’plasma-formationfront’, v. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186xiiList of Figures1.1 Overview of the occurrence of neutral plasmas - Someexamples for neutral plasmas are shown at their respectiveposition in the density-temperature diagram. The scaling isdouble-logarithmic. The  = 1 line separates thermal plas-mas from plasmas in the correlated regime. . . . . . . . . . . 22.1 Hund’s cases: Coupling of rotation and electronic mo-tion - J is the total angular momentum, N is the angularmomentum of the nuclear rotation. K is the total angularmomentum apart from spin. L is the total electronic orbitalangular momentum and ⇤ it’s projection on the molecularaxis. S is the total electronic spin and ⌃ it’s projection onthe molecular axis. In Hund’s case (d), N becomes R, a goodquantum number. (Figure (c) does not show the coupling ofS and K to J.) Credit: J.P. Morrison . . . . . . . . . . . . . . 92.2 NO molecular orbital diagram - Only the open shell elec-tron levels are displayed. The excitation pathway for !1 pho-tons is indicated. . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 NO level diagram for X, A and Ry-states and tran-sitions A X, Ry A. - Labeling information for energylevels and transitions are found in the text. . . . . . . . . . . 133.1 Classical field ionization of a Rydberg atom . . . . . . 214.1 Concept of circular logic on which plasma theory isbased . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24xiiiList of Figures4.2 Collision in phase-space - Two particles collide in one-dimensional phase-space. t1, t2, t3 are subsequent time steps.As a result of the collision, both particles assume new trajec-tories which can be seen as the annihilation of the old andthe creation of a new respective trajectory. . . . . . . . . . . 324.3 Di↵erential scattering cross-section for large and smalldeflections - Large-angle scattering events only occur for im-pact parameters smaller than b⇡/2, i.e. for initial trajectoriescrossing trough the inner circle area. . . . . . . . . . . . . . . 345.1 Bird-view schematic of our experiment table . . . . . 545.2 Continuum free-jet expansion - Left: A schematic sketchshowing the regions of an unobstructed supersonic gas ex-pansion from a converging nozzle (adapted from: [83]). Right:Schematic of our high vacuum chamber with indicated supersonicmolecular beam. . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3 Thermodynamic values for the hydrodynamic flow -Red line = u/umax; blue line = T/T0; green line = ⇢/⇢0;yellow line = P/P0 (or from top to bottom) . . . . . . . . . . 625.4 Two-body collision number - Mean-number of collisionsexperienced by a molecule at point z as it travels from z toinfinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.5 Rotational temperature of nitric oxcide - The upper-most curve shows data from a typical 1+1’ REMPI scan ofthe transition spectrum between the 2⇧1/2 ground state andthe A2⌃+ first excited state in NO. The data was recorded byintersecting the molecular beam with a tuneable laser, 7.5 cmdownstream of the skimmer, and accelerating freed electronstowards a MCP detector. The inverted plots beneath aretheoretical predictions for above transition spectra for di↵er-ent rotational populations of the molecular states. For bettercomparison, three theoretical predictions with temperaturesof 2.5 K, 5 K and 10 K are given. . . . . . . . . . . . . . . . . 66xivList of Figures5.6 Geometry, used in the calculation of F(r, l) . . . . . . . 705.7 Top: Experimental setup for the imaging of the molecular jet.The jet enters the experiment chamber through the skimmerand impacts on a MCP stack after a free-flight of 468mm.Free electrons are created, multiplied and viewed as photonsby a CCD camera. Bottom-Left: CCD response of molecu-lar beam impacting on phosphor screen detector (at a distanceof 468mm from the skimmer) without laser light present;Bottom-Right: Vertical summation over image pixel andfit, fitted Gaussian width is 4.35mm. . . . . . . . . . . . . . . 725.8 Left: Random distribution of 10,000 particles in an area cor-responding to the skimmer orifice. Each particle has x and yvelocity components sampled from a Mexwell-Boltzmann dis-tribution of 0.6 K. Middle: The same distribution evolvedover a flight-time corresponding to the distance skimmer-detector. Each particle-trajectory originates from the particlepoint of origin within the red circle, i.e. all trajectories areoutward. Right: Radial plot and fit of evolved density dis-tribution (middle). I obtain a Gaussian width of ⇠4.53 mm(cf. Figure 5.7). . . . . . . . . . . . . . . . . . . . . . . . . . . 746.1 Creation of a NO Rydberg ellipsoid - The gas mixtureof NO and He exits the nozzle and cools as it expands intothe vacuum. Only the coldest part of the beam passes theskimmer. This part slightly expands as it continues, which isdue to the remaining thermal motion of the gas particles. It isoverlapped by two counter-propagating lasers that create anellipsoid of NO Rydberg molecules which continues to travelwith the molecular beam. . . . . . . . . . . . . . . . . . . . . 78xvList of Figures6.2 Plot of peak density vs. Gaussian beam width as afunction of beam temperature for a system with 2.5 cm nozzle-skimmer distance and at a point 7.5 cm downstream fromthe skimmer. For a temperature of 0.7 K, the graph yields adensity of 1.62 ·1014 cm3 NO molecules at a Gaussian widthof 750 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.1 Density distribution over n-levels - The top-left figureshows the n-level distribution after the Penning process for⇢0Ry = 1 · 1012 cm3 and n0 = 50. State n=50 has been cuto↵, its value being ⇢n=50(t = 0) = 0.67 · 1012 cm3. . . . . . . 947.2 Evolution of electron temperature and density - Thefigures show the identical dataset as Figure 7.1, with the left-hand figures displaying an expanded timescale. The densityplots show electron, total Rydberg and deactivated neutraldensities (blue, green, red). . . . . . . . . . . . . . . . . . . . 957.3 Electron density and temperature evolution for statesn=40, 50, 60 and initial density ⇢Ry = 1, 0.9, 0.8,...,0.1·1012cm3 (for each figure top to bottom). . . . . . . . . . . . . . 967.4 Plasma avalanche times for states n=40, 50, 60 (blue,green, red or top to bottom) and varying initial density ⇢0Ry. . 988.1 Illustration of plasma in the shell picture . . . . . . . . 1028.2 Plasma density along the z-axis for a spherical system forinitial electron temperatures of Te = 5, 20, 100 K. Each fig-ure shows the early evolution of plasma for time steps of 0,1, 2 and 3 µs (top to bottom curves). The top figures dis-play simulation results for a system without DR, the bottomsimulations include DR. . . . . . . . . . . . . . . . . . . . . . 1078.3 Shell positions and shell velocities of shells with initialradii of 1 and 3 with (red) and without (blue) DR in anultracold plasma with Te = 20 K plasma. . . . . . . . . . . . 108xviList of Figures8.4 Plasma density along the z-axis for a spherical system with-out DR at time t = 3 µs. Upper curves: without initial hy-drodynamic expansion. Lower curves: with k(0) = 0.0193µs1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098.5 Shell position of 50 K electron temperature plasma withinitial widths of x = 750 µm and y = z = 200 µm. Left:shells with initial radii of 1 and 3 along x and y/z axesfor the self-similar case. (the locus of circles underlying theshell-model result plot values obtained using Eq. (8.7)) Rightbottom: evolution of 1 shells along x and y/z axes for adissociating system. Right top: evolution of 3 shells alongx and y/z axes for a dissociating system. . . . . . . . . . . . . 1109.1 Schematic of our 1D detection scheme - The molecularbeam enters from the left along the center-line of the detector.It passes through a fine grid plate G1. As the jet traversesthe distance between plates G1 and G2, it is intersected witha nano-second laser pulse. The excited volume continues totravel with the molecular beam to reach detection grid G2.G1 and G2 are grounded, G3 holds a positive voltage. Upontransit through grid G2, the system encounters an electricfield that accelerates negatively charged particles towards aMCP detector. The charge impact is amplified and read outvia a detector anode. . . . . . . . . . . . . . . . . . . . . . . 1169.2 E-drawing of the moving-grid detector - The detectoris mounted on the 10” diameter backflange of our vacuumchamber. FMP1 and FMP2 represent front-mounting plate1 and 2, respectively. Detection is analogous to the ZEKEscheme in Figure 9.1. The entire detection carriage, includingG2, FMP2, MCPs and anode can be translated along the z-dimension, thus varying the time after which the laser excitedvolume transits detection grid G2. . . . . . . . . . . . . . . . 117xviiList of Figures9.3 Typical plasma signal observed by ZEKE or moving-grid detector - Laser excitation take place in the field-freeregion between grids G1 and G2. We excite to n0=50. Thefirst peak at ⇠ 200 ns represents prompt electrons that arecreated during the plasma formation process. The secondpeak at ⇠ 8 µs gauges the charge density and the spatialdimensions of the plasma volume as it traverses grid G2. . . . 1199.4 Resonance in the plasma signal observed scanning !2 -The (late peak) plasma signal is averaged over one second andthen integrated to yield an intensity measure. This signal isrecorded as !2 is scanned with a step-size of 0.001 nm. Theassigned peaks correspond to the nf -Rydberg series whichconverges to the rotational level, N+ = 2, in the 1⌃+ vi-brational ground state of NO+. The ionization potential isexpected to be at ⇠ 327.58 nm. . . . . . . . . . . . . . . . . . 1209.5 Field ionization by a linear voltage ramp - Our Ryd-berg ensemble is prepared at n0=50. We measure the systemresponse to a linear voltage ramp (forward-bias for electrons)between G1 and G2. The field is raised from 0 to ⇠200 V/cmin ⇠400 ns. Each figure displays 3000 single-measurements ina contour plot, sorted according to total electron yield. Theapplication of the ramp is systematically delayed by steps of⇠200 ns. Overall, we observe the transition of Rydberg gasto plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259.6 Summation and binning of ramp-delay data - Eachcurve represents one of the ramp delays. I integrate overeach of the 3000 oscilloscope traces and sort the resulting to-tal into bins. The 10ns-delay data (blue) appears closest to anormal distribution while 200, 400, ..., 1000 ns delays (green,red, ..., yellow) move progressively closer to low count. Thisprogression illustrates the second-order plasma decay due toDR. The top-right corner bar plot illustrates the total loss ofsignal over the first microsecond evolution. . . . . . . . . . . . 128xviiiList of Figures9.7 Evolution of the plasma peak - Plasma waveform cap-tured by the moving-grid detector as a function of flight time.The system is prepared in the n0=50 Rydberg state. G1holds a +2V positive charge to deflect the prompt peak (lineat t=0). The di↵erent traces are vertically o↵set for betterclarity. The Figure shows how the plasma expand and decay,as it evolves in time. . . . . . . . . . . . . . . . . . . . . . . . 1299.8 Plasma evolution characterized through Gaussian fits- We characterize our dataset through Gaussian fitting pa-rameter: width  (top), area A (middle), peak hight I0 (bot-tom). Displayed error bars represent 95% confidence bounds. 1319.9 Comparison of experiment and simulation results: ex-perimental data points as in Figure9.8; self-similar non-sphericalVlasov curve (cf. Eq. (8.7)) for initial conditions from table7.1 and 1.9 K electron temperature. . . . . . . . . . . . . . . 1339.10 Our detection scheme: As our plasma volume transits thedetection grid, a forward bias extracts electrons and acceler-ates them onto the MCP. Our oscilloscope trace, as collectedby a metal anode, is one-dimensional and represents the flowof electrons through the x-y-plane at the detector entrance grid.1349.11 Detector response to a unitary plasma sphere . . . . . 1359.12 Density evolution and corresponding detector responseover the first µs: I have simulated our system using myshell model formalism and initial condition based on Table7.1. The initial electron temperature was set to 2 K. Abovefigures show the evolution in charged particle density along z.Figures below correspond to the expected detector response(Gaussian fits included). . . . . . . . . . . . . . . . . . . . . . 137xixList of Figures9.13 Comparison of experiment and simulation results: Topto bottom - detection simulation for dissociating plasma withinitial conditions from table 7.1 and 15, 7.5 and 1.9 K initialelectron temperature. The black curve represents 3D shellmodel data without DR at 1.9 K electron temperature. Itserves as proof of accuracy for the computation. Experimen-tal data points taken from Figure 9.8. . . . . . . . . . . . . . 1399.14 Comparison of experiment and simulation results -Computation results for Te(0)=1.9 K and densities ⇢(0) =1 · 1010, 1 · 1011, 1 · 1012 cm3 (green, blue, red) compared toexperimental data from Figure 9.8. Top: Evolution of distri-bution width along z. Bottom: Evolution of area measuredalong z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14010.1 Schematic of plasma-tv detector in di↵erent configu-rations - The BURLE detector is mounted at the back ofour vacuum chamber. It consists of a set of MCPs and afiberoptic phosphor screen. The trajectory of our skimmedmolecular beam is normal to the plane of detection (MCPs).We record timing and x/y position of particle impact on theMCPs. In configuration a), the distance d between laser andMCPs is 453 mm. Configuration b) extends this distance tod=732 mm. Configuration c) shortens the free flight distanceof the illuminated volume to d<21 mm. Here, detection gridG2 extracts spatial information of plasma electrons (see text). 14610.2 Schematic of plasma-tv detector in configuration c) -This figure shows an e-drawing of our electronic lens system.The molecular beam enters from the left. Vacuum chamberviewports allow for laser access between plates G1 and G2.The illuminated volume transits G2 and loosely bound elec-trons are accelerated towards the MCP detector at the endof the chamber. . . . . . . . . . . . . . . . . . . . . . . . . . 147xxList of Figures10.3 Charged particle trajectories within electrostatic lensarray - This figure was created with the aid of SimIon. Itshows charged particle trajectories in real-space (top) as wellas potential energy space (bottom) for detector voltages inTable 10.1. The detector is set to project a magnified realimage of the charge distribution transiting G2 onto the MCPs.14810.4 Example image PlasmaTV detector - The left-hand im-age shows the phosphor screen as seen by our CCD camera,mounted in the back of the vacuum chamber. In the right-hand image, we’ve placed a blackout around CCD cameraand detector. The image shows the lighting-up of the phos-phor screen, as we create a free electron flux through grid G2.Magnification amounts to a factor of ⇠25. . . . . . . . . . . . 15010.5 Detector response recorded after preparing the Ry-dberg gas volumes with the following set of approximateinitial densities and selected initial principal quantum num-ber: (left) 3⇥ 1011 cm3 and n0 = 32, (centre) 3⇥ 1011 cm3and n0 = 65, (right) 7 ⇥ 1011 cm3 and n0 = 78. Here,we have positioned the detector at the longer flight distancemeasured as 747 mm from the skimmer wall, yielding a flighttime in all cases of 420 µs after laser excitation. All figuresrepresent averages over 250 CCD images. . . . . . . . . . . . 151xxiList of Figures10.6 Distributions of electron density in x, y and z followingexcitation to Rydberg gasses with initial principal quantumnumbers of n0 = 32 and n0 = 65 and propagation over thedistances indicated from the point of laser interaction to thedetector plane. (top) Distribution of the electron density overthe horizontal cross-beam coordinate x, integrated in y and z.(centre) Distribution of the electron density over the verticalcross-beam coordinate y, integrated in x and z. (bottom)Distribution of the electron density over the molecular beampropagation coordinate z, integrated in x and y, displayedas the waveform of the detector anode signal as a functionof time. Smooth red curves represent Gaussian (y and z) orsum of Gaussian (x) fits. . . . . . . . . . . . . . . . . . . . . 15210.7 Normalized distributions of y, z integrated electrondensity as function of x, collected using a an !1 pulse energyof 1.75 µJ with long flight path for selected initial principalquantum numbers, n0f(2), in the range from 78 to 28. . . . 15510.8 x, y plasma images collected using a long flight path witha single initial n0f(2) principal quantum number of 58 andvarying laser pulse energies from 1.75 µJ to 4.24 µJ. . . . . . 15510.9 Top: Relative velocity, v˘x, with which mesoscopic volumeelements of the plasma charge distribution separate along x asa function of Rydberg gas initial principal quantum numberat constant density of 3 ⇥ 1011 cm3, and as a function ofRydberg gas density at a constant principal quantum numberof n0 = 58. Bottom: Penning fraction as a function ofRydberg gas initial principal quantum number at constantdensity of 3 ⇥ 1011 cm3, and as a function of Rydberg gasdensity at a constant principal quantum number of n0 = 58,as determined by Eq. (10.1). . . . . . . . . . . . . . . . . . . 158xxiiList of Figures10.10Relative velocity, v˘x, with which lobes of the chargedistributions separate along x as a function of Penningelectron density, for plasmas formed both at constant !1 pulseenergy with varying principal quantum number, n0 (circles)and at constant quantum number with varying !1 pulse en-ergy (squares). . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.11Left: Self-similar expansion of Gaussian ultracold plasmas:(grey lines) Ambipolar expansion of a model Gaussian plasmacore ellipsoid of cold ions and Te = 180 K electrons withy(0) = z(0) = 83 µm and x(0) = 250 µm. Ions rapidlyattain ballistic velocities, @ty = @tz = 272 m s1 and@tx = 132 m s1. (blue line with red data points) Experi-mental measure of @tz(t) fit by Vlasov model for a Gaussianspherical expansion for Te = 5 K. Right: Electron signal asa function of flight time to G2 observed in the moving gridapparatus diagrammed in Figure 9.2 with a constant reversebias on G1 of 1.20, 0.60, 0.48 and 0.24 V (top row), and for-ward bias of 0.0, 0.12, 0.24 and 0.36 V (bottom row),all with a flight distance to G2 of 56 mm. The Rydberg gaswas prepared at n0=50. . . . . . . . . . . . . . . . . . . . . . 16110.12Detector response in configuration a) - Our system isprepared at n=64. MCP input is at +7 V, MCP output at1.6 kV and Screen at 3.5 kV. We record 100 single imagesand corresponding oscilloscope traces. Top: Averaged oscil-loscope traces. Bottom left: Averaged CCD image and defi-nition of top-corner and centre measurement areas. Bottomright: For each of the 100 images, I sum over the total pixelresponse of area B and sort the images accordingly. I thenintegrate pixel in the centre area A in the vertical and stackthe obtained traces. The obtained graph highlights the factthat the stronger signal B the stronger bifurcation in signal A. 167xxiiiList of Figures10.13Detector response in configuration c) - Our system isprepared at n0=57. Out of 100 single shot images, I displayexample captures of 3 families of CCD camera responses (1stline / 2nd line / 3er & 4th line). The bottom left oscilloscopetrace corresponds to the 3rd family. Discussion in text. . . . . 17010.14Time sequence of plasma bifurcation - At the core ofthe Rydberg ellipsoid, plasma self-assembles into a hot state,distinct from the surrounding outer layers. Ambipolar expan-sion quenches electron kinetic energy and accelerates the coreas a shock though the outer layers, causing bifurcation viaion-ion collisions and resonant charge exchange. . . . . . . . 17211.1 Self-similar ambipolar expansion of a 200 K plasma -Initial condition according to Table 11.1 with Te(t = 0) = 200K. Top: Evolution of plasma width according to self-similarmodel. Dimensions x and y,z are colour-coded blue and red.Bottom: Evolution of electron temperature. . . . . . . . . . . 17711.2 Extrapolation of ensemble expansion based on ex-perimental data from Table 10.2 - Dimensions x,y,z arecolour-coded blue, red, green. Dots represent experimentallymeasured distribution widths at 225 µs and 421 µs evolution.The lines represent fits of ambipolar expansion curves to therespective two data points in each dimensions. X marker plotthe expected thermal width-evolution of a non-interacting’marked’ expansion. . . . . . . . . . . . . . . . . . . . . . . . 179A.1 Birdview-schematic of the experimental setup . . . . . 203A.2 Schematic of the vacuum chamber . . . . . . . . . . . . 206A.3 Dimensionality of the PlasmaTV detector in configu-ration c) along the z-axis . . . . . . . . . . . . . . . . . . 209A.4 Layout of the PlasmaTV output coupler . . . . . . . . 210xxivList of AbbreviationsIn order of appearance:UNP ultracold neutral plasmaMOT magneto-optical trapDIH disorder-induced heatingUMP ultracold molecular plasmaNO nitric oxideN nitrogenO oxigenIP ionization potentialRD radiative decayPD predissociationPQN principal quantum numberTBR three body recombinationDR dissociative recombinationFI field ionizationHe heliumREMPI resonance enhanced multi-photon ionizationCCD charged coupled deviceMCP multichannel plateFWHM full width at half maximumRy RydbergODE ordinary di↵erential equationZEKE zero electron kinetic energyFMP front mounting plateSFI selective field ionizationITO indium tin xidegrnd. electrical groundICD interatomic (intermolecular) Coulombic decayxxvList of SymbolsOver the course of this thesis, I will often designate the three dimensions inspace by subscript k=x,y,z. Subscripts ↵ = i, e refer to ions and electrons,respectively. The boldface typeset indicates a vector. Due to the wealthof variables in this document, I use subcategories spectroscopic symbol andglobal symbol.Spectroscopic variables:L total orbital angular momentum` orbital angular momentumML projection of L on molecular axis⇤ absolute of ML⌃,⇧,, ... spectroscopic numbering,⇡, , ... spectroscopic numberings, p, d, ... spectroscopic numberingS total spins spinMS projection of S on molecular axis⌃ absolute of MSJ total angular momentum⌦ total electric angular momentumprojection on molecular axisN angular momentum of nucleusR angular momentum of nucleus in Hund’s case d)F1, F2 spectroscopic labelsP , p transition J/K = +1xxviList of SymbolsQ, q transition J/K = 0R, r transition J/K = -1! photon classifierB⌫ rotational constantGlobal variables: Coulomb coupling parametere electron (elementary charge)e Euler’s number⇡ mathematical constant ⇡✏0 Vacuum permitivityaws Wigner-Seitz radiuskB Boltzmann constantT Temperatureh Planck constant~ reduced Planck constantr nabla - vector di↵erential operatorM , m massµ reduced massx, y, z spatial dimensionsr position vectort timer, R radius wave functionW energyn principal quantum numberR1 Rydberg constantRy Rydberg energy` quantum defectk rate constant⌧ life timea0 Bohr radiusxxviiList of Symbols⇢ DensityE electric fieldP pressureu mean velocityv velocity electric potentialQ, q electric charge(r) delta functionD Debye lengthf particle distribution functionF forceN particle numberb impact parameter˜ cross sectionF flux[P ] stress tensor hydrodynamic velocity parameter Gaussian widthCp heat capacity at const. pressureR gas constanth heat capacity ratiog gravitational constanth enthalpy per unit massa isentropic sound speedM mach numberD nozzle diameterZ collision ratef˜ probability distribution total flowl lengthZ˜r partition functionP˜ powerI intensityxxviiiList of SymbolsE(r) Erlang distribution functionfP Penning fractionCcut cut-o↵ condition width, change in a variable# numberV volumeG1, G2, G3 grids within experimental apparatus holding potentialP1, P2 plates within experimental apparatus holding potentialC1, C2, C3 cyllinders within experimental apparatus holding potentialxxixAcknowledgementsI would like to express my profound gratitude to Professor Edward Grant, forhaving been much more than my research supervisor, but a mentor through-out my journey at UBC.I further like to acknowledge the help of all whom I had the privilegeof calling colleagues: visiting professor James Keller; post doctoral fel-lows Chris Rennick, Nicolas Saquet and Josh Gurian; my fellow studentsJonathan Morrison, Hossein Sadeghi, Mahyad Aghigh, Rommel Oliveira,Luke Melo, Jaimie Ortega-Arroyo, Paul Godin, Julian Yiu, Anneke Kruyen,Jachin Hung, Chelsea Cayabab and Rafael Haenel.A special thanks to the Department of Chemistry Electronics and Ma-chine Shops for all their help.Thank you all for six happy and fulfilled years in Vancouver!xxxDedicationTo my parents.xxxiChapter 1General introductionPlasma, also referred to as the fourth aggregate state, is by far the mostcommon state of visible matter in the universe. It exists in hot surroundingsand is created by ionizing collisions between particles or by high energyradiation fields in interstellar space.The field of plasma physics was founded in the early 20th century whenisolated researchers developed an interest in ionized gas phenomena. Earlyresearch focused on the e↵ects of ionospheric plasma on long-range short-wave radio propagation and gaseous electron tubes in the pre-semiconductorera of electronics. In the 1950s, plasma physics became a central field ofinterest in the large-scale research e↵orts to produce fusion energy. Thisgoal has been pursued ever since, nowadays in multi-national research facili-ties like ITER (International Thermonuclear Experimental Reactor). Otherimportant research e↵orts that build on the field of plasma physics includesolar- and astrophysics, as well as combustion engine and propulsion technol-ogy. Plasmas appear in technical applications such as metal manufacturing,surface treatments, lithography and plasma display panels. In the near fu-ture, advances in medical research could replace the unbeloved dentist drillby a painless plasma plume [45]. Because of their great importance in as-trophysics and fundamental sciences, as well as modern technology, plasmashave for some time been of great interest to researchers.Two central quantities by which plasmas can be characterized are tem-perature and density. This is illustrated in Figure 1.1, where a variety ofnatural and artificial plasmas are displayed. A distinction can be made be-tween thermal plasmas ( < 1) and Coulomb correlated plasmas ( > 1),where  denotes the so called Coulomb coupling parameter, which is1Chapter 1. General introductionFigure 1.1: Overview of the occurrence of neutral plasmas - Someexamples for neutral plasmas are shown at their respective position in thedensity-temperature diagram. The scaling is double-logarithmic. The  = 1line separates thermal plasmas from plasmas in the correlated regime.determined by the dimensionless ratio =e2/4⇡✏0awskBT.Here, e is the electron charge, and aws is the Wigner-Seitz radius, whichrelates to the particle density ⇢ by 4/3⇡a3ws = 1/⇢. The condition  > 1refers to a change in plasma behaviour, when densities and temperatures2Chapter 1. General introductionof the charged particles reach regimes where the average electrostatic po-tential energy exceeds the average thermal energy and correlation e↵ectsoccur. For example, plasmas with  > 1 can form liquid-like states andnegative-pressure environments [54, 56], and highly correlated plasmas at ⇡ 180 have even been predicted to undergo phase transitions to formbody-centered-cubic crystals [25], which was later experimentally observed[16]. E↵ects like these presumably occur to some degree in the core ofJovian-type planets, stars and laser implosion experiments [39].For some time now, researchers have tried to achieve high correlation inexperimental plasma systems. Systems with solid-like particle distributionsare impractical, as their ionization, while remaining in the correlated tem-perature frame, is dicult to achieve. Due to this, experimentalists focusedtheir attention on cold, low-density gases. One of the earliest approachesto experimentally study plasma in the correlated regime was in a cryogenicafterglow experiment at an initial gas temperature of 4.2 K, electron tem-perature of 10 K and density of ⇠ 1024 m3 [21].More recently, a new means of creating and studying ultracold neutralplasma (UNP) in a laboratory was introduced by Rolston and coworkersin an experiment at NIST in 1999 [42]. Rolston’s experiment combined thecooling power of a magneto-optical trap (MOT) with a narrow-band laserionization to create a plasma, which was colder than any formerly observed.His experiment demonstrated that a sample of metastable xenon atoms,held at a density as high as 2 ·109 cm3 and cooled to ⇠ 10 µK, when photoionized, would subsequently evolved to form a plasma. This plasma has thespecial feature that, because of the very slow ion motion, the initial amountof electron kinetic energy is only determined by the tuning of the ionizinglaser, i.e. by the di↵erence between photon energy and binding potential.It is therefore not necessary to heat the whole system to achieve collisionalionization, as the narrow-band laser deposits energy right into the free elec-tron kinetic energy bath. While the newly formed ions carry on with themomentum of the former neutrals, the electrons initially obtain only the3Chapter 1. General introductionexcess energy of the laser. Such systems, being created far from a state ofequilibrium, exhibit surprisingly long life times. In his experiment, Rolstonobserved a lifetime of ⇠ 100 µs and demonstrated that initial plasma elec-tron temperatures as low as 10 mK were achievable. This is considerablylower than any cold plasma occurring in nature. This fact, together withthe method of creation, led Rolston to name his system an ’ultracold neutralplasma’.Subsequent experiments extended this method to a variety of di↵erentatoms, such as strontium [55], rubidium [88], calcium [18] and caesium [69].Latter experiment with caesium also showed that ultracold plasmas can formspontaneously from dense ultracold Rydberg ensombles.UNPs facilitate ideal tabletop experiments for the study of many-bodysystems with strong inter-particle forces. For example, studying these sys-tems can help astro-physicists to better understand the dynamics of star-clusters [17]. Naturally, Coulomb correlation became a central topic in UNPresearch.The goal of achieving high orders of correlation in tabletop experiments,however, received a set-back with the discovery of disorder-induced heat-ing (DIH) in UNPs [44]. DIH takes place in the first moments after plasmaformation. It describes the release of potential energy of the spatially uncor-related ion distribution into thermal motion, thus limiting ion equilibriumtemperatures to several degree Kelvin [43].In 2008, our research group demonstrated that it is possible to create anultracold molecular plasma (UMP) from a⇠ 1 K cold molecular Rydberggas of nitric oxide [50]. We form this ensemble by intersecting a supersonicmolecular beam with laser light, driving resonant transitions in nitric oxideto populate a high-Rydberg state. This Rydberg ensemble spontaneouslytransitions to plasma. We named our system ultracold in reference to thefield of ultracold neutral plasmas. Our experiment showed that, by utilizingthe beam method, it is possible to create ultracold plasma with initial den-4Chapter 1. General introductionsities three orders of magnitude larger than those created in MOT systems.Higher densities yield, as seen in the definition of the Coulomb couplingparameter, , higher correlations. Additionally, our approach enables us toperform research on many-body molecular plasma systems, which is scarcelypossible in MOT systems. Here, new processes like two-body electron-cationdissociative recombination appear on the scene, which are able to contributerelaxation channels which do not exist in MOT plasmas [66, 71].The UMP we create in the laboratory shows surprising characteristics.We observe the spontaneous breaking of symmetry of the developing ensem-ble and the arrest of plasma relaxation dynamics with a subcomponent ofthe system, expressed through a lack in signal decay and thermally drivenexpansion. What are the fundamental microscopic mechanisms leading tothis unexpected behaviour? Could this be an e↵ect of Coulomb coupling? Iwill pursue these and other question over the course of this thesis.I have composed my thesis in four parts. Part I - Fundamentals, willreview nitric oxide spectroscopy as well as Rydberg and plasma theory -all three are central concepts in UMP physics. Part II - Formation of amolecular Rydberg ensemble, will introduce our experiment. I will put anemphasis on the discussion of the physics of supersonic beam systems, whichis central to characterizing the initial state of our system. This part will con-clude with a statement of the initial phase-space distribution function of ourRydberg ensemble. Part III - Theory work, will take a theoretical approachto discuss the internal conversion of Rydberg gas to plasma and its subse-quent expansion. Part IV - Experimental work, will introduce our meansof plasma detection and discuss our observations within the framework ofParts I through III.5Part IFundamentals6In this first part of my thesis, I will review some concepts fundamental toour field of research. In our experiment, we cool nitric oxide in a supersonicjet. Two-resonant photon transitions drive ground-state molecules into thehigh-Rydberg manifold. Internal conversion of the Rydberg ensemble formsan ultracold molecular plasma.Chapter 2 will discuss the spectroscopic properties of nitric oxide (NO).Following that, Chapter 3 reviews the fundamental features of Rydberg sys-tems and their interaction with external perturbers. Chapter 4 will intro-duce important concepts of plasma theory and discuss some of their implica-tions to UNP research. Subsequent parts of my thesis will frequently drawon the knowledge base derived over the following pages.7Chapter 2Spectroscopy of nitric oxideThe following is a brief introduction into the spectroscopic properties of ni-tric oxide.1 In our experiment, we excite NO from the X ground state viathe intermediate A-state into the high Rydberg manifold. I will limit thisdiscussion to these quantum states. Vibrations do not play a role in ourexperiment and are not discussed. Neither is the e↵ect of nuclear spin. Ifollow the notation in Herzberg [32].The motion of electrons in a diatomic molecule takes place within thecylindrical field of force set by the intermolecular axis. A precession of totalelectronic orbital angular momentum (without spin), L =Pi `i, takesplace about this axis. Only the projection of L on the molecular axis,with components ML(~/2⇡), is a constant of motion. ML can take valuesML = L,L  1, L  2, ...,L. Projection states di↵ering only in the signof ML have the same energy. Since L is not a good quantum number formolecules, the following notation has been established:⇤ = |ML| = 0, 1, 2, ..., L = ⌃,⇧,,, ... (2.1)The representative numbering of ⇤ through Greek capital letters is anal-ogous to the mode of designation for atoms. ⇧,,, ... states are doublydegenerate as ⇤ = | ±ML|. States ⇤ = ⌃ are non-degenerate.The spin states of the individual electrons form a total spin S =Pi si. In⌃ states and in absence of external fields or rotation, S is uncoupled from themolecular axis. For states ⇤ 6= 0, precession of L about the molecular axis1Excerpted from Herzberg’s Spectra of Diatomic Molecules [32], Brown et. al. Chem-istry: The Central Science [12] and Abraham’s review paper on nitric oxide spectroscopy[10].8Chapter 2. Spectroscopy of nitric oxideFigure 2.1: Hund’s cases: Coupling of rotation and electronic mo-tion - J is the total angular momentum, N is the angular momentum ofthe nuclear rotation. K is the total angular momentum apart from spin.L is the total electronic orbital angular momentum and ⇤ it’s projectionon the molecular axis. S is the total electronic spin and ⌃ it’s projectionon the molecular axis. In Hund’s case (d), N becomes R, a good quantumnumber. (Figure (c) does not show the coupling of S and K to J.) Credit:J.P. Morrisoncauses an internal magnetic field in the same direction. This in turn causesS to precess about the molecular axis with constant components MS(~/2⇡).The notation,⌃ =MS = S, S  1, S  2, ...,S (2.2)has been established. The total electronic angular momentum about theintermolecular axis is called ⌦. It is obtained by addition of ⇤ and ⌃.⌦ = |⇤+ ⌃| (2.3)Angular momentum caused by nuclear rotation of the molecular core isrepresented through quantum vector N. There is no quantum number asso-ciated with N. The total angular momentum of the system - combination ofelectron spin, electronic orbital angular momentum and nuclear rotation - isalways designated J. Di↵erent classifications for di↵erent modes of couplingfor J were first introduced by Hund. Figure 2.1 shows Hund’s cases (a), (b)and (d), which play a role in our experiment.9Chapter 2. Spectroscopy of nitric oxideHund’s case (a) assumes that the interaction of nuclear rotation withthe electronic motion ⌦ is weak. Thus, J, constant in magnitude and direc-tion, is a resultant formed by the nutation of ⌦ and N about J. A di↵erentway of understanding Hund’s case (a), is that the precession of L and Sabout the internuclear axis is much faster than above mentioned nutation.In Hund’s case (b), the electronic spin S is only very weakly coupledto the molecular axis. This is the case when ⇤ = 0 (absence of internalmagnetic field) or even if ⇤ 6= 0 in the case of particularly light molecules.Thus, ⌦ is not defined. Now, ⇤ (if nonzero) and N together form theresultant K.K = ⇤,⇤+ 1,⇤+ 2, ... (2.4)is the total angular momentum apart from spin. K and S together formresultant J.J = (K + S), (K + S  1), (K + S  2), ..., |K  S| (2.5)Hund’s case (d) refers to the case where the coupling between L andthe nuclear axis is very weak while that between L and the nuclear rotationis strong. This is usually the case for molecular high Rydberg states. Here,the angular momentum of nuclear rotation is called R (rather than N) andhas magnitudepR(R+ 1)~/2⇡. Possible values for R are 0, 1, 2, ... .Vector addition of R and L yield K, which can have values:K = (R+ L), (R+ L 1), (R+ L 2), ..., |R L| (2.6)The angular moments K and S together form total angular momentum J.In general (except for K<S), each level with a given K consist of 2S+1 sub-components.Nitric oxide has a total of fifteen electrons and forms an electronicground state in the (1)2(2)2(3)2(4)2(5)2(1⇡)4(2⇡)1 configuration, asshown in Figure 2.2. NO has eleven valence electrons but only the ⇡⇤2pxelectron is unpaired. Thus, total orbital angular momentum projection is10Chapter 2. Spectroscopy of nitric oxideFigure 2.2: NO molecular orbital diagram - Only the open shell electronlevels are displayed. The excitation pathway for !1 photons is indicated.⇤ = 1 and total spin is S = 1/2. The NO ground state is (mostly) Hund’scase (a). Spin-orbit interaction yields a multiplet ⌦ = ⇤ + ⌃ = 3/2, 1/2.The notation for the resulting states are:2⇧3/2 and2⇧1/2Figure 2.3 shows the NO level diagram and our excitation pathway. As typ-ical for Hund’s case (a), spin-orbit splitting between ⌦ = 3/2 and ⌦ = 1/2is large compared to the rotational spacing. As mentioned previously, levelswith ⇤ 6= 0 are double degenerate. Interactions between nuclear rotationand L causes this degeneracy to lift and is called ⇤-type doubling. As aresult, each J value splits into two components. Designations e and f labelthe rotationless parity as positive and negative, respectively.The first excited state in NO has the configuration (1)2(2)2(3)2(4)2(5)2(1⇡)4(6)1 and corresponds to a 2⌃ electronic state. Since ⇤ = 0, thisstate is Hund’s case (b). For S=1/2 the multiplicity is 2. Similar to abovecase, molecular rotation induces a small internal magnetic field which splits11Chapter 2. Spectroscopy of nitric oxidethe spin degeneracy. The spectroscopic labels F1(K) and F2(K) refer tocomponents with J=K+1/2 and J=K-1/2, respectively.For any electric dipole transition, the following selection rules apply:J = 0,±1, with the restriction J = 0) J = 0Parity: only pos⌦ negS = 0The first two rules are rigorous. A very cold (<1K) spectrum of the NO Xto A-state transitions consists of four lines:pQ11(1/2) : J“ = 1/2,K“ = 1, F1f ) J ‘ = 1/2,K‘ = 0, F1eqR11(1/2) : J“ = 1/2,K“ = 1, F1e) J ‘ = 3/2,K‘ = 1, F1eqQ21(1/2) : J“ = 1/2,K“ = 1, F1e) J ‘ = 1/2,K‘ = 1, F2frR21(1/2) : J“ = 1/2,K“ = 1, F1f ) J ‘ = 3/2,K‘ = 2, F2fTransitions are designated P, Q, R for changes in J of +1, 0 , -1; p, q, r forchanges in K. The subscripts 11 (or 21) label transitions to F1 (or F2) fromF1.In our experiment, we tune laser !1 to the qQ11(1/2) transition (⇠226nm)to populate the rovibronic ground-state of the NO A-state. Subsequentphoton absorption of !2 light populates the high-Rydberg manifold be-tween principal quantum numbers 35 to 80. The ionic core of such Ryd-berg molecules has configuration (1)2(2)2(3)2(4)2(5)2(1⇡)4 and cor-responds to 1⌃.12Chapter 2. Spectroscopy of nitric oxideFigure 2.3: NO level diagram for X, A and Ry-states and transitionsA X, Ry A. - Labeling information for energy levels and transitions arefound in the text.13Chapter 3Rydberg physicsThe physical description of highly excited valence electron of an atom ormolecule can be approximated through hydrogen wave functions. This ispossible because the electric field of the system core, as seen from greaterdistance, is shielded through the inner electrons such that it looks similarto that experienced by the electron in the hydrogen atom. Highly excitedsystems, for which this is the case, are referred to as Rydberg atoms ormolecules. They have special characteristics like an increased response toelectric or magnetic fields as well as long radiative lifetimes.3.1 Rydberg moleculesTo first order, a Rydberg system is identical with atomic hydrogen, whosetime-independent Schro¨dinger equation in the absence of external fields is:✓~2r22me e24⇡✏0r◆ (r, ✓,) = E (r, ✓,) (3.1)Here, r is the gradient operator, ~ is the reduced Plank constant ~ = h/2⇡,me is the rest mass of the electron, ✏0 is the vacuum permittivity and ris the Rydberg electron radial distance from the system core. Eq. 3.1 isbest solved in spherical coordinates (see for example [19]), yielding the wellknown hydrogen wave function  n`m and quantized energies:Wn = m2ee48✏20h21n2= Ryn2(3.2)Ry is the Rydberg unit of energy (1Ry = hcR1) and n is called the principalquantum number. The angular momentum quantum number ` = 0, 1, ..., n 1143.1. Rydberg moleculesin wave function  n`m determines the magnitude of angular momentum ofthe Rydberg electron and magnetic quantum number m = `, ..., ` it’s pro-jection (on an arbitrary z-axis).Only Rydberg systems with high values for n and ` are well representedby Eq. (3.1). Low angular momentum states, however, can penetrate andpolarize the electronic shell structure of the ionic core. Here, the Rydbergelectron experiences a larger positive force from the system core when it’swave function penetrates the inner electronic shells. Such states tend tobe closer bound and the Rydberg electron has reduced energy. To accountfor this, a parameter called quantum defect was introduced through whichan e↵ective quantum number, ⌫ ⌘ n  `, is defined. The quantum defectdepends on angular momentum `. For nitric oxide, ` takes values of s =1.21, p = 0.73, d = 0.05 and `>3  0.01 [11]. Thus, the Rydberg systemenergy spectrum becomes,WRydn =WIP Ry(n `)2 , (3.3)where WIP is the ionization threshold of the respective atom or molecule.Based on the solution of Eq. (3.1), many properties of Rydberg systemscan be derived. Table 3.1 summarizes some important trends with regard toprincipal quantum number scaling. A detailed discussion of Rydberg physicsis found in [28].Property n-dependenceBinding energy n2Energy between adjacent n states n3Orbital radius n2Geometric cross section n4Polarizability n7Dipole moment n2Radiative lifetime n3Table 3.1: Some general properties of Rydberg atoms153.2. Dynamic processes in dense Rydberg gases3.2 Dynamic processes in dense Rydberg gasesThe study of dynamic processes in dense Rydberg gasses has been a ’hottopic’ in our laboratory over the recent years. We were in particular in-terested in creating a model which could accurately describe the internalconversion of a dense Rydberg gas into a plasma. These e↵orts were firstled by my colleagues Nicolas Saquet and Jonathan Morrison [52, 77]. Theywere further developed by Jachin Hung and Hossein Sadeghi [38] and morerecently myself. The following shall serve as an introduction into the topic.Angular momentum mixingThe properties of Rydberg molecules like dipole moment and predissocia-tion lifetime are strongly dependent on the angular momentum state of themolecule. Rydberg molecules created within our experiment are subjectto a number of interactions with their environment. Prominent perturbersare stray-fields, black-body-radiation and collisions with neutrals, ions andelectrons. The result of these interactions is a statistical mixture of thenearly degenerate angular momentum-state manifolds. This greatly dilutesthe core-penetrating low-l angular momentum states and boosts Rydberglifetimes by typically an order of magnitude. For Rydberg states in NO,this e↵ect has been observed and studied by [11, 87]. For our system, weexpect l-mixing on a sub-nanosecond timescale.Lifetimes of Rydberg statesThere are two processes by which diatomic Rydberg molecules decay in theabsence of a collision partner, radiative decay and predissociation:M⇤ kRD!MX M⇤ kPD! A+B (3.4)M⇤ is the Rydberg molecule, MX is the molecule in the ground state andA/B are atomic systems. NO Rydberg states decay primarily by predissocia-tion [65]. Here, the Rydberg electron collides with the many-electron molec-ular core, leading to an internal transition of the molecule to an unstable163.2. Dynamic processes in dense Rydberg gasesexcited state. The molecule dissociates predominantly into neutral atoms.Excess energy goes into the motion of these fragments. The frequency ofsuch collisions scales with the Bohr frequency and core penetration dependsstrongly on electron angular momentum `. Murgu et. al. [53] have collectedthe `-dependent rate coecients for NO in atomic units:Table 3.2: Predissociation rates for NO (atomic units)ks 1.4 · 102/(2⇡n3)kp 4.6 · 102/(2⇡n3)kd 2.9 · 102/(2⇡n3)kf 1.2 · 103/(2⇡n3)k`>3 3 · 105/(2⇡n3)Here, I have set k`>3 = kg. As discussed previously, we expect angularmomentum mixing on a sub-nanosecond timescale. Assuming equipartitionof states over available quantum levels m` yields a rate constant for thepredissociation of NO Rydberg molecule with PQN n:kPD(n) =P`(2`+ 1)k`n24.13 · 1016s12⇡n3(3.5)The first term accounts for the statistical weight of `-levels. Using Equation(3.5), Table 3.3 computes several values for the predissociation lifetime,⌧n = 1/kPD(n), of NO Rydberg states.Penning ionizationPenning ionization is an ionization process between two neutral atoms ormolecules. Applied to our experiment, it refers to a dipole-dipole interactionbetween two neighbouring Rydberg molecules, which results in the ionizationof one partner and the de-excitation of the other to a lower energy state.It’s symbolic representation is:M⇤(n0, `) +M⇤(n0, `) !M+ +M⇤(n‘, `‘) + e (3.6)173.2. Dynamic processes in dense Rydberg gasesTable 3.3: Predissociation lifetimes ⌧n for NO⌧10 0.05 ns⌧20 1.54 ns⌧30 11.1 ns⌧40 44.1 ns⌧50 125 ns⌧60 286 ns⌧70 565 ns⌧80 1.00 µs⌧90 1.64 µsHere, M+ is the molecular ion and e a free electron. (n0, `) and (n‘, `‘)are principal quantum number and angular momentum for initial and finalstates, respectively.A theoretical study by Robicheaux [67] investigated the Penning processbetween two cold Rydberg atoms using a classical trajectory Monte Carlomethod. The simulation was conducted for a fully mixed `-manifold. Anionization probability of 90% within 800 Rydberg periods was found forcollision partners separated by less than 1.8 · 2n2a0. Here, a0 is the Bohrradius. He found that in such interaction the deactivated partner,M⇤(n‘, `‘),would transition to a distribution state n‘ / n5 and n‘max = n0/p2 due toenergy conservation.Charged particle collisions with Rydberg moleculesAs discussed above, dense Rydberg gasses can achieve a level of ionizationvia Penning interactions. This, in turn, leads to new dynamics between theremaining Rydberg states and Penning ions and electrons. The three mostprominent examples, valid for both atomic and molecular Rydberg systems,are:Impact ionizing collisions between a free electron and a Rydberg183.2. Dynamic processes in dense Rydberg gasessystem. In such collisions the free electron transfers kinetic energy to thebound Rydberg electron, leading to ionization. ni is the initial principalquantum number.M⇤(ni) + ekion!M+ + e + e (3.7)Three body recombination (TBR) is the reverse process of the electron-Rydberg ionizing collision. Here, two free electron and an ion interact, yield-ing to the capture of one electron by the ion. A new Rydberg state is formedand the excess energy is carried away by the second electron. nf is the finalprincipal quantum number.M+ + e + e kTBR! M⇤(nf ) + e (3.8)n-changing collisions between free electron and bound Rydberg states.In such interaction, the free electron either gains or looses energy, leavingthe Rydberg system behind in lower or higher n-state, respectively.M⇤(ni) + ekni,nf! M⇤(nf ) + e (3.9)A theoretical study of these three interactions was done by Mensbachand Keck [46] and more recently further refined by Pohl et. al. [64]. Thefollowing rate coecients kion, kTBR and kni,nf were obtained from fits toMonte Carlo simulation data:kion(ni) =11(Ry/kBT )1/2k0e✏i✏7/3i + 4.38✏1.72i + 1.32✏ikTBR(nf ) =11(Ry/kBT )1/2k0e✏f✏7/3f + 4.38✏1.72f + 1.32✏fn2f⇤3⇢ee✏fkni,nf = k0✏5/2i ✏3/2f✏5/2>e(✏i✏<) 22(✏> + 0.9)7/3+9/2✏5/2> ✏4/3!(3.10)193.3. Field ionization of Rydberg moleculesHere, k0 = e4/(kBTpmeRy), ✏i/f = Ry/n2i/fkBT , ✏ = |✏f  ✏i|, ✏> =max(✏i, ✏f ), ✏< = min(✏i, ✏f ) and ⇤ =ph2/2⇡mekBT is the thermal deBroglie wavelength. ⇢e is the electron density, me the electron mass and kBis the Boltzmann constant.There is one more important electron-Rydberg interaction that appliesonly to the case of molecular Rydberg states. An atomic Rydberg ion col-liding with a free electron cannot recombine (radiationless) in a two-bodyprocess due to energy conservation.2 Molecules, however, have more internaldegrees of freedom and are equipped to absorb energy. Thus, a molecular ioncan bind a free electron in a two-body collision. This typically leads to aninternal transition of the now neutral molecule to an unstable excited state.The result is molecular dissociation. This process is called dissociativerecombination (DR). For nitric oxide, it is represented by:NO+ + e kDR! N⇤ +O⇤ (3.11)Here, N⇤ and O⇤ are neutral nitrogen and oxygen in a possible state ofexcitation. The rate constant, kDR, has been thoroughly characterized byscattering theory calculations and experimental measurements [81]. We use:kDR = 6⇥ 106 T1/2epK · cm3s(3.12)3.3 Field ionization of Rydberg moleculesCharacteristic for Rydberg systems is the relatively loosely-bound valenceelectron, which is susceptible to field ionization for fields of typically a fewhundred Volts per cm. In a classical model, ionization occurs when anexternal electric field lowers the potential barrier of a bound electron enoughfor the electron to escape.The hydrogen potential along zˆ, for a system perturbed by an uniform2The cross-section for recombination accompanied by photon emission is negligible indense plasma.203.3. Field ionization of Rydberg moleculesFigure 3.1: Classical field ionization of a Rydberg atomexternal field E = Ezˆ, becomes:W (z) = ⇣z Eze where ⇣ = e24⇡✏0(3.13)This potential has a saddle point at zsp =p⇣/Ee with a reduced bindingenergyW (zsp) = 2p⇣Ee. When this energy reduction becomes larger thanthe classical binding energy, WB = Ry/n2, the system ionizes. Settingthese two energy values equal yields the value EFI for the field strengthrequired to field ionize Rydberg electron with principal quantum number n:EFI =W 2B4⇣e=Ry24⇣en4(3.14)Eq. 3.14 states the classical ionization limit for atomic hydrogen withoutconsidering the Stark e↵ect. The Stark e↵ect describes the shifting of energylevels as a consequence of external electric fields. For Rydberg systems,these shifts can be significant due to their large dipole moments. It turns213.3. Field ionization of Rydberg moleculesout that, other than expected, red, or down shifted, Stark states ionize nearthe classical ionization limit while blue, or upshifted, states only ionize athigher fields.[28] It is possible to express the increase in binding energy ofextreme red, high-n Stark states due to the linear stark e↵ect via WStarkB =p16/9WB. Thus, Eq. 3.14 becomes:EStarkFI =4Ry29⇣en4(3.15)I rewrite above equation in terms of binding energy, expressed in wavenum-bers, and field, expressed in V/cm, and obtain:WB[cm1] = 4.59 ·pE[V/cm] (3.16)22Chapter 4Plasma theoryThis chapter aims to give an introduction into plasma theory. One goal ofthis thesis is to obtain a formalism linking the microscopic properties of ourplasma system to macroscopic observables. We can only approximate ourreal system values and it is important to keep track of and understand all theassumptions that are necessary to derive a theoretical kinetic representationof our system.For this reason, the first part of this chapter will review some generalrelations in plasma theory. Important concepts like the Debye length andquasi-neutrality, as well as plasma kinetic theory will be introduced. Sec-tion 4.2 will derive a theory framework on which I will draw in Part III ofmy thesis to model our system. Section 4.3 examines the applicability of’classical’ plasma theory in UNPs.4.1 Introduction to plasma theory1It is important to note that not all ionized gases can be called a plasma,because there is always some degree of ionization in any gas.2 F. F. Chendefines a plasma as ”a quasineutral gas of charged and neutral particleswhich exhibits collective behaviour”.[15] To understand this definition, it isimportant to examine what is meant by ’collective behaviour’ and ’quasi-neutrality’.1For a more extensive account, it is referred to [7], [15] or [29].2The Saha equation gives the amount of ionization to be expected in a gas in thermalequilibrium. For ordinary air at room temperature the fractional ionization is ⇢ionized⇢neutral =10122 (see [15]).234.1. Introduction to plasma theoryFigure 4.1: Concept of circular logic on which plasma theory isbasedThe following analysis will show that plasma theory can be based on aconcept of circular logic. One can start by requiring quasi-neutrality and byapproximating the plasma as a fluid, and subsequently derive the conceptof ’Debye shielding’ based on those assumptions. Debye shielding in turncan be used to show that a plasma will act to remain in a quasineutral stateand that it can be described in a fluid approximation. I will investigate thismore closely in the following.Debye shieldingConsider a finite-temperature plasma consisting of a statistically large num-ber of electrons and ions, distributed uniformly and having temperatures,Te and Ti, respectively. Plasma species do not need to be in thermal equilib-rium with each other, they are, however, usually assumed to be equilibratedwithin one species. Each species, henceforth denoted as ↵, can be regardedas a fluid3 with density ⇢↵, pressure P↵ = ⇢↵kBT↵ and a mean velocity u↵.3The fluid approximation is a description in which particles move on streamlines asdetermined by an electric mean field and a pressure gradient. It only holds for a nearlycollisionless plasma where collisions may be neglected to first order. Origin and quality ofthis assumption will be discussed shortly.244.1. Introduction to plasma theoryThe fluid equation of motion for each species ism↵du↵dt= q↵E 1⇢↵rP↵, (4.1)where m↵ and q↵ are the particle mass and charge and E is the electricfield. The given plasma is now subjected to a ’very slow’ perturbation sothat the following assumptions hold: the inertial term d/dt is negligible asthe perturbation is ’very slow’; inductive electric fields are negligible so thatE ⇡ r, where  is the electric potential; the plasma species remain inthermal equilibrium. With this, Eq.(4.1) reduces to0 ⇡ ⇢↵q↵r kBT↵r⇢↵, (4.2)which can be solved to yield⇢↵ = ⇢↵0 · exp(q↵/kBT↵), (4.3)where ⇢↵0 is a constant. Suppose that the discussed perturbation origi-nates from a single additional test particle with charge, qT , that is slowlyinserted into the initially unperturbed, spatially uniform, neutral plasma(i.e. (t = 0) = 0). The origin of the coordinate system shall be incidentwith the test particle. Upon insertion, particles with the same charge as thetest particle will be slightly repelled and particles with the opposite chargeslightly attracted. These slight displacements will result in a small, non-zeropotential which will be determined by the superposition of the test parti-cle potential with the potential of the plasma particles that have moved.The response of plasma particles towards the test particle is called shieldingor screening, because the displacements reduce the e↵ectiveness of the testparticle field. The change in  is calculated using Poisson’s equation, wherethe cloud contribution is self-consistently represented by Eq.(4.3). Thus,Poisson’s equation is given by:r2(r) =  1✏0[qT (r) + qi⇢i(r) + qe⇢e(r)] (4.4)254.1. Introduction to plasma theoryThe perturbation due to a single test particle is assumed to be small so that|q↵|⌧ kBT↵, in which case Eq.(4.3) becomes ⇢↵/⇢↵0 = 1 q↵/kBT↵ andtherefore:r2(r) =  1✏0[qT (r) + (1 qe(r)kBTe)⇢e0qe + (1 qi(r)kBTi)⇢i0qi] (4.5)The assumption of initial neutrality means that ⇢e0qe+⇢i0qi = 0, and aboveequation reduces tor2(r) 12D(r) = qT✏0(r), (4.6)where the e↵ective Debye length is defined as12D=X↵12D,↵(4.7)and the species Debye length is:2D,↵ =✏0kBT↵⇢↵0q2↵(4.8)Equation (4.6) can be solved to give(r) =qT4⇡✏0rer/D , (4.9)which has the general form of a Yukawa potential but is in this contextcommonly referred to as Debye-Hu¨ckel potential. For r ⌧ D this potentialis identical to the potential of the test particle in vacuum. For r  Dthe potential (r) approaches zero because the test charge is completelyscreened by the surrounding charged particle cloud. With this, D can beseen as the spherical distance at which the shielding by the plasma cloudlessens the particles potential by the factor 1/e.It is important to realize that the test particle could be any given particlein the plasma and therefore, Eq.(4.9) is the time-averaged e↵ective potentialof any selected particle in the plasma system.264.1. Introduction to plasma theoryThis analysis is based on the assumption that there is a macroscopicallylarge number of shielding particles in the cloud. Therefore, one requiresthat the system is much larger than one Debye length and that there is astatistically large number of particles in each Debye sphere, i.e. one requiresthat43⇡3D · ⇢0  1. (4.10)Quasi-neutralityOne of the initial conditions required in the derivation of Debye shieldingwas the condition of quasi-neutrality. In the moment of ionization, everyplasma system is quasineutral. The above requirement means that the ionand electron densities remain approximately equal, i.e. that ⇢e ⇡ ⇢i =⇢ for all times. It is possible to show that quasi-neutrality is indeed agood approximation for systems with a Debye length much smaller than thesystem size.Consider an initially neutral plasma with temperature Te and calculatethe largest radius rmax of a sphere that could spontaneously become depletedof electrons due to electron thermal motion.4 Optimal depletion means thatthe electrons have a velocity distribution, such that they all leave the sphereradially and come to a rest on the sphere’s surface, before they are pulledback in by the positive field of the ionic charge. One way to calculate theenergy stored in this artificial system is by determining the energy in theelectrostatic field that is produced by the ions. This electrostatic energywas initially zero and only exists due to the developed charge separation. Itmust be equivalent to the work done by the electrons.The energy density of an electric field, E, is ✏0E2/2. The ion chargedensity in a sphere of radius r is Q = 4⇡⇢er3/3 and the electric field atradius r is Er = Q/4⇡✏0r2 = ⇢er/3✏0. The ionic electrostatic field energy in4In this calculation, ion motion is neglected and the ions are approximately representedby a uniform background charge. This is a valid approximation, because the electronmobility vastly exceeds that of the ions due to the large mass ratio.274.1. Introduction to plasma theorythe electron-depleted sphere of radius rmax is thus:W =Z rmax0✏0E2r24⇡r2dr = ⇡r5max2⇢2e245✏0(4.11)The initial electron thermal kinetic energy was:Wkinetic =32⇢kBTe · 43⇡r3max (4.12)Equating energies yieldsr2max = 45✏0kBTenee2or rmax ⇡ 7D. (4.13)This quick analysis shows that it is theoretically possible for a plasma oftemperature Te to evolve into a state where a volume 4/3⇡(7D)3 sponta-neously loses all its electrons. Since the initial conditions for this to happenare statistically extremely improbable, expected regions of non-neutralityare much smaller and unlikely to exceed one Debye length. Thus, an initiallyquasineutral plasma with a microscopic Debye length remains quasineutral.Kinetic theory and Vlasov’s equationA good description of ultracold plasmas can be obtained by kinetic theorywith the introduction of the phase space distribution functionf↵(x,v, t). This function describes how particles are distributed in bothphysical and velocity space. The number density of particles of species ↵ inphysical space is defined by⇢↵(r, t) =Zf↵(r,v, t)dv, (4.14)and the mean (fluid) velocity of the particles is given by:u↵(r, t) =1⇢↵(r, t)Zvf↵(r,v, t)dv (4.15)284.1. Introduction to plasma theoryIn both equations the integrals range from 1 to +1.To follow the evolution of a system with a known initial distributionfunction, one needs to obtain an equation that describes the changes tothis function as time progresses. This can be done by invoking Liouville’stheorem. The theorem asserts that the phase-space density is constant alongthe trajectories of the system - that is that the number of particles, as onefollows a group of particle through phase-space, is constant with time.The theorem holds for conservative systems within a Hamiltonian for-malism, and for collisionless particles that share the same Hamiltonianfunction. It is not applicable for systems of di↵erent species but can be usedfor each species separately. The constraint of a ’conservative system’ meansthat if the system undergoes any series of changes, and is then brought back(in any manner) to its original state, then the whole work done is zero.The basic idea behind this theorem relies on an interesting feature ofphase-space. For a conservative Hamiltonian system, a pair of simple rela-tions completely determines particle evolution:drdt= vdvdt=Fm(4.16)Knowing a particle’s starting point in phase-space predefines its whole tra-jectory. Liouville’s theorem is based on the fact that phase-space trajectoriesof independent particles do not intersect. If two trajectories intersect at onepoint, then that point represents a starting point for the further develop-ment of these trajectories. Both trajectories would then develop the sameway - in both directions in time - in that case they would not be intersecting,but identical.Consider a single particle on a trajectory in phase-space, and define aninfinitesimal vicinity around this particle at a point t0 in time. As timeadvances, one can follow this particle and all other particles that started inthis infinitesimal volume. All particles in this volume share similar initialconditions and will be subjected to identical forces - they will stay close294.1. Introduction to plasma theorytogether as time evolves. Due to the fact that phase-space trajectories don’tintersect, it is impossible that a trajectory - once in this volume - leaves it,because this would mean that it would have to intersect with a trajectoryon the volume surface. Therefore, the number or density of particles in thevicinity of one point, which is moving in phase-space according to the equa-tion of motion, is constant in time.To enforce this theorem in three-dimensional phase-space, one must en-sure the conservation of the total number of particles,N =Zf(r,v, t)drdv, (4.17)within a phase-space volume element. This is done by invoking that thetotal time derivative must vanish:0 =dNdt=Z(@f@t+@f@r@r@t+@f@v@v@t)drdv (4.18)The above equation must hold for every volume element in phase-space,therefore:@f@t+@f@r@r@t+@f@v@v@t= 0 (4.19)It is easy to rewrite this equation by implementing the equations of motion,Eq. (4.16), which yields@f@t+ v · @f@r+Fm· @f@v= 0, . (4.20)Equation (4.20) is called the ’Vlasov equation’ after A. A. Vlasov who wasthe first to formulate it in this form.If one continues to investigate kinetic theory, one can show that fluidtheory, as it was used to derive the Debye screening length, can be found asa specialization of the above formalism. More specifically, the plasma fluidequation of motion, Eq. (4.1), is obtained by taking the first moment of Eq.304.1. Introduction to plasma theory(4.20), i.e.: Zv[@f@t+ v · @f@r+Fm· @f@v]dv = 0) m↵du↵dt= q↵E 1⇢↵rP↵ (4.21)In this, the circle of reasoning closes as fluid theory and quasi-neutrality cannow be used to introduce the Debye length.In the above discussion of plasma theory, the fact that particle collisionsseem forbidden in the kinetic theory approach was set aside. Is it in factreasonable to introduce Liouville’s theorem to describe the evolution of aplasma system?Collisionless plasma?The kinetic theory approach is based on Liouville’s theorem for collisionlessparticles. Why is it that collisions threaten the validity of this theorem?Figure 4.2 shows the trajectories of two (neutral) particles in one-dimen-sional phase-space. t1, t2, t3 are subsequent steps in time. Both particleshave an initially positive velocity, however one of the particles is movingmuch faster. At t = t2 the faster particle bumps into the slower, therebytransferring kinetic energy. The collision results in a sudden change in thevelocity of both particles that can be regarded as an end of the phase-spacetrajectory at one point (annihilation) and the creation of a new one at adi↵erent point (creation). This corrupts the conservation of phase-spacevolume and therefore Liouville’s theorem.For charged particle systems, one can argue that collisions that result inan apparent annihilation of a phase-space trajectory are infrequent enoughso that they can be neglected.Consider a classical many-body system of plasma particles where theelectrostatic forces, F↵,p, upon which the various charged particles p react,are known. These forces change the particle trajectories in phase-space.314.1. Introduction to plasma theoryFigure 4.2: Collision in phase-space - Two particles collide in one-dimensional phase-space. t1, t2, t3 are subsequent time steps. As a resultof the collision, both particles assume new trajectories which can be seen asthe annihilation of the old and the creation of a new respective trajectory.They are in general composed of a macroscopic, or slowly varying part, to-gether with a microscopic, or rapidly varying part due to short-range inter-particle forces, i.e. collisions. This first part is termed macroscopic becauseit will be approximately the same for all particles occupying the same regionin phase-space. The fundamental assumption of the kinetic theory approachis that the macroscopic forces are dominant over the microscopic and phase-space volume is approximately conserved.To examine this statement, one can consider the change to the path of atest particle of charge qT and massmT , as it is injected with velocity vT intothe plasma. This test particle will undergo a number of random collisionswith the plasma particles, thereby altering its momentum and energy. The324.1. Introduction to plasma theoryintensity of a collision can be characterized by the deflection angle ✓, whichis given bytan(✓/2) =qT q↵4⇡✏0bµv2rel, (4.22)with reference to the Rutherford scattering problem in the center-of-massframe. Here, µ1 = m1T + m1↵ is the reduced mass, b is the impact pa-rameter, vrel is the initial relative velocity between the test particle and theplasma particle in the respective scattering event and q↵ and m↵ denote thecharge and mass of the latter.To di↵erentiate between macroscopic and microscopic exertion of influ-ence onto the test particle, it is common to separate scattering events intotwo groups: small-angle collisions where ✓ < ⇡/2 and large-angle collisionswhere ⇡/2  ✓  ⇡. The impact parameter of a ⇡/2 collision can be foundusing Eq. (4.22):b⇡/2 =qT q↵4⇡✏0µv2rel(4.23)Figure 4.3 shows an example for a ⇡/2 scattering event and a small-anglescattering event with respective cross-sections. The total cross section forall large-angle scattering events is:˜large ⇡ ⇡b2⇡/2 = ⇡(qT q↵4⇡✏0µv2rel)2 (4.24)Grazing (small-angle) collisions occur much more frequently but withless of an impact on the test particle. To compare the respective cumula-tive e↵ect of small- and large-angle collisions, it is possible to construct ane↵ective small-angle collision cross-section.For this, the first step is to sum over a number of small-angle deflectionsand calculate at which point the cumulative e↵ect is equivalent to a large-angle deflection, i.e.:✓2large ' ⇡2/4 !=NXi=1✓2i,small = Ft⇤ ·Z Db⇡/22⇡b[✓(b)]2db (4.25)334.1. Introduction to plasma theoryFigure 4.3: Di↵erential scattering cross-section for large and smalldeflections - Large-angle scattering events only occur for impact parame-ters smaller than b⇡/2, i.e. for initial trajectories crossing trough the innercircle area.2⇡bdb is the circle area between b and db. ✓(b) is the deflection at im-pact parameter b according to Eq.(4.22). With this, the integration distin-guishes between di↵erential cross-sections at di↵erent impact parameters.F = ⇢P vrel is the apparent flux of particles where ⇢P is the plasma den-sity. t⇤ is the unknown variable in above equation and has dimension oftime. The lower integration limit is the impact parameter limit for large-angle collisions, b⇡/2, and the upper limit is the Debye length since particlesfurther away are screened.Solving Eq.(4.25) for t⇤ gives the time needed by small-angle collisions toachieve the cumulated e↵ect of one (⇡/2) large-angle collision. At the sametime, t1 = ˜F (cross-section times particle flux), can be seen as the rateat which collisions occur. With this, Eq.(4.25) can be rearranged to give,˜⇤, the cross-section for cumulative grazing collisions that are equivalent to344.1. Introduction to plasma theorya large-angle scattering event:˜⇤ =Z Db⇡/28⇡1b(qT qP2⇡✏0µv2relb)2db ⇡ ⇡ln✓Db⇡/2◆˜large (4.26)With this equation, it becomes apparent that if D  b⇡/2, then the pathof the plasma particles is mostly determined by the cumulative e↵ects ofmacroscopic particle interactions rather than by head-on collisions. This iswhat was meant by ’collective behavior’. For systems in which above condi-tions apply, it is possible to neglect large-angle perturbation by short-rangeparticle interactions and employ Liouville’s theorem as an approximation ofthe system.5The requirement D  b⇡/2 can be rearranged to give 6⇡⇢3D  1, whichis similar to the condition found in 4.1. Introducing the Coulomb couplingparameter, =e2/4⇡✏0awskBT, where aws = (3/4⇡⇢)1/3 (4.27)is the Wiegner-Seitz radius that gives the average distance between particles,the very same relation can be written as:p323/2 1 (4.28)Collisionless plasma theory therefore requires that  ⌧ 1. It is in fact thebreakdown of standard plasma theory and the lack of an adequate descrip-tion of correlated many-body systems, which makes UNPs so intriguing.This will be discussed further in section 4.3.1.Returning to the previous analysis, one can conclude that plasma kinetictheory does not follow each particle on its complicated path through space5Choosing ⇡/2 as the deflection angle that separates macroscopic from microscopicinteractions seems somewhat arbitrary. This, however, is of no consequence as the loga-rithmic dependence in Eq. (4.26) makes any choice of similar order basically equivalentin the final result.354.2. Dynamics in two-species neutral plasmasbut instead averages over the region where binary collisions occur and thentreats a particle and its immediate surroundings as subject to an overallcollective mean-field that is created by plasma particles positioned furtheraway.Instead of knowing the exact location of all N = Ne + Ni particles insix-dimensional phase-space by solving 6N equations (4.16), it is the re-quirement ofp3/4/3/2  1 that allows for the employment of a statisticaltreatment by utilizing ⇢, u↵ and T↵.64.2 Dynamics in two-species neutral plasmasWithin the collisionless approximation, this section establishes a formalismthat allows us to model the expansion and decay of a non-spherical plasmasystem into vacuum. We wish to extend above analysis, represented byVlasov’s equation, Eq. (4.20), to a dissociating plasma. The evolution ofsuch system is described by the Boltzmann equation, a generalization ofVlasov’s equation, formulated to include dissociative recombination (DR) ofelectrons and ions:@f↵@t+ v · @f↵@rm1↵@f↵@vq↵@(r)@r=✓@f↵@t◆DR(4.29)Here, m↵ and q↵ designate once more the mass and charge of particle species↵ = e, i, for electrons and ions respectively. The force term from Eq. (4.20)is expressed via the electrostatic potential (r):F = q↵@(r)@r(4.30)6A more rigorous discussion of the kinetic theory of plasmas starts with a classicalHamiltonian to derive a general form of Liouville’s equation for the N-particle phase-spacedensity. It is possible to obtain a hierarchy of density functions (BBGKY-Hierarchy - afterBogolyubov, Born, Green, Kirkwood and Yvon) through integration. The one-particlephase-space probability function f↵ is obtained through neglecting of higher orders in thishierarchy.364.2. Dynamics in two-species neutral plasmas4.2.1 Moments of the Boltzmann equationI begin my analysis with the zeroth and first moments of the Boltzmannequation. To calculate the zeroth moment of Eq. (4.29), I integrate over dv:Z@f↵@tdv+Zv@f↵@rdvZm1↵@f↵@vq↵@(r)@rdv =Z ✓@f↵@t◆DRdv (4.31)The first integral, Z@f↵@tdv =@@tZf↵dv =@⇢@t, (4.32)is reduced to the density as the time derivative is placed in front of theintegral. I apply quasineutrality, i.e. ⇢e ⇡ ⇢i ⌘ ⇢, in the last step.The second term can be rearranged to giveZv@f↵@rdv =@@rZvf↵dv =@@r(⇢u↵), where u↵ =Rvf↵dvRf↵dv. (4.33)I rearrange the last integrand on the left side using the divergence theorem:Zm1↵@f↵@vq↵@@rdv =  q↵m↵@@rZ@@vf↵dv =  q↵m↵@@rZ v=1Svf↵d2v| {z }=0(4.34)This term vanishes as the phase-space density is zero for v ! 1. Theright-hand side of Eq. (4.31) can be evaluated to give:Z ✓@f↵@t◆DRdv =✓@⇢@t◆DR= kDR · ⇢2 (4.35)Here, in the first step, I integrate the phase space distribution functionover velocity to obtain the variation of density with time, equivalent to theright hand side of Eq. (4.31). The second step accounts for the fact thatdissociative recombination is a second order kinetic process, dependent onthe product of the (equal) ion and electron densities. kDR is the electron-ion dissociative recombination rate constant. With this, the zeroth order374.2. Dynamics in two-species neutral plasmasmoment of the Boltzmann equation becomes the equation of continuity:@⇢@t+@@r(⇢u↵) + kDR · ⇢2 = 0 (4.36)To find the first moment of the Boltzmann equation I multiply Eq. (4.29)by m↵v, and integrate over dv. I obtain:m↵Zv@f↵@tdv+m↵Zv(v@f↵@r)dvq↵Zv@@r@f↵@vdv =Zm↵v✓@f↵@t◆DRdv(4.37)The first term can be rewritten asm↵Zv@f↵@tdv = m↵@@t(⇢u↵) = m↵@⇢@tu↵ +m↵⇢@u↵@t, (4.38)where u↵ = ⇢1Rvf↵dv is the local hydrodynamic velocity. The third termof Eq. (4.37), integrated by parts, yields:q↵Zv@@r@f↵@vdv = q↵@@rZv@f↵@vdv = q↵@@rZf↵dv = q↵@@r⇢ (4.39)Now, I restructure the second integral of Eq.(4.37) to give:Zv(v@f↵@r)dv =@@rZvvf↵dv =@@r(⇢vv) (4.40)Here, we use the fact that v is an independent variable in phase space. Theover-bar indicates an averaged quantity. The last equality holds because theaverage of a quantity is 1/⇢ times its weighted integral. I can now separatethe velocity, v, into two parts, the average (fluid) velocity and a thermalvelocity:v = u↵ + (v u↵) (4.41)) @@r(⇢vv) =@@r(⇢u↵u↵) +@@r(⇢(v u↵)(v u↵)) + 2 @@r(⇢u↵(v u↵))(4.42)The fluid velocity u↵ is already an averaged quantity and the average ofthe local thermal velocity (v u↵) is zero, so the last term vanishes. The384.2. Dynamics in two-species neutral plasmasquantity m↵⇢(v u↵)(v u↵) is precisely defined as the stress tensor [15][P]↵ = m↵⇢(v u)(v u) = m↵⇢Z{(v u↵)⌦ (v u↵)}f↵dv, (4.43)where a ⌦ b indicates the dyadic product of two vectors a and b. Theremaining term in Eq.(4.42) can be written as@@r(⇢u↵u↵) = u↵@@r(⇢u↵) + ⇢u↵@u↵@r= u↵@⇢@t u↵kDR · ⇢2 + ⇢u↵@u↵@r,(4.44)where I use the first moment, the equation of continuity. The right handside in Eq. (4.37) yields:Zm↵v✓@f↵@t◆DRdv = m↵✓@⇢@t◆DRu↵ + ⇢m↵✓@u↵@t◆DR= u↵kDR · ⇢2(4.45)Step one is once more equivalent to the first part on the right-hand side ofEq. (4.37). The last step uses Eq. (4.35), and the fact that electron-ionrecombination does not impose a force on the system as charge and globalmomentum are conserved.Collecting these results, I now express Eq.(4.37) in a form that considerselectrons and ions separately. The first moment of the Boltzmann equationgives:me⇢@ue@t+me⇢ue@ue@r+@@r[P]e  e@@r⇢ = 0 (4.46)mi⇢@ui@t+mi⇢ui@ui@r+@@r[P]i + e@@r⇢ = 0 (4.47)4.2.2 QuasineutralityEquations (4.36), (4.46) and (4.47) represent the Boltzmann equation in arestructured form. They describe how particles must act upon pressure andfield gradients to conserve phase-space volume. My goal is to express theelectrostatic potential, , by means other than Poisson’s equation [41]. By394.2. Dynamics in two-species neutral plasmassubtracting Eq.(4.47) from Eq.(4.46) and rearranging the result, I find [61]:e@@r=@ue@t  @ui@t + ue @ue@r  ui @ui@r1/me + 1/mi+@@r([P]eme [P]imi )⇢/me + ⇢/mi(4.48)This is equivalent to Poisson’s equation. The first term on the right-hand side of this equation describes the charge separation in the plasma.Di↵erences between electron and ion densities and charge currents give riseto this term. The second term describes the e↵ect of the thermal pressureon the particle densities.Once more, I apply the quasi-neutrality condition and allow for electronrelaxation. Owing to the much smaller electron mass, the electron distribu-tion function relaxes on a much shorter time scale than the characteristicrate at which the ion density changes. This, in combination with quasi-neutrality, allows me to write the change of electron velocity in time andspace as @ue@t ⇡ @ui@t and @ue@r ⇡ @ui@r . On these grounds, the first term inEq.(4.48) can be neglected with respect to the the second. The expressionfor the electrostatic potential  becomes:e@@r⇡@@r([P]eme [P]imi )⇢/me + ⇢/mi(4.49)4.2.3 EquilibrationIn later Chapters of this work, I will derive the phase-space distributionfunction for the plasma formed in our experiment. I will find, that the pre-cursor plasma volume travelling within our skimmed supersonic molecularbeam develops distinct velocity in the dimensions perpendicular and parallelto the direction of propagation, characterized by T?, (Tx and Ty) and Tk(Tz). Upon ionization, these temperatures refer to NO+ velocity distribu-tions in the plasma. To first order, we can neglect the collisional couplingof ion motion over orthogonal axes.This does not apply to the electron velocity distribution. The electronsin the plasma volume move with a speed on the order of ⇠ 104µm/µs andundergo a very large number of collisions. They certainly possess the means404.2. Dynamics in two-species neutral plasmasto transfer kinetic energy between coordinate axes directions. In the follow-ing, I develop a treatment incorporating complete equilibration of the freeelectrons.I begin by taking a closer look at the pressure tensor. For the specialcase of a Maxwellian velocity distribution, drifting at a given velocity u,the o↵-diagonal elements of the pressure tensor vanish and one obtains thesimple expression [15]:[P]↵ =264 ⇢↵kBT↵,x 0 00 ⇢↵kBT↵,y 00 0 ⇢↵kBT↵,z375I want to find a simplified expression for the electric field in Eq.(4.49).To incorporate electronic equilibration, I recognize that the electron and iontemperatures in a typical ultracold plasma system di↵er by no more thantwo orders of magnitude, while mi > 50000 · me. This leads to the finalexpression for the electrostatic potential [43]:e@@r⇡@@r([P]eme [P]imi )⇢/me + ⇢/mi⇡@@r([P]eme)⇢/me⇡ kBTe⇢1@⇢@r(4.50)In the third step, as mentioned before, I account for electron-electron colli-sions by the assumption of instant equilibration between the very fast elec-trons.It is interesting to note that with Eq.(4.50), I reduce the term thatcouples the moments of the Boltzmann equation to a simple expressionthat depends only on the density distribution and the electron tempera-ture. The ion-electron coupling remains governed by the requirement ofquasi-neutrality. As seen in Eq.(4.50), the electrostatic force that drives theplasma expansion is determined only by the electronic thermal energy andthe normalized density gradient. This simple form is only possible becausethe Maxwellian velocity distribution allows for a fluid treatment and theunwinding of the pressure tensor.A full description of the system requires one final expression. Eqs. (4.46)414.2. Dynamics in two-species neutral plasmasand (4.47) express the first moment of the Boltzmann equation in terms ofthe newly found electrostatic field, Eq. (4.50), separated into its coordinates:me⇢@ue@t+me⇢ue@ue@r= 0 (4.51)mi⇢@uk@t+mi⇢uk@uk@rk+ kB@⇢@rk(Te + Ti,k) = 0, (4.52)where I recognize the Maxwellian nature of the ion temperature (with Ti,x =Ti,y = 0) and deem possible contributions of @rTi negligible. The subscriptk = x, y, z stands for the di↵erent Cartesian coordinates. I find that theelectron velocities are una↵ected by density gradients and are only coupled toions through quasi-neutrality while ion motion evolves according to plasmashape.4.2.4 Global energyTo explicitly recognize the equilibration of the electron thermal energy overthe coordinate axes, I set Te,x(t) = Te,y(t) = Te,z(t) ⌘ Te(t). For fu-ture reference, I require an expression for the total energy of the system,Wtot = Wkin + Wpot. The macroscopic electric potential energy, Wpot, isnegligible in a classical plasma, where the Coulomb correlation parameter, = q2/4⇡✏0awskBT ⌧ 1. Electrons and ions shield each other as longas the approximation of quasi-neutrality holds. Lacking a formalism to dealexplicitly with a strongly coupled plasma, I treat the system as uncorrelatedand neglect Wpot. Thus, I represent the total energy by:Wtot =Wthermalkin +Whydrodynamickin =32NkBTe(t) +Xk12NkBTi,k(t) +Xp12miu2p (4.53)where the second term sums contributions to the ion kinetic energy over thethree coordinate axes, k = x, y, z and N ⌘ Ne = Ni represents the number424.2. Dynamics in two-species neutral plasmasof particles of each species. up is the hydrodynamic velocity of ion p. Theequation above neglects the hydrodynamic velocity of the electrons owingto their small mass. Recalling that self-similarity requires a hydrodynamicvelocity parameter, k = uk/rk = ˙k/k, the third term rearranges to give:Xp12miu2p =12miXkXpu2p,k =12miXkZ2kr2k⇢i dr=12miXkZ2kr2k⇢i,0exp✓ r2k22k◆exp✓ r2k022k0◆exp✓ r2k0022k00◆dr=12miXk2⇡k0k002k⇢i,0 ·Zr2kexp✓ r2k22k◆drk| {z }=p2⇡3k=12miNXk2k2k(4.54)To solve this equation, I reduce it to a Gauss integral using integration byparts. In the last step I recognize that,N = (2⇡)3/2xyz⇢0. (4.55)The expression for the total energy then becomes:Wtot =32NkBTe(t) +12NXk⇥kBTi,k(t) +mi˙2k⇤(4.56)The total energy of the plasma changes over time owing to dissociative re-combination and energy transfer between plasma ions and Rydberg moleculesby resonant charge exchange. For present purposes, I neglect the latter. Thisthen leads to a balance of temperature decrease and hydrodynamic velocityincrease, such that dW = (@W/@N)dN [70], that is:dWtotdt=12kBN˙"3Te(t) +XkTi,k(t)#+12miN˙Xk˙2k (4.57)Taking the total time derivative of Eq. (4.56) gives:434.3. Plasma theory applied to UNPsdWtotdt=12kB"N˙ 3Te(t) +XkTi,k(t)!+N 3T˙e(t) +XkT˙i,k(t)!#+12mi"N˙Xk˙v2k + 2NXk˙k¨k#(4.58)Equating (4.58) and (4.57), yields an expression for the time derivative oftemperature: "3T˙e(t) +XkT˙i,k(t)#= 2mikBXk˙k¨k (4.59)I will return to these important equations of plasma theory when I beginto construct computation models for our UNP in Part III.4.3 Plasma theory applied to UNPsHaving derived the fundamental concepts and equations of plasma theory,I wish to discuss the special case of ultracold neutral plasma within theframework of plasma theory.4.3.1 Kinetic theory in molecular ultracold plasmaIt is important to state that the description of our system proceeds only inthe classical regime, despite the very low temperatures of ultracold plasmasystems. The term ’ultracold’ is meant to separate the field of ultracoldneutral plasma from the field of ’cold plasma’, in which electron tempera-tures still range to typically 103 degree Kelvin. We use it to show the closeresemblance of our molecular system to UNPs.Over the course of this work, it will be shown that our initial chargedparticle densities are in the range of 1012 cm3 and our electron and iontemperatures are larger than one Kelvin. Assuming spherical coordinates,our density predicts an electronic Fermi temperature,TF =~22mekB(3⇡2⇢e)2/3, (4.60)444.3. Plasma theory applied to UNPsof 4.2 mK, orders of magnitude below the estimated system temperature.Based on this quick analysis, I will base the description of our system kinet-ics solely on classical mechanics.Most plasma systems are accurately described by a fluid theory. Thisapproximation allows for a description of all relevant dependent variables,such as density, fluid velocity and pressure as functions of space and timeonly. This simplified treatment becomes possible because within the fluid ap-proximation velocity distributions are implicitly assumed to be Maxwellianabout some mean. The need to know each particle velocity is replaced by anoverall temperature. The development of a Maxwellian distribution, how-ever, implies the particles can transfer energy and therefore equilibrate theirkinetic energies on a timescale that is short compared to the time at whichwe observe the system. This is obviously not the case for an expansion ofa small plasma volume into vacuum. In fact, we enter a regime where itbecomes problematic to refer to a system temperature or for the case of ionseven a species temperature at all.By now, it has become apparent that the Coulomb coupling parameter plays an important role in plasma systems.  itself only depends on thedensity, ⇢, and on the temperature T .To determine , we must rely on indirect measurements and predictions.The system density and ion temperature can be estimated through analy-sis of the molecular beam. Later in this work, an ’electron temperature’value will be obtained from a direct fit of the solution of Vlasov’s equationto the time-dependent plasma width as well as through computation models.But is it justified to employ Boltzmann’s equation? And can our systemeven be referred to as a plasma? Following the analysis of Section 4.1, itis feasible to set  = 0.3 as an upper limit for a thermal plasma. This isbecause: = 0.3 ) ˜⇤˜large⇡ ⇡ · ln(p323/2) ⇡ 5 (4.61)454.3. Plasma theory applied to UNPs = 0.3 means that the path of a plasma particle is governed five timesstronger by the collective field than by large-angle collisions and allows theapproximate treatment of collisionless kinetic theory for thermal plasmas.We have pushed our experimental system into a regime where the con-cepts derived for thermal plasmas, together with Chen’s definition (see 4.1),begin to fail.   0.3 does however not mean that the concepts of Debyescreening or quasi-neutrality become unsound. One has to remember thatthese concepts were derived in a density frame as opposed to a particle frame.High Coulomb correlation simply means that the average particle distanceexceeds the classical Debye length and this can lead to new e↵ects. Ultra-cold Plasma researchers therefore settle for a less strict definition of plasmaand only require that the Debye length of the system is much smaller thanthe system itself.In our research, we are bound for now to approximate our system, how-ever cautious, through thermal plasma concepts. This approach is encour-aged by our experimental results, as our system exhibits prominent plasmacharacteristics. For example, we have subjected our plasma volume to apulsed electric field of an amplitude as low as 3 V/cm, which is far smallerthan that required to field-ionize the Rydberg state into which the systemwas initially excited [50]. This field is able to extract a small signal of elec-trons. Pulses of amplitudes as high as 200 V/cm, however, fail to destroythe plasma signal. These observations can be interpreted as follows: the sys-tem exhibits a loosely-bound surface charge of electrons, at the same timea Debye screening length much smaller than the diameter of the plasma isshielding the system core. This is strong evidence of plasma behaviour.I conclude that it is possible to refer to our system as a plasma. Inlack of a more suitable theory, I will use Boltzmann’s equation as a firstapproximation to describe our plasma. This is done in full knowledge thatit might not extend to properly describe all aspects to our system.464.3. Plasma theory applied to UNPs4.3.2 Correlation e↵ectsIn the following, I will give an example how the onset of correlation mightchange the nature of a plasma. The following model uses Debye-Hu¨ckel the-ory as described by Murillo [54].7To employ this theory, one assumes an infinite electronplasma in a neutralizing and uniform ionic background with density ⇢.8 Theorigin of the coordinate system shall be fixed onto one electron. Sphericalsymmetry of the particle distribution is assumed. The description of thissystem through the one-dimensional Poisson equation and the assumptionof a linearized Boltzmann distribution yields the Debye-Hu¨ckel potential,(r) =e4⇡✏0· er/Dr, (4.62)where the Debye length is given by D =p✏0kBTe/e2⇢, as before. Thederivation is analog to Section 4.1. The Debye-Hu¨ckel potential is an inter-polation between the exact solutions at r = 0 and r !1. Using Eq. (4.62),one can now calculate the potential felt by each particle, excluding its owncharge:limr!0(r) e4⇡✏0r=  e4⇡✏0D(4.63)The right-hand side of above equation states the energy between one parti-cle and the remainder of the plasma. Summation over all N free electrons(without double counting) gives a factor of N/2 and the total energy be-comes:WDH =32NkBTe  N2e24⇡✏0D=32NkBTe✓1 1p33/2e◆, (4.64)Eq. (4.64) shows that a decrease in the total energy can occur as a result7In this reference, Debye-Hu¨ckel theory is derived in the cgs-system. For conformity,this discussion was transferred into mks-units.8A study by T. Pohl showed ion correlation to have an negligible e↵ect on plasmaexpansion ([63]).474.3. Plasma theory applied to UNPsof correlations. This reduction is due to interactions of the electrons withthe opposite-signed background. Debye theory provides an easy access tothe complex attributes of correlated plasmas. Above findings conform qual-itatively with the observed Wigner crystallization of highly coupled plasmas[25]. The quantitative e↵ect of linearization of Boltzmann’s equation in thederivation of the Debye-Hu¨ckel potential (cf. 4.1), which turns the potentialinto an interpolation between the exact solutions at r = 0 and r ! 1, onthe quality of Eq. (4.64) is dicult to assess.4.3.3 Disorder induced heatingIn the introduction to my thesis, I have already eluded to the fact thatdisorder-induced heating is an important mechanism in UNP research. Re-searchers in our field strive to create highly correlated plasma in the labora-tory. MOT plasmas, prepared at densities of ⇢ ⇡ 1 ·109 cm3 and low initialion and electron temperatures, Ti=10 µK and Te=0.1 K, seemed ideal toachieve high levels of correlation, i ⇡ 2.7 · 106 and e ⇡27. Kuzmin andO’Neil [44], however, point out that these initial temperature values are meremeasures of initial particle kinetic energy and not equilibrium temperatures.Molecular-dynamics simulation supported by experimental data [85] showrapid ion heating in MOT plasmas during equilibration over the first 250ns after plasma formation. The origin of this heating is the fact that UNPsare created from a gas of spatially uncorrelated atoms. During the earlyevolution of the plasma, the charged particles move towards equilibrium po-sitions, transferring potential energy into (thermal) kinetic energy. A crudefirst-order approach to estimate the energy released into the thermal bathduring equilibration is found by equating thermal energy and the Coulombinteraction between neighbouring ions [43]:kBTDIH ⇡ e24⇡✏0aws(4.65)For above example system, TDIH becomes ⇠ 2.7 K yielding a value  ⇡ 1.484.3. Plasma theory applied to UNPsMolecular ultracold plasma is also formed by excitation of an uncor-related gas. Application of above model yields heating on the order ofTDIH=27 K for initial plasma densities of ⇢ = 1 · 1012 cm3. We arguethat the Penning ionization process at our densities yields a degree of pre-correlation [71]. This process selects for pairs of excited molecules closetogether. The deactivated Penning partners predissociate, depleting theleading edge of the distribution of nearest-neighbor distances. I will returnto the process of DIH in Part IV.49Part IIFormation of a molecularRydberg ensemble50In order to form an ultracold neutral plasma (UNP) in the laboratory, onerequires a cold and dense ensemble of atoms or molecules and a laser ex-citation pathway to bound or continuum states near the ionization limit.Within our research field, the use of magneto-optical traps (MOTs) to coolatomic ensembles has become the method of choice. Our approach di↵ers,in that we form a UNP within the expansion of a supersonic beam.A supersonic molecular beam presents a dynamic experimental envi-ronment, in which gas expanding from a high-pressure reservoir attains asub-Kelvin temperature in a local frame that moves with relatively highvelocity in the laboratory. This di↵ers substantially from the conditions en-countered after laser cooling of a gas of atoms in a MOT. The supersonicbeam environment yields two distinct advantages for the study of ultracoldplasma physics: Charge-particle densities in beams can exceed those attain-able in a MOT by orders of magnitude. This medium can also supportthe formation of molecular ultracold plasmas, introducing interesting andpotentially important new degrees of freedom that govern the dynamics oftheir evolution.Charged particle density distributions play a fundamental role in defin-ing the properties of a plasma system. The initial state of the excitationvolume sampled when forming a plasma in a molecular beam depends on aset of complex conditions underlying the expansion. This complicates theinitial phase-space distribution of a beam plasma, whereas the excitation ofan atomic gas trapped in a MOT yields a well defined plasma phase-spacedistribution, in which initial density and ion temperature follow from sim-ple analysis of the trapping potential. The beam plasma, while allowing forstraightforward detection, requires a more sophisticated characterization ofinitial conditions. For example, few experimental methods exist that canprovide an accurate absolute measure of either neutral or ion densities.This part of my thesis provides an analysis that addresses this apparentdisadvantage. In the following, I characterize the initial phase-space distri-bution function of a molecular Rydberg ensemble within the lower symmetry51geometry formed by the intersection of lasers and molecular beam. This en-semble is the precursor of our UNP, and thus of fundamental importance toour work.Chapter 5 will discuss the physics of our supersonic molecular beam indetail. Chapter 6 considers the laser excitation of cold NO in the jet andcombines the findings of Part II to state a distribution function for NORydberg states in our experiment.52Chapter 5Molecular Beam GasDynamicsIn this chapter, I construct a theoretical description of the molecular beamthat depends solely upon the measured reservoir gas pressure and the knownsystem dimensions. This discussion is vital because it defines the importantcharacteristics of a nozzle expansion that allow its characterization in phase-space. In some cases, I compare predictions to values measured by experi-ment. I argue that the good agreement obtained in those cases for which acomparison is possible, confirms the essential accuracy of the hydrodynamicmodel presented below.5.1 Experimental setup overviewFigure 5.1 gives a bird-view schematic of our experimental setup. Our ex-periment table top is 4 by 10 feet and supports a high vacuum chamberpumped down to a vacuum pressure of ⇠ 108 torr. The chamber containsa supersonic-beam apparatus, consisting of nozzle and skimmer, as discussedin the next section. Horizontal viewports on the chamber allow us to in-tersect the molecular beam in between two detection plates (optional) withlaser light.The laser light consists of two coincident nano-second light pulses, !1and !2 (cf. Chapter 2). Both originate from pumped dye laser and arefrequency doubled. The intensity of !1-light is attenuated via a set of Glan-Thompson polarizer. We take great care to optimize the shape of !1 with a50µm-aperture spatial filter, before sending it on to the chamber. A dichroicmerges !1 and !2 prior to transit through the viewports.535.2. The supersonic beamFigure 5.1: Bird-view schematic of our experiment tableA more complete account of the dimensions and components (incl. lightsources) of our experimental setup can be found in Appendix A.5.2 The supersonic beamA supersonic beam apparatus typically consists of a gas reservoir, an evac-uated chamber and a nozzle, through which the gas enters the evacuatedspace. As the gas flows into the vacuum, isentropic expansion transformsrandom thermal energy into directed laboratory kinetic energy, and the jetcools in the moving frame. Provided that the ratio between reservoir pres-sure, P0, and background chamber-gas pressure, Pb, exceeds a certain criticalvalue, the flow becomes supersonic [83]. This is crucial for a simple descrip-tion of the beam. An experimental chamber cannot have infinite dimensions,and the beam must encounter obstacles as it expands. Information about545.2. The supersonic beamthese boundary conditions travel back to the source of the jet by the meansof pressure waves, which propagate at the local speed of sound, and forcethe beam to adjust. This information, however, cannot reach the part ofthe beam that propagates with a supersonic velocity. Atoms and moleculesadvancing with higher velocity sweep away back-reflected pressure waves.This collision forms a shock front that defines a region of supersonic gas ve-locity known as the zone of silence. This is illustrated this on the left-handside in Figure 5.2.Figure 5.2: Continuum free-jet expansion - Left: A schematic sketchshowing the regions of an unobstructed supersonic gas expansion from aconverging nozzle (adapted from: [83]). Right: Schematic of our high vacuumchamber with indicated supersonic molecular beam.A high-vacuum molecular beam system usually employs two chambers,a source chamber and an experimental chamber, which are separated by adi↵erential wall fitted with a skimmer. This arrangement creates a narrowbeam that propagates in the low background pressure of the experimentalchamber. The skimmer orifice forms this beam by sampling the super-sonic flow region. Particles within it experience no e↵ect of the downstreamconditions, so their theoretical description follows that of an unobstructedexpansion. In our experimental system, all processes of interest take placein a di↵erentially pumped experimental chamber within this zone of freeexpansion.555.3. Gas properties5.3 Gas propertiesA gas reservoir held at a pressure of 500 kPa supplies an expansion mixture ofnitric oxide (NO) seeded at a ratio of 1:10 in helium. The pumping capacityof the vacuum system substantially exceeds the flow of gas produced by thepulsed nozzle operating with an orifice diameter of 500 µm and an openinginterval of 400 µs. I therefore assume that the gas establishes a stagnationpoint in this reservoir, at which the overall particle velocity relative to thereservoir wall is zero. At this point, the supply gas has a stagnation pressure,P0, equal to the release pressure, and a stagnation temperature, T0, equalto room temperature.Turbomolecular pumps, attached to both chambers, maintain a vacuumin the apparatus. With the pulsed beam running at a frequency of 10 Hz,we measure a source chamber background pressure of Pb ⇡ 0.001 Pa.While the use of helium as a carrier gas increases the beam velocity andimproves cooling, it also adds complexity to the hydrodynamic descriptionof the system. Real gas mixtures present complications, and it is customaryto introduce simplifications. Thus, I will assume an ideal gas mixture at allpoints in the beam, and neglect the very small e↵ects of di↵usive separation,non-continuum velocity slip and temperature slip.At a temperature of 293 K, nitric oxide has a constant pressure molarheat capacity of CNOp = 29.88 J mol1 K1, very close to 7/2 R, whereR is the gas constant [1]. This state quantity changes with the cooling ofthe beam as the number of thermally accessible rotational levels decreases.The population of higher rotational levels in nitric oxide collapses at a char-acteristic rotational temperature of ⇠ TNOrot = 2.46 K, reducing the heatcapacity to 5/2 R [33]. The gas reaches this phase-transition at a point inthe expansion, where the vast majority of collisions has already taken place.Additionaly, NO accounts for only 10% of the mixture. Accordingly, I sim-plify the model by fixing CNOp to its room temperature value. The mass ofnitric oxide is mNO = 30.006 g/mol. Helium has a heat capacity of 12.471J K1 mol 1 and a mass of mHe = 4.0026 g/mol.As an ideal gas mixture, I can describe the expanding mixture of He and565.4. Continuum free-jet expansionNO in terms of its mean mass M , mean heat capacity Cp and mean heatcapacity ratio h, given respectively by:M =XiXimi = 0.9 ·mHe + 0.1 ·mNO = 6.603 gmol(5.1)Cp =XiXiCmolpi = 21.7Jmol ·K (5.2)h = (1 R/Cp)1 = 1.62 (5.3)Xi is the particle fraction, where i represents NO or He. Table 5.1 summa-rizes the state properties of the stagnation mixture.Table 5.1: State properties of the seeded supersonic expansion stagnationmixtureTemperature, T0 293 KPressure, P0 500 kPaNozzle diameter 0.5 mmSkimmer diameter 1 mmheat capacity ratio, h 1.625.4 Continuum free-jet expansionI now derive the equations for the flow variables for a gas mixture with theabove state properties, within the continuum free-jet approximation. In thisapproximation, the particle jet is considered to be stationary, adiabatic andcompressible. All state properties below refer to those of the ideal mixture,as defined above.The opening of the nozzle forms a pulsed beam. However, the durationof this pulse exceeds the time required for the leading edge of the jet tocross the dimensions of the chamber. Such conditions satisfy the continuumbeam approximation, so I can assume stationary flow [83].Neglecting the e↵ects of viscosity and shock waves, which is a good575.4. Continuum free-jet expansionapproximation in the zone of silence, I can model the isentropic flow usingBernoulli’s equation [76]:g · y + u22+ h = const. (per unit mass) (5.4)Here, u is the fluid flow speed at a point on a streamline, g is the gravitationalacceleration, y is the elevation and h is the enthalpy per unit mass. If theflow is irrotational, i.e. r⇥ u = 0, the constant in Eq. (5.4) has the samevalue for all streamlines (homoentropic flow). For our short times and lowdensities and small spatial length scales, gravitational e↵ects are completelynegligible. Thus, I simplify Eq. (5.4) to yield:u22+ h = const. (5.5)For an infinitesimal variation,d(u22) + dh = d(u22) +Rmhh  1dT = 0, (5.6)in which I represent dh for an ideal gas by,dh = CpdT =Rmhh  1dT (5.7)Integrating Eq.(5.6) between distinct points of the flow field, i.e. from thestagnation point (P0, ⇢0, T0, u0) to (P, ⇢, T, u), yields:u22+Rmhh  1T =u202+Rmhh  1T0 = const. (5.8)Introducing the isentropic sound speed a =p(dP/d⇢)0 =qhRTM , whichis defined as the propagation of a small disturbance where M is the molarmass of the gas, I rewrite Eq (5.8),u22+a2h  1 =u202+a20h  1 = const., (5.9)585.4. Continuum free-jet expansionwhere a0 =qhRT0M . I define the Mach number, M = u/a, and recognizethat M0 = 0 at the stagnation point. Now rearranging Eq. (5.9) yields thesonic velocity, a:a =a0(1 + h12 M2)1/2(5.10)As a / T 1/2, I can writeT =T0(1 + h12 M2), (5.11)and, applying the Poisson adiabatic equations of state, PP0 = (TT0)hh1 and⇢⇢0= ( TT0 )1h1 , Eq. (5.11) yields an equation of state for the local pressure:P =P0(1 + h12 M2)hh1(5.12)I can define particle number density, ⇢, simply in terms of the ideal gasequation of state:⇢ =NV=PkBT, (5.13)and, returning to the hydrodynamic beam velocity u, I write:|u| =M · a = M · a0(1 + h12 M2)1/2(5.14)For the maximum beam velocity, I find:umax = limM!1|u| = limM!1a0( 1M2 +h12 )1/2= a0r2h  1 (5.15).Eqs. (5.11-5.14) represent the hydrodynamic flow equations that char-acterize the expansion. Using parameter values defined above by T0, h andM in Table 5.1, Eq. (5.15) yields a theoretical quantity, a0 = 773 m/s forthe speed of sound at the nozzle. The speed of sound subsequently drops asthe beam cools. The gas exiting the nozzle quickly accelerates to approach595.5. The method of characteristicsa center-line beam velocity of umax = 1387 m/s.A photodiode monitoring the rising edge of the !1 laser pulse triggersdata acquisition. The time between this start pulse and the signal waveformproduced when the illuminated volume reaches the detector yields a time offlight over a known distance, and thus determines the velocity of the beam.Present conditions yield an experimental velocity of uexpmax = 1447± 30 m/s.The di↵erence of ⇠ 4.5% between the measured and predicted velocityvery likely arises from a deviation in the commercially prepared gas mixturefrom its nominal 1:10 NO to He composition. It could be said that the veloc-ity determined in this way provides an exceedingly accurate in situ measureof expansion gas mixture composition. Other factors such as source temper-ature variation, a more precise accounting of the rotational energy releasefrom the relaxation of the NO heat capacity, or NO–He speed di↵erentialsarising from velocity slip have far less consequence. I therefore maintain thatthe correspondence between predicted and measured molecular beam veloc-ity assures the reliability with which this hydrodynamic picture determinesother properties of this free-jet expansion.Returning to the flow equations, Eqs.(5.11 - 5.14), it is important tonotice that I can account for all four thermodynamic quantities at any pointof any streamline in terms of the local Mach number M.5.5 The method of characteristicsI proceed now to employ the approximations introduced above to describethe supersonic flow in the skimmed expansion starting with the followingset of fluid mechanics equations [84, 91].Conservation of mass: r(⇢ · u) = 0 (5.16)Conservation of momentum: ⇢u ·ru = rP (5.17)Conservation of energy: h0 = const. (5.18)Equation of state: P = ⇢RMT (5.19)605.6. Non-equilibrium e↵ectsThermal equation of state: dh = CpdT (5.20)Substituting Ma for u, yields a set of hyperbolic partial di↵erential equa-tions for a supersonic flow that can be solved numerically by the method ofcharacteristics.This approach determines particular curves in space along which thecomponent equations of the partial di↵erential equations can be combined toyield ordinary di↵erential equations. Ashkenas and Sherman [4, 57], providea function that accurately fits the variation of the Mach number, within thezone of silence as a function of the nozzle distance z for M > 5.5:M = A(z  z0D)h1  12(h + 1h  1)/[A(z  z0D)h1], (5.21)The quantity z measures the distance from the nozzle orifice along the cen-treline of the beam. D is the nozzle exit diameter and constants A and z0are fitting parameters obtained numerically. The heat capacity ratio of ourexpansion, h = 1.62 conforms closely with that of pure helium, h = 1.67,for which Ashkenas and Sherman find z0/D = 0.075 and A = 3.26. I adoptthese parameter values as an approximation, and solve the hydrodynamicflow equations, Eqs.(5.11 - 5.14), for points along the centerline of the beam.Figure 5.3 plots results of this calculation. Here I express T , ⇢ and Pas quantities in proportion to their values at the stagnation point and giveu as a fraction of its long-distance limit umax. Note that the beam velocityreaches its maximum within a travel distance of ⇠ 1 mm, accompanied byrapid declines in gas temperature, density and pressure.5.6 Non-equilibrium e↵ectsThe foregoing derivations describe an idealized gas that maintains energyequilibration over all levels. Energy transfer takes place by means of particle-particle collisions. I have already shown that the temperature and the par-ticle density of the beam both decrease, as the gas expands into the sourcechamber. It is instructive to examine how this process a↵ects the collisionfrequency and the completeness of energy equilibration.615.6. Non-equilibrium e↵ectsFigure 5.3: Thermodynamic values for the hydrodynamic flow - Redline = u/umax; blue line = T/T0; green line = ⇢/⇢0; yellow line = P/P0 (orfrom top to bottom)To assess this in reasonable detail, I calculate the mean-number of hard-sphere, two-body collisions for a particle at a specific point along the cen-treline flight path. In the following description, I determine the collisionnumber applying an approximation, in which gas particles are representedby spheres of radius r0. As helium is the chief constituent of the gas, Iassume that r0 equals the distance, at which the Lennard-Jones potentialfor helium atoms,W = 4✏{( ˜LJr)12  ( ˜LJr)6}, (5.22)becomes repulsive. This occurs at r0 = 2.64 A˚ [59] and yields a hard-spherecross section, ˜He = ⇡r20 = 21.9 A˚2.This calculation focuses on a single particle that travels with the velocityv2 across a group of particles that all have the same velocity v1. I placethe origin of the coordinate system at the center of the test particle. Thisparticle now sees the other particles as a group travelling with a relativevelocity, vrel = |v1  v2|. Among those particles, ones that have centres625.6. Non-equilibrium e↵ectsthat pass through a circle area of diameter d = 2r0, that lies at the centerof the coordinate system and is perpendicular to vrel, collide with the testparticle. This gives rise to a collision frequency,Zv1,v2c = ⇡d2vrel⇢, (5.23)where ⇢ is the gas number density. Eq. (5.23) neglects the fact that otherparticles can transfer momentum to the test particle, thereby changing v1and thus vrel. I can account for this by a statistical approach. To do this,I assume that a collision changes v2 into v02, and that v02 is determinedby a Maxwellian probability distribution f˜ . Additionally, I want to allowgroups of particles with di↵erent velocities v1 to take part in the collision.v1 shall also be sampled from a Maxwell-Boltzmann distribution. Assumingspherical coordinates, the z-axis being coincident with v2, I find the mean-magnitude of the relative velocity to be:< vrel >=Z Z qv21 + v22  2v1v2cos✓f˜(v1)f˜(v2) dv1dv2 (5.24)A solution to this integral is,< vrel >=r16RT⇡M=p2 < v >, (5.25)where < v > is defined as [76]:< v >=R10 vf˜(v)dvR10 f˜(v)dv(5.26)For temperature T (z) and density ⇢(z), I find the two-body mean-collisionrate per unit time as:Zc(z) = 16 ˜HerRT (z)⇡M⇢(z) (5.27)For example, the collision rate Zc(z) at the entrance orifice of a skimmer atz = 3.5 cm distance from the nozzle, is ⇠ 175 000 collisions per second.635.6. Non-equilibrium e↵ectsI transform this characteristic property of the expansion to define aquantity with greater utility. Let Zcdt represent the number of collisionsundergone by one particle during the time-interval dt. The distance that istraveled by the particle during this time interval is dz = v(z)dt. Therefore,Zc(z)/v(z)dz represents the number of collisions experienced by a particlethat travels the infinitesimal distance between z and dz. With this, I calcu-late the number of remaining collisions, Zr, that a particle at a given pointz will experience until it reaches infinity:Zr(z) =Z 1zZc(z0)v(z0)dz0 (5.28)Figure 5.4: Two-body collision number - Mean-number of collisionsexperienced by a molecule at point z as it travels from z to infinityFigure 5.4 shows that Zr(z) decreases drastically over the course of the645.6. Non-equilibrium e↵ectsexpansion. Near the nozzle, each particle takes part in hundreds of collisions,but after only 1 2 cm of flight, collisions become much less frequent. Thisis the region where the rotational heat bath, determined by the occupationof the rotational energy states which depends on Trot, decouples from thetranslational temperature bath, described through Ttrans.This decoupling occurs because energy does not transfer between therotational states of two colliding molecules in every single collision. Typicalrotational collision numbers, i.e. the mean-number of collisions a moleculeundergoes before changing rotational state, is known to be about 5-20 forN2 and O2 [13]. Assuming a comparable number for NO, which has similarmass and size, I estimate from Fig. 5.4, that molecules in the beam stopexchanging rotational energy at a distance of about 0.6 cm to 1.3 cm afterthe nozzle. Comparing this with Fig. 5.3 yields a temperature of roughly1.6 4.1 K in this region.In summary: we can expect a quantity that relates to the rotational tem-perature, as determined by the Boltzmann population of rotational energylevels, to decouple (or freeze out) from the still-falling equilibrium transla-tional temperature at 1.6 4.1 K.Our experimental setup allows a facile measurement of the rotationaltemperature via the spectrum of the X2⇧  A2⌃+ transition. Figure 5.5compares theoretical predictions9 for the temperature-dependent A-statespectrum with a plot of a !1 1+10 REMPI scan. We find that the calculatedtemperature for rotational freezing accords very well with that measured bythe 1 + 10 REMPI scan. This is an important result as we can concludethat the overall beam temperature reaches ⇠ 2.5 K, at a point where eachparticle experiences an average of a dozen more collisions. These remainingcollisions, while not likely to further relax rotational energy, continue to coolthe translational temperature of the beam. Thus, we can expect Trot Ttrans.Not only do the rotations freeze out, but we also expect that the particletranslational temperature itself will stop declining and separate from the9We employ PGOPHER, a Program for Simulating Rotational Structure, C. M. West-ern, University of Bristol, http://pgopher.chm.bris.ac.uk655.6. Non-equilibrium e↵ectsFigure 5.5: Rotational temperature of nitric oxcide - The uppermostcurve shows data from a typical 1+1’ REMPI scan of the transition spec-trum between the 2⇧1/2 ground state and the A2⌃+ first excited state in NO.The data was recorded by intersecting the molecular beam with a tuneablelaser, 7.5 cm downstream of the skimmer, and accelerating freed electronstowards a MCP detector. The inverted plots beneath are theoretical predic-tions for above transition spectra for di↵erent rotational populations of themolecular states. For better comparison, three theoretical predictions withtemperatures of 2.5 K, 5 K and 10 K are given.665.7. Sudden freeze and point-source approximationcalculated equilibrium temperature at a point, where collisions begin tocompletely cease. I will discuss this in the following section.The above description of hard-sphere collisions and temperature decou-pling from equilibrium mechanics represents a crude approximation of theprocesses in a real pulsed supersonic molecular beam system. Without suchsimplification, it is very hard to make statements about the exact regionof freezing and its extent. However, it is not as much the location of thefreezing surface but the understanding of this process that will be importantfor the following sections.5.7 Sudden freeze and point-sourceapproximationIn this part of the molecular beam description, I derive an expression forthe density distribution of the gas particles after the point of skimming. Ibegin by introducing two approximations that enable this calculation.The last section argues that the closing of equilibration channels causethe translational temperature to freeze when particles cease to collide. Thisis a continuous process. But, as a simplification, I shall introduce the sud-den freeze approximation [3]. This model will assume the existence of adiscrete freezing surface across all streamlines. Upstream of this surface, Iwill describe the translational temperature by equilibrium mechanics, andassume the temperature downstream to be frozen. I will further assumethat this surface is located at the entrance of the skimmer.10 Particles thatpass through the skimmer orifice continue to travel in a free flow with theover-all velocity, u, and relative velocities, v  u, which I determine fromthe calculated beam temperature at the skimmer entrance. At this point,we need not further consider the properties of He, as its role of bu↵er gasceases with the end of particle collisions.10This approximation is necessary to obtain a simple solution for the particle densitydistribution in the experimental chamber. Earlier calculations predict that ⇠ 2 collisionsremain on average for a particle once it passes the skimmer at a distance of ⇠ 3 cm fromthe nozzle (cf. Figure 5.4). I neglect these remaining collisions in following discussion.675.7. Sudden freeze and point-source approximationTo further simplify the calculations, I also assume that all particles enterthe experimental chamber originating from a single point, coincident withthe centre of the skimmer entrance. This so-called point-source approxima-tion [76] holds as long as we concern ourselves with planes at distance l,where l is much larger than the diameter of the skimmer.I start this description at the point of skimming at zskimmer = zs. As areminder, this is the freezing surface where equilibrium flow merges into freemolecular flow and the gas past this point is determined by the previouslycalculated (and now frozen) values for T (zs) and ⇢(zs), where I multiplyour previous density by 0.1 and only consider NO. The particle velocitiesare approximated by a Maxwell-Boltzmann distribution function which isnormalized to the beam density:f˜(v) = ⇢s✓m2⇡kBTs◆3/2e m2kBTs (vu)2, (5.29)Here:⇢s = ⇢(zs) =Zf˜(v)dv u =< v >= ⇢1sZvf˜(v)dv (5.30)Ts = T (zs) =23kBZm2|v u|2f˜(v)dv (5.31)It is possible to split the velocity v into a component parallel to the beamcenterline, vk, and a component in the perpendicular direction v?.v u = vk  u+ v? (5.32)) (v u)2 = u2{(vku 1)2 + v2?u2} (5.33)This definition allows me to separate the distribution function into paralleland perpendicular components where,f˜(vk, v?) = ⇢sf˜kf˜?, (5.34)685.7. Sudden freeze and point-source approximationin which:f˜k = (m2⇡kBTs)1/2e mu22kBTs (vk/u1)2=Ssup⇡F eS2s (vk/u1)2f˜? = (m2⇡kBTs)e mu22kBTs (v?/u)2=S2su2⇡eS2s (v?/u)2Ss = u/p2kBTs/mMolecules at the entrance of the skimmer now contribute to the total density,ns, with velocity components vk and v?, in the intervals (vk, vk + dvk) and(v?, v? + dv?) as,d3⇢s = f˜(vk, v?)dvkd2v? (5.35)I write the intensity (or flux density) di↵erential in the velocity componentsat the skimmer as:d3F = vkd3⇢s = vk⇢sf˜kdvkf˜?d2v? (5.36)The intensity di↵erential in the sole component vk isdF = vk⇢sf˜kdvkZ 11Z 11f˜?d2v? = vk⇢sf˜kdvk, (5.37)and the total flux density per area and along the beam line, follows as,F =Z 10dF = ⇢2sS2s⇡{eS2s2S2s+p⇡2Ss(1 + erf(Ss))} (5.38)Appendix G.1 of reference [76] solves this integral. The total flow throughthe skimmer of area A is now:s = A · F (5.39)I next derive an expression for the density distribution past the skimmer.For this, I must define a new set of coordinates ✓, l, r, according to Figure5.6. In these coordinates, I find695.7. Sudden freeze and point-source approximationFigure 5.6: Geometry, used in the calculation of F(r, l)d3v = dvkd2v? = dvkv?dv?d = (rdrl2)v2kdvkd, (5.40)where I use v? = vk tan(✓) ⇡ vkr/l. It follows thatd3⇢s = f˜(vk, v?)d3v = ⇢sf˜k(vk)f˜?(vk, r)rdrdl2v2kd2vk, (5.41)where:f˜?(vk, r) =S2s⇡u2eS2s (vkrul )2(5.42)The flow rate di↵erential in the variables vk, r and  must conform with theflow rate at the point of skimming. In the new coordinates, this is given by:d3s = Avkd3⇢s = Af˜kf˜?nsrdrdl2v3kdvk (5.43)The total flow across the part of the ⇡-plane (see fig.5.6), which lies betweenr and dr, is obtained by integrating over . From cylindrical symmetry, it705.7. Sudden freeze and point-source approximationfollows that d2s =R 2⇡0 d3s = 2⇡d3s. With this:d2s2⇡rdr= A⇢sf˜kf˜?l2v3kdvk = dF(r, l) (5.44)The above expression represents the flux density of particles through a cir-cular element between r and r+dr that lies in a plain orthogonal to z and atdistance l from the skimmer. The integration of this expression [76] yieldsthe desired density distribution:F(r, l) = A⇢su2⇡3/2l2Sse⌘2⇠2(1+⌘2)2{p⇡⇠(⇠2+3/2)(1+erf(⇠))+e⇠2(1+⇠2)}(5.45)Here ⌘ = r/l and ⇠ = Ss(1+⌘2)1/2. F(r, l) has the dimension of cm2 s1.To convert flow into density, I divide by the velocity vk. Plotting the result,using the molecular beam parameters given above and choosing a distancel from the skimmer much greater than its diameter, indicates the Gaussiannature of this function.My analysis shows that I can approximate F(r, l)/vk by a Gaussian if Iam willing to accept an error on the order of 1%:⇢Gaussl (r) = ⇢0l · er222l , (5.46)Here, constants, ⇢0l and l, depend on parameters that I obtain numericallyby fitting to F(r, l)/vk. This yields a very simple mathematical form for theNO density distribution.Let us compare these findings with experimental observation. Figure5.7 displays a measurement made by collecting the signal produced by aneutral particle beam incident on an imaging detector mounted 468 mmdownstream from the skimmer.11 This measurement uses a nozzle-skimmerdistance of 35 mm. Considering these flight parameters, together with thestagnation reservoir properties listed in Table 5.1 in the collision model ofSection 5.6, I predict a temperature of 0.46 K at the point of skimming.11A full description of the detector employed in this experiment can be found in Ap-pendix A.715.7. Sudden freeze and point-source approximation0       20      40      60 0       20      40      60mmmm0       20      40      60mm0       20      40      60mmω1 + ω2High voltage"& read outMCPIndium tin oxide"(ITO) anodePhosphorCCD"CameraFigure 5.7: Top: Experimental setup for the imaging of the molecular jet.The jet enters the experiment chamber through the skimmer and impactson a MCP stack after a free-flight of 468mm. Free electrons are created,multiplied and viewed as photons by a CCD camera. Bottom-Left: CCDresponse of molecular beam impacting on phosphor screen detector (at adistance of 468mm from the skimmer) without laser light present; Bottom-Right: Vertical summation over image pixel and fit, fitted Gaussian widthis 4.35mm.725.8. Phase-space distribution of nitric oxideUpon passing through the skimmer, a particle in the beam has an averageof 2.7 collisions left on its way to infinity, which I neglect in this analysis.For above conditions, I predict a beam width of 3.8 mm at the detectorwhich is 13% narrower than the measured width of Figure 5.7.From the experimentally measured width, we can estimate that the NOparticles impacting the detector in Figure 5.7 have a temperature of 0.6 Kin the moving frame as they pass through the skimmer. Given the num-ber of assumptions in this analysis, this correspondence between calculatedand measured properties of the beam suggest that the model presented herereflects the important aspects of the beam physics and yields a good de-scription of the system employed in these experiments.5.8 Phase-space distribution of nitric oxideAs a final step, I calculate the velocity distribution of the beam particles at adistance l beyond the skimmer. Here, I assume that the beam propagates ina continuous fashion along the z-axis and, after the skimmer orifice collisionsare no longer present.Considering a moving-frame coordinate system that travels with thebeam velocity uk, I expect that the particles in the experimental chamberwithin this frame conserve their Maxwellian velocity distribution, owing tothe absence of collisions. Note in particular that fast particles, i.e. particleswith velocities that originate in the wings of the distribution, leave the vol-ume sooner than slower particles. As fast particles leave a volume elementalong z, fast particles from the leading and trailing volume elements enterand the overall velocity distribution, and with it Tz = Tparallel, remains thesame along the z-axis.A di↵erent situation describes the velocity distribution perpendicular tothe beam centreline. In the point-source approximation, a particle stream ofintensity s enters the experimental chamber. In the same moving frame asabove, faster particles quickly deviate from the centreline and exit a givenvolume element sooner than slower moving particles. To a first approxima-tion, the position of a particle within a plane in x-y coordinates directly735.8. Phase-space distribution of nitric oxideFigure 5.8: Left: Random distribution of 10,000 particles in an area corre-sponding to the skimmer orifice. Each particle has x and y velocity compo-nents sampled from a Mexwell-Boltzmann distribution of 0.6 K. Middle:The same distribution evolved over a flight-time corresponding to the dis-tance skimmer-detector. Each particle-trajectory originates from the par-ticle point of origin within the red circle, i.e. all trajectories are outward.Right: Radial plot and fit of evolved density distribution (middle). I obtaina Gaussian width of ⇠4.53 mm (cf. Figure 5.7).depends its initial velocity component, v?. This is illustrated in Figure 5.8and holds for distances much larger than the skimmer diameter.I can assign each point in space a hydrodynamic velocity, u?(r), givenbyu?(r) = u?(l, r?) =r?time to reach l=r?luk, (5.47)which represents the velocity directed away from the centerline. If a particleis found at point r, it must exhibit the velocity u?(r).This analysis does not mean that the Maxwellian distribution is lost.It means that the velocity spread for all particles in any x-y-plane is justthe same as it was at the point of skimming. The velocity at an infinitelysmall volume and at an arbitrary point r (after the skimmer and within thepoint-source approximation), however, is just u?. The temperature T?(r),i.e. the width of the velocity distribution perpendicular to the stream lineand at point r in the experimental chamber, quickly approaches zero owingto this mechanism.745.8. Phase-space distribution of nitric oxideI thus write an expression for the phase-space density of the NO particlesin the volume of a thin plane-perpendicular section at a distance l from theskimmer as:fNOl (r,v) = ⇢0l · exp(r2?22l) · exp(mNO(vk  uk)2kBTk) · (v? u?(l, r?)) (5.48)75Chapter 6The Rydberg ensemble6.1 Excitation processThe last chapter has developed a description of nitric oxide molecules seededin a supersonic molecular beam of helium. In the chapter to follow, I utilizethe characteristics of this beam to obtain the initial plasma particle phase-space distribution functions.The spectroscopic characteristics of Nitric oxide have been discussed atlength in Chapter 2. To estimate the initial density of a photopreparedRydberg gas, I begin by calculating the fraction of NO molecules that popu-late the lowest lying rotational ground state level, from which the excitationprocess originates.In Hunds case (a), the J 00 levels of 2⇧ NO split into manifolds of ⌦ =12 (F1) and ⌦ =32 (F2), separated by the spin-orbit splitting, which forthis system is about 119.8 cm1 [1]. Rotational cooling in the supersonicexpansion relaxes population to the lowest few levels of the F1 manifold. Toa good approximation, the rotational energy of levels in the 2⇧1/2 manifoldincreases as:Wrot(N00) = BvN 00(N 00 + 1) (6.1)where Bv = 1.67 cm1 [1].12Statistical thermodynamics relates the fractional populations in these12The NO X-state is described (mostly) though Hund’s case a). I use notation N todescribe the molecular core rotation despite the fact thatN is not a good quantum number.766.1. Excitation processlevels to the local rotational temperature,P (N 00) =2(2N 00 + 1)eWrot(N 00)/kBTZ˜r(6.2)whereZ˜r =XN 002(2N 00 + 1)eWrot(N00)/kBT . (6.3)Under molecular beam conditions, the partition function sum extends overonly a few levels of F1. The factor of two in the degeneracy reflects ⇤-doubling of the rotational levels in the ⇧ state into positive and negativeparity components.The foregoing molecular beam model as well as our general experimentalexperience accord with a rotational temperature range from TR = 1.5 to 4K. Over this range, the predicted fractions of the NO population occupyingF1 levels with angular momentum of the nuclear rotation of N 00 = 0 and 1vary as follows.Table 6.1: Fractional population in F1 rotational states of 2⇧ NO as afunction of rotational temperature in the molecular beamTR N 00 = 0 N 00 = 11.6 K 0.87 0.132.5 K 0.68 0.304.1 K 0.48 0.44In most cases, we find spectroscopically measured intensities consistentwith TR ⇡ 2.5 K. At this rotational temperature, 68 percent of the nitricoxide in the illumination volume occupies the rotational ground state. Halfof these molecules have the (-) parity required for a dipole-allowed transitionto the rotational level N 0 = 0 of the A 2⌃+ v0 = 0 state that represents thefirst step of our excitation sequence (cf. Figure 2.3).Thus, we tune the output, !1, of the first of two dye lasers to ⇠226 nm,which excites the (-) parity component of the ground state NO X 2⇧ 12J 00 =1/2 molecules to the J 0 = 1/2 (+) level of the A 2⌃+ v0 = 0 state via776.1. Excitation processFigure 6.1: Creation of a NO Rydberg ellipsoid - The gas mixtureof NO and He exits the nozzle and cools as it expands into the vacuum.Only the coldest part of the beam passes the skimmer. This part slightlyexpands as it continues, which is due to the remaining thermal motion ofthe gas particles. It is overlapped by two counter-propagating lasers thatcreate an ellipsoid of NO Rydberg molecules which continues to travel withthe molecular beam.the pQ11(1/2) transition. A second laser, !2, tuned to a wavelength in theinterval between 327 nm and 330 nm, drives this population to a single,high lying NO Rydberg level of principal quantum number, n, in the rangebetween 30 and 80, selected from manifolds of Rydberg states, all with totalangular momentum neglecting spin of K = 1. The short predissociationlifetimes of all other accessible Rydberg states in practice limits this processto the nf series.13 The two-colour, laser-crossed molecular beam intersectionforms an ellipsoidal excitation volume schematically diagrammed in Figure6.1.To calculate the A-state density distribution within this ellipsoid, I as-sume that the molecular beam expands to a negligible degree over the widthof the illuminated region in z. The mathematical form of the A-state distri-13This is further discussed in Chapter Excitation processbution then follows through convolution of cylindrical molecular beam andcylindrical laser geometry:⇢Astate(x, y, z) = ⇢0Astate · exp✓ x222NO y222NO y222!1 z222!1◆= ⇢0Astate · exp✓ x222x y222y z222z◆where x = NO, y =s2NO · 2!12NO + 2!1, z = !1(6.4)I now endeavour to estimate the A-state peak density ⇢0Astate. The 1+1REMPI spectrum of NO presents a set of transitions. Choosing the ground-to-ground rovibronic NO X-A resonance, we decrease the !1 laser intensityby means of two polarizers. We do this to minimize contributions from hotelectrons produced by 2-!1-photon ionization. For most experiments, weo↵set the polarizer by ⇠ 50 degrees, which yields an !1 intensity of 1.75µJ per pulse. Assuming a perfect Gaussian laser shape, I can calculate thepeak intensity of the beam from the average laser power, P˜ :P˜ = Ipeak ·Zexp✓ y222◆dyZexp✓ z222◆dz = Ipeak · 2⇡2) Ipeak = P˜2⇡2 (6.5)I take the Gaussian rms-width of the laser intensity distribution to be1 ⇡ 0.2 mm, as determined from razor-blade tomography. For a singlepulse of 10 ns, the peak power is Ipeak = 0.7 mW/cm2. From this photonflux, I calculate the density of molecules excited to the NO A-state, ⇢Astate,by:⇢Astate = ⇢GS · abs · F (6.6)F = Ipeak · E1Here, F is the photon-flux and ⇢GS is the ground state density at the796.1. Excitation processcentre of the spatial distribution. abs is the photon absorption cross-section.The energy of a 226 nm photon is W = hc/ = 8.79 · 1019 J. Based onmeasurements of Bethke [9], Hippler and Pfab estimate 5 · 1016 cm2 as thee↵ective cross-section of a typical rovibronic transition of the (0, 0) bandfor NO in a comparable experiment [34]. I will adopt this value. For aphoton flux of 7.9 · 1022 s1 cm2, Eq. (6.6) calculates an A-state densityof,⇢Astate = ⇢GS · 0.4 (6.7)The factor 0.4 approaches saturation and I employ a simple rate equationmodel to account for this. I also introduce an ionization channel through˜ion, representing the cross-section for one-photon ionization from the A-state. This cross-section was measured by Zacharias, Schmiedl and Welgeto be ˜ion = 7·1019 cm2 for 266 nm light [30], and I approximate this valueto hold for our transition. I solve the following set of equations numerically,dNesdt= ˜abs ·Ngs · F (6.8)dNiondt= ˜ion ·Nes · F, (6.9)where Ngs, Nes and Nion are the number of molecules in the transitionground state, excited state and ionized state respectively.The A-state population, exterior to the point of maximal intensity, isassumed to depend linearly on the laser intensity. The overlap of the lowintensity, Gaussian !1-laser profile and the Gaussian NO density distribu-tion within the molecular beam creates a prolate ellipsoid of A-state NOmolecules. The !2 laser is much stronger in intensity, saturating the transi-tion from the A-state into the Rydberg level. Furthermore, since the FWHMof !2 has about 3 times that of !1, the spatial Rydberg density is deter-mined by !1 alone and matches the ellipsoidal distribution of initial A-statepopulation.806.2. Phase-space distribution of the Rydberg ensemble6.2 Phase-space distribution of the RydbergensembleI now summarize the results from above discussion. For this purpose, I con-sider a setup with a 2.5 cm nozzle-skimmer distance and at 7.5 cm skimmer-laser distance. The model above (Chapter 5) predicts a beam temperatureat the skimmer of 0.7 K, with a remaining collision number of 4.7. By thepoint of laser-overlap, the beam will have developed a Gaussian width of750 µm with a peak density of 1.62 · 1014 cm3 NO molecules.Let us put these last values into context. Section 5.7 develops a methodto determine the density distribution of NO within the experimental cham-ber. A prediction using this method underestimates apparent beam tem-perature by 13% compared to the quantity we determine experimentally (cf.Figure 5.7). Thus, we ought to view 0.7 K as a lower bound for the actualbeam temperature. Figure 6.2 details typical variations in NO density anddistribution width as a function of beam temperature. We can see that un-derestimating temperature to a small extent leads to a small overestimate indensity and small underestimate in width. For the purpose of this tutorial,I will continue with the theoretical values derived above.At an assumed !1 laser intensity of 1.75 µJ and a Gaussian laser widthof 0.2 mm I calculate an A-state peak density of 1.5 · 1013 cm3 and an ionnumber of 5 · 109 cm3 produced in the first excitation step. Saturation ofthe !2 transition promotes half of the A-state density to Rydberg molecules.I will now generalize the phase space distribution of the Rydberg gas.Here, I include provision for the conversion of thermal energy into radial816.2. Phase-space distribution of the Rydberg ensemblestreaming motion.fRy(r,v) ⇡ ⇢Ry·exp✓mNO(vx  x · x)22kBTx x222x◆·exp✓mNO(vy  y · y)22kBTy y222y◆·exp✓mNO(vz  z · z)22kBTz z222z◆ (6.10)The above equation recognizes the linear dependence in the hydrodynamicvelocity (cf. Eqs. (5.47) and (7.11)) and writes uk(l, k) = k · k, where I usethe hydrodynamic velocity parameter k.I set initial conditions according to the discussion above, where: Tz = 0.7K, and Ty = Tx ⇡ 0 K (as a result of the point source approximation).x = NO = 0.75 mm while z = !1 = 0.2 mm. Eq. (6.4) yields the valuey = 0.193 mm for NO = 0.75 and !1 = 0.2 mm. I elect to simplify thefollowing consideration by treating our plasma as cylindrically symmetric,i.e. y = z ⇡ 0.2 mm.Figure 6.2: Plot of peak density vs. Gaussian beam width as a func-tion of beam temperature for a system with 2.5 cm nozzle-skimmer distanceand at a point 7.5 cm downstream from the skimmer. For a temperatureof 0.7 K, the graph yields a density of 1.62 · 1014 cm3 NO molecules at aGaussian width of 750 µm.826.2. Phase-space distribution of the Rydberg ensembleI derive a value for each ˙k by considering the rate of expansion aboutthe respective k-axis. At the instant of Rydberg formation, the distributionof ion velocities in z has a width determined by Tz and a first moment ofzero. Thus, the initial ensemble has no expansion velocity about z in themoving frame, ˙z(t = 0) = 0. The quantities, ˙x(t = 0) and ˙y(t = 0) havenonzero values owing to the small underlying divergence of the molecularbeam. I obtain 0x = l from Eq. (5.46) and use the relationlvk=0x˙x(6.11)to obtain ˙x and similarly ˙y as:˙y(t) =0y0x· ˙x(t) (6.12)The initial hydrodynamic velocity parameter is defined by k(t) = ˙k(t)/k(t).Table 7.1 summarizes the initial conditions of our Rydberg ensomble as laserTable 6.2: Initial conditions imposed on Rydberg molecule position andmomentum based on underlying molecular beam and excitation scheme.Below values in combination with equation 6.10 give a full account of thephase-space distribution of our Rydberg ensemble.Rydberg ensemble initial conditionsx 750 µmy 200 µmz 200 µmx 0.0193 µs1y 0.0193 µs1z 0 µs1Tx ⇠0 KTy ⇠0 KTz 0.7 K⇢Ry 7.5 · 1012 cm3excitation has concluded. The initial peak density of Rydberg states, ⇢Ry,836.2. Phase-space distribution of the Rydberg ensembleshould be understood as an upper boundary, as it was derived for an idealsystem.84Part IIITheory work85The previous part of my thesis developed the hydrodynamics of super-sonic expansion in terms of the functional characteristics of a di↵erentiallypumped molecular beam apparatus. I characterized the Rydberg gas phase-space volume created by the convolution of laser and molecular beam distri-bution functions. Chapter 6.2 concluded with a statement of the phase-spacedistribution function of the Rydberg ensemble formed in our experiment.Over the course of part III, I will consider the evolution of this Rydbergensemble in time. This will be done in two steps. Chapter 7 will discuss theinternal conversion of our Rydberg gas to plasma via Penning ionization andelectron-Rydberg collisional avalanche. As I will show, this plasma forma-tion process is reaching a quasi-equilibrium on a nano-second timescale. Thechapter will conclude stating the initial state of the phase-space distributionof plasma ions and electrons. Following this, Chapter 8 will shift timescalesto consider ambipolar expansion and decay of the plasma ensemble over tensof micro-second.While seeking to take full account of the spatial asymmetry and chem-ical reactivity encountered in the molecular beam ultracold plasma, I willstop short of considering the e↵ects of Coulomb correlation. Ion and elec-tron densities and temperatures, estimated by gauging the model developedbelow against the results consistently produced by molecular beam ultra-cold plasma experiments, point to plasma evolution through a regime ofstrong coupling. The means by which the plasma reaches such conditionsand the e↵ect of strong coupling on plasma observables remains to be ad-dressed. The toolset described in the following provides a solid, zeroth-orderplatform from which to consider the evolution to a state of strong couplingunder the non-spherical, reactive conditions of the molecular beam ultracoldplasma.Chapter 7 will deploy a computational method originally derived byJonathan Morrison and Nicolas Saquet [79], and later developed by JachinHung and Hossein Sadeghi [37], all previous members of my research group.Similarly, Chapter 8 deploys a method first developed by Hossein Sadeghi for86a high-symmetry plasma in the absence of thermal or hydrodynamic motionof the ions [70]. I have developed both methods further. I am grateful to mycolleagues for sharing their code and wish to acknowledge their contributionto the following two chapters.87Chapter 7Early-time dynamics in anUNPAt a distance of 7.5 cm beyond the skimmer, where our lasers cross themolecular beam, the thermal energy of the nitric oxide in the beam haslargely relaxed to become directional kinetic energy and collisions haveceased (cf. Chapter 5). This changes after the laser pulses, !1 and !2excite about 10 percent of this population to form a Rydberg gas. A nitricoxide molecule, excited to a high lying Rydberg state, let us say, n0 = 50,has an electronic orbital diameter of 2n20a0, in this case approximately 260nm.At the core of the Rydberg ellipsoid, where the initial Rydberg gasdensity is as high as ⇢0 ⇠ 1012 cm3, the mean distance between excitedmolecules is defined by a Wigner-Seitz radius, (3/4⇡⇢0)1/3 = aws of ⇠ 500nm. As a consequence of overlapping Rydberg orbitals, we can expect thatin our experiment, plasma formation is fundamentally linked to Penningionization.According to Raubicheaux (cf. Section 3.2), Rydberg atoms as closetogether as 1.8 · 2n2a0 ionize in 90% of the cases in less than 800 Rydbergcycles. For example, by exciting into n0 = 50, 800 Rydberg periods require2⇡n3 ⇡ 15 ns (atomic units). Thus, plasma formation in dense Rydbergsystems is a nano-second process.An ionized Rydberg molecule will continue to move with the velocity ofthe former neutral. Movement of NO⇤ and NO+ is based on two concepts.887.1. Plasma evolution from a Rydberg gasAlong the z-axis, molecules have mean velocity components according tothe thermal spread v =pkBTbeam/mNO ⇡ 14 nm/ns. At the same time,molecules, sitting for example at a radial distance of one standard deviation,x, from the molecular beam centre line, have a radial velocity componentof v? = x ·x = 14 nm/ns. Overall, molecular movement on a nano-secondtimescale is very slow compared to the inter-particle distance. Based on thisanalysis, I will neglect NO core movement in a first-order approximation, asI consider plasma formation at the nano-second timescale.7.1 Plasma evolution from a Rydberg gasThe following model is based on the concepts of Penning ionization andcharged particle collisions with Rydberg molecules. These concepts, alongwith the respective rate coecients were discussed in Chapter 3.Penning ionizationLet us consider a uniform Rydberg gas of density ⇢0Ry. In order to find thenumber of Penning partners within a critical distance of rc = 1.8 ·2n20a0, onecan consult the density-dependent Erlang distribution of nearest neighbourdistances:dE(r) = 4⇡⇢0Ryr2e4⇡3 ⇢0Ryr3dr (7.1)For densities as high as 1012 cm3, an appreciable fraction of NO⇤ moleculesfalls within the critical distance, where Penning ionization occurs sponta-neously. To gauge the fraction of molecules, termed fP , that form pairsseparated by critical distance, rc, I integrate Eq. 7.1.fP =Z rc0dE(r) = 1 e 4⇡3 ⇢Ryr3c (7.2)From this number, I estimate the density of Penning electrons:⇢0e = fP · ⇢Ry · 1/2 · 0.9 (7.3)897.1. Plasma evolution from a Rydberg gasThe factor, 1/2, accounts for the fact that a Penning interaction involvingtwo Rydberg molecules produces one Penning electron and one deactivatedRydberg state. I multiply by 0.9 to comply with Raubichaux’s 90% ion-ization. The principal quantum numbers of deactivated Penning partners(Pi ⇢deacRy,i = ⇢e) distribute in proportionality to n5 over states with n fromthe ground state to n0/p2, the highest final state in an interaction thattransfers sucient energy to ionize a molecule in state n0. The drop in Pen-ning partner binding energy determines the Penning electron temperature.The density of unaltered Rydberg states after Penning ionization is:⇢unalteredRy = (1 fP · 0.9) · ⇢Ry (7.4)At a Rydberg density of ⇢0Ry = 1012 cm3 and for n0 = 50, the Penningprocess supports as much as 16% electron-ion pair formation. The firstPenning electrons escape from the gas volume. This creates a net positivespace charge that begins to trap the remaining electrons. The escape ofelectrons ceases as soon as the depth of the trapping potential becomesequal to the electron kinetic energy. It is possible to estimate the number ofescaping electrons, by calculating the excess charge that is required to trapelectrons of thermal energy 3/2kBTe, within a Gaussian sphere of dimension [17]:Ni Ne = 32kBTe4⇡✏0e2r⇡2= 2250 ⌧ Ni, (7.5)for  = 200 µm and an electron temperature of 100 K (this temperatureconforms with results found later in this chapter). I find that the small frac-tion of ⌧ 1 % lost electrons can account for a trapping potential which isstrong enough to confine the remainder and leads to a good approximationof quasi-neutrality for our plasma.I assume that the velocities of the trapped electrons equilibrate instantlyon the timescale of the experiment to define a time-evolving global electrontemperature with an initial value T 0e . The magnitude of this value follows907.1. Plasma evolution from a Rydberg gasfrom energy conservation in the Penning process:T 0e = Ryn20⇢Ry +Ryn20⇢unalteredRy +XnRyn2⇢deacRy,n!/✓32⇢0ekB◆(7.6)For above values, I find an initial electron temperature of T 0e ⇡ 40 K.The electron and ion densities, the Rydberg distribution as well as theelectron temperature serve as seed conditions for the evolution of our Ryd-berg system to plasma.Ionization avalancheEvolution to plasma proceeds in a dynamically complex sequence of events.Fast-moving Penning electrons impact isolated Rydberg molecules to start acollisional ionization cascade. For densities formed by saturated !1 and !2transitions in the core of the Rydberg gas ellipsoid, the processes of Penningionization and avalanche occur on the same timescale as laser excitation. Tosimplify the present description, I neglect any coupling between the dynam-ics of photoexcitation, Penning ionization and electron-impact avalanche.Above discussion allows me to find post-Penning process vaules ⇢unalteredRy ,⇢deacRy,n, ⇢0e and T0e for initial conditions ⇢0Ry and n0. I now compile ⇢unalteredRyand values ⇢deacRy,n into a new array ⇢0n = ⇢(n, t = 0). Eqs. (7.7, 7.8) and Eq.(7.10) construct a rate equation model for the evolution of atomic Rydberggas to plasma, incorporating the processes and rate coecients introducedin Section 3.2:ddt⇢n = kion(n)⇢n⇢e + ktbr(n)⇢3e +Xni 6=nk(ni, n)⇢ni⇢e Xnf 6=nk(n, nf )⇢n⇢e(7.7)ddt⇢e =Xnkion(n)⇢n⇢e Xnktbr(n)⇢3e (7.8)Please note that ⇢i = ⇢e in the quasi-neutral approximation. The evolution917.2. Simulation resultsof the electron temperature follows from energy conservation:32kBTe⇢e RyXn⇢nn2Wtot(t) = 0 (7.9)The derivative of above equation yields:ddtTe = XnRy⇢˙nn2 1.5kBTe⇢˙e!/(1.5kB⇢e) (7.10)For above computation, array ⇢n is in principal infinite as n = 1, ...,1.I constrict accessible values of n to an interval between nmin and nmax.Intuitively, one should set nmin = 1. Low-n Rydberg states, however, aremost likely to decay via predissociation (see Table 3.3) or radiative decay.A possibility to account for low-n decay in a first-order approach is to setnmin > 1 and implement a decay path M⇤(n  nmin))Mdeac in the com-putation. Here,Mdeac refers to either the ground state or a dissociated stateof the molecule, in either case a state no longer taking part in the systemdynamics. Neither radiative decay nor predissociation influence the elec-tron temperature and Eq. 7.10 remains unaltered. To improve the speedof computation, I elect not to consider dissociative recombination as a de-cay mechanism in the early time dynamics simulation. This approximationyields an acceptable error for evolution times ⌧ 1µs.Morrison et. al. [52] have studied the e↵ect of boundary conditions forplasma formation in NO and determined nmin = 10 as adequate. They setboundary nmax = [pRy/kBTe] according to the thermal energy of electrons.The square brackets indicate a rounded value. I will adopt these boundariesin the following.7.2 Simulation resultsIn this section, I will discuss ionization avalanche computation results forseveral values of ⇢0Ry and n0. As previously, I begin with ⇢0Ry = 1 ·1012 cm3927.2. Simulation resultsand n0 = 50.Rydberg manyfold decayFigure 7.1 displays the evolution of n-level density with time. The top-leftbar diagram shows the Rydberg manifold right after the Penning process.Of the initial Rydberg density ⇢0Ry in level n=50, 16% has been ionized andthe same fraction was de-excited to levels below n=36. 68% of Particles donot take part in the Penning process and remain in level n=50.As time progresses, n-changing electron-Rydberg collisions fan out theinitial distribution. In these interactions, a hop to a neighboring state ismore likely than to a state further away. A general trend of the Rydbergmanifold down towards low-n states becomes apparent as kn,n1 > kn,n+1.There are three loss mechanisms of Rydberg density considered in ourmodel. The first is electron impact ionization. The second is via level-hopping to nmax at which point the Rydberg state is considered ionized. Andthe third is level-hopping to n < nmin and decay to an inactive state (groundstate or dissociated state). Repopulation is possible through three-body-recombination. The evolution of the Rydberg manifold is largely completedafter 100 ns.Electron temperature and density evolutionFigure 7.2 shows the evolution of Rydberg density and temperature. Start-ing with the Penning temperature of ⇠ 40 K, the system initially cools overthe first nanosecond. This is due to cold electrons entering the distributionvia level-hopping to nmax and subsequent ionization. This ionization pro-cess ceases after the first few nanoseconds and is the main source of freeelectrons and ions.The cascade of the Rydberg manifold down to low-n states is releasinglots of thermal energy to the electrons. After the initial quench in tempera-ture, the electrons heat rapidly to a temperature approaching ⇠ 200 K. Theheating slows as the bulk of the Rydberg manifold reaches nmin and decays(at ⇠ 20 ns - see Figure 7.1). Slower heating via electron-Rydberg collisions937.2. Simulation resultsFigure 7.1: Density distribution over n-levels - The top-left figure showsthe n-level distribution after the Penning process for ⇢0Ry = 1 · 1012 cm3and n0 = 50. State n=50 has been cut o↵, its value being ⇢n=50(t = 0) =0.67 · 1012 cm3.947.2. Simulation resultsFigure 7.2: Evolution of electron temperature and density - Thefigures show the identical dataset as Figure 7.1, with the left-hand figuresdisplaying an expanded timescale. The density plots show electron, totalRydberg and deactivated neutral densities (blue, green, red).continues to take place up to ⇠ 1 µs evolution, until all low-n levels havedecayed via nnim.Evolution at di↵erent densities and PQNsFigure 7.3 extends the previous computation to values n0 = 40, 50, 60 anddi↵erent initial densities ⇢0Ry = 1,0.9,...,0.1·1012 cm3. Several trends areapparent in the graphs:• For the same PQN value, normalized electron densities approach thesame upper limit.957.2. Simulation resultsFigure 7.3: Electron density and temperature evolution for statesn=40, 50, 60 and initial density ⇢Ry = 1, 0.9, 0.8,...,0.1·1012 cm3 (for eachfigure top to bottom).967.2. Simulation results• The lower the initial Rydberg density, the lower the initial electrondensity and in turn the longer the decay of the Rydberg manifold (i.e.plasma formation).• Plasmas formed from larger PQNs have a larger Penning fraction andthus higher initial electron density. They avalanche faster than plas-mas formed at the same density but lower PQN.For the evolution of electron density in time the following holds: Themain growth in density is briefly delayed until the level-hopping Rydbergmanifold reaches nmax. The growth later ceases as the bulk of the manifolddecays to lower n-levels. This defines a point of inversion, POI, of thedensity evolution. POI is mathematically well defined and I will consider itsdouble value as the plasma rise time, trise = 2 · tPOI . This coincides with⇠ 90% total ionization and shall serve as a measure for the time it takesfor a Rydberg system to evolve to plasma. Figure 7.4 displays rise times forplasmas formed at di↵erent ⇢0Ry and PQN.Ionization avalanche for our experimental conditionsOur research group has developed above formalism to simulate the earlytime dynamics in an ultracold neutral plasma. I will now use this modelto derive the initial plasma properties for our experimental system, that isfor ⇢0Ry = 7.5 · 1012 cm3. In the following, I will consider the case of laserexcitation to Rydberg state n0 =50.The phase-space distribution of the Rydberg ensemble formed in ourexperiment was summarized in Section 6.2, Table 6.2. The formed plasmaoccupies the extent of the former Rydberg volume. I set Ry,k = i,k basedon the fact that di↵erent density values achieve the same fractional ionizationfor a given PQN (cf. Figure 7.3). From quasi neutrality it follows that⇢i = ⇢e = ⇢ and i,k = e,k = k. The ions are created with the velocityvectors of the former neutrals (Ry,k = i,k and TRy,k = Ti,k).In this model, electrons are assumed to thermalize instantaneously. Theydo not take on any hydrodynamic velocity characteristics and are fullydescribed in momentum space by electron temperature Te. For values of977.2. Simulation resultsFigure 7.4: Plasma avalanche times for states n=40, 50, 60 (blue, green,red or top to bottom) and varying initial density ⇢0Ry.⇢0Ry = 7.5 · 1012 cm3 and n0 =50, I find ⇢e ⇡ 6 · 1012 cm3, Te appoching⇠ 200 K and trise ⇡ 1 ns. All results are sumerized in table 7.1.Once more, the initial electron density should be understood as an orderof magnitude estimate based on the predicted Rydberg density. I reformulateEq. 6.10, the phase-space distribution function of the Rydberg ensemble, toyield the distribution function for plasma electrons and ions.987.2. Simulation resultsTable 7.1: Initial conditions imposed on plasma particle position and mo-mentum based on underlying molecular beam and excitation scheme. Belowvalues in combination with equation 7.11 give a full account of the phase-space distribution of our plasma.Plasma initial conditionsx 750 µmy 200 µmz 200 µmi,x 0.0193 µs1i,y 0.0193 µs1i,z 0 µs1Ti,x 0 KTi,y 0 KTi,z 0.7 Kn0 50⇢0Ry 7.5 · 1012 cm3e,k(0) 0 µs1⇢ ⇠ 6 · 1012 cm3Te ⇠200 Kfi(r,v, t) = ⇢·(vx  i,x)(vy  i,y)·exp✓mNO(vz)22kBTi,z x222x y222y z222z◆fe(r,v, t) = ⇢·exp✓ me2kBTe(v2x + v2y + v2z)◆·exp✓ x222x y222y z222z◆(7.11)Note that the thermal velocity distribution in x and y has collapsed to adelta-function for Ti,k ! 0.The next section details a hydrodynamic approach to simulate the long-time phase-space evolution of a plasma with the charged-particle distributionfunctions described above.99Chapter 8Long-time dynamics in anUNPIn our experiment, we form a plasma with phase-space distribution resem-bling Eq. 7.11, at initial conditions, Table 7.1. The last section discussedthe process of plasma formation at sub-microsecond timescales and for fixedmolecular core positions. I now shift gears and will study the evolution ofplasma including expansion for timescales up to tens of microseconds. Forthis purpose, I will reset the time origin (value t=0). In the following t0refers to the point in time at which the previous section ended, the pointwhere the formation of plasma is complete.Section 8.1 will derive an analytic expression for the expansion of a non-dissociating plasma into vacuum. Section 8.2 will extend this discussion toderive a computation model for plasma expansion under the influence ofdissociative recombination. Simulation results will be discussed in Section8.3.8.1 Analytic solution to plasma expansionOne can describe the free expansion of a non-dissociating plasma in simpleterms via the Vlasov equations. Setting the right-hand side of the Boltzmannequation, Eq. (4.29), to zero, I obtain the electron and ion Vlasov equationsas:@fe@t+ v@fe@r+m1e@fe@ve@@r= 0, (8.1)@fi@t+ v@fi@rm1i@fi@ve@@r= 0 (8.2)1008.1. Analytic solution to plasma expansionDorozhkina and Semenov have derived an exact self-similar solution ofthe three-dimensional Vlasov equations, Eqs. (8.1) and (8.2), for electronsand ions with self-consistent electric fields,e@@r= kBTe⇢1@⇢@r, (8.3)as in Eq. (4.50). The result describes how a spatially localized plasmavolume expands into vacuum in the quasi-neutral approximation. Theirsolution yields [23, 24]:fi / exp Xkr2k22k(t)!exp Xkmi(vk  k(t)rk)22kBTi,k(t)!(8.4)fe / exp Xkr2k22k(t)!exp Xkme(vk  k(t)rk)22kBTe,k(t)!(8.5)Temperatures, rms-radii and the hydrodynamic velocity parameters withinthis solution satisfy,T↵,k(t) = T↵,k(0) · 2k(0)2k(t)k(t) =˙k(t)k(t)2k(t) = (k(0) + ˙k(0)t)2 + c2kt2 c2k =kBmi(Te,k(0) + Ti,k(0)) (8.6)using once again, the condition that mi  me.Eqs. (8.4) and (8.5) conform with the initial conditions of the molec-ular beam plasma system. However, a closer look at Eqs. (8.6), revealsthat this description fails to include a means to transfer energy between thecoordinates axes of the system. I introduce coupling between these coordi-nates once again by explicitly recognizing rapid electron equilibration, i.e.Te,x(t) = Te,y(t) = Te,z(t) ⌘ Te(t).I combine the equation of continuity, Eq. (4.36), for kDR = 0, withEqs. (8.6), the first moment of Boltzmann’s equation (Eq. 4.52) and thepreviously found expression for the total energy (Eq. 4.56). This, for thespecial case of a Gaussian density distribution, leads to a set of ten ordinary1018.2. Expansion under the influence of dissociative recombinationdi↵erential equations that include electronic equilibration and accuratelyapproximate the Gaussian ellipsoidal plasma in the case of absence of a lossmechanism [43]:˙k(t) = k(t)k(t) ˙k(t) =kB(Te(t) + Ti,k(t))mik(t) 2k(t)T˙e(t) = 23Te(t)Xkk(t) T˙i,k(t) = 2k(t)Ti,k(t) (8.7)Note that the initial temperature of the molecular beam detailed in Table7.1, Ti,x(t) and Ti,y(t), start out at 0 K and remain so throughout the ex-pansion. Ti,z(t) begins at 0.7 K and falls. Terms linking Te(t) with theevolution of x(t), y(t) and z(t) couple electron thermal energy into allthree dimensions of the expansion.In the following section, I will derive a model that incorporates expansionunder the influence of particle loss, in our case dissociative recombination.Above set of rate equation, derived analytically, will be useful to test themore complex computational method.8.2 Expansion under the influence of dissociativerecombinationFigure 8.1: Illustration of plasma in the shell pictureThe presence of DR precludes an exact self-similar solution of the corre-sponding Boltzmann equations. So, I numerically model the expansion ofthe system with reference to a set of equal density surfaces, leading to a1028.2. Expansion under the influence of dissociative recombinationshell or onion model description. This is illustrated in Figure 8.1.I follow the discussions in [73] and [70], extending the analysis fromcylindrical and spherical symmetry to model an ellipsoidal plasma with fi-nite initial ion temperatures and radial velocities. The shells in this plasmamodel expand on account of initial hydrodynamic velocities as well as theambipolar conversion of electronic and ionic temperatures into directionalkinetic energy. The charged particle number per shell changes only due toDR. Three body recombination to Rydberg states and subsequent predisso-ciation only plays a minor role in the long-term evolution of plasma and isin the following neglected.Initial conditions:To define the initial conditions of this model, I first establish a global regionof interest by introducing a computational cut-o↵, Ccut, found by multi-plying the Gaussian rms-width along each axis by 5, Ccutk = 0k · 5. Theinitial shell width follows from division of Ccut by the number of shells,shellk = Ccutk /#shell. The surface of the j-th shell is now defined byr2x(R0x,j)2+r2y(R0y,j)2+r2z(R0z,j)2= 1, (8.8)where the semi-principal axis lengths are defined as R0k,j = j · shellk . Idetermine the density of shell j by evaluating,⇢0j = ⇢peak0 exp (R0k,j)22 · 0k!, (8.9)using the same prescription for each axis k = x, y, z. The initial hydrody-namic velocity components of each shell are determined by u0k,j = R0k,j · 0k .I find the shell volume using,Vj =43⇡"YkRk,j YkRk,j1#, (8.10)1038.2. Expansion under the influence of dissociative recombinationdetermining the initial number of ions by N0j = ⇢0j · V 0j . The initial condi-tions, for the system of di↵erential equations I intend to solve include,N0j , R0x,j , R0y,j , R0z,j , u0x,j , u0y,j , u0z,j , Ti,x(0), Ti,y(0), Ti,z(0), Te(0), (8.11)where subscripts j point to an array of size #shell.Di↵erential equations:My ODE integrator advances the system in steps starting from the initialconditions defined by Eq. (8.11). After each step, Eq. (8.10) determinesthe new volume of each shell, j, and ⇢j = Nj/Vj yields the correspondingdensity. Dissociative recombination reduces the number of ions in the jthshell according to Eq. (4.36) expressed as:@Nj@t= kDR ·N2j /Vj (8.12)The radius and volume of each shell evolves in time according to the hydro-dynamic velocity of the ions:@Rk,j(t)@t= uk,j(t) (8.13)Equation (4.52) formulated for a uniform uk(rk) within each shell, j, de-scribes the temporal development of the hydrodynamic velocity in each case:@uk,j@t=kBmi⇢j@⇢j@rk· (Te + Ti,k) (8.14)Eq. (8.14) defines two important characteristics of these shell-model ex-pansion dynamics. First, note that the change in velocity in any coordinatedirection depends on the corresponding density gradient, @r⇢. This gradientvaries with axis direction according to initial conditions. However, the quan-tity @r⇢j changes smoothly, moving from one major axis to another on anyarbitrary shell. Thus, I can represent the force as it changes the ellipsoidalshape of any shell j by the gradients, @r⇢j , determined at the principal axis1048.2. Expansion under the influence of dissociative recombinationpositions Rk,j from evolving plots of ⇢k versus r in each axis direction.Note that Eq. (8.14) also holds that the electron and ion temperaturescombine to drive the expansion. Energy conservation gives rise to the samee↵ect in the Vlasov case, where it appears in the time variation of temper-ature, Eq. (4.59), and in the analytic solution for the time dependence ofk, Eq. (8.7). While this description fully couples the electron temperaturein three dimensions, I allow the ion temperature to act as an energy reser-voir for expansion only according to its initial coordinate axis distribution,Ti,x(0) = Ti,y(0) = 0 and Ti,z(0) = 0.7 K.I rewrite Eq. (4.59) as it reflects the conversion of electron and ionthermal energy to hydrodynamic ion radial expansion energy."3@Te@t+Xk@Ti,k@t#=  2mikBPj NjXk,jNjuk,j@uk,j@t(8.15)Here, I have translated Eq. (4.59) into the shell picture by realizing that inthis frame, the hydrodynamic energy represented by Eq. (4.54) becomes:1/2miNXk2k2k ) 1/2miXk,jNju2k,j (8.16)Secondly, I assume that the ion thermal energy contribution to the hydrody-namic expansion energy is in proportion to its contribution to the thermalsum. Thus, I gauge the thermal relaxation of electrons and ions in eachdegree of freedom by:@Te@t=  TeTe + 1/3Pk Ti,k1324 2mikBPj NjXk,jNjuk,j@uk,j@t35 (8.17)@Ti,k@t=  Ti,kTe + Ti,k24 2mikBPj NjXk,jNjuk,j@uk,j@t35 (8.18)In summary, Eqs. (8.12), (8.13), (8.14), (8.17) and (8.18) representa complete set of di↵erential equations modelling the evolution of plasma1058.3. Simulation resultsundergoing DR, based on initial condition (8.11).8.3 Simulation resultsSpherical expansion and the e↵ect of dissociative recombinationI begin by applying the 3D shell model methodology described above tosimulate the self-similar expansion hydrodynamics of a spherical Gaussiannitric oxide plasma, for which I can compare the numerical results withcorresponding analytical solutions. To represent the free expansion of thistest plasma, I choose an initial Gaussian width of x = y = z = 200 µm,and set the initial charged particle density to ⇢0e = ⇢0i = ⇢0 = 6 · 1012 cm3.I wish to develop an intuition for the behaviour of plasma to di↵erentelectron temperature values. The upper three panels in Figure 8.2 showthe shell model evolution of Gaussian radial density distributions of a nitricoxide plasmas with three selected electron temperatures, Te = 5, 20, 100K. These model plasmas lose no ions to dissociative recombination, and theshell-model solution conforms exactly with the time-dependent Gaussiansolutions of the Vlasov equations.Note that in the representation of Figure 8.2, expansion appears mostevidently in the transfer of charged particles from the comparatively smallvolume of the plasma core to occupy the much larger and much lower densityplasma domain at larger r. This dispersion of NO+ plasma mass occursmuch faster at higher temperature. On the scale of the initial peak density,the charge distribution spreads to become barely observable after 3 µs forTe = 100 K. Experimental observations of nitric oxide plasma evolution showevidence of a peaked electron signal reflecting a spatially localized plasmaeven after hundreds of microseconds [50, 73, 78].The lower frames of Figure 8.2 compares the shell model evolution of aplasma that undergoes dissociative recombination described by a rate con-stant (cf. Section 3.2):kDR = 6 · 106T1/2e cm3 s1, (8.19)1068.3. Simulation resultsFigure 8.2: Plasma density along the z-axis for a spherical system forinitial electron temperatures of Te = 5, 20, 100 K. Each figure shows theearly evolution of plasma for time steps of 0, 1, 2 and 3 µs (top to bottomcurves). The top figures display simulation results for a system without DR,the bottom simulations include DR.In the presence of dissociative recombination, an initially Gaussian plasmadeparts from a state of self-similar expansion. At low electron temperaturein particular, the rapid recombination of NO+ with electrons in the densecentre of the plasma flattens the core of the spatial distribution. This changereadily appears in a plot of density as a function of radial distance, r. Fig-ure 8.3 shows that by flattening the distribution function at its core, DRsignificantly retards the rate at which the shell with an initial radius of 1expands.However, as suggested above, an overwhelming majority of the electronsin the plasma sphere occupy the sizeable volume at larger r. Here, disso-ciative recombination proceeds at a rate that is slow compared with expan-sion. As evident in Figure 8.3, DR has a much smaller e↵ect on the rate atwhich radial velocity grows in the shell at 3. In an apparatus that collects1078.3. Simulation resultsFigure 8.3: Shell positions and shell velocities of shells with initial radiiof 1 and 3 with (red) and without (blue) DR in an ultracold plasma withTe = 20 K plasma.electrons released as a spherical plasma transits a perpendicular grid, theself-similar hydrodynamics of this Gaussian tail obscures the experimentalobservation of retarded expansion in the core, if present [70].Figure 8.2, shows how the e↵ect of DR diminishes substantially at higherelectron temperature. Generally speaking, the rate of plasma expansionprovides a reliable gauge of electron temperature, while the rate at whichDR causes plasma decay depends both on electron temperature and localdensity. In the absence of DR, a Gaussian plasma of a given radius expandsat the same rate regardless of its overall density.E↵ect of thermal velocity components of the molecular beamThe molecular beam hydrodynamics model developed in Section 7.2 de-scribes the intrinsic divergence of the plasma volume in x and y owing tothe initial conditions of the molecular beam. This divergence adds slightlyto the radial expansion of a spherical plasma measured in those coordinate1088.3. Simulation resultsdimensions. Similarly, though the illuminated volume has no hydrodynamicdivergence in the z dimension, the molecules move back and forth alongthe axis of propagation with the residual translational temperature of thesupersonic expansion, Tk, and this a↵ects the rate of expansion measured inz.Figure 8.4: Plasma density along the z-axis for a spherical system withoutDR at time t = 3 µs. Upper curves: without initial hydrodynamic expansion.Lower curves: with k(0) = 0.0193 µs1 .Examining the plasma dimensions at an evolution time of 3 µs and atelectron temperatures of 5, 20 and 100 K, I find that the plasma createdwith initial divergence expands just a bit faster. Figure 8.4 displays thedensity distributions of spherical plasmas without DR for cases that assumeno initial hydrodynamic divergence, compared with plasma expansions thatbegin under conditions of k(0) = 0.0193 µs1, for k = x, y, and longitudinal(z-axis) temperature, Tk = 0.7 K, all as described in Table 7.1. Initialdivergence increases the expansion rate, but the net e↵ect is very small.The initial values of k(0) reflect a gas temperature of 0.7 K at the pointof skimming. The e↵ect produced is barely discernible when the electrontemperatures exceeds a few degrees Kelvin.A similar situation exists if I treat the ions as an energy bath with atranslational temperature determined by the temperature of the NO pre-cursor in the beam. Any such energy must stream into expansion. However,1098.3. Simulation resultsfor large Te compared to Ti resulting e↵ect is very small. Overall, I concludethat the e↵ect of ion thermal motion on the hydrodynamics of our molecularbeam plasma are negligible for any Te  0.7K.Non-spherical expansion, including the e↵ect of dissociativerecombinationI finally consider the shell-model expansion of an ellipsoidal plasma volumewith DR, compared with that of one without DR, described both by shellmodel and analytically. Figure 8.5 details the shell model expansion of aplasma volume that begins with x = 750 µm and y = z = 200 µm andan electron temperature of 50 K, with and without DR. On the left, we seethe self-similar expansion of shells at 1 and 3 , compared with the Vlasovdescription (cf. Section 8.1) of expansion in the long and short axes of theGaussian ellipsoid.Figure 8.5: Shell position of 50 K electron temperature plasma with initialwidths of x = 750 µm and y = z = 200 µm. Left: shells with initialradii of 1 and 3 along x and y/z axes for the self-similar case. (thelocus of circles underlying the shell-model result plot values obtained usingEq. (8.7)) Right bottom: evolution of 1 shells along x and y/z axes for adissociating system. Right top: evolution of 3 shells along x and y/z axesfor a dissociating system.1108.3. Simulation resultsFigure 8.5 makes an important prediction. The higher density gradientsalong y and z cause more thermal energy flow into the expansion alongthese coordinates. In the self-similar case on the left, the Gaussian width inthe short-axis coordinate directions, y and z overtakes that in the long-axisdirection, x after 8 µs. By this point, ion acceleration has ceased and theexpansion has become ballistic. The plasma volume proceeds from prolateto become spherical at this point and then evolves further as an oblateellipsoid. In the absence of DR, all shells cross this point at the same time;the expansion remains self-similar.On the right, we see the e↵ect of DR, which occurs mainly in the high-density core of the plasma. The flattening of the spatial distribution retardsexpansion in the core, and with reduced radial velocity, the shell represent-ing a 1 ion density becomes spherical at a much later time, ⇠15 µs. The3 shell, with a spatial distribution that is much less a↵ected by DR ex-pands with long- and short-axis ballistic velocities much like the self-similarexample.111Part IVExperimental work112In Part II of my thesis, I introduced the components involved in preparingthe initial state of our system under study. This discussion led to a de-scription of the phase-space distribution function of the molecular Rydbergensemble formed in our experimental apparatus. Chapter 7 in Part III in-troduced a rate equation model detailing the internal conversion of a denseRydberg gas to plasma. Following this, Chapter 8 developed a simulationmodel suited to the expansion hydrodynamics of a plasma with the initialconditions of our experiment. Therein, dissociative recombination competeswith ambipolar expansion to shape the charged particle distribution. Elec-tron temperature regulates rate processes in both, plasma formation andevolution.The application of Boltzmann’s equation, as discussed in Section 4.3.1and applied in Chapter 8, is applicable to systems with a Coulomb couplingparameter  < 0.3. For a charged particle density of ⇢ = 1 · 1012 cm3, acoupling of  < 0.3 or  > 1 is achieved for temperatures T>90 K and T<27K, respectively. Based on the analysis of plasma formation in Chapter 7,we expect our system to establish initial electron and ion temperatures ofT0e ⇡200 K and T0i ⇡1 K. These values yield 0e ⇡ 0.13 and 0i ⇡ 27. Thee↵ects of disorder-induced heating, as discussed in Section 4.3.3, have so farnot been considered. DIH is expected to further limit i to values close toone. Overall, the approximate description of our plasma system throughBoltzmann’s equation, represented by the shell expansion model developedin Chapter 8, seems feasible.We are an experimental group and have studied ultracold molecularplasma in the laboratory for the past ten years. We have developed pre-viously discussed theoretical framework and computational methods to aidus in the interpretation of our experimental results. A full conceptual un-derstanding of our system should lead to the convergence of theory modelsand experimental data. So far, this has not been the case. The search forthe root of this discrepancy lies at the centre of Part IV.113In the following, I will introduce our schemes of plasma detection anddiscuss our observations. Chapter 9 will introduce our first generation ofdetectors and discuss our observations of plasma evolution in 1D. Chapter10 will introduce our new 3D tomographic imaging machine - the PlasmaTV.The development of this new detector has been the central task of my PhD.I have elected to split considerations in Chapters 9 and 10 with referenceto our knowledge-base before and after the introduction of the PlasmaTV.This will highlight the paradigm-shift brought about by this new methodof studying plasma. Chapter 11, with the title ’Arrested relaxation’, willconclude my thesis. Here, I collect the findings of this thesis and speculateon future research avenues.114Chapter 9Plasma detection in 1D9.1 The ZEKE and moving-grid detectorsOur research group began work in the field of ultracold neutral plasmas in2006. Our original experiments were performed with a ZEKE (Zero ElectronKinetik Energy) spectrometer with modified potentials. This apparatus isshown in Figure 9.1 and consists of three grid plates, G1 to G3, a stack ofMCPs and an anode. Typical operation voltages are listed in Table 9.1.As detailed in Part II of this thesis, we create a molecular Rydberg vol-ume that moves within a supersonic beam expansion. Laser excitation takesplace in the relatively field free region between G1 and G2. The Rydbergvolume, as it travels, has a few microseconds time to evolve before reachingdetection grid G2. Upon transition through G2, this volume encounters aforward-potential, typically >100 V/cm, which extracts loosely bound elec-trons and accelerates them towards the stack of 2.54 cm diameter MCPs.Here, impacting electrons are multiplied at a total gain of 1 · 107 and col-Element VoltageG1 variableG2 grnd.G3 100VMCP front 200VMCP middle 1200VMCP back 2200VAnode 2400VTable 9.1: This table details typical voltages applied to the components ofthe ZEKE detector - see Figure 9.11159.1. The ZEKE and moving-grid detectorsFigure 9.1: Schematic of our 1D detection scheme - The molecularbeam enters from the left along the center-line of the detector. It passesthrough a fine grid plate G1. As the jet traverses the distance between platesG1 and G2, it is intersected with a nano-second laser pulse. The excitedvolume continues to travel with the molecular beam to reach detection gridG2. G1 and G2 are grounded, G3 holds a positive voltage. Upon transitthrough grid G2, the system encounters an electric field that acceleratesnegatively charged particles towards a MCP detector. The charge impact isamplified and read out via a detector anode.lected at the anode. A capacitive output coupler monitors the supply ofvoltage to the anode and allows us to view time-resolved charge impact asan oscilloscope trace. Transit time of our excited volume through G2 is onthe order of one microsecond. For the purpose of our experiments, electronextraction and detection are instantaneous.In 2009, we introduced a new detector that has since been at the centreof our experimental work.14 This detector is based on the ZEKE spec-trometer with the added feature, that the plane of detection can moveup and down the streamline of the molecular beam. This is achieved bymeans of a bellows-isolated linear actuator driven by an animatics inte-grated SM23165D Smart Motor. The detector assembly travels on 1/2 inchmounting rods via linear bearings along a range of ⇠ 10 cm. By moving the14Design and construction of this new detector was spearheaded by Chis Rennick, atthe time a post-doctoral fellow in our group. A more complete account of the design andoperation of ZEKE and moving-grid detectors can be found in [26, 49].1169.2. Observations and discussionFigure 9.2: E-drawing of the moving-grid detector - The detector ismounted on the 10” diameter backflange of our vacuum chamber. FMP1and FMP2 represent front-mounting plate 1 and 2, respectively. Detectionis analogous to the ZEKE scheme in Figure 9.1. The entire detection car-riage, including G2, FMP2, MCPs and anode can be translated along thez-dimension, thus varying the time after which the laser excited volumetransits detection grid G2.detector up and down the molecular jet streamline, we vary the durationover which the excited volume can evolve prior to detection at G2.9.2 Observations and discussionOver the following pages, I will discuss key observations through use of our1D detection systems.1179.2. Observations and discussion9.2.1 Rydberg spectrumAs discussed in Chapter 6.2, we prepare a cigar-shaped Rydberg volume,moving with the velocity of our molecular beam. The experiment is preparedaccording to Part II, yielding initial conditions stated in Table 6.2. Thedetector voltages conform with table 9.1, with grid G1 on ground.Nitric oxide spectroscopy was reviewed in Chapter 2. We populate NOquantum-states near the ionization continuum through double-resonant laserexcitation. When calibrating our laser system, we first set !2 to ⇠ 321 nmand !1 to ⇠ 226 nm. One !1 and one !2 photon combined are energeticenough to ionize the NO ground state. We ensure that neither of the twolaser beams in itself is able to multi-photon-ionize our sample by observingthe detector response as we vary the intensity of one laser while the otheris blocked. Following this, we scan for the desired intermediate transitionbetween rotational ground state and rovibronic excited state by 1+1’ reso-nance enhanced multi photon ionization (REMPI) signal analysis with !1.15Excitation, by means of an intermediate state, boosts the creation of ionsand we obtain a transition spectrum between the X2⇧1/2 ground state andthe A2⌃+ first excited state of NO. We apply a small forward bias betweenthe plate G1 and grid G2 to collect the liberated electrons.We tune to the pQ11(1/2)-line, representing the transition between thelowest rovibronic level of the NO ground state and the lowest rovibronic levelof the A-state, to select A2⌃+ (⌫ 0 = 0 K 0 = 0 J 0 = 1/2) as the intermediatestate of the excitation process.With !1 fixed to populate this state, we next reduce the !2-photonenergy below the ionization potential, replace the forward bias between G1and G2 with a reverse bias of ⇠ 500 mV/cm and scan !2 for a Rydbergresonances.A typical resonance signal, as we observe it on the oscilloscope, is dis-played in Figure 9.3. We see an electron prompt peak after ⇠ 250 ns, orig-inating from Penning ionized electrons that leave the illuminated volume15This was displayed in Figure 5.5.1189.2. Observations and discussionFigure 9.3: Typical plasma signal observed by ZEKE or moving-grid detector - Laser excitation take place in the field-free region betweengrids G1 and G2. We excite to n0=50. The first peak at ⇠ 200 ns representsprompt electrons that are created during the plasma formation process. Thesecond peak at ⇠ 8 µs gauges the charge density and the spatial dimensionsof the plasma volume as it traverses grid G2.shortly after the process of laser excitation (cf. Chapter 7). These promptelectrons leave a small charge imbalance behind. In a first-order model, thisimbalance traps the remaining electrons and forces them to travel with theslow ionic volume. We observe an impact signal that is measured as theilluminated volume transits G2, ⇠ 8 µs after excitation. At this point, thequasi-free electrons are extracted by the electric field between G2 and G3and accelerated towards the MCP.Having obtained a signal, we integrate over the plasma peak (secondpeak) as we scan our Rydberg state distribution (!2-scan). This is shownin Figure 9.4. We average ten shots before taking a step of 0.001 nm in !2-wavelength. The x-scale is vacuum-corrected. We record the peak positionsof the Rydberg series. Least-square fit of the Rydberg formula (Eq. 3.3 fora series of PQNs) to the measured series allows us to assign IP and quantumdefect to the series as well as the principal quantum number to each peak.1199.2. Observations and discussionFigure 9.4: Resonance in the plasma signal observed scanning !2- The (late peak) plasma signal is averaged over one second and then in-tegrated to yield an intensity measure. This signal is recorded as !2 isscanned with a step-size of 0.001 nm. The assigned peaks correspond to thenf -Rydberg series which converges to the rotational level, N+ = 2, in the1⌃+ vibrational ground state of NO+. The ionization potential is expectedto be at ⇠ 327.58 nm.DiscussionThere are two curiosities found in the data displayed in Figure 9.4: First:The observed Rydberg spectrum is toward higher wavelength dominatedby one series. The extrapolated ionization potential of this series confirmswith the N+ = 2 rotational level of the 1⌃+ vibrational ground state ofNO+. Second: Toward lower wavelength, the signal response rises, reflect-ing an increase in state density, combined with more ecient avalanche toplasma, moderated by a declining electronic transition moment. The res-onance peaks at wavelength !2=327.85 nm, which lies ⇠0.3 nm below theIP in photon energy space. The signal decays quickly for higher energies.1209.2. Observations and discussionThe shape of our spectrum is by no means obvious and we have only re-cently understood the underlying dynamics. I will delay their discussionuntil Chapter 10.A discussion of the dominance in f -series requires further spectroscopicconsiderations. As discussed in Chapter 2, Rydberg states conform withHund’s case (d). The A-state is Hund’s case (b). In both states, K, thetotal angular momentum apart from spin, is a good quantum number andselection rule,K = 0,±1,applies. For any dipole-transitions, the selection rule J = 0,±1 holdsrigorously. The total angular momentum of the A-state is J = K +S = 0+1/2 = 1/2. The transition K=0 ! K=0 is forbidden because of selectionrule S=0 and the fact that the absorption of one photon must change theangular momentum by one unit of ~. Thus, the populated Rydberg statemust have values J = 3/2 or 1/2, K=1 and S=1/2. Table 9.2 lists possiblecombinations of angular momentum coupling for accessible Rydberg states.Selection rules alone cannot explain the dominance of the f -wave inFigure 9.4. The reason is two-fold: Dixit et. al. [22] have conducted a the-oretical study of the rotational branching ratios in the resonance-enhancedmultiphoton ionization of NO via the A-state. The valence electron of theA-state, due to interactions with the non-spherical molecular potential, isnot excited into a p-wave, as expected for the atomic case, but into a stateof coupled `-waves. Their results show that p- and f -waves dominate in thiscoupled state based on shape-resonance.A look at Table 9.2 reveals that p-waves approach NO+ in either R=0 orR=2 core states. f -waves approach NO+ in Rydberg molecule core statesR=2 or 4.16 Performing pulsed field ionization experiments in a similar setupto ours, Vrakking and Lee [87] found that the dominante Rydberg series in16Note, that the Rydberg system is described in Hund’s case d), the ion, however, inHund’s case b). In accordance with Herzberg’s notation, I will refer to the molecular corerotation with variable R for the Rydberg system and variable N for the ion.1219.2. Observations and discussionTable 9.2: Possible angular momentum coupling for Rydberg states popu-lated from the ground rovibronic level of the NO A-state.J = R + L + S character1/2 1 0 -1/2 s-wave1/2 0 1 -1/2 p-wave1/2 2 -1 -1/2 p-wave1/2 -1 2 -1/2 d-wave1/2 3 -2 -1/2 d-wave1/2 -2 3 -1/2 f -wave1/2 4 -3 -1/2 f -wave1/2 -3 4 -1/2 g-wave1/2 5 -4 -1/2 g-wave-etc-3/2 1 0 +1/2 s-wave3/2 0 1 +1/2 p-wave3/2 2 -1 +1/2 p-wave3/2 -1 2 +1/2 d-wave3/2 3 -2 +1/2 d-wave3/2 -2 3 +1/2 f -wave3/2 4 -3 +1/2 f -wave3/2 -3 4 +1/2 g-wave3/2 5 -4 +1/2 g-wave3/2 -4 5 +1/2 h-wave-etc-NO (excited from the A-state) are np(R = 0), converging to 30,522.443cm1, and nf(R = 2), converging to 30,534.394 cm1. Their results agreewell with our findings.The p-series only begins to emerge for energies higher 36f(R = 2). Asdiscussed in Chapter 7, internal conversion of Rydberg gas to plasma takesseveral nanoseconds. For long wavelengths, collisional cross-sections of Ryd-berg molecules become small and conversion to plasma takes longer. But thelifetimes of the `-waves are very short for states ` < 3. For example, beforethe onset of `-mixing, states excited to n=50 have lifetimes ⌧ s50 = 1.36 ns,⌧p50 = 0.41 ns, ⌧d50 = 0.56 ns and ⌧f50 = 15.8 ns (cf. Table 3.2). This explains1229.2. Observations and discussionthe absence of the p-series for longer wavelength.9.2.2 Plasma formationThe operation principle of our detector can make it hard for us to di↵er be-tween the signal response of a plasma transiting G2, from that of a Rydbergvolume. In the case of a plasma, electrons bound by the ionic space chargeare extracted by the field between G2 and G3 and swept into the detector.In the case of the Rydberg gas, and for suciently strong voltage on G3,field-ionization removes electrons upon transit through G2 and sweeps theminto the detector. The e↵ective signal is similar.There are two indicators in Figure 9.3 which lead us to believe that weare indeed observing a plasma: 1) The prompt peak appearance is unjusti-fied in case of a Rydberg gas. 2) The lifetime of ⇠8 µs is unjustified for aRydberg gas prepared at n <100 (see Section 3.2). In the following, I willdiscuss an experiment which serves as further proof of concept for plasmaformation as discussed in Chapter 7.Figure 9.5 displays the electron signal as a function of electric field ob-tained by applying a voltage ramp in between grids G1 and G2. Our Ry-dberg gas was prepared in the following way: Laser !1 has ⇠ 8 µJ powerand we determined for this dataset a FWHM of 2.57 mm via razor-blade-tomography (Gaussian width of !1 = 1.1 mm (no teslescope on !1)). Forthe measurement we closed an iris on !1 to a diameter of 1.7 mm, onlyallowing the maximum intensity through. This created a cylinder of A-statemolecules in the molecular beam with only one dimension (x-axis) beingGaussian and the other two dimension approximately constant in density.Based on the laser width and power, I calculate an A-state peak density of3.11 · 1012 cm3 (cf. Chapter 6.1).Laser !2 has ⇠ 2 mJ power and a FWHM of ⇠ 3 mm, much larger than!1. We allow for a 200 ns delay between lasers !1 and !2 (approximateNO life-time) to reduce the Rydberg gas density by a factor of ⇠1/e [40].Assuming saturation of !2, we expect to populate a Rydberg state peak1239.2. Observations and discussiondensity of 5.7 · 1011 cm3 (or mean density of ⇠ 2.6 · 1011 cm3).17!2 is tuned to prepare this Rydberg ensemble in state 50f(R = 2). Weobserve the system response to a 3 kV field ramp (forward-bias detectionof electrons). The first 1/3 of the ramp is approximately linear and reaches1 kV after ⇠500 ns. The distance between G1 and G2 is set to ⇠4.55 cmusing the moving-grid detector. Figure 9.5 shows the response of the illumi-nated volume to the linear ramp from field-strength 0 to 200 V/cm. The sixdi↵erent figures represent delays in the application of the field of 10, 200,400, ..., 1000 ns (accuracy ⇠ ±10 ns). Each (time-delay) figure compiles3000 single-measurements, sorted vertically in the post-data-analysis by to-tal electron yield.DiscussionThe first thing we notice in viewing Figure 9.5 is a variation in responsedespite constant experimental conditions. This is caused by the interplaybetween molecular beam and lasers, which prepares the Rydberg ensemblein each measurement with slight variations in shape and density. Densersystems have a higher electron yield. Our sorting of the 3000 measurementsby electron yield creates a contour-plot with increasing density along they-axis. I assume that the densities in Figure 9.5 are distributed around amean value. The horizontal traces displayed near the middle of the y-axisshould correspond closely to this mean. I refrain from labeling this axis asI expect a non-linear spread around the mean density.Our analysis of this data is straightforward. Electrons pulled o↵ theilluminated volume at low field strength indicate the presence of a weakly-bound plasma surface-charge. Electron signal at higher fields represents fieldionization of Rydberg states.17In a Gaussian distribution, 50% of particles sit within a distance of 1.26 from thecentre. At this distance, the density has a value of ⇠0.45 times the peak density.1249.2. Observations and discussionFigure 9.5: Field ionization by a linear voltage ramp - Our Rydbergensemble is prepared at n0=50. We measure the system response to a linearvoltage ramp (forward-bias for electrons) between G1 and G2. The field israised from 0 to ⇠200 V/cm in ⇠400 ns. Each figure displays 3000 single-measurements in a contour plot, sorted according to total electron yield.The application of the ramp is systematically delayed by steps of ⇠200 ns.Overall, we observe the transition of Rydberg gas to plasma.1259.2. Observations and discussionOur measurements at 10 ns ramp-delay displays two distinct Rydbergfeatures with an onset near ramp voltages E⇡62 V/cm and E⇡114 V/cm.Based on the discussion in the last section, we expect to predominantlyexcite Rydberg states in the np(R = 0)- and nf(R = 2)-series. A quickspectroscopic calculation shows that our !2-photons which populate Ryd-berg state 50f(R = 2) also populate 59p(R = 0). E=114 V/cm conformswith the minimum field necessary to diabatically ionize 50f -Rydberg statesto NO+(N = 2). Here, I used the following relation:4.12 ·pE = Ry(n l)2 (9.1)Eq. 9.1 derives from Eq. 3.16. The factor 4.12 is a fitted parameter andreplaces the hydrogenic factor 4.59 from Eq. 3.16 [26]. E is in V/cm andRy=109737.3 cm1. E=62 V/cm confirms with the field necessary to ionizethe 59p-state approaching NO+(N = 0).In studying this data at a 10 ns ramp-delay, we note that for a fractionof runs (⇠ 20%), clear plasma character can already be observed. As weincrease the delay between !2 and application of the ramp, the transitionfrom Rydberg to plasma moves down the y-axis to lower densities. Overall,this behaviour agrees very well with our plasma rise-time model for a spreadof initial densities, as discussed in Chapter 7. The data clearly displays thetransition of a Rydberg gas to plasma.I wish to derive an estimate of the initial Rydberg densities in this data.In other words, at which time-delay display ramp-traces at medium y-axishight (mean of density distribution) over 50% plasma characteristics. Whilethese traces show clear Rydberg character for an early ramp, a very crudevisual analysis shows that plasma character becomes dominant after ⇠200ns.Let’s compare this value to a theoretical prediction. Applying the com-putational model from Chapter 7, I find that trise ⇡200 ns for n0=50 con-firms with a mean density of ⇢Ry0 = 5·1010 cm3 (or peak density of ⇠ 1·10111269.2. Observations and discussioncm3).Traces in the high-density region of the data appear to have transformedto plasma on the nanosecond timescale. This is typical for initial Rydbergdensities of ⇠ 1 · 1012 cm3. Overall, I estimate that the data displayedin Figure 9.5 is spread over two orders of magnitude in peak density from⇠ 1 · 1010 cm3 to ⇠ 1 · 1012 cm3 with a mean at ⇠ 1 · 1011 cm3. Fortypical measurements with coincident laser excitation, i.e. no dealy between!1 and !2, I expect values multiplied by the Euler factor.Another approach for analyzing this data-set is displayed in Figure 9.6.Here, I compute the total signal of each of the 3000 traces for each of the6 time-delays. I sort the resulting totals into bins and display the evolu-tion of signal distribution over time-steps 10, 200, 400, ..., 1000 ns. Theblue curve in Figure 9.6 (10 ns-delay) shows an approximately normal dis-tribution skewed slightly to lower total electron signal. One can understandthis in the following way: For each run, our experiment prepares a Ryd-berg volume with a density sampled from a distribution with a mean anda spread. The interplay of molecular beam and lasers are responsible forthis shot-to-shot variation. Even during the population of Rydberg statesthrough !2, higher order processes begin to take place. Penning ionization,dissociative recombination and electron-Rydberg collisions are second orderprocesses in density and constitute a loss-mechanism for signal electrons.At a given !1!2 delay, Rydberg systems formed at higher density sustainhigher-rate second and third-order loss processes. These processes depletethe fraction of traces with large total signal, yielding the observed skew ofthe distribution to lower total signal.Overall, the ramp-field delay data discussed in this section agrees verywell with the theoretical concepts developed in Chapter 7.1279.2. Observations and discussionFigure 9.6: Summation and binning of ramp-delay data - Each curverepresents one of the ramp delays. I integrate over each of the 3000 oscil-loscope traces and sort the resulting total into bins. The 10ns-delay data(blue) appears closest to a normal distribution while 200, 400, ..., 1000ns delays (green, red, ..., yellow) move progressively closer to low count.This progression illustrates the second-order plasma decay due to DR. Thetop-right corner bar plot illustrates the total loss of signal over the firstmicrosecond evolution.1289.2. Observations and discussionFigure 9.7: Evolution of the plasma peak - Plasma waveform capturedby the moving-grid detector as a function of flight time. The system isprepared in the n0=50 Rydberg state. G1 holds a +2V positive charge todeflect the prompt peak (line at t=0). The di↵erent traces are verticallyo↵set for better clarity. The Figure shows how the plasma expand anddecay, as it evolves in time.9.2.3 Plasma expansion and decay in 1DSo far, our discussion was restricted to experiments with a fixed distancebetween the point of laser excitation and the detection grid G2. In thefollowing, I will introduce a dataset won through use of the moving-griddetector.The experiment is once more prepared according in initial conditions inTable 6.2. The detector voltages conform with Table 9.1, with a potentialof +1.2 V on grid G1. !2 is fixed to populate the n=50 Rydberg level. Thecreated plasma is now characterized by the shape of the plasma peak. Weare interested in the change of its shape, as the plasma evolves in time.We bring our movable detector close to the region of illumination andstart a scan in which only the detector position, relative to the region of1299.2. Observations and discussionplasma formation, is varied. Figure 9.7 displays the change of the plasmasignal as the detector is moved along the stream line of the molecular beam.This allows the plasma an evolution timespan between ⇠ 1.5 µs and ⇠ 40µs. The separate signal traces are averaged over 10 shots and have beenvertically o↵set for clarity. The small positive voltage on G1 reduces theprompt peak, as initial electrons are pulled away from the detector.DiscussionTwo immediate observations can be drawn from Figure 9.7: (1) As theplasma evolves in time, it decays as indicated by the decline in plasma peakamplitude. (2) During its evolution, the plasma expands as indicated by theincrease in signal width.A closer study of the data reveals that the individual traces agree re-markably well with Gaussian waveforms:II0,(z) = I0 · exp✓ z222◆(9.2)To aid the characterization of this data, I fit Eq. 9.2 to the signal tracesand extract fitting parameters I0 and . The relative area of the plasmasignal is A =p2⇡I0. I obtain 95% confidence bounds for I0 and  throughmy fitting routine. The error of area A is obtained using error propagation.The results of this characterization is displayed in Figure 9.8.Our data displays the broadening of the plasma to five times it’s ini-tial z-width over 40 µs evolution. The evolution can be separated into tworegions, before and after the ⇠10 µs mark. As discussed in Chapters 4, 7and 8, our plasma forms from a dense Rydberg gas via internal conversionon a nanosecond timescale. In its initial state, the electrons carry almostall the kinetic energy of the system. In their attempt to leave the plasma,they pull on the ions and thus transfer energy into the radial motion of theions. This mechanism is called ambiploar expansion and continues until the1309.2. Observations and discussionFigure 9.8: Plasma evolution characterized through Gaussian fits- We characterize our dataset through Gaussian fitting parameter: width (top), area A (middle), peak hight I0 (bottom). Displayed error barsrepresent 95% confidence bounds.1319.2. Observations and discussionsystem kinetic energy is almost entirely captured within the radial motionof ions. The top graph in Figure 9.8 shows the acceleration of ions typi-cal for ambipolar expansion. After ⇠10 µs, the acceleration stops and theexpansion becomes ballistic. In our data, we observe the decrease of thedistribution peak hight to become linear as well as a levelling out of thedistribution area around this point.The self-similar Vlasov pictureNext, I will conduct a comparative study of theory and experimental re-sults. I wish to compare our experimental observations with simulationresults based on the formalism developed in Part III in order to draw someimportant conclusions.I begin by comparing our experimental data to a non-dissociating plasmaexpansion in the self-similar Vlasov picture, as given by Eqs. 8.7. I use ini-tial conditions from Table 7.1, leaving the electron temperature as a loosefitting parameter in order to reproduce our data. This is done in Figure9.9. I find that our experimental expansion curve is closely reproduced bysetting the initial electron temperature to Te(0)=1.9 K. This is in strongcontrast to our expected temperature value, Te(0) ⇡200 K, which was basedon molecular dynamics calculation (see Section 7.2).The observation of this unexpectedly slow expansion of beam plasmareaches as far back as our first experiments. Our search for the root of thisdramatic mismatch between theory prediction and experimental observationhas been going on for many years. MOT plasmas do not seem to exhibitthis behaviour, leading us to suspect that the molecular nature of our systemmight be the root of the very slow expansion. The primary characteristicof molecular compared to atomic plasma is dissociative recombination. Ideveloped the expansion model discussed in Section 8.2 in order to allow fora direct comparison between molecular plasma theory and our experimentaldata.1329.2. Observations and discussionFigure 9.9: Comparison of experiment and simulation results: exper-imental data points as in Figure9.8; self-similar non-spherical Vlasov curve(cf. Eq. (8.7)) for initial conditions from table 7.1 and 1.9 K electron tem-perature.Plasma expansion under the influence of dissociativerecombinationI wish to compare simulation results from my dissociating shell model to ourexperimental data curve. To do so, I first need to take a closer look at themethod by which our detector collects plasma information. One weaknessof our current detection scheme is that our detector only extracts relativecharged particle density measurements along the z-dimension while it inte-grates over x and y (cf. Figure 9.10). This represents a loss of informationand complicates comparison between data and simulation.I can illustrate this e↵ect by asking, what response a plasma sphere ofconstant density, sent through our detector, would create. This sphere is1339.2. Observations and discussionFigure 9.10: Our detection scheme: As our plasma volume transits thedetection grid, a forward bias extracts electrons and accelerates them ontothe MCP. Our oscilloscope trace, as collected by a metal anode, is one-dimensional and represents the flow of electrons through the x-y-plane atthe detector entrance grid.defined as:⇢(r) =8<:⇢0 for r  10 for r > 1 (9.3)As this sphere passes into our detector, the entrance grid works as a samplerof charge. For a unitary density sphere, the obtained signal at any point intime is given by the convolution of sphere and grid. As seen in Figure 9.11,this is a circle area with radius R =p1 z2, z being the distance from thecentre of the plasma. The response of the detector is now given byResponse / ⇢0⇡ ⇥R2 = ⇢0⇡ ⇥ (1 z2) (9.4)This example illustrates how a spherical object of unitary density is turnedinto a quadratic signal and helps us understand our detector response.Above, I compared our experimental data to a self-similar expansion. Inthis, I took advantage of the fact that we measure along a symmetry axis1349.2. Observations and discussionFigure 9.11: Detector response to a unitary plasma sphereand that a Gaussian density distribution, integrated over two dimensions,remains Gaussian. This is not the case for a dissociating plasma.To obtain a means to compare and draw conclusions from calculations, Isimulate our detector response for 3D shell model data in a way comparableto the unitary density sphere example. The output of my 3D shell modelsimulation yields shell densities ⇢j(t), shell positions Rk,j(t), shell velocitiesuk,j(t) and electron temperature Te(t) for shells j at time-step t and alongdimension k = x, y, z.I select a single time-step t and fit a ’SmoothingSpline’ to density overshell positions to reconstruct full density information for our charged particledistribution. Next, I define segments of equal distance, Z1,..,m,..,M startingat Z1 = 0, along the positive z-axis (for an ellipsoidal centred coordinatesystem). I compute ⇢(Zm) and subsequently find Xm and Ym for which⇢(Xm) = ⇢(Ym) = ⇢(Zm). This defines a new shell system of arbitraryexactness.The Zm’s will serve me as bins for a particle count. Bin m is definedto sit in between two planes, both parallel to the x-y-plane, at position Zmand Zm+1. I want to count the particles within each bin to reconstruct ourdetector response. My m shell surfaces are now defined as:x2X2m+y2Y 2m+z2Z2m= 1 (9.5)1359.2. Observations and discussionI want to find the intercept of my shells with the planes at positions Zm. Iselect a single m and find the intercept of shells n = m+ 1, ..,M as ellipsesgiven through:x2X2n+y2Y 2n= 1 Z2mZ2n(9.6)The individual ellipses are fully defined through their major and minor radii:xm,n =sX2n ·✓1 Z2mZ2n◆(9.7)ym,n =sY 2n ·✓1 Z2mZ2n◆(9.8)I can now construct the area between two consecutive ellipses asAm,n = ⇡ · (xm,n · ym,n  xm,n1 · ym,n1), (9.9)from which I calculate the the number of particles within bin m asNm =XnAm,n · ⇢(Zn) ·, (9.10)where  is the bin width, i.e. the distance between Zm and Zm+1. Com-pleting this process for all bins results in a half peak, which when mirrored,represents the response of our detector to a system obtained by the 3D shellmodel.I now apply my detector simulation formalism to 3D shell model dataand compare the result to experimental data. In a first step, I consider thefirst ⇠1 µs of evolution of a 2 K electron temperature dissociating plasmaprepared at a peak density of ⇢ = 6 · 1012 cm3.18 Figure 9.12 displays thedensity distribution along z for my dissociating shell model plasma for four18This density conforms with the theory prediction from Chapter 7. As seen in Sub-section 9.2.2, indirect plasma density measurements fall short of this value by an order ofmagnitude. Nonetheless, I will begin my discussion with the case of a high-density plasmato draw some important conclusions.1369.2. Observations and discussiontime steps in the early evolution of the system. I have chosen a time at whichexpansion is still at an acceleration state while DR is highly dominant in thecore of the plasma. Dissociation is proportional to ⇢2 and T1/2e (cf. Eqs.8.12 and 8.19). One can see how the initially Gaussian distribution flattensat the peak and evolves away from the Gaussian form. The flattening of thepeaks is accompanied by a fall in peak density from initially ⇠ 6 ·1012 cm3down to ⇠ 0.2 · 1012 cm3 within the first 1µs. The lower graphs in FigureFigure 9.12: Density evolution and corresponding detector responseover the first µs: I have simulated our system using my shell model for-malism and initial condition based on Table 7.1. The initial electron tem-perature was set to 2 K. Above figures show the evolution in charged particledensity along z. Figures below correspond to the expected detector response(Gaussian fits included).9.12 represent the traces I would expect to see on the oscilloscope, if plasmasas simulated by the shell model formalism would pass into our detector. Onecan observe that the flat distribution heads vanish as the detector model in-tegrates simulated plasma slices over x and y. There is a simple explanationto this as illustrated on a spherical shell plasma undergoing DR: the flat-1379.2. Observations and discussiontening of the density distribution is due to the ⇢2-dependence of dissociativerecombination. As it turns out, the part of the plasma most a↵ected byDR lies within the 1 radius of the density distribution but contains lessthan 20% of the total particle number. The plasma wings, however, remainnear-Gaussian and constitute the bulk of charges. Our method of detectionintegrates over plasma slices and we are unable to discern core-dissociationwithout further tools.I fit Gaussian curves to my simulated detector response in order to makethese DR e↵ects more visible. One can note in particular, that the plasmawidth grows over the first 1 µs in steps of 34, 15, 11 µm, i.e. apparent growthslows down. This is counterintuitive as one would have expected ambipolarexpansion to accelerate the system. As it turns out, this apparent initialexpansion is a remnant of DR, which flattens our density distribution leav-ing our detector response slightly non-Gaussian. This misleads our fittingroutine towards observing an increase in width while ambipolar expansiononly begins to accelerate the system. Figure 9.13 illustrates this e↵ect moreclearly by comparing the apparent detector width of dissociating shell plas-mas at temperatures of 15, 7.5 and 1.9 K over a lifetime of 35 µs. The blackcurve on top of the data points represents shell model calculations with DRdeactivated and serves as a test for the accuracy of the simulation rutine.Figure 9.13 discourages the notion that DR could be the origin for thevery slow expansion in our experimental data. Our riddle persists. Theoverall expansion of our system, that is the slope of a line through the datapoints in Figure 9.13, places our initial electron temperature at about ⇠2K. Such low temperature, however, would cause DR to reshape our detec-tor response and become visible through deviation from Gaussian form asindicated by the accelerated growth right after plasma formation as well aslarger error bars. We observe neither in our experimental data.One possible explanation remains within the framework constructed inChapter 8. In the following, I will explore the possibility of an initial plasma1389.2. Observations and discussionFigure 9.13: Comparison of experiment and simulation results: Topto bottom - detection simulation for dissociating plasma with initial condi-tions from table 7.1 and 15, 7.5 and 1.9 K initial electron temperature. Theblack curve represents 3D shell model data without DR at 1.9 K electrontemperature. It serves as proof of accuracy for the computation. Experi-mental data points taken from Figure 9.8.density much lower than 1 · 1012 cm3. As DR is a second-order process inplasma density, low initial densities could explain our failure to observe itin our data. I will fix the initial electron temperature in my computationto Te(0) =1.9 K and consider di↵erent initial density values ⇢(0) = 1 · 1010,1 · 1011, 1 · 1012 cm3. Figure 9.14 compares the expected width and areaevolution for this computation to the experiment data.Studying the shape progression of the expansion curves in Figure 9.14,it becomes apparent that system peak densities lower than 1 · 1010 cm3 areable to reproduce our experimental results even under the influence of DR.1399.2. Observations and discussionFigure 9.14: Comparison of experiment and simulation results -Computation results for Te(0)=1.9 K and densities ⇢(0) = 1 · 1010, 1 · 1011,1 · 1012 cm3 (green, blue, red) compared to experimental data from Figure9.8. Top: Evolution of distribution width along z. Bottom: Evolution ofarea measured along z.1409.3. Chapter summaryHowever, a look at the area evolution for the low density system simulationsshows vast disagreement from the experimental results.In summary, I must conclude that the theoretical model developed inChapter 8 is unable to reproduce our experimental results, likely due to asystematic error in our assumptions.9.3 Chapter summaryOver the course of Chapter 9, I introduced our 1D detection scheme. InSection 9.2.1, I discussed an experiment in which we excited a molecularvolume via the NO A-state to the high-Rydberg manifold. We allowed thisvolume to travel with the supersonic beam towards detection grid G2 andobserved the following:• Our illuminated volume travels with the velocity of the molecular beamand generates signal at our detector for specific !2-energies.• A spectroscopic analysis shows that the resonant !2-energies conformwith the NO nf(R = 2)- and np(R = 0)-Rydberg energies.• Signal was observed down to PQNs of 30f(R = 2) and 39p(R = 0).This we understand in the following way: Plasma formation is fuelledby the Penning process. This process depends on Rydberg orbital size(/ n2) and lifetime of Rydberg states. nf -Rydberg states live muchlonger than np-states.Section 9.2.2 discussed selective field ionization measurements performedon our illuminated volume. The volume was prepared in Rydberg level50f(R = 2). A field-ramp was applied at di↵erent !2 to ramp time delays.I found that:• Our !2-laser populates degenerate Rydberg states 50f(R = 2) and59p(R = 0). We observe two dominant Rydberg features in our SFImap conforming with these two state binding energies.1419.3. Chapter summary• Our SFI map tracks the conversion of the illuminated volume fromRydberg gas to plasma. I was able to confirm our conceptual under-standing of plasma formation, as discussed in Chapter 7. A quantita-tive analysis put the Rydberg volume under study at an initial densityof 1 · 1011cm3.• Thus, typical Rydberg systems formed without !1 to !2 time-delayform at ⇠ 2.7 · 1011cm3.Lastly, Section 9.2.3 discussed expansion data obtained with our moving-grid detector. Once again, the system was prepared in state 50f/59p. Mydiscussion of this dataset yields the following results:• We are able to observe the system evolution over a timespan of up to45µs.• Our results show a system with a rapid decay of ⇠60% of its chargecarriers over the first ⇠5µs of evolution. Afterwards, decay appearsdeactivated.• The plasma exhibits a very slow expansion characteristic, suggestinginitial electron temperatures of only⇠2 K based on conversion of initialelectron temperature to radial ballistic ion motion during ambipolarexpansion.While we were able to learn a great deal about our experimental system,several fundamental questions remain. I summarize these in the following:• In Section 9.2.1, I could not explain the fact that the plasma resonancepeaks ⇠0.3 nm prior to reaching the nf(R = 2) ionization energy.• Molecular dynamics calculations for the process of plasma formationpredicted equilibrium electron temperatures on the order of ⇠200 K(cf. Chapter 7). An indirect temperature measurement through ex-pansion velocity obtained a value of ⇠2 K. Based on the findings inthis chapter, I cannot explain the mismatch.1429.3. Chapter summary• The decay of our plasma does not conform with expectations. Theexpected flattening of the plasma density distribution due to DR wasnot observed. The shut o↵ is too abrupt.There seems to be a link between the latter two open questions which addsinsight towards finding a solution. One can argue, that the absence of DRpoints towards a further contradiction in initial electron temperature. Thelow electron temperature causes DR to appear prominently in our simula-tion (kDR /pTe). Hot electrons are fast and not so easily captured by theNO ion. Thus, we have two indications for the presence of hot electrons andit seems prudent to consider other mechanisms of energy dissipation whichmight explain the observed slow expansion despite hot initial electrons.As discussed in Section 9.2.3, our 1D detectors integrate plasma distri-bution in y and z, which constitutes a loss of information. The search forthis unknown energy dissipation mechanism fuelled the development of ournew 3D plasma detector, discussed in the next chapter.143Chapter 10Plasma detection in 3DIn the following, I will introduce our new 3D imaging detector, which hasacquired the name ’PlasmaTV’ within our research group. With this detec-tor, we employ direct imaging to observe the dynamics of plasma expansionin x, y and z. Our findings provide evidence for surprising new channels ofenergy disposal that operate in the particular geometry of our laser-crossedmolecular beam plasma apparatus.10.1 The plasma-tv detectorThe key component of our new detector is a Chevron Model 3075FM de-tector assembly by BURLE ELECTRO-OPTICS INCORPORATED. Thisassembly consists of an 8 inch vacuum flange with a fiberoptic window of76.7 mm diameter at its center. Stacked on the vacuum side of the windowsits an indium tin oxide (ITO) layer with a phosphor coating. Mounted inparallel to this fiberoptic phosphor screen, and with a few millimetre dis-tance, are two Microchannel plates (MCPs). The active area of this detectorhas a 75mm diameter. A schematic of it is shown in Figure 10.1.Charged particles impacting on the MCPs create an electron cascadewithin the channels. This cascade exits the MCPs at the back and is ac-celerated towards the ITO anode. Typical voltages are MCPin = -200 Vto +200 V, MCPout = MCPin + 1.8 kV and Anode = MCPout + 2 kV.The electron cascade impacts on the phosphor, causing luminescence. Werecord the spatial information of such charged particle impacts using a CCDcamera. Additionally, we are able to record the temporal information via acapacitive output coupler, much like we do with our 1D detectors (cf. Chap-14410.1. The plasma-tv detectorter 9).19Our vacuum chamber is equipped with magnetic mu-metal shielding,fitted to the inside diameter of the chamber walls. Similar to previous ex-periments, we illuminate a volume of the molecular beam 150 mm after theskimmer and employ our BURLE detector in one of three configurations (cf.Figure 10.1):• a) We mount the detector onto a 10” to 8” zero-length-adapter, whichcloses our vacuum chamber. This creates an unobstructed flightpathof our molecular beam within the experiment chamber. The distancebetween skimmer tip and detector measures 468mm in this configura-tion. We detect particle impact on the MCP.• b) We use the previous configuration but add an 11” vacuum cham-ber extension, which allows for a total distance of 747 mm betweenskimmer and detection plane.• c) We employ an array of vertical grids (G1, G2) combined with anelectronic lensing system (plates and cylinders: P1, P2, C1, C2, C3)as displayed in Figures 10.1, c and 10.2. In this setup, we form ourRydberg volume between plates G1 and G2. The volume travels onlya short distance (max. 21mm) towards detection grid G2. Upon tran-sit through G2, loosely bound electrons are extracted by a forwardpotential (typically formed by 1kV on plate P2). Extracted electronscontinue on a trajectory towards the BURLE detector. The electroniclensing system is set to recreate a true image of the charge distributionat the detection plane G2, with a magnification factor of typically 10to 25.2019We detect an image that represents our systems density distribution in x/y whileintegrating over z. Simultaneously, we collect the charge impact on the detector to obtaininformation of the density distribution in z while integrating over x/y. Please note,that the obtained 3D distribution only approximates the full density information, as thisintegration process incorporates a loss of information.20The initial design of this electron optics array was done by Nicolas Saquet, at the timea post-doctoral researcher in our lab.14510.1. The plasma-tv detectorFigure 10.1: Schematic of plasma-tv detector in di↵erent config-urations - The BURLE detector is mounted at the back of our vacuumchamber. It consists of a set of MCPs and a fiberoptic phosphor screen.The trajectory of our skimmed molecular beam is normal to the plane ofdetection (MCPs). We record timing and x/y position of particle impacton the MCPs. In configuration a), the distance d between laser and MCPsis 453 mm. Configuration b) extends this distance to d=732 mm. Con-figuration c) shortens the free flight distance of the illuminated volume tod<21 mm. Here, detection grid G2 extracts spatial information of plasmaelectrons (see text).14610.1. The plasma-tv detectorFigure 10.2: Schematic of plasma-tv detector in configuration c) -This figure shows an e-drawing of our electronic lens system. The molecularbeam enters from the left. Vacuum chamber viewports allow for laser accessbetween plates G1 and G2. The illuminated volume transits G2 and looselybound electrons are accelerated towards the MCP detector at the end of thechamber.The design and work with configuration c) is more complex comparedto configurations a) and b). We based the dimensions of the electronic lensarray on simulations performed with the SimIon software. Typical voltagesof configuration c) are listed in Table 10.1. Figure 10.3 shows an instructivedisplay of particle trajectories within the lens array. The detector is setto project a magnified real image, of the charge distribution transiting G2,onto the MCPs.Figure 10.4 gives an example of the working of our new detector. Weplace a CCD camera in the back of our vacuum chamber and adjust distanceand angle until the image is focussed on the fiberoptic window of the BURLEdetector. The known window plate dimensions allow us to compute the infocus CCD pixel width. For measurements, we put a blackout around cameraand detector and view the lighting-up of the phosphor screen due to chargedparticle impact on the MCPs. The right-hand side of Figure 10.4 shows thecamera capture for an electron burst created between G1 and G2. For this14710.1. The plasma-tv detectorFigure 10.3: Charged particle trajectories within electrostatic lensarray - This figure was created with the aid of SimIon. It shows chargedparticle trajectories in real-space (top) as well as potential energy space(bottom) for detector voltages in Table 10.1. The detector is set to projecta magnified real image of the charge distribution transiting G2 onto theMCPs.14810.2. Observations and discussionElement VoltageG1 variableG2 grnd.P1 20 VP2 1 kVC1 grnd.C2 2 kVC3 50 VMCP front 200 VMCP back 2 kVAnode 4 kVTable 10.1: This table details typical voltages applied to the components ofthe PlasmaTV detector - see Figures 10.1, c and 10.2image, applied voltages conform with Table 10.1 and -300 V on G1. Weobserve magnification of the detection grid G2 of a factor 25.10.2 Observations and discussionThe following three subsections will discuss our findings employing the Plas-maTV detector in configurations a), b) and c). Sections 10.2.1 and 10.2.2closely follow recently published work, references [80, 82]. Recent qualitativeresults obtained through use of our new detector in configuration c) will bediscussed in Section Plasma bifurcationExperimentalExperimental data discussed in this chapter was won by preparing a Rydbergensemble as discussed in Part II of my thesis. Laser pulses !1 and !2 crossthe molecular beam at a position 143 mm beyond the skimmer, correspond-ing to a detection chamber flight time of t = 100 µs. The hydrodynamics ofthe skimmed supersonic expansion yield a beam at this point with a Gaus-sian radial distribution about its propagation axis that has a FWHM of 3.214910.2. Observations and discussionFigure 10.4: Example image PlasmaTV detector - The left-hand imageshows the phosphor screen as seen by our CCD camera, mounted in the backof the vacuum chamber. In the right-hand image, we’ve placed a blackoutaround CCD camera and detector. The image shows the lighting-up ofthe phosphor screen, as we create a free electron flux through grid G2.Magnification amounts to a factor of ⇠25.mm and a peak density of NO of 2.43 ⇥ 1013 cm3. The molecules have avelocity spread parallel to the beam defined by a longitudinal temperature,Tk = 0.6 K. Phase-space cooling reduces the perpendicular temperature byan amount proportional to the divergence, defined by r? = v?t.The fraction of NO molecules excited to the A-state varies linearly with!1 laser pulse energy. A rotatable Glan-Taylor polarizer and spatial filterin the optical train of !1 form a beam with a variable pulse energy between1.75 and 4.24 µJ. Assuming a saturated !2 transition, we estimate thatthe corresponding initial average density of the resulting Rydberg gas variesfrom 3⇥ 1011cm3 to approximately 7⇥ 1011 cm3.After the skimmer, the molecular beam travels without obstruction toreach the detector, as diagrammed in Figure 10.1. Interchangeable 10 inchdiameter conflat flange tubes determine the total, field-free, skimmer-to-detector flight distance. The use of a short tube section defines a distanceof 468 mm (configuration a). The longer section a↵ords an experimentalchamber flight distance of 747 mm (configuration b).15010.2. Observations and discussionThis plasma flies freely to impact the grounded input face of the MCPstack. To determine the width of this signal in z, we apply 1.8 kV tothe back if the detector, bias the ITO anode to a potential of 4.3 kV andrecord an oscilloscope trace reflecting the number of electrons incident onthe x, y plane of the detector as a function of time. The nozzle opens at afrequency of 10 Hz for a duration of 300 µs. The camera records images ofthe x, y distribution of electrons integrated in z by collecting photons for 30ms following each laser shot. Both camera images and oscilloscope tracesdiscussed in this section represent averages of 250 shots.Results:Spatial Distribution of Charge in the Ultracold Plasma as aFunction of Rydberg Gas Principal Quantum Number        -20       0      20         -20       0      20x mmx mm         -20      0      20y mm         -20      0      20y mmn0 = 32 n0 = 65        -20       0      20x mm         -20      0      20y mmn0 = 78Figure 10.5: Detector response recorded after preparing the Ryd-berg gas volumes with the following set of approximate initial densitiesand selected initial principal quantum number: (left) 3 ⇥ 1011 cm3 andn0 = 32, (centre) 3 ⇥ 1011 cm3 and n0 = 65, (right) 7 ⇥ 1011 cm3 andn0 = 78. Here, we have positioned the detector at the longer flight distancemeasured as 747 mm from the skimmer wall, yielding a flight time in allcases of 420 µs after laser excitation. All figures represent averages over 250CCD images.Figure 10.5 shows x, y particle density distribution detector images recordedafter preparing Rydberg gas volumes with initial principal quantum num-bers, n0 = 32, n0 = 65 and n0 = 78 in the f series. We allow these volumes15110.2. Observations and discussionElectron signal (arb.)        -20       0        20x mm        -20       0        20x mmElectron signal (arb.)        -20       0        20y mm        -20       0        20y mmElectron ignal (arb.)Time (µs)0            100          200          300          400          500n0 = 32z = 325 mmz = 604 mmz = 325 mm z = 604 mmElectron signal (arb.)        -20       0        20x mm        -20       0        20x mmElectron signal (arb.)        -20       0        20y mm        -20       0        20y mmElectron signal (arb.)Time (µs)0            100          200          300          400          500n0 = 65z = 325 mm z = 604 mmz = 325 mmz = 604 mmFigure 10.6: Distributions of electron density in x, y and z followingexcitation to Rydberg gasses with initial principal quantum numbers of n0 =32 and n0 = 65 and propagation over the distances indicated from the pointof laser interaction to the detector plane. (top) Distribution of the electrondensity over the horizontal cross-beam coordinate x, integrated in y andz. (centre) Distribution of the electron density over the vertical cross-beamcoordinate y, integrated in x and z. (bottom) Distribution of the electrondensity over the molecular beam propagation coordinate z, integrated in xand y, displayed as the waveform of the detector anode signal as a functionof time. Smooth red curves represent Gaussian (y and z) or sum of Gaussian(x) fits.to propagate a distance of 604 mm from the point of laser interaction. Then0 = 32 and n0 = 65 images were collected with the !1 attenuating po-larizer set to 50. This a↵ords a laser pulse energy of 1.75 µJ, well belowthe threshold for saturating the initial X2⇧ to A 2⌃+ transition. For thepurposes of obtaining the n0 = 78 image in Figure 10.5, we set the polarizerto 0, yielding a laser pulse energy of 4.24 µJ which nearly saturates the !115210.2. Observations and discussiontransition.When these illuminated volumes strike the detector, they produce imagesthat di↵er in shape from the Gaussian radial distribution of the molecularbeam in several important ways. The laser intersects the molecular beam toform a prolate ellipsoid. We can determine the size of this initial ellipsoidfrom the measured radial profiles of !1 and the molecular beam at the pointof intersection. Comparing the dimensions of this volume to the distributionsin Figure 10.5, we can determine that the excited density diverges faster thana correspondingly marked volume of the unperturbed molecular beam.Figure 10.6 plots distributions of electron density in x, y and z obtainedfor n0 = 32 and n0 = 65 measured after propagation distances of 325 and 604mm. Note for n0 = 32, the charge distribution in the x dimension exceedsthe neutral beam divergence by more than a factor of two. The expansion ofthe n0 = 32 image in the y coordinate increases faster than the calculateddivergence of an element of the core of the molecular beam marked to havethe initial 1 mm FWHM y dimension of the !1 laser beam. The width ofthe charge distribution measured along the axis of propagation, z, growsfaster than y. Note as well that the distribution evolves in y to exhibit anaccelerated, non-Gaussian tail.For n0 = 65, the plasma evolves in y much like it does for n0 = 32, show-ing again non-Gaussian tails in the expansion along ±y. Note, however, thatthe distribution extends over a much wider distance in x, and exhibits a dis-tinctive low-density hole in the centre, on the axis of the molecular beam.Table 10.2 details the measured widths in x, y and z following short andlong flight paths for n0 = 32 and 65.Gaussian waveforms fit to the anode current signals in Figures 10.6 es-tablish arrival time di↵erences for long and short flightpaths of 194.1 µs forn0 = 32 and 194.9 µs for n0 = 65. The average of this flight time di↵er-ence divided by the distance di↵erence of 279 mm yields a beam velocity,vbeam=1434 m s1. With this beam velocity, the di↵erence in the x separa-tion of the density maxima measured for n0 = 65 determines that the lobes15310.2. Observations and discussionTable 10.2: Experimental measures of FWHM in x, y and z dimensions forexcited volumes striking the imaging detector following short (unprimed)and long (primed) flight times, as derived from Gaussian fits to intensitydistributions pictured in Figure 10.6. All values are expressed in mm. Notethat we can determine unperturbed x0 = 10.25 and y0 = 3.3, from theshort flight path divergence of the beam, marked simply by its intersectionwith !1. We further predict z0 = 6.9 based on the z-dimension ther-mal velocity of ’marked’ molecules as well as !1-laser size. We convert thetemporal widths in z and z0 to spatial widths using vbeam.x y z x0 y0 z0n0 = 32 20.42 6.25 16.08 31.71 12.14 28.77n0 = 65 - 6.17 17.72 - 10.61 32.12in these charge density images separate with a ±x velocity of 74 m s1.The cross-beam, x expansion of the prolate excitation volume variessystematically with the Rydberg gas initial principal quantum number cho-sen by tuning !2. Figure 10.7 plots normalized distributions of the electronsignal as a function of x for selected nf(2) principal quantum numbers from28 to 78, all recorded with an !1 pulse energy of 1.75 µJ.Contour plots corresponding to these images show that y remains con-stant as function of n0. However, the distribution along x, which beginswith a single mode at n0 = 28, clearly widens and splits to a degree thatgrows with increasing initial principal quantum number.Spatial Distribution of Charge in the Ultracold Plasma as aFunction of Rydberg Gas DensityReference to Figure 10.7 shows that, with a laser pulse energy of 1.75 µJ, theplasma formed from a Rydberg gas for which n0 = 58 bifurcates in x to showa slight bimodal character. Rotating the polarizer in 10 steps to reduce itsde-alignment with !1, increases the !1 pulse energy and changes the spatialdistribution for n0 = 58. As shown in Figure 10.8, the distribution along xdevelops distinct modes, and these separate with increasing !1 pulse energyfrom 1.75 µJ to 4.24 µJ. The highest pulse energy approaches the point of15410.2. Observations and discussion78 73 68 65 62 6058 56 53 50 48 4644 42 39 37 35 3332 31 30 29 280    20   40   60 0    20   40   60 0    20   40   60 0    20   40   60 0    20   40   60 0    20   40   60 0    20   40   60 0    20   40   60 0    20   40   60 0    20   40   60 0    20   40   60 -20 0  20x mm-20  0  20x mm-20  0  20x mm-20   0  20x mm-20 0   20x mm-20   0   20x mmElectron signal (arb.)Figure 10.7: Normalized distributions of y, z integrated electrondensity as function of x, collected using a an !1 pulse energy of 1.75 µJwith long flight path for selected initial principal quantum numbers, n0f(2),in the range from 78 to 28.saturating the !1 transition. For an !1 energy of 3.74 µJ, the separationevident here for n0 = 58 exceeds that observed for n0 = 78 at 1.75 µJ.Figure 10.8: x, y plasma images collected using a long flight path with asingle initial n0f(2) principal quantum number of 58 and varying laser pulseenergies from 1.75 µJ to 4.24 µJ.15510.2. Observations and discussionDiscussion:Electron Impact Avalanche and the Evolution to PlasmaThe volume of Rydberg gas formed by double-resonant excitation evolves toplasma in two steps. Within the time of the !2 laser pulse, pairs of Rydbergmolecules that happen to form within a critical intermolecular separation, rc,Penning ionize to yield an ion plus free electron and a deactivated Rydbergmolecule. This initial ionization forms a space charge that traps the majorityof electrons. These electrons collide with the remaining Rydberg moleculesin the charged volume. Electron-Rydberg collisions drive transitions in theRydberg manifold, and after a time that depends on the density of Penningelectrons, ⇢e, an electron-impact avalanche begins. This avalanche quicklyforms an ultracold plasma quasi-equilibrium of ions, electrons and Rydbergmolecules.The density of promptly formed Penning electrons, ⇢e, depends on theinitial density of the Rydberg gas, ⇢0, and the selected initial principalquantum number, n0, to a degree determined by the Erlang distributionfunction of nearest-neighbour distances, integrated from 0 to the criticalvalue, rc.⇢e(n0, ⇢0) =0.924⇡⇢20Z rc0r2e4⇡3 ⇢0r3dr (10.1)The plasma formation mechanism was discussed in detail in Part III of thisthesis.Dynamics of plasma expansion in the plane perpendicular tolaser propagationThe plasma formed by electron impact avalanche at lower values of n0 andlower !1 laser pulse energy grows in major and minor axis dimensions at arate that exceeds the divergence of the molecular beam by about a factor oftwo. In the previous chapter, I have analyzed the electron signal waveformobtained by collecting the integrated charge in successive x, y planes as theplasma volume transits a perpendicular grid to gauge the rate of expansionalong the short axis aligned with the molecular beam propagation direction,15610.2. Observations and discussionz. This expansion conforms with the hydrodynamics of an ambipolar ex-pansion driven by the pressure of an electron gas with a temperature in therange of ⇠2 K.For all values of n0 and !1 laser pulse energy, the expansion rate in theshort axis dimension, y, measured crudely here by the di↵erence between theimage widths in y for short and long flight times, accords with observationsof z expansion seen both in earlier experiments, and for present conditionsin the oscilloscope traces at the bottom of Figures 10.6. This short axis ex-pansion does not vary significantly, either with increasing principal quantumnumber or increasing !1 laser pulse energy.Dynamics of plasma expansion in the direction of laserpropagationIn contrast with the unvarying short-axis expansion dynamics observed hereand in previous experiments, the cross-beam, long-axis dimension of theplasma volume evolves dramatically, bifurcating to form separating lobes ofcharge density. Inspecting Figures 10.7 and 10.8, we see that the velocityof separation, v˘x, grows with either an increase in the level of Rydbergexcitation or initial laser pulse energy.In separate experiments, we have studied the saturation of the NO A-state population as a function of the !1 energy. These studies show thatthe interval of !1 intensities used in the present experiment fall in the upperend of the regime of linear absorption. Under conditions of NO A-statesaturation, we estimate a Rydberg gas density ⇢0 ⇡ 1⇥ 1012 cm3 [74].For lower laser powers, in the linear regime, we associate !1 laser pulseenergies with Rydberg gas density as shown in Figure 10.9. Thus, assuminga near-saturation density of 7⇥1011 cm3 for the Rydberg gas prepared usingan !1 laser pulse energy of 4 µJ, we can estimate a Rydberg gas density of3⇥ 1011 cm3 in the first frame of Figure 10.8, and all of the measurementsshown in Figure 10.7, which were obtained at laser pulse energy of 1.75 mJ.The experimental images provide striking evidence that the ellipsoidalplasma volume forms repelling plasma volumes. In the top frames of Figure15710.2. Observations and discussion0 20 40 60 80 100 120 140 0 2E+11 4E+11 6E+11 8E+11 Relative Velocity (ms-1 ) Rydberg Gas Density (cm-3) 0 0 1 2 3 4 Laser Pulse Energy (µJ) 0 20 40 60 80 100 120 25 35 45 55 65 75 Relative Velocity (ms-1 ) Principal Quantum Number 0.E+00 5.E+10 1.E+11 25 35 45 55 65 75 Penning Electron Density (cm-3)  Principal Quantum Number 0 5E+10 1E+11 1.5E+11 2E+11 0 2E+11 4E+11 6E+11 8E+11 Penning Electron Density (cm-3) Rydberg Gas Density (cm-3) 0 2 6 4 8 x1011 0 2 6 4 8 x1011 Figure 10.9: Top: Relative velocity, v˘x, with which mesoscopic volumeelements of the plasma charge distribution separate along x as a functionof Rydberg gas initial principal quantum number at constant density of 3⇥1011 cm3, and as a function of Rydberg gas density at a constant principalquantum number of n0 = 58. Bottom: Penning fraction as a functionof Rydberg gas initial principal quantum number at constant density of3 ⇥ 1011 cm3, and as a function of Rydberg gas density at a constantprincipal quantum number of n0 = 58, as determined by Eq. (10.1).10.9, we see clearly that the recoil velocity of these separating lobes growslinearly with an increase in Rydberg gas initial principal quantum number,or initial Rydberg gas density.Raising n0, increases the orbital radius of the excited NO molecules in theRydberg gas. Increasing the density decreases the average distance betweenRydberg molecules, and more importantly, increases the fraction of nearestneighbour distances that fall within a particular radius.The question now remains, whether a single property of the Rydberggas varies with a change in orbital radius (depends on n0) or a change inaverage inter particle distance (depends on ⇢0) in such as way as to causethis uniform variation in v˘x.15810.2. Observations and discussionReferring to Eq. (10.1), we note that pairs of NO Rydberg moleculeswith nearest-neighbour separations that fall within a critical distance instan-taneously interact to form a population of Penning electrons. The densityof this population, ⇢e, rises both with n0 and with ⇢0. The lower frames ofFigure 10.9 draw upon Eq. (10.1) to plot the variation of ⇢e with n0, for afixed Rydberg gas density of ⇢0 = 3 ⇥ 1011 cm3, and the variation of ⇢ewith ⇢0 for a fixed principal quantum number of n0 = 58.Referring to this information, we can recast recoil velocities observed asfunctions of n0 and ⇢0 into a universal form that represents the variation invx with a change in the Penning electron density, ⇢e. Figure 10.10 showsthat the simple, self-consistent choice of 3⇥ 1011 cm3 as the Rydberg gasdensity produced by double-resonant excitation using an !1 pulse energy of1.75 µJ, linearly relates v˘x to ⇢e over a factor of three in recoil velocity andmore than an order of magnitude in Penning electron density.0 20 40 60 80 100 120 140 160 0 5E+10 1E+11 1.5E+11 2E+11 Relative Velocity (ms-1 ) Penning Electron Density (cm-3) 0.5 1.5 1.0 2.0 x1011 Figure 10.10: Relative velocity, v˘x, with which lobes of the chargedistributions separate along x as a function of Penning electron density,for plasmas formed both at constant !1 pulse energy with varying principalquantum number, n0 (circles) and at constant quantum number with varying!1 pulse energy (squares).15910.2. Observations and discussionPlasma recoil velocity as a function of Penning electron densityWhether increased by raising the initial principal quantum number or in-creasing the initial Rydberg density, we see that, v˘x, the recoil velocity ofplasma segments separating along the long-axis dimension, x, scales uni-formly with Penning electron density, ⇢e. This quantity reflects the popula-tion of electrons and ions that appears instantaneously on the timescale ofthe plasma frequency.We represent ⇢e in Figures 10.9 and 10.10 by its average value in theRydberg gas volume, as determined by the orbital radius (n20a0) taken to-gether with the average spacing ([3/4⇡⇢0]1/3). But, while double-resonantexcitation selects a uniform value of n0 in the Rydberg gas, the laser-crossedmolecular beam illumination forms a non-uniform excited state density dis-tribution. The charge distributions pictured in Figure 10.5 for n0 = 32with a short flight path and lower !1 pulse energy clearly show that theellipsoidal Rydberg gas produced by the intersection of a Gaussian !1 laserbeam and Gaussian molecular beam peaks decidedly in the centre. Thisa↵ects the distribution of plasma electrons that forms on the timescale ofthe laser pulse.10.2.2 Ultracold plasma hydrodynamics in threedimensionsThe last section introduced the concept of plasma bifurcation as a means toconvert core electron thermal energy into mesoscopic recoil. In this section,I wish to further discuss the process that facilitates this energy transfer pro-cess and propose a qualitative mechanism leading to plasma bifurcation.Plasma expansion as a measure of electron temperatureElectron temperature values exceeding 100 K represent a signature feature ofavalanche ionization to form a dense ultracold plasma. In a freely expandingplasma, the ambipolar pressure of this electron gas causes a radial accelera-tion of the ions. The Vlasov equations describe this acceleration analytically16010.2. Observations and discussionFigure 10.11: Left: Self-similar expansion of Gaussian ultracold plasmas:(grey lines) Ambipolar expansion of a model Gaussian plasma core ellipsoidof cold ions and Te = 180 K electrons with y(0) = z(0) = 83 µm andx(0) = 250 µm. Ions rapidly attain ballistic velocities, @ty = @tz = 272m s1 and @tx = 132 m s1. (blue line with red data points) Experimentalmeasure of @tz(t) fit by Vlasov model for a Gaussian spherical expansionfor Te = 5 K. Right: Electron signal as a function of flight time to G2observed in the moving grid apparatus diagrammed in Figure 9.2 with aconstant reverse bias on G1 of 1.20, 0.60, 0.48 and 0.24 V (top row), andforward bias of 0.0, 0.12, 0.24 and 0.36 V (bottom row), all with aflight distance to G2 of 56 mm. The Rydberg gas was prepared at n0=50.16110.2. Observations and discussionfor the self-similar expansion of a spherical Gaussian plasma. We and oth-ers have extended these hydrodynamics both analytically and numericallyto non-spherical plasma distributions (see Part III of this thesis).In the limit of saturated !1 excitation, laser-crossed molecular beamillumination creates an ellipsoidal volume of Rydberg gas with a peak densityas high as ⇠ 1012 cm3. A coupled rate-equation model (c.f. Figures 7.2and 7.3) predicts avalanche in a regime of this density on a nanosecondtimescale to form an electron gas with an initial temperature, Te(0) = 180K. The pressure of this electron gas drives the expansion of the ions. Thesteeper charge-density gradient in the y, z coordinate directions causes therate of radial expansion in the short axis plane to substantially exceed thatalong the cross-beam x axis of the ellipsoid.Grey curves in Figure 10.11 diagram the self-similar ambipolar expan-sion of a Gaussian plasma ellipsoid of initially cold ions and Te = 180 Kelectrons in a subspace with y(0) = z(0) = 83 µm and x(0) = 250 µm.In a Vlasov model, electrons of this temperature accelerate the distributionof NO+ ions to a z expansion rate, @tz, of 272 m s1. This radial veloc-ity, which approaches 20% of the laboratory speed of the molecular beam,greatly exceeds the observed expansion rate of the ultracold plasma in z de-termined experimentally by the growing widths of waveforms such as thosedisplayed in Figure 9.7. A typical set of late-peak Gaussian widths yieldsvalues of z(t) plotted as the red data points in Figure 10.11.Clearly, the system travelling with the velocity of the supersonic beamhas little in common with this hot plasma predicted by rate-equation models.The employment of our moving-grid detector at a fixed G1 to G2 distanceof 56 mm, while varying the potential on G1, yields new insights. Datadisplayed in the right frame of Figure 10.11 provides clear evidence for thepresence of two distinct domains of ultracold plasma evolution. The forwardbias between G1 and G2 reveals points toward an avalanche of a high-densityRydberg gas that produces a di↵use, rapidly expanding plasma of energeticions and electrons. A reverse bias of 480 mV cm1 or more sweeps theseelectrons away, isolating the slowly expanding waveform of the ultracold16210.2. Observations and discussionplasma. But, the signal of electrons in advance of the late peak waveformpersists up to a reverse bias as great as 250 mV cm1, signifying an initialfast-component Te of 200 K.21How does the observation of a rapidly expanding plasma domain of en-ergetic electrons and ions figure into the process of plasma bifurcation?Plasma evolution in a Rydberg gas of non-uniform densityWe propose the following process for ensemble bifurcation: Laser-crossedmolecular beam excitation forms an ellipsoidal Rydberg gas in which den-sity distributes di↵erently in x, y and z. The local density of ground-stateNO does not vary in the molecular beam propagation direction, z. Thus,the Rydberg gas density distribution in this coordinate simply reflects thenarrow Gaussian intensity profile of the laser beam. The Rydberg gas hasthe broader Gaussian width of the molecular beam in the x coordinate di-rection, defined by the propagation of the laser. The profiles of the laserbeam and molecular beam combine to form a steeper gradient of Rydberggas density in y, determined by the product of these Gaussians.The superimposed distributions of laser intensity and molecular beamnumber density create a Gaussian ellipsoid of NO⇤ Rydberg molecules. Thedensity distribution peaks at the core of this volume. There, rate-equationsimulations predict avalanche on a nanosecond timescale, forming a localpopulation of free electrons in a quasi equilibrium with initial temperatureTe(0) as high as 180 K. Such a high electron temperature must drive plasmaexpansion.For the purposes of a conceptual model, we represent the density distri-bution of this core plasma by means of a set of concentric ellipsoidal shells.To simplify, we will neglect both the thermal motion of the ions and anyprocess that leads to neutralization, such as dissociative or three-body re-combination, so that the number of charged particles in each shell remainsconstant.21This experiment was performed by my colleague Rafael Haenel. My thanks to himfor allowing me to use his data in my thesis.16310.2. Observations and discussionIn the quasi-neutral approximation, an electric potential gradient repre-sents a force, erk,j(t), which accelerates the ions in shell j in directionk according to (see Chapter 8): em0rk,j(t) =@uk,j(t)@t=kBTe(t)m0⇢j(t)⇢j+1(t) ⇢j(t)rk,j+1(t) rk,j(t) (10.2)The instantaneous velocity, uk,j(t) determines the change in the radial co-ordinates of each shell, rk,j(t),@rk,j(t)@t= uk,j(t) = k,j(t)rk,j(t) (10.3)which in turn determines the shell volume and thus its density, ⇢j(t).The electron temperature supplies the thermal energy that drives thisambipolar expansion. Ions accelerate and Te falls according to:3kB2@Te(t)@t=  m0Pj NjXk,jNjuk,j(t)@uk,j(t)@t(10.4)where we have defined an e↵ective ion mass that recognizes the redistributionof ion momentum by free-electron-mediated resonant ion-Rydberg chargeexchange, which occurs with a very large cross section [62]:m0 =✓1 +⇢⇤j (t)⇢j(t)◆m (10.5)The initial avalanche in the high-density core of the ellipsoid leaves fewRydberg molecules, so this term has little initial e↵ect. Rydberg moleculespredominate in the lower-density wings. There momentum sharing by chargeexchange assumes greater importance.As indicated by the high-temperature expansion curves in Figure 10.11,core ions reach ballistic velocities in just a few hundred nanoseconds. Thisquenches electron temperature. After expanding less than 50 µm, the motionof the core ellipsoid becomes a flux of ballistic ions and very cold electrons.16410.2. Observations and discussionMuch of this population streams out of the Rydberg gas volume in they and z directions to produce the early signal pictured in Figure 10.11.But, a substantial fraction of these expanding ions and electrons remain inthe volume of Rydberg gas, streaming in ±x. Here, ion-Rydberg chargeexchange has two important consequences:(1) A rapid sequence of electron transfer processes act to redistributethe directed momentum of the ions over the entire population of Rydbergmolecules and ions [6]. The heavy particles relax to correlated positions [72],and the two volumes stream as a whole in opposite directions.(2) Charge exchange increases the e↵ective inertial mass of the ions,which acts to retard the ambipolar expansion of the plasma forming in thebifurcated volumes.10.2.3 Recent workIn the following, I include two recent experiments and their qualitative dis-cussion. These experiments serve as proof of concept for my discussion thusfar and identify families of experiments in which we plan to conduct majorwork in the future.Experiment 1:Experimental: Figure 10.12 displays data obtained with the 3D-imagingdetector in configuration a). Here, we have tuned our lasers to prepare Ry-dberg molecules at PQN 64. Our detector voltages were set to the followingvalues: MCP input = +7V, MCP output = 1.6 kV and Screen = 3.5 kV.The positive input on the MCP is unusual as we tend to set this potentialeither to ground or slightly negative to achieve a better signal-to-noise ra-tio. We record 100 single images and corresponding oscilloscope traces withconstant experimental conditions.The bottom left image in Figure 10.12 displays the pixel average of these100 shots. We observe strong bifurcation of the plasma ellipsoid and notetwo peculiarities in the image. The first is a strong charge particle impactbetween 10 and 11 o’clock in the image. The second peculiarity is an asym-16510.2. Observations and discussionmetry in signal strength between the two bifurcated volumes. The top signaltrace in Figure 10.12 displays the corresponding oscilloscope measurement.Discussion: I define two distinct measurement areas in image and os-cilloscope trace. Henceforth, signal A refers to particles travelling with thevelocity of the molecular beam. These particles impact on the detector be-tween 200 and 250 µs after laser excitation and light up the phosphor withinthe central measurement area. I have discussed signal A, the arrested sub-component, at length in the last section. Signal B appears ⇠170 µs priorto signal A and corresponds to particle impact on the top-left corner of thephosphor screen.What is the origin of signal B? In the current configuration, our detec-tor has a positive input face and repels positive charges. I can rule out ionimpact. We observe that the variation of potentials on the MCP input face,between +7 V and +25 V, moves signal B on a line from the observed posi-tion to the image centre.22 Examining the electric field environment withinthe vacuum chamber gives me further insights. The high voltage vacuumfeedthroughs which apply potential to our detector are located, as seen bythe camera, in positions 11 o’clock (screen) and 10 o’clock (MCP out). Atthe time of this measurement, these feedthroughs were not shielded.23 In-terplay between the large MCP input area and these connectors creates aninhomogeneous field, channeling negative charges emitted by the evolvingRydberg volume toward the top-left corner of the detector. This inhomoge-neous field is also responsible for the asymmetry between the left-hand andright-hand bifurcating volumes.We observe an interesting correlation between signal B and the mag-22I claim correspondence between signals B on the oscilloscope and screen based oncircumstantial evidence. Varying the voltage on the MCP input face, I can move signal Bo↵ the screen, which results in it’s disappearance on the scope.23Usually, we install a shield which screens these potentials.16610.2. Observations and discussionFigure 10.12: Detector response in configuration a) - Our system isprepared at n=64. MCP input is at +7 V, MCP output at 1.6 kV and Screenat 3.5 kV. We record 100 single images and corresponding oscilloscope traces.Top: Averaged oscilloscope traces. Bottom left: Averaged CCD image anddefinition of top-corner and centre measurement areas. Bottom right: Foreach of the 100 images, I sum over the total pixel response of area B andsort the images accordingly. I then integrate pixel in the centre area A inthe vertical and stack the obtained traces. The obtained graph highlightsthe fact that the stronger signal B the stronger bifurcation in signal A.16710.2. Observations and discussionnitude of bifurcation. For this, I consider each of the 100 measurementsindividually. For each image, I extract the total pixel count of the top-leftmeasurement area (displayed in Figure 10.12) as well as a signal trace, ob-tained by integrating signal A in dimension y. I next sort the signal tracesin dependence of signal B and display my findings in the bottom-right ofFigure 10.12. I learn that the extent of bifurcation is intimately linked tothe magnitude of signal B. The stronger signal B the more bifurcation is ob-served. This result fits well into the concept of ballistic core ions streaminginto the distribution wings.Outlook: At present, we are planing a systematic study of signal B.For this, we need a better characterization and control of the field environ-ment in our experiment. At what point in time is B polarized? How highis the required field? Signal B appears to behave very much like the plas-mas observed in MOT experiments for direct ionization (not via Rydbergmanifold). Does our system become all B for higher photon energies, assuggested by Figure 9.4?Experiment 2:Experimental: We prepare our Rydberg ensemble at n0=57. Our detectoris in configuration c) and we set detector voltages to: G1=1.2 V / G2=grnd./ P1=20 V / P2=500 V / C1=grnd. / C2=2 kV / C3=50 V / MCPin=200V / MCPout= 1.5 kV / Screen=3 kV. Our molecular beam is illuminated⇠1.7 cm before grid G2. We record 100 single images and the correspondingoscilloscope traces. A ’by eye’ analysis of the images shows that we canclassify these images into three categories. The first category of imagesdisplay a cigar shaped volume transiting imaging plane G2. It is possible todiscern eight to ten vertical grid lines and one horizontal line in the images.This allows us to estimate the volume size at G2 to 4mm to 5mm in x-width and about 1mm in y-hight (grid spacing .5mm). We find that 36 outof 100 images fit this first category. The second category shows the samevolume, but with a distinct hole of varying size in the distribution. This16810.2. Observations and discussionhole is located predominantly towards the centre of the volume. We find32 images in this category. The third category displays images similar tocategory two, but with a distinct signal appearance at the centre of the hole.We recognize that our detection grid G2 imprints its shape onto this hole-centre-signal. The oscilloscope signal corresponding to category three tellsus that the hole-centre-signal appears 0.5µs prior to the rest of the cigarshaped volume. For the other two categories, the scope trace was purelyGaussian. 32 images fit this last category.Figure 10.13 displays examples of category one in the first row, exam-ples of category two in the second row and examples of category three inthe third and fourth rows. The bottom left subfigure shows the oscilloscoperesponse for figures of the last category.Discussion: Grids G1 and G2 are separated by a distance of 20.75 mm.A voltage of +1.2 V on G1 yields a field of 0.58 V/cm (G2 at ground).Loosely bound electrons are pulled away from detection grid G2. Positiveions can reach G2 but not the positively charged input face of the MCP.Our plasma forms in this environment.Viewing the first row (category) of images in Figure 10.13, we recognizethe ellipsoidal shape of our illuminated volume. The corresponding oscillo-scope trace shows that this signal travels with the velocity of the molecularbeam to arrive at the detector after ⇠12.5 µs. The shape of the detected vol-ume does not seem to be perturbed by the small external field. We recognizeplasma state A from experiment 1.Images of category 2 (second row) display a distinct hole at the centreof the ellipsoidal distribution. Possible reasons for this observation are: a)The core has decayed/dissociated to neutrals which cannot be field ionizedby the potential between G2 and P2. b) The core has developed into a statewhere it can be polarized by the small external field. Electrons are pulledaway from G2 toward G1.Images of category 3 seem to support the latter hypothesis. Here, weobserve a strong signal originating at the location of the missing core in16910.2. Observations and discussionFigure 10.13: Detector response in configuration c) - Our system isprepared at n0=57. Out of 100 single shot images, I display example cap-tures of 3 families of CCD camera responses (1st line / 2nd line / 3er &4th line). The bottom left oscilloscope trace corresponds to the 3rd family.Discussion in text.17010.3. Chapter summarycategory 2. Note, that this core-signal completely disappears for voltageson G1 higher than 2V. This core signal di↵ers distinctly from the imagesof distributed charge. The oscilloscope trace for category 3 images tells usthat this signal arrives up to 500 ns earlier at the detector. This can beexplained via two mechanisms: 1) Particles responsible for this signal haveexcess kinetic energy compared to the molecular beam. They are faster andarrive earlier. 2) The polarizing field between G2 and P2 does not abruptlybegin at G2 but reaches a small distance through the G2 mesh into the regionbetween G1 and G2. Detection through polarization takes place earlier.Likely is that both mechanisms are present. The round shape of the ob-served core-signal pores, as opposed to the quadratic shape of the grid, canbe explained via mechanism 2. At the same time, the bottom-right imageshows this core-signal extending beyond the original hight of the ellipsoid iny, suggesting the presence of excess kinetic energy. It stands to reason thatthis core-signal corresponds to plasma state B in experiment 1. We con-firm that, for a yet unknown reason, our plasma develops into two distinctmacroscopic regions.Outlook: The characterization of the nature of the arrested componentlies at the centre of our current work. Our new 3D detector in configurationc) allows us to study the early time behaviour of our system. Is a microscopicsystem variable or perhaps a race-condition responsible for the macroscopicboundary between regions A and B? Systematic experiments, similar to theabove, might lead us to understand this boundary, which in turn allows usto understand the observed state of arrest.10.3 Chapter summaryImaging of our ensemble density distribution at long evolution times showthat the ellipsoidal ultracold plasma formed in a supersonic beam by avalancheionization in a Rydberg gas of nitric oxide can spontaneously break symme-try, to channel a substantial fraction of its excess energy into the cross-beamlaboratory motion of opposing plasma volumes. As discussed in Section17110.3. Chapter summaryFigure 10.14: Time sequence of plasma bifurcation - At the core of theRydberg ellipsoid, plasma self-assembles into a hot state, distinct from thesurrounding outer layers. Ambipolar expansion quenches electron kineticenergy and accelerates the core as a shock though the outer layers, causingbifurcation via ion-ion collisions and resonant charge exchange.10.2.1, the dynamics of this process can be controlled by selecting the initialdensity and principal quantum number of the Rydberg gas. These obser-vations point to a new physics of ultracold plasma relaxation, in which thethermal energy of avalanche-heated electrons transforms first into radial mo-tion of electrons and ions, and then into the recoil energy of plasma lobes.Section 10.2.2 discussed the observation of two plasma components, onehot and one cold. Based on this observation, I proposed a qualitative mech-anism explaining plasma bifurcation. Here, core avalanche produces a cloudof initially hot electrons and cold ions. Ambipolar expansion quenches elec-tron kinetic energy and accelerates core ions to velocities of as much asone-fifth the laboratory speed of the molecular beam. These ions stream inthe ±x direction, into the wings of the Rydberg gas. There, recurring reac-tions that exchange charge between NO+ ions and NO⇤ Rydberg moleculesequilibrate velocities and change ion motion within the gas to ±x motionof gas volumes in the laboratory. In the wings, momentum redistributionowing to continuing charge transfer with the residual high-Rydberg popula-tion retards axial expansion. By redirecting electron energy from ambipolaracceleration to ±x plasma motion, NO+ to NO⇤ charge exchange dissipateselectron thermal energy and preserves density. This sequence of events lead-ing to plasma bifurcation is shown in Figure 10.14.17210.3. Chapter summaryLastly, I introduced two recent experiments and discussed my qualitativefindings. These are summarized in the following:• I confirm the development/self-assembly of our system into 2 macro-scopic states (A and B), observed as early as 12 µs into system evo-lution. These two states are distinguished by their appearance timeon the oscilloscope as well as their impact location on the phosphorscreen.• We know about state A that it:– Is quasi-neutral and travels with the velocity of the molecularbeam.– It survives field pulses exceeding 100 V/cm, applied 500 ns after!2 [51].– Does not show signs of decay after evolution times of 5-10 µs (cf.Section 9.2.3).– Makes up the bifurcated wings.• We know about state B that it:– Reacts much stronger to electric fields.– Appears to be polarizable a few microseconds into its evolutionby fields as low as 0.5 V/cm.– Appears to have excess kinetic energy.• We confirmed that bifurcation acts as a link between system states Aand B. The larger the observation of state B the more bifurcation.Overall, experiments 1 & 2 seem to confirm our current concept on plasmaformation. That our Rydberg system is able to develop two distinct macro-scopic domains which are linked through bifurcation. Future work in ourgroup will focus on the detailed study of the interplay of domains A and B.In the concluding remarks of the last chapter, I stated three fundamentalquestion which have for years been at the centre of our work: 1) Why does17310.3. Chapter summaryour plasma signal vanish completely prior to reaching the IP? 2) Why don’twe observe indirect e↵ects of hot electrons? 3) Why does the signal decayshut o↵? Based on the findings of this chapter, I can explain the first twopoints in the following:• The Rydberg scan displayed in Figure 9.4 captures the transit of thelong-lived plasma component through detection grid G2. For highPQNs, this component bifurcates and develops a macroscopic recoilvelocity. At some point, further increase in PQN pushes these bifur-cated volumes out of the detection aperture of our detector, resultingin the observed disappearance of the signal. The question remains,whether a long-lived plasma component even exists at !2-energies nearthe IP.• We do observe indirect e↵ects of hot electrons. However, other thanexpected these e↵ects manifest through bifurcation rather than am-bipolar expansion in z.Our observation of two distinct domains of plasma is of fundamental im-portance. Two questions naturally arise: What aspect of the Rydberg gasrelaxation dynamics distinguishes these two domains, which separate so ev-idently in the molecular beam propagation direction, z? How do these dy-namics shut o↵ signal decay in a highly unstable system?174Chapter 11Arrested relaxationIn previous chapters of my thesis, up to Section 10.2.3, I have discussed es-tablished and for the most part published and reviewed material. In the fol-lowing chapter, my last chapter, I will break with this narrative and discussup-to-date research topics, which still raise controversy within our researchcommunity. At the heart of the following discussion lies our observation ofthe shut-o↵ of signal decay within the long-lived plasma component whichtravels with the velocity of the molecular beam. I will quantify our observa-tions with regard to this phenomenon and speculate on the correspondingmicroscopic system state.11.1 Comparison of model calculations andexperimentPart III of this thesis represents a toolbox which allows predictions of ourRydberg ensemble evolution within idealized theory models. In the follow-ing, I will select experimental data from the previous chapter, and predictensemble evolution based on initial experimental conditions. Comparisonof simulation and experimental results will highlight shortcomings of ourcurrent computational model as well as understanding.Initial conditionsFor the following, I will select experimental data discussed in Section 10.2.1,more specifically the dataset discussed on the left-hand side of Figure 10.6.Here, I prepared our Rydberg system in PQN 32 and probe its density distri-bution after 225 and 421 µs evolution in 3 dimensions. I choose this dataset17511.1. Comparison of model calculations and experimentdue to the lack of apparent bifurcation. The initial conditions at which thissystem was prepared are collected from Section 10.2.1 and summarized inTable 11.1. I adjust the initial Rydberg gas peak density to reflect my find-ings in Section 9.2.2.Table 11.1: Initial conditions corresponding to experimental data discussedin Figure 10.6 (left-hand side).Rydberg ensemble initial conditionsFWHMx ⇠3.2 mmFWHMy ⇠1 mmFWHMz ⇠1 mm⇢Ry ⇠ 2.7 · 1011 cm3n0 32Evolution according to model calculationsChapter 7 developed a plasma formation model by which a uniform-densityRydberg gas undergoes Penning ionization to yield a seed density of quasi-free electrons. Subsequent electron-Rydberg collisions model the avalancheprocess turning the Rydberg ensemble to plasma. Computation results bythis model for densities of 100%, 80%, ... , 20% of ⇢Ry = 2.7 ·1011 cm3 andinitial PQN of 32 are summarized in Table 11.2.Table 11.2: Plasma rise-time calculations for uniform density Rydberg sys-tems. Density in [1011 cm3].% [⇢0Ry] ⇢Ry trise [ns] Te(2 · trise) [K] ⇢(2 · trise)100 2.7 ⇠160 ⇠200 2.180 2.16 ⇠214 ⇠200 1.760 1.62 ⇠308 ⇠200 1.2740 1.08 ⇠514 ⇠200 0.8520 0.54 ⇠1200 ⇠200 0.4317611.1. Comparison of model calculations and experimentFigure 11.1: Self-similar ambipolar expansion of a 200 K plasma -Initial condition according to Table 11.1 with Te(t = 0) = 200 K. Top: Evo-lution of plasma width according to self-similar model. Dimensions x and y,zare colour-coded blue and red. Bottom: Evolution of electron temperature.17711.1. Comparison of model calculations and experimentNote that plasma formation for this n0 = 32 system is much delayed(compared with higher PQN systems) due to the small Rydberg orbitalsize, yielding a much lower initial Penning density. Nonetheless, this systemis predicted to transition to plasma with ⇠ 200 K hot quasi-free electronson a microsecond timescale.In the following, I will take system dimensions from Table 11.1 and as-sume an electron temperature of 200 K as plasma initial conditions. Section8.1 provided a set of di↵erential equations, solving self-similar plasma ex-pansion in the absence of dissociative recombination.24 Figure 11.1 plotsresults obtained though this model. This simulation shows how, over thefirst 10 µs, ⇠90% of electron thermal energy is transformed into ion hy-drodynamic motion. The density gradient plays an important role in thisenergy redistribution process and acceleration along y/z exceeds that alongx. As a result, the system turns from prolate, via spherical shape after ⇠ 7.5µs, to oblate. After 425 µs evolution, the simulated system exhibits a size of⇠125x275x275 mm FWHM, exceeding the size of our experiment chamber.Observed experimental expansionOur molecular Rydberg ensemble, formed via 2-photon resonant transitionsin nitric oxide, is prepared far from thermal equilibrium. Each Rydbergmolecule receives over 9 eV excess energy through photon absorption. Modelcalculations, supported by experimental observations (cf. Section 9.2.2),show the transition of this ensemble to plasma. Our calculations predictthat a substantial fraction of the Rydberg excess energy flows into elec-tron thermal motion which subsequently drives ambipolar expansion. Ourexperimental observations, however, do not confirm this prediction.24At 200 K electron temperature, electrons travel very fast resulting in weak electron-ion recombination. This allows me to employ the simple modal from Section 8.1. Theinitial thermal as well as hydrodynamic motion of the ions is a result of the < 1 K thermalmotion of supersonic beam molecules at the point of skimming. I neglect this residualmotion in comparison with the initial electron temperature.17811.1. Comparison of model calculations and experimentFigure 11.2: Extrapolation of ensemble expansion based on experi-mental data from Table 10.2 - Dimensions x,y,z are colour-coded blue,red, green. Dots represent experimentally measured distribution widths at225 µs and 421 µs evolution. The lines represent fits of ambipolar expansioncurves to the respective two data points in each dimensions. X marker plotthe expected thermal width-evolution of a non-interacting ’marked’ expan-sion.Figure 11.2 plots the measured ensemble width after 225 and 421 µs fora system prepared at PQN 32. X marker predict the ensemble width at225 µs, assuming that the !1-laser simply ’marked’ the beam distribution,which subsequently evolves only according to the unperturbed thermal andhydrodynamic particle trajectories (cf. Section 6.2). Our experiment clearlyshows that some energy (⇠1 K) has been channelled into hydrodynamic ex-pansion. This value, however, falls ⇠99.5% short of the prediction from thelast subsection and represents a clear discrepancy between simulation andobservation. It conforms precisely with the very slow rates of expansion,determined for short times of flight in our moving grid machine (data fromFigure 9.8). The slow expansion rate of this component in all three spa-tial dimensions and over its entire trajectory puts a significant limit on the17911.1. Comparison of model calculations and experimentfinal kinetic energy of the excited molecules or ions.25 Expansion modelsin the ellipsoidal geometry of the system call for initial temperatures (Ryd-bergs/ions plus electrons) no more than 2 degrees Kelvin. If not channeledinto expansion, where are the ⇠9 eV excess energy per molecule?Experimental evidence for the arrest of decay channelsFigure 9.8 traces the number of signal generating particles as a functionof flight time observed for an ultracold plasma produced in a beam of NOseeded in helium. Here, a system with initial Rydberg gas density of⇠ 3·1011cm3 falls with the loss of its hot component and dissociative recombinationafter 10 µs to ⇠40% of its initial particle number. Thereafter, the lossof signal apparently stops, and the number of extracted electrons remainsconstant for as long as we can observe it in the moving grid apparatus.26The z waveforms in Figure 10.6 show that the longer flight-path imagingapparatus forms an anode waveform at 421 µs with comparable area thanthe one we observe with the detector positioned for a time of flight of 225µs. The persistent magnitude of this signal suggests that the correspondingRydberg  NO+ electron system lives a millisecond or more.Thus, our bifurcated domain exhibits remarkable chemical stability withrespect to neutral fragmentation on a millisecond timescale. Charge separa-tion persists, despite ultrafast predissociation channels that exist for nitricoxide molecular Rydberg states with lower angular momentum at all prin-cipal quantum numbers, and ecient electron Rydberg inelastic collisionsthat rapidly scramble populations in n and `. Our long-time observations ofnitric oxide molecular ultracold plasma evolution, thus appear to indicate astate of arrested relaxation.25In this formulation, I am being purposefully vague. As previously mentioned, ourdetector does not di↵er between a plasma electron and a high-Rydberg electron as longas it can be extracted from the ensemble to create a cascade in the MCP pores.26In similar fashion, after an initial period of dissipation, the evolving NO-Rydbergensemble entrained in a slower beam of argon survives undiminished out to 180 µs [31].18011.2. The inadequacy of ’classic’ models11.2 The inadequacy of ’classic’ modelsOur experiment clearly establishes the existence of a cold, durable compo-nent that forms from an elliptical Rydberg gas as it evolves to plasma. Wehave observed this durable component in cigar shape, morphing into twobifurcated volumes with a ±x recoil velocity that depends systematically onthe initial density of the Rydberg gas, ⇢0, and its selected principal quan-tum number, n0 (see Section 10.2.1). Rates of expansion suggest ensembletemperatures comparable to disorder induced heating. Rydberg/ion radialvelocities reflect little more kinetic energy than might be available throughspatial correlation (cf. Section 4.3.3). This durable component exhibitsremarkable chemical stability with respect to neutral fragmentation on amillisecond timescale.It remains now to speculate on a final state that best accounts for theproperties of this system. I consider four possible quasi-stationary statesprecedented by our current understanding of Rydberg gas – ultracold plasmadynamics:A ’hot’ classical plasma: This is the most logical system state andconfirms with predictions by model calculations in Chapter 7 of this thesis.Our SFI experiments (Figure 9.2.2) support this scenario. The evolution ofour Rydberg ensemble to plasma, however, must be accompanied by electronheating to temperatures exceeding 100 K (cf. Table 11.2). Fast electrons,in turn, should drive plasma expansion as discussed for example in Section11.1. Based on the observed lack of system expansion, we must dismiss thisscenario.A ’cold’ classical plasma: As a thought experiment, let’s consider thepremise of a ’cold’ plasma despite the lack of concept on how such a systemmight emerge. Early in its flight path from z = 0 to 600 mm, a classicalplasma with the properties suggested by our long-time system behaviourwould exist in a regime of strong coupling with respect to both ions andelectrons. Ecient three-body recombination and Rydberg relaxation wouldconsume ions, heat electrons and drive the system to exhibit a fast expansion,which we do not see. Furthermore, Section 9.2.3 considered the shape-shift18111.2. The inadequacy of ’classic’ modelsof the distribution due to high-order density dependent decay channels andfound that such mechanism should be apparent in our probing of the densitydistribution. Thus, we identify three inconsistencies in this scenario.A quenched system of long-lived high-Rydberg states: Such aquasi-stationary state accords with selective field ionization measurementsthat collect the entirety of the electron signal at exceedingly low voltage.Let’s derive an estimate of the peak density of the experimental system con-sidered in Section 11.1. Figure 11.2 fits expansion curves through the datapoints derived experimentally. After 15 µs evolution, the fitted curves sug-gest a system size of FWHMx=3.42 mm, FWHMy=1.07 mm and FWHMz=1.4 mm. Assuming an initial peak density of ⇢0Ry = 2.7 · 1011 cm3 andaccounting for a signal loss of 60% (cf. Section 11.1) over the first 15 µsevolution, I predict a remaining peak density of,⇢(15µs) =0x0y0z0x0y0z· ⇢0Ry · loss factor = 6.7 · 1010 cm3, (11.1)where k represents the distribution width along dimension k. Under com-parable conditions in a MOT, Penning interactions [68] and interatomicCoulomb decay [2] drive electron-impact avalanche on a timescale no slowerthan tens of microseconds [89]. Conventional relaxation would again heatelectrons and produce an expansion that fails to materialize in our experi-mental observations. Even after 100 µs evolution, above argument yields aremaining peak density of ⇢(100µs) = 2.2 · 109 cm3, which, according tothe model from Chapter 7 and assuming a PQN of 90 turns plasma withina microsecond.A relaxed system of deeply bound Rydberg states: Such a systemmight resist Penning ionization on the observation timescale so-far employedin our bifurcated plasma experiments. However, studies show that NO Ryd-berg states of low principal quantum number predissociate on a nanosecondtimescale (see Table 3.3). Our volumes show no evidence of such dissipa-tion. Moreover, Rydberg molecules with deep binding energies would showa characteristic SFI spectrum not seen in our experiments.18211.3. Future workThe process of elimination suggests unconventional or ’new’ physics asresponsible for the arrest of relaxation dynamics. The apparent experimentalcondition of arrested relaxation in our experiment naturally raises the ques-tion whether many-body and/or quantum mechanical coupling somehowconstrains the transport of matter and energy. An incoherent mechanismmay yet emerge, but our experimental observations substantially narrow thepossibilities that can rely on conventional kinetics.11.3 Future workThis thesis does not intend to solve the open question, which microscopicstate might explain our macroscopically observed system behaviour. Nonethe-less, the following section will introduce avenues of research which our groupcurrently pursues.How & when does arrest emerge?Part II of this thesis went to great length to describe the initial state of ourexperiment. Which mechanism could facilitate the transition of our initialsystem state into the arrested domain? What can we learn through thisquestion?Hypothesis 1: Bifurcation as quench?The results of selective field ionization experiments, such as those picturedin Figure 9.5, suggest that the state of the bifurcated plasma in its earlyevolution is one of high-Rydberg molecules and electrons weakly bound toa space charge of NO+ ions. We observe the prompt expansion of electronsand accelerated ions that initiates the process of bifurcation (see Chapter10). On simple electrodynamics grounds, these initial ions reach ballisticvelocities in a few hundred nanoseconds, after travelling only a few tens ofmicrometers. This ambipolar transfer of the thermal energy of the electronsto the radial energy of the ions quenches Te to the order of Ti.18311.3. Future workBallistic ions with velocity components in ±x expand into the wingsof the elliptical Rydberg gas. Resonant charge exchange transfers ion mo-mentum to Rydberg molecules. This process recurs, creating a uniform ±xvelocity field in which ions, cold electrons and Rydberg molecules all streamtogether. For higher principal quantum numbers at higher Rydberg gas den-sities, the observed recoil velocity of bifurcating plasma volumes approacheshalf the initial radial velocity of the ambipolar accelerated core ions. Thissuggests a medial expansion of ions and electrons that advances to meet anapproximately equal density of cold Rydberg molecules.Internal energy of the system in all coordinates flows to the relative mo-tion of bifurcating volumes, quenching the internal state of the plasma. Astheir relative velocities approach zero, ions and Rydberg molecules naturallymove to positions of minimum potential energy. These spatial correlationsdeplete the leading and trailing edges of the initially random distributionof nearest neighbours in the Rydberg gas [72], forming a random, three-dimensional network in which the distribution of NO+/NO+ distances, r,(referring both to Rydberg molecules and bare ions) peaks sharply at aWigner-Seitz radius. We can thus imagine that the plasma self-assemblesto form a correlated spatial distribution of intermolecular distances dictatedentirely by internal forces. Thus, bifurcation quenches the ensemble of ionsand electrons to an annealed domain of low energy, perhaps to an ultracoldregime.Hypothesis 1 favours the notion of ballistic ions streaming into a Rydberggas based on an argument of conservation of momentum along dimension x.Ions, streaming along dimension x into a Rydberg gas, undergoing resonantcharge exchange, are able to contain a large fraction of their momentum inx. This is not true for ballistic ions streaming into stationary ions, where amajority of collisions result in large angle deflections, transferring momen-tum from dimension x into y. Table 10.2 shows that the bifurcated plasmais as narrow or even more narrow in y as the non-bifurcated one. Thus, theensemble wings must predominantly contain Rydberg molecules in order to18411.3. Future workcontain the momentum along x.27In recent work, we have identified two inconsistencies between abovemodel and our experimental observations: 1) In Figure 10.5, we observeour arrested volume (prepared at n0=32) after 420 µs in ellipsoidal shape.Above model requires the flow of ballistic ions into a Rydberg gas, leadingto bifurcation as quench. Why do we observe arrest in a non-bifurcated sys-tem? 2) In order for ballistic core ions to stream into Rydberg wings, theseions must overtake the plasma-formation in the wings. Table 11.3 considersthe plasma formation times of our system and computes a velocity for a’plasma-formation front’ travelling outward though our Rydberg volume. Ifind that this velocity exceeds that of the ballistic ions (vion ⇡ 200m/s) byat least an order of magnitude, which represents an inconsistency to abovemodel.The first inconsistency could be circumvented through postulation of amuch smaller core streaming into the surrounding Rydberg gas, leading toarrest without causing bifurcation. We have plans to study this issue furtherby gating our 3D detector to obtain sliced images of our density distribu-tion. A solution to inconsistency two, however, would require much delayedavalanche times in the system wings. This in turn leads to problems asRydberg predissociation lifetimes only extend to hundreds of nanoseconds.Overall, this suggests that either the information pertaining to arrested re-laxation travels faster than the ballistic ions, or that we should considerarrest as a mechanism that occurs independent from the bifurcating core.Hypothesis 2: Density dependent self-assembly?High Rydberg systems have been widely studied in the context of light-matter interactions governed by exceedingly strong dipole-dipole interac-tions [35, 60]. Laser excitation to Rydberg states gives rise to a num-27A solid state nature of the arrested plasma, i.e. the formation of a macroscopic lattice,could also explain the observed observation of momentum conservation along x, but alsoleads us further into the realm of speculation.18511.3. Future workTable 11.3: I consider a spherical Rydberg volume with peak density, ⇢0 =2.7 ·1011 cm3, and FWHM of 3.2 mm. I model the system through ten non-interacting shells with densities ⇢0, 0.9 · ⇢0, 0.8 · rho0,...,0.1 · ⇢0. I calculateeach shell centre position, x, and divide by the computed avalanche time foreach shell, t (cf. Chapter 7) to find the velocity of the ’plasma-formationfront’, v.shell % [⇢0] ⇢ [1011 cm3] x [mm] t [ns] v [m/s]1 1 2.7 0 16 02 0.9 2.43 0.62 19 2081003 0.8 2.16 0.91 22 947844 0.7 1.89 1.15 27 480225 0.6 1.62 1.38 33 376656 0.5 1.35 1.60 45 188867 0.4 1.08 1.84 61 149878 0.3 0.81 2.11 91 89779 0.2 0.54 2.44 170 417210 0.1 0.27 2.92 410 1994ber of important phenomena of current interest in quantum optics andmany-body quantum dynamics, including entanglement, dipole blockadeand electromagnetic-induced transparency [8, 27, 36, 75], as well as coherentenergy transport, optical bistability and directed percolation [5, 14, 47, 48].We speculate, that under our conditions, each Rydberg molecule under-goes an excitonic interaction with its nearest neighbour [20, 58, 86], eithera Rydberg molecule in a di↵erent high-n, high-` electronic state or an ionin the field formed by the quasi-continuum of electrons bound to multiplecharge centres. Importantly, every such interaction in the bifurcated en-semble randomly pairs two molecules in di↵erent excited states to define aunique, resonant close-coupled interaction.The question arises whether the quasi-particles defined by these pairwiseinteractions can constrain the global evolution of the system in its spatial orenergetic coordinates. Modelling many-body ensembles of resonant spins,Yao and coworkers [90] argue that, in three dimensions, 1/r3 dipole-dipole18611.3. Future workinteractions give rise to sub-di↵usive transport. We speculate that a compa-rable mechanism might be responsible for the observed arrest of relaxationchannels.Thus, as an alternative to hypothesis 1, I consider a Rydberg systemwhich develops an arrested state independent of bifurcation but based onthe ratio between the size of the Rydberg molecules and the average distancebetween Rydberg cores. The hypothesis is simple. In order for arrest todevelop, Rydberg molecules must have orbital diameters comparable to theinter-particle spacing, but without a significant overlap of orbitals. If theRydberg gas density is too low, the molecules are too far apart. If the densityis too high, Rydberg molecule overlap is too large, shutting o↵ arrest.Within the framework of hypothesis 2, I can envision the following mech-anism forming the two observed domains: In the core of our Rydberg ensem-ble, density is highest and a significant number of molecules have overlap-ping orbitals, leading to Penning ionization (ICD). Similar to the orderingin a Rydberg blockaded gas, the Penning process constitutes an orderingof the plasma ions and remaining Rydberg molecules, by removing nearestneighbours from the respective distribution [71]. Thus, the randomness ofour Rydberg ensemble is lowest at its core and highest in the outer layers.This, combined with the well know fact that localization, whether classicalor quantum in nature, is highly dependent on disorder (randomness) couldlead to an explanation of the observed system behaviour. Randomness mightconstitute a boundary between delocalized and localized transport withinour Rydberg ensemble, leading to the suppression of avalanche to plasmawithin the ensemble wings.Questions clearly remain: Can our system truly self-assemble to form aglass or localize, at least in the coordinates of electron binding energy? Doesthe relaxation of varying systems to a common density signify a universalmacroscopic physics? Does a network of quantum mechanical interactionsexplain the slow dynamics of the nitric oxide molecular ultracold plasma?187Chapter summaryHypothesis 2, much like hypothesis 1, is pure speculation. It has enteredmy considerations because it solves the two inconsistencies of hypothesis 1- yet di↵erent issues arise. Clearly, at this point in time, further systematicexperiments as well as computation models are necessary to proof, disproveor develop above hypotheses.Chapter summaryIn the context of my findings in Chapters 2 through 10, this last thesis chap-ter compared experimentally observed and ’expected’ ensemble evolution toderive an inconsistency, highlighted by the arrest of relaxation channels. Ispeculated on the nature as well as formation process of the arrested do-main. Based on a process of elimination, Section 11.2 called for new physicsto explain the arrest of relaxation pathways. The ensuing discussion pro-posed two hypotheses for the transition of the Rydberg ensemble to thestate of arrest: Bifurcation as quench and density dependent self-assembly.I discussed pros and cons of each hypothesis.188Summary and OutlookOver the course of this thesis, I have given a comprehensive account of thework in our research group. The complexity of the system under study,together with the fact that the field of ultracold molecular plasmas is stillyoung, motivated a first principal discussion. Part I, Fundamentals, intro-duced the spectroscopic character of nitric oxide, our molecular ion, andfollowed with a review of Rydberg physics. Next, I discussed the fundamen-tal assumptions of ’established’ plasma theory and motivated the theoreticaland experimental study of correlated plasma, due to the breakdown of saidassumptions in a regime of Coulomb correlation.Part II, Formation of a molecular Rydberg ensemble, began by intro-ducing our experimental setup. I described how the convolution of twolaser pulses with the cold molecular beam drives resonant transitions inNO, resulting in the preparation of a Gaussian ellipsoidal Rydberg ensem-ble. Here, the study of well established methods to describe supersonic beamexpansions allowed for the characterization of the nitric oxide distributionand subsequently of the Rydberg molecule distribution in a moving-framephase-space representation.Part III, Theory work, discussed the microscopic evolution of this Ry-dberg ensemble to plasma, via Penning ionization, electron-Rydberg colli-sional heating as well as ionization avalanche, on a nanosecond timescale.This was followed by a discussion of macroscopic plasma evolution, incor-porating plasma expansion and decay, on a microsecond timescale. Thequalitative discussions in Part III were aided by computation results of rate-equation model calculations.Part IV, Experimental work, introduced our methods of plasma detec-tion. I discussed various experimental datasets within the context of previ-189Summary and Outlookous chapters. My analysis showed, that (under certain conditions) our Ry-dberg ensemble unexpectedly develops two distinct macroscopic domains.These domains are distinguished by their polarizability as well as their lo-cality within the plasma. The first domain appears at the system core anddisplays a character similar to that of plasma discussed in Part III. Thesecond domain travels with the velocity of the supersonic beam and is dis-tinguished through the apparent arrest of microscopic relaxation channels.I found that both domains are linked via the process of plasma bifurcation.Part IV concludes with speculations on the nature of the arrested domainas well as its mechanism of formation.My hope is that this thesis provides a strong theoretical foundation toour area of research and helps to establish this young field of ultracold molec-ular plasmas. In proving a fundamental knowledge of the plasma precursor,the underlying Rydberg ensemble, I aim to attract the interest of other re-searchers into our work and gain support with the theoretical diculties weare facing.We believe that the molecular beam environment, introducing higherdensities and additional evolution channels, causes a distinction betweenthe fields of ultracold molecular plasma and ultracold atomic plasma. Theanalog computer in our laboratory, the studied Rydberg ensemble, solvesour well defined many-body system in a way we do not yet understand. Thediscrepancy between our conceptual understanding of this many-body sys-tem and our observation of nature provides us with the motivation as wellas the justification for our continued work. Based on the process of elimi-nation, led up to over the course of this thesis and discussed in Chapter 11,we consider new physics to describe the evolution of our system, drawingfrom quantum mechanics as well as the immature field of correlated plasmaphysics. An incoherent mechanism may yet emerge, but our experimentalobservations substantially narrow the possibilities that can rely on conven-tional kinetics.190Summary and OutlookThere is a great deal left to discover. At present, we are working on max-imizing our control of the electric field environment within our experimentchamber. The observed polarizability of the non-arrested plasma domainallows for a systematic study of this state, which may lead to an improvedunderstanding of the macroscopic boundary condition between the two do-mains. This, in turn, leads to a better understanding of the arrested domain.Our electronic lens aided 3D imaging system, the ’plasma-TV’ in con-figuration c), has only recently come online. Initial results with this newdetector, discussed in Section 10.2.3, hint on the vast research possibili-ties of this machine. A systematic mapping of the boundary between thearrested and non-arrested domains will certainly yield new insight. Further-more, a gating of this detector through voltage pulses on P2 or the MCPsallows us to record sliced images of the plasma density distribution. We havealready begun experimenting with this tool. Lastly, the reconfiguration ofthe plasma-TV detector for velocity-map-imaging of the plasma domains isconceivable.In addition to our present detection methods, we plan to develop a mov-ing phosphor screen detector, merging the capabilities of the plasma-TV inconfigurations a)/b) with those of the moving-grid detector.More up to date, we are working to employ di↵erent carrier gases thanhelium or di↵erent molecules than nitric oxide in our experiment. 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Zucrow and J. Ho↵man. Gas Dynamics, Vols. I and II. Wiley, NewYork, 1976.201Appendix AHardwareAppendix A of my PhD thesis is meant to give a detailed description of ourhardware implementation. Within my first year of working in our researchgroup, I took on the task to build up a new experimental setup, beginningwith the moving and connecting of three empty optical tables. Since then,I have made countless decisions with respect to the experimental setup, likechoosing one electronic part, method or technical realization over another.The following will be a valuable toolset for someone with interest in joiningthe field of molecular ultracold plasma in an experimental capacity.Our group currently works with two independent experimental setups.These setups only di↵er slightly in manufacturer or generation of the equip-ment. In both systems, we form an ultracold molecular plasma within asupersonic beam expansion. The main di↵erence between our two setups isthe way of detection. The older setup is hosting what we call the ’movablegrid detector’ (Sec. 9.1) while the new setup I’ve build makes up the ’Plas-maTV experiment’ (Sec. 10.1), referring to a tomographic plasma imagingsystem. For the purposes of this thesis, I will only describe the new setupas well as the new generation detector in detail. For more information onthe older setup, including ZEKE and moving-grid detectors, please refer toolder theses from our group [26, 49].A.1 OverviewFigure A.1 gives a schematic overview of the components on the optical ta-bles that make up the PlasmaTV experiment. The PlasmaTV setup consistsof three optical tables. The tables are bolted together to minimize relativemovement. The bottom table is 3 by 10 foot and holds two laser sources.202A.1. OverviewFigure A.1: Birdview-schematic of the experimental setup203A.2. Laser and opticsWe di↵er between two light-paths, !1 and !2. On the middle table, also 3by 10 foot, we frequency double and color filter our laser pulses. A photodiode records accurate timing information on !1 for triggering purposes.Two periscopes raise the laser beams to final hight for transit though ourvacuum chamber. The last table, the detector table, is 4 by 10 feet with a16 inch by 5 feet central cut-out which holds the vacuum chamber, housingmolecular beam and detector. Prior to transit through this chamber, laserbeam !1 is attenuated, spatially cleaned and merged with !2. A translationstage supporting the last 90 degree prism before the chamber allows us tomove the light pulses horizontally across the diameter of our viewport.Further measurement equipment like oscilloscope, timing unit, mol. beamcontrol unit and measurement PC are sitting in racks above and below theoptical tables. The individual component descriptions follow.A.2 Laser and opticsFigure A.1 shows a schematic of the optical layout on our laser table. Thesetup realizes two laser pathways, !1 and !2. The unfocused beams overlapwithin the experimental section of our vacuum chamber.A.2.1 Pump- and dye-lasersThe !1-beam originates within a Scanmate Pro C-400 laser by CoherentLambda Physik. This system incorporates both, pump and dye laser, in acompact setup. The pump laser is a Brilliant b Nd:YAG system by Quan-tel, operated in pulsed mode. Part of the beam, at 1064 nm, is doubled andsubsequently mixed with the fundamental to generate the third harmonic,which is split form the background of the first and second fundamental. Thisassembly yields an output beam of ⇠ 180 mJ/pulse at a wavelength of 355nm. This beam pumps the dye cavity of the ScamMate system. We useCoumarin 460 in methanol as a dye solution to generate a tuneable laseroutput at ⇠ 452 nm with a power of ⇠ 11 mJ/pulse.204A.2. Laser and optics!2 follows the same concept, except it originates in the second harmonicof a Continuum Powerlite 8000 DLS Nd:YAG laser. The used dye in thesecond step is DCM in a Continuum ND6000 dye laser. The respective laserintensities and wavelengths are ⇠ 640 mJ/pulse at 532 nm after the YAG,allowing for ⇠ 200 mJ/pulse at ⇠ 652 nm after the dye laser. We typicallyattenuate the pump laser by means of a flash-lamp/Q-switch delay to limitour dye laser output to ⇠ 100 mJ/pulse in order to protect subsequentoptics.A.2.2 Frequency-doubling and colour filteringBoth dye-laser output beams are frequency doubled in a single-pass InradAutotracker III unit and subsequently pass through a 4-prism colour filterwhich only allows the doubled light through. At this point, the beam powerof !1 at ⇠ 226 nm is less than 2 mJ/pulse. !2 at ⇠ 326 nm retains a powerof ⇠ 20 mJ/pulse.A.2.3 Additional optics• We further reduce the !1-beam by a set of Glan-Thompson polariz-ing crystals, which are rotated with respect to the laser polarizationaxis. Typical !1 powers used in our experiments vary between 1.5 and15µJ.• Before merging our laser beams, we send !1 through a spatial filtermodel KT310 by the manufacturer Thorlabs. The filter mounts a 50µmpinhole to produce a clean, spatially uniform, Gaussian beam.• We merge our laser beams using a dichroic filter.• In some experiments we utilize an UV lens Galilean telescope tocollimate our !1-beam after the spatial filter and before the merging.205A.3. The vacuum chamberA.3 The vacuum chamberA.3.1 Chamber dimensionsFigure A.2: Schematic of the vacuum chamberOur vacuum chamber is a 10 inch diameter chamber separated by theskimmer wall into two compartments, the source chamber and the experi-mental chamber. The horizontal chamber dimensions of the three connectedstainless steel segments, as illustrated in Figure A.2, are 11inch, 16inch and15inch. The skimmer wall mounts a nickel electro-formed molecular beamskimmer with an orifice diameter of 1mm, supplied by BEAM DYNAMICS,INC..The experimental chamber allows the access of laser beams through fusedsilica view port windows. Pumping access for our turbo molecular vacuumpups is via two 8” ports below the chamber. The top of our chamber allowselectrical access to the experiment chamber via a vacuum feedthrough pass-ing 8 SHV connectors into the chamber. We use the access port above thesource chamber to monitor our vacuum pressure.206A.3. The vacuum chamberA.3.2 Mu-metal shieldingOur vacuum chamber incorporates magnetic shielding via 1mm thick mu-metal plates, fitted to the inside wall of the experiment chamber. In con-figuration c), a shield-cap of 7.25” diameter with a 2.66” diameter hole sitson the hight of G2 and closes the shield towards the source chamber. 20mm diameter pores above the pump port allow for pump access through theshield. This creates a cylindrical shield, closed to one end. The mu-metalsheets were cut and vacuum annealed by Magnetic-Shield Corporation. Werolled the sheets after delivery.A.3.3 VacuumBoth, our source and experimental chamber, are di↵erentially pumped bya combination of a PFEIFFER VACUUM HiPace 700 turbo pump anda Welch 1397 Duo-Seal roughing pump. We monitor our source chamberpressure via a BOC Edwards WRG-SL gauge using Pirani technology for theupper pressure range, with a seamless switch over to an inverted magnetronfor the lower range. We typically achieve vacuum pressure in the range of109torr. This pressure drops by two orders of magnitude when we run ourbeam valve.A.3.4 The supersonic beamA General Valve Series 9 nozzle with a 0.5 mm diameter orifice creates asupersonic jet of nitric oxide, seeded at 10% in helium, that expands intothe experimental vacuum. The nozzle has an opening interval of 350µs ata 10Hz pulse count. The gas mixture is a commercial product from Parax-iar Distribution Inc. with a certified gas ratio of 9.97% and an analyticalaccuracy of ±2%.The beam cools by progressively converting thermal energy into direc-tional kinetic energy. It traverses the source chamber and its core encountersa skimmer wall with a 1 mm diameter skimmer at a distance of 25 to 35mm downstream of the nozzle. The coldest part of the particle jet entersthe experimental chamber.207A.4. The PlasmaTV detectorA.4 The PlasmaTV detectorThe PlasmaTV detector was introduced in Chapter 10. It consists of aChevron Model 3075FM detector assembly by BURLE ELECTRO-OPTICSINCORPORATED, integrated into our experiment in one of three configu-rations (cf. Figure 10.1). In the following, I will discuss subcomponents ofthis detector in more detail.A.4.1 The chevron detectorManufacturer specifications: ”The Chevron Model 3075FM detector as-sembly contains two Imaging Quality Extended Dynamic Range Long-LifeMicrochannel Plates and a fiberoptic phosphor screen with P20 phosphormounted to an 8.0” vacuum flange. The fiberoptic is frit sealed into theflange to form a vacuum seal that is tight to a leak rate of 10-10 cc/secof helium. The detector assembly is mounted in stainless steel hardwarewhich is bakeable to 300 C. The vacuum feedthroughs are mounted radi-ally on the flange to facilitate viewing and access to the fiberoptic phosphorscreen. The flange mounted assembly is equipped with one BSHV vacuumfeedthrough rated up to 7.5 kV (for the phosphor screen) and three SHVvacuum feedthroughs rated up to 5 kV (two for the MCPs and one spare).”The MCPs are further characterized by a quality diameter of 75 mm,centre-to-centre spacing of 32 µm, pore size of 25 µm and 8 degree biasangle. A 0.1 µm thick gold coating is applied to the input and output face.At 2 kV potential applied over the MCPs, electron gain is a minimum of4 · 106.A.4.2 The lens elementThe purpose and function of the electronic lens element has been discussedin Section 10.1. Figure A.3 displays the dimensionality in z of the PlasmaTVdetector in configuration c). The di↵erent lens elements are stacked ontothree 1/2 inch stainless steal rods, which are heat shrunk into a 10 to 8inch zero length adapter. The di↵erent lens elements are isolated from each208A.5. Electronicsother as well as the mounting rods via 1mm thick non-conducting spacers.Plates P1 and P2 have 4mm and 5mm thickness and a hole diameter of10mm and 8mm, respectively. Grid plate G2 has 2.5” diameter with 4holes, each 0.125” diameter on a 1.061” radius. The integrated mesh has90% transparency and measures 50 lines per inch, stainless steel. G1 has thesame dimensions without a mesh. Both plates, G1 and G2, were suppliedby Jordan TOF Products, Inc..Figure A.3: Dimensionality of the PlasmaTV detector in configura-tion c) along the z-axisA.5 ElectronicsA.5.1 PlasmaTV voltage supply and output couplerWe set voltage potentials on our PlasmaTV lens element as well as the MCPinput via an in-house build variable voltage supply. To supply MCP outputand anode/screen potentials to the chevron detector, we use a Burle ModelPF1054 Dual Output Power Supply with a voltage range 0 to 3kV and 0 to8kV and up to 500µA current for both outputs.Figure A.4 shows a schematic of our signal output coupler. The displayed209A.5. Electronicsbox is connected directly to the BSHV vacuum feedthrough of the chevrondetector. The detector anode/screen is charged though a low-pass filterwhich blocks high-frequency components originating in the power supply.Particle impact on the charged chevron detector creates a high frequencysignal which passes though our high-pass filter to be viewed with an oscil-loscope. We employ this dual-filter scheme to protect our oscilloscope linefrom voltages spikes (e.g. power outage) on the HV in line.28Figure A.4: Layout of the PlasmaTV output couplerA.5.2 OscilloscopeWe use a Agilent Technologies DSO5052A Oscilloscope with a bandwidthof 500 MHz and 4 GSa/s to view the detector signal. We typically triggervia a photo diode, located after the Continuum ND6000 dye laser.A.5.3 TimingBoth pulsed laser beams are synchronized and overlap temporally and spa-tially in the experimental chamber where they intersect with a particle jet.The timing of the flash lamps and Q-switches of both laser units, aswell as of the nozzle and camera, are performed by a BNC - Model 57528Please note that the anode/screen does not represent a 50⌦ input impedance. Thus,our detection circuit incorporates an impedance mismatch which leads to strong ’ring-ing’ especially for brief (nanosecond) signal bursts. The plasma impact, however, is amicrosecond event and we obtain a well-defined impact characteristic.210A.5. Electronicspulse/delay generator.A.5.4 CameraIn order to capture the lighting-up of our phosphor screen, we employ aUEye UI-224xSE-M Camera with the following specifications: sensor type- CCD monochrome; exposure mode - electronic global shutter; resolution- 1280 x 1024 pixel; pixel clock range - 5 - 30 MHz; frame rate in triggermode with 1ms exposure time - 14 fps. We reset the active area to 1024 x1024 pixel and software bin to 512 x 512 pixel size images stored in unsigned8-bit integers.We have designed an aluminum camera mount that attaches to thechevron detector. The camera is adjusted to focus on the edges of thefiberoptic window. Detector and camera are surrounded by a blackoutscreen.211Appendix BComputation source-codeIn the following, I will publish the Matlab source-code I developed, discussedand applied in Part III of my thesis. Once more, I’d like to acknowledge thecontributions of my coworkers, Nicolas Saquet, Jonathan Morrison, HosseinSadeghi and Jachin Hung, who were kind enough to share their code so Iwould not have to start from scratch.B.1 Early-time dynamics model1 %% This function evaluates the plasma rise time for a given Ry density, n02 % I set n min as lower decay boundry. States reaching levels <= n min3 % are considered to decay immediately via PD45 % den0 = Ry atom density [umˆ3] or [1e12 cmˆ3]6 % n0 = initial PQN7 % t final = final computation time [ns]89 % Use like: [t,nden,eden,aden,Te,˜]=msw runplasmasim atom(0.3,50,500)1011 function [t,nden,eden,deac,Te,y0]=msw runplasmasim atom(den0,n0,t final)1213 %%%%%%%%%%%%%%%%%%%%14 % Penning fraction %15 %%%%%%%%%%%%%%%%%%%%1617 % This function calculates the inital seed electrons1819 % n are initial PQNlevels before Penning ionization20 % den is Ryatom density in 10ˆ12 pcc212B.1. Early-time dynamics model2122 % eden is the # of electrons produced23 % rden are the remaining PQNlevels after Penning ionization2425 function [PenningFraction eden rden]=penningfraction(n0,den0)2627 a0=5.2917721092e5; % bohr radius in um28 Rn0=n0.ˆ2*a0;29 % radius of Rydb. atom by bohr model using semiclassical method30 Rmax=1.8*(Rn0*2);31 % Robicheaux paper, within this distance, 90% penning ionize (˜15ns)3233 PenningFraction=zeros(length(n0),1);3435 % Calculates number of Penning partners based on Erlang distribution:36 for i=1:length(n0)37 % proportion between 0 and Rmax  integral is solved analytically38 PenningFraction(i)=1exp(4*pi*den0*Rmax.ˆ3/3);39 end4041 % 90% ionization within certain time (assume rest noninteractive)42 eden=PenningFraction/2*den0*.9;43 % the dens. of electrons produced is half44 rden=(1PenningFraction*0.9)*den0;45 % this is remaining density of rydbergs4647 end4849 %%%%%%%%%%%%%%%%%%%%%%50 % Initial conditions %51 %%%%%%%%%%%%%%%%%%%%%%5253 % Set constants and lower cutoff for PQN distribution:54 kB = 1.3806504e23; % #m2 kg s2 K1;J K1; 8.62e5 #eV K1,55 Ry = 2.179872e18; % #J: Rydberg energy in J5657 firstn=1;58 n min=10;59 numlev=100; % this is the number of n levels initially considered60 deac=0; % start with all Rydberg states213B.1. Early-time dynamics model61 nl=(firstn:firstn+numlev1)'; % array of accessible n levels6263 tspan=linspace(0,t final,500);6465 % Sets initial conditions for electron and nlevel distributions:66 % no initial electrons > calc. Penning seed electrons (Robicheaux)6768 [PF,eden,rden]=penningfraction(n0,den0);6970 % Redistributes the Penning partners over lower n's:71 f=@(x)5.95*x.ˆ5;72 % This is the penning fraction distribution73 np=firstn:fix(n0/sqrt(2));74 % Array of n states allowed after Penn ion75 ind=1:length(np);76 % This is the distribution of penning fraction77 nden=nl*0;78 nden(ind)=eden*f(np/n0)/sum(f(np/n0));79 % dividing by the sum normalizes the function8081 nden(nl==n0)=rden; % set n0 to rden8283 deac=sum(nden(1:n min)); % allow n<=n min to decay to aden84 nden(1:n min)=zeros(n min,1);8586 % Set initial temperature: (Robicheaux 2005 JPhysB)8788 T penning=(Ry*den0/n0ˆ2 + Ry*rden/n0ˆ2 + ...89 Ry*sum(nden(ind)./nl(ind).ˆ2) )*1/(3/2*kB*eden);90 % set according to energy conservation919293 %%%%%%%%%%%%%%%94 % rate coeffs %95 %%%%%%%%%%%%%%%9697 function [ni,nf,II,minn,maxn,diffsn]=buildns(nl)9899 % Is needed to reevaluate the rate coeff.100 a=length(nl);214B.1. Early-time dynamics model101 II=ones(a,a); % will be matrix of ij=1 and ii=0102 ni=zeros(a,a);103 nf=zeros(a,a);104 minn=zeros(a,a);105 maxn=zeros(a,a);106 for i=1:a107 for j=1:a108 ni(i,j)=nl(i); % an array of initial states109 nf(i,j)=nl(j); % an array of final state110 minn(i,j)=min(ni(i,j),nf(i,j));111 %find min of init and final state potential problem112 maxn(i,j)=max(ni(i,j),nf(i,j));113 %find max of init and final state114 if i==j115 II(i,j)=0; % set to 0 for same ini and final states116 end117 end118 end119 diffsn=abs(1./ni.ˆ21./nf.ˆ2);120 % difference in energy between the 2 states (no units)121122 end123124 function ktbr=kTBR(n,T) % threebodyrecombination125126 % TBR rates output units in umˆ6 ns1127 kB = 1.3806504e23;% #m2 kg s2 K1;J K1; 8.62e5 #eV K1,128 emass = 9.1093822e31;% #kg129 h = 6.6260690e34;% #m2 kg / s 4.13e15 #eV s130 Rydhc = 2.179872e18;% #J: Ryd [cm1] > [J]131132 epsi=Rydhc./(power(n,2.0)*kB*T);133 lmbd=1e6*h./sqrt(2.0*pi*emass*kB*T);134135 ktbr=kION(n,T).*power(n,2.0).*power(lmbd,3.0).*exp(epsi);136 % unit umˆ6 ns1137 ktbr(isnan(ktbr))=0;138 % take care of computation error for values matlab139 % sees as too small and turns NaN140 ktbr(˜isfinite(ktbr))=0;215B.1. Early-time dynamics model141 % take care of computation error for values matlab142 % sees as too large and turns inf143144 end145146 function kion=kION(n,T) % ionizing collisions147148 % output unit is umˆ3149 kB = 1.3806504e23;% #m2 kg s2 K1;J K1; 8.62e5 #eV K1,150 Rydhc = 2.179872e18;% #J: Ryd [cm1] > [J]151 epsi=Rydhc./(power(n,2)*kB*T); % find reduced initial energy152 kion=11*sqrt(Rydhc./(kB*T)).*kNOT(T).*exp(epsi)./...153 (power(epsi,2.33)+4.38*power(epsi,1.72)+1.32*epsi);154155 end156157 function out=knnp(ni,nf,II,minn,maxn,diffsn,T) % rate for n to n'158159 % output units is umˆ3 nsˆ1160 % use in conjunction with [ni,nf,II,minn,maxn,diffsn]=buildns(nl);161162 kB = 1.3806504e23;% #m2 kg s2 K1;J K1; 8.62e5 #eV K1,163 Rydhc = 2.179872e18;% #J: Ryd [cm1] > [J]164165 eps i=Rydhc./(power(ni,2.0)*kB*T);166 eps f=Rydhc./(power(nf,2.0)*kB*T);167 max eps i=Rydhc./(power(minn,2)*kB*T);168 min eps i=Rydhc./(power(maxn,2)*kB*T);169170 diffs=Rydhc.*diffsn./(kB*T); % scale dffsn properly171172 out=II.*(kNOT(T).*power(eps i,5/2).*power(eps f,3/2)./...173 power(max eps i,5/2))...174 .*exp((eps imin eps i)).*((22./(power(max eps i+0.9,7/3)))...175 +(4.5./(power(max eps i,2.5).*power(diffs+1II,4/3))));176177 end178179180 %%%%%%%%%%%%%%%%216B.1. Early-time dynamics model181 % Calculations %182 %%%%%%%%%%%%%%%%183184 y0=[nden;eden;deac;T penning]; % set initial value for ODE185186 ncrit=@(T)round(sqrt(Ry/(kB*T))); % to calculate nmax187188 [ni,nf,II,minn,maxn,diffsn]=buildns(nl);189 % is needed to calculate the rate coeffs190191 progress = waitbar(0,'Progress...'); % shows progress of computation192193 function dy=eqrateode(t,y)194195196 % Select valiables:197 nden=y(1:numlev); % pick out density over distribution of n198 eden=y(numlev+1); % pick out electron density199 deac=y(numlev+2); % pick out density of radiatively decayed atoms200 T=y(numlev+3); % pick out temperature201202 nc=ncrit(T); % calculates n max with this temperature203204 % Adjusts max allowed n:205 if nc>=nl(1) && nc<nl(end)206 index=find(nl==ncrit(T));207 elseif nc<nl(1)208 index=1;209 else210 index=numlev;211 end212213214 % Evaluate the updated rate terms:215216 d tbr=zeros(numlev,1);217 d tbr(1:index)=kTBR(nl(1:index),T)*edenˆ3; % units [umˆ3 nsˆ1]218219 d ion=kION(nl,T).*nden*eden; % units [umˆ3 nsˆ1]220217B.1. Early-time dynamics model221 % rate for transfer from n to n', units [umˆ3 nsˆ1]222 k n np=knnp(ni,nf,II,minn,maxn,diffsn,T);223 d n np=sum(k n np,2).*nden*eden;224225 % rate for transfer from n' to n, units [nsˆ1]226 k np n=zeros(numlev,numlev);227 for i=1:index % only to levels <= nc228 k np n(i,1:numlev)=k n np(1:numlev,i).*(nden(1:numlev));229 end230 d np n=sum(k np n,2)*eden; % units [umˆ3 nsˆ1]231232 % transfer from n's above ncrit(T) to eden233 k n npion=zeros(numlev,1);234 if index<=numlev235 k n npion(1:index)=sum(k n np(1:index,index+1:numlev),2)...236 .*nden(1:index); % units [nsˆ1]237 end238 d n npion=sum(k n npion)*eden; % units [umˆ3 nsˆ1]239240 % Evaluate time derivatives:241242 d eden=sum(d iond tbr)+d n npion;243244 d nden=d tbrd iond n np+d np n;245246 dT=(Ry*sum(d nden./nl.ˆ2)1.5*kB*T*d eden)/(1.5*kB*eden);247248 % Implements PD for levels n <= n min:249 d deac=sum(d nden(1:n min));250 % deac is the number of Ry's predissociated251 d nden(1:n min)=zeros(n min,1);252253254 dy=[d nden;d eden;d deac;dT];255256 waitbar(t/t final);257258 end259260 %%%%%%%218B.2. Long-time dynamics model261 % ODE %262 %%%%%%%263264 options=odeset('reltol',1e8);265 [t,y]=ode23(@(t,y)eqrateode(t,y),tspan,y0);266267 nden=y(:,1:numlev);268 eden=y(:,numlev+1);269 deac=y(:,numlev+2);270 Te=y(:,numlev+3);271272 close(progress);273274 endB.2 Long-time dynamics model1 %% This function evaluates plsma evolution within the 3D shell model2 % The input is collected in array in.* (see below)3 % Output: [N; R x; R y; R z; u x; u y; u z; Ti x; Ti y; Ti z; Te]4 % spatial dimensions are in [um]5 % time dimensions are in [us]67 function [yout] = msw expansionmodel 3D ion(in)8910 %%%%%%%%%%%%%%%%%%%%%%11 % Initial conditions %12 %%%%%%%%%%%%%%%%%%%%%%1314 % Define input:15 sig0 x=in.rx; % initial plasma width in um (sigma0 x)16 sig0 y=in.ry; % initial plasma width in um (sigma0 y)17 sig0 z=in.rz; % initial plasma width in um (sigma0 z)18 Te0=in.Te0; % initial electron temperature in K19 shells=in.shells;20 % total number of shells used to evaluate gaussian profile21 Ccut x=in.distance x;22 % the final xboundary of plasma usually 5 times initial width219B.2. Long-time dynamics model23 Ccut y=in.distance y;24 % the final yboundary of plasma usually 5 times initial width25 Ccut z=in.distance z;26 % the final zboundary of plasma usually 5 times initial width27 tspan=in.tspan';28 % the time array which will be used to evaluate time integration of ODE29 rho0=in.n0; % peak density of plasma in [umˆ3]30 kB=1.3806488e23; % kB Boltzmann constant SI unit31 mi=4.982629691838400e26; % mi Mass of one molecule of NO in Kg32 kBm=kB/mi;33 % save time on calculation of kB/mi in ODE loop34 kDR1kelvin=in.kDR1kelvin; % DR constant at 1 kelvin in umˆ3/us35 kDRpower=in.kDRpower;36 % power of DR temperature dependence [unitless]37 gamma0 x=in.u0 x; % HD velocity x in um/us38 gamma0 y=in.u0 y; % HD velocity y in um/us39 gamma0 z=in.u0 z; % HD velocity z in um/us40 Ti x0=in.Ti x; % initial ion temperature in x41 Ti y0=in.Ti y; % initial ion temperature in y42 Ti z0=in.Ti z; % initial ion temperature in z4344 % Create shells:45 % array of initial kshell radius between 0 to 5 * r046 % divided to number of shells47 % avoiding zero shell radius by shifting the array48 R0 x=linspace(0,Ccut x,shells)';49 R0 x=R0 x+R0 x(2)R0 x(1);50 R0 y=linspace(0,Ccut y,shells)';51 R0 y=R0 y+R0 y(2)R0 y(1);52 R0 z=linspace(0,Ccut z,shells)';53 R0 z=R0 z+R0 z(2)R0 z(1);5455 rho=rho0*exp(R0 x.ˆ2/(2*sig0 xˆ2));56 % rho is array of density at xaxis shell boundry5758 % Find array of initial HD velocity:59 % array that initialized the velocity (um/us) of ions at each shell60 u0 x=R0 x.*gamma0 x;61 u0 y=R0 y.*gamma0 y;62 u0 z=R0 z.*gamma0 z;220B.2. Long-time dynamics model6364 % this function calculates the volume between each shell.65 function [volume]=volume3D(x,y,z)6667 xm1=[0;x(1:end1)];68 ym1=[0;y(1:end1)];69 zm1=[0;z(1:end1)];70 volume = 4/3*pi*((x.*y.*z)  (xm1.*ym1.*zm1)); % area of ellipsoid7172 end7374 % Find shell info:75 vol=volume3D(R0 x,R0 y,R0 z);76 N=rho.*vol;7778 % Define ion temperature:79 Ti0 x=Ti x0*ones(shells,1);80 Ti0 y=Ti y0*ones(shells,1);81 Ti0 z=Ti z0*ones(shells,1);8283 %%%%%%%%%%%%%%%%84 % Calculations %85 %%%%%%%%%%%%%%%%8687 % Define initial conditions:88 y0=[N; R0 x; R0 y; R0 z; u0 x; u0 y; u0 z; Ti0 x; Ti0 y; Ti0 z; Te0];8990 % Define fit parameter:91 fo = fitoptions('method','SmoothingSpline','SmoothingParam',1e1);92 ft = fittype('smoothingspline');9394 % Define inner function:95 function [dy du]=expansionmodelinnerfunction(t,y)96 % this function takes time and ODE variable and calculates dy/dt97 % (called dy) and du/dt (optional: called du)98 % this format is used by mode ODE integratores99100 % fprintf('%f\n',t); % uncomment to see the progress of ODE101102 % Give proper name to variables from the VERCAT y221B.2. Long-time dynamics model103 N=y(1:shells);104 R x=y(shells*1+1:shells*2);105 R y=y(shells*2+1:shells*3);106 R z=y(shells*3+1:shells*4);107 u x=y(shells*4+1:shells*5);108 u y=y(shells*5+1:shells*6);109 u z=y(shells*6+1:shells*7);110 Ti x=y(shells*7+1:shells*8);111 Ti y=y(shells*8+1:shells*9);112 Ti z=y(shells*9+1:shells*10);113 Te=y(shells*10+1);114115 % Calculate some parameters base on the updated values116 vol=volume3D(R x,R y,R z);117 rho=N./vol;118 kDR=kDR1kelvin*(Teˆ(kDRpower));119120 % Change in N due to DR121 dN=kDR*rho.ˆ2.*vol;122123 % Calculating expansion force along x:124 rxnp=[flipud(R x); R x];125 nnp=[flipud(rho); rho];126 % building up a density profile on either side of r=0127 % to get a symmetric fit128129 cfx=fit(rxnp,nnp,ft,fo);130 % fit density over distance, in order to calculate expansion force131132 fx=differentiate(cfx,R x)./rho;133 % part of expansion force that is calculated from density profile134135 du x=(kBm*(Te+Ti x)).*fx;136137 dR x=u x;138139 % Calculating expansion force along y:140 rynp=[flipud(R y); R y];141142 cfy=fit(rynp,nnp,ft,fo);222B.2. Long-time dynamics model143144 fy=differentiate(cfy,R y)./rho;145146 du y=(kBm*(Te+Ti y)).*fy;147148 dR y=u y;149150 % Calculating expansion force along z:151 rznp=[flipud(R z); R z];152153 cfz=fit(rznp,nnp,ft,fo);154155 fz=differentiate(cfz,R z)./rho;156157 du z=(kBm*(Te+Ti z)).*fz;158159 dR z=u z;160161 % Update Te and Ti in terms of transfer of thermal162 % to directional kinetic energy:163 % Change in total temp:164 dT x=2*(u x.*du x)/(3*kBm);165 dT y=2*(u y.*du y)/(3*kBm);166 dT z=2*(u z.*du z)/(3*kBm);167168 % Change in ion temp:169 dTi x=Ti x./(Te+Ti x).*dT x;170 dTi y=Ti y./(Te+Ti y).*dT y;171 dTi z=Ti z./(Te+Ti z).*dT z;172173 Ti mean=sum(N.*(Ti x+Ti y+Ti z))/(3*sum(N));174175 dTe=Te/(Te+Ti mean)*sum(N.*(dT x+dT y+dT z))/(3*sum(N));176 % change in el. temp.177 % has to be scaled by particle number per shell178179 % Update y180 dy=[dN; dR x; dR y; dR z; du x; du y; du z; dTi x; dTi y; dTi z; dTe];181182 end223B.3. Simulate detector response to 3D Gaussian ellipsoid183184 odeoptions=odeset('RelTol',1e13);185 handler=@(t,y)expansionmodelinnerfunction(t,y);186187 %%%%%%%188 % ODE %189 %%%%%%%190191 % Solve ODE:192 [˜,yout]=ode113(handler,tspan,y0,odeoptions);193194195 % To have matlab check if electron temp. evolution is ok:196 for j=(shells*7+1):(shells*10+1)197198 T=yout(1,j);199 for i=2:length(tspan)200 Tp1=yout(i,j);201 if Tp1 > (T+0.01)202 sprintf('Error in the evolution of temperature! row: %i column: %i',i,j)203 break204 end205 T=yout(i1,j);206 end207 end208209210 endB.3 Simulate detector response to 3D Gaussianellipsoid1 % This function simulates detector response for the shell model plasma.2 % It takes the output of msw expansionmodel 3D ion and simulates the3 % response of our detector for a given plasma. The output of this function4 % is equivalent to plasma observed in our detector!56 % Input: ([N; Rx; Ry; Rz; ux; uy; uz; Tix; Tiy; Tiz; Te],timestep)224B.3. Simulate detector response to 3D Gaussian ellipsoid7 % timestep: select (solution) in OutExpMod3D for which to simulate response8910 function [grapth out,sig,peak] = SimDetecResp3D ion(OutExpMod3D,timestep)1112 % Extract parameters of output msw expansionmodel 3D ion:1314 [rows,columns]=size(OutExpMod3D);15 shells=(columns1)/10;1617 N=OutExpMod3D(timestep,1:shells)';18 R x=OutExpMod3D(timestep,shells*1+1:shells*2)';19 R y=OutExpMod3D(timestep,shells*2+1:shells*3)';20 R z=OutExpMod3D(timestep,shells*3+1:shells*4)';2122 % takes function from within msw expansionmodel 3D ion:23 vol=volume3D(R x,R y,R z);2425 % Find rho with reduced computational error:26 rho=N./vol;27 rho=[flipud(rho);rho]*1e6;2829 % Define fit parameter and turn shell30 % model to continuous distribution:31 fo = fitoptions('method','SmoothingSpline','SmoothingParam',1e1);32 ft = fittype('smoothingspline');3334 dens profile x=fit([flipud(R x);R x],rho,ft,fo);35 dens profile y=fit([flipud(R y);R y],rho,ft,fo);36 dens profile z=fit([flipud(R z);R z],rho,ft,fo);373839 % We'll slice up the plasma and count the particles per40 % slice as the system traverses the detection grid:414243 % Calculate geometies:444546 % Find new z shells:225B.3. Simulate detector response to 3D Gaussian ellipsoid47 bin pos z=0:15:R z(end);48 new dens=dens profile z(bin pos z);49 bin number=size(bin pos z);50 bin number=bin number(2);5152 % Find new y shells:53 shell pos y=zeros(1,bin number);5455 for i=2:bin number % shells start at 2nd index56 desiredValue = new dens(i);57 objective = @(pos) dens profile y(pos)  desiredValue;58 desiredPos = fzero(objective, bin pos z(i));5960 shell pos y(i)=abs(desiredPos);61 % because of sym. sometimes finds neg. value62 end6364 lookforerror=max(find(isnan(shell pos y)));65 if isempty(lookforerror) == 0; n=1:lookforerror; shell pos y(n)=0;66 fprintf('NaN error at y=%i\n',max(n));67 end6869 % Find new x shells:70 shell pos x=zeros(1,bin number);7172 for i=2:bin number % shells start at 2nd index73 desiredValue = new dens(i);74 objective = @(pos) dens profile x(pos)  desiredValue;75 desiredPos = fzero(objective, bin pos z(i));7677 shell pos x(i)=abs(desiredPos);78 % because of sym. sometimes finds neg. value79 end8081 lookforerror=max(find(isnan(shell pos x)));82 if isempty(lookforerror) == 0; n=1:lookforerror; shell pos x(n)=0;83 fprintf('NaN error at x=%i\n',max(n));84 end8586226B.3. Simulate detector response to 3D Gaussian ellipsoid87 % Do integration (bin count):888990 response=zeros(bin number,2);9192 response(:,1)=bin pos z;93 dRz=bin pos z(2); % bins are equal distance9495 for i=2:bin number96 Rz=bin pos z(i1);97 correction=1(Rzˆ2./bin pos z(i:end).ˆ2);98 % takes care of ellipsodal geometries99100 % Define 2D ellipse corrected for z:101 Rx=sqrt(correction.*shell pos x(i:end).ˆ2);102 Rxm1=[0,Rx(1:end1)];103 Ry=sqrt(correction.*shell pos y(i:end).ˆ2);104 Rym1=[0,Ry(1:end1)];105106107 % gives area of ellipse shells108 Area=pi*(Rx.*RyRxm1.*Rym1);109110 % gives # of particles per shell111 Number shell=dRz*Area'.*new dens(i:end);112113 % gives # of particles for whole bin114 Number=sum(Number shell);115 response(i1,2)=Number;116 end117118119 x=[flipud(response(:,1));response(:,1)];120 y=[flipud(response(:,2));response(:,2)];121 grapth out=[x,y];122123 GaussFit=fit(x,y,'gauss1');124 coeff=coeffvalues(GaussFit);125 ci = confint(GaussFit,0.95);126227B.3. Simulate detector response to 3D Gaussian ellipsoid127 sig=coeff(3)/sqrt(2);128 sig up=ci(5)/sqrt(2);129 sig down=ci(6)/sqrt(2);130131 peak=coeff(1);132133 sprintf('sigma = %.6f (%.6f  %.6f) us',sig,sig up,sig down)134135 end228


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